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9797
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Jan 9, 2024
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Final Exam
FIN9797: Options Markets, Liuren Wu, Fall 2008
Assume zero interest rates (
r
) and zero dividends (
q
) wherever applicable. All options are
European.
1. Consider stock with a current price (
S
t
) of $50 and a constant annualized return volatility (
σ
) of 40%. The
stock price evolves according to the geometric Brownian motion.
(a) (10) Based on the above assumption, we can writer the risk-neutral log stock return dynamics as ln
S
T
/
S
t
=
(
r
-
q
-
1
2
σ
2
)(
T
-
t
) +
σ
(
W
T
-
W
t
)
. The return is normally distributed, what are mean and variance of
the return over a time horizon of three months (
T
-
t
=
3
/
12)? [
in concrete numbers
]
(b) (10) Using the approach discussed in class, construct a
two-step
binomial tree to approximate the stock
price dynamics, with each step being
3 months
. List the stock price at each node at three and six months.
(c) (10) Compute the risk-neutral probability of going up an going down at each step. [T
he probability of
going up is the same at each node. So just calculate once.
]
(d) (10) Based on the binomial tree, compute (i) the current value and (ii) the delta of a
call
option on the
stock with a maturity of
three months
and a strike price of
$55
. [
Just one step
]
(e) According to the Black-Scholes formula, the delta of the call option at $55 strike and three-month
maturity is 0
.
3532. I have also computed
N
(
d
2
) =
0
.
2821. Compute:
i. (10) The Black-Scholes (i) value and (ii) vega of the call option.
ii. (15) The Black-Scholes (i) value, (ii) delta, and (iii) vega of a
put
with the same strike and maturity.
iii. (5) Compared to the 3-month $55 strike call, should the following call options have higher or lower
delta? (i) a call at $55 strike and 1-year maturity. (ii) a call at $60 strike and 3-month maturity.
2. You have a portfolio of options on the same stock, with a delta 10 million and vega
-
400 million.
(a) (5) If the stock price suddenly falls by one dollar while the volatility does not change, how much do you
expect your portfolio value to change?
(b) (5) If the stock price does not change but the volatility suddenly goes up by one percentage point (0
.
01,
or 1%), how much do you expect your portfolio value to change?
(c) (10) If you want to alter your risk exposure using (i) the underlying stock and (ii) a put option with a
delta of
-
0
.
5 and a vega of 10. How many of these two contracts do you need to balance your portfolio
to delta and vega neutral?
3. (10) Comparing to a normal distribution benchmark, investors expect that the stock has a higher (risk-adjusted)
probability of generating large positive returns and a lower probability of generating large negative returns.
Plot the implied volatility (
IV
) at a fixed maturity as a function of the strike (
K
) that reflects this view of the
investors. [
A schematic plot, no numbers needed.
]
1
Final Exam
FIN9797: Options Markets, Liuren Wu, Spring 2008
Assume a continuously compounding dollar interest rate of 5% for all maturities, whenever applicable.
1. Consider a non-dividend paying stock, with a current price (
S
t
) of $100 and a constant annualized return
volatility (
σ
) of 20%. The stock price evolves according to the geometric Brownian motion as assumed by
Black, Scholes, and Merton. We want to price a three-month (
T
-
t
=
3
/
12) European call option on the
stock with a strike price (
K
) of $105.
(a) (5) Is this option in the money, out of the money, or at-the-money?
(b) (10) Using the approach discussed in class, construct a two-step binomial tree to approximate the
stock price dynamics, with each step being 1.5 months.
(c) (10) Compute (i) the current value and (ii) the delta of the call option based on the two-step binomial
tree (using any approach you like)?
(d) (15) Compute (i) the current value, (ii) the delta, and (iii) the vega of the call option based on the
Black-Scholes formula. I have:
N
(
d
1
) =
0
.
3772 and
N
(
d
2
) =
0
.
3398.
(e) (10) Based on your calculated delta and vega numbers from the Black-Scholes formula, how much
will the call option value change approximately if the stock price goes up by $1? How much will the
call value change if the stock return volatility increase from 20% to 21%?
(f) (15) Given the assumptions, calculate the stock return’s volatility (standard deviation) over horizons
of three months, one year, and two years, respectively.
2. (20) Assume that you have a portfolio with delta 30 and vega 400, and you want to alter your delta and
vega exposure using (i) the underlying stock and (ii) a delta-neutral straddle with a vega of 20.
(a) How many of these two contracts do you need to balance your portfolio to delta and vega neutral?
(b) Suppose you want to constrain the delta exposure of your portfolio to be within
±
10 and the vega
exposure to be within
±
100.
What are the minimum number of contracts you need to get your
portfolio within your target exposure range?
3. (15) We have three over-the-counter quotes on one-year dollar-yen options (dollar is home currency): (i)
delta-neutral straddle at 15%, (ii) 25-delta risk-reversal at
-
2%, and (iii) 25-delta butterfly spread at 1%.
(a) Which of the three contracts has the highest delta exposure? Which of the three contracts has the
highest vega exposure?
(b) Based on the quotes, do you think the one-year risk-neutral dollar-yen return distribution is positively
skewed, negatively skewed, or symmetric? Why (which quote gives you the information)?
(c) Based on the quotes, do you think the one-year risk-neutral dollar-yen return distribution has fatter,
thinner, or the same tails as a normal distribution does?
1
Final Exam
FIN9797: Options Markets, Liuren Wu, Spring 2009
1. Consider stock with a current price (
S
t
) of $100 and a constant annualized return volatility (
σ
) of 20%. The
stock does not pay dividends. A risk-free zero-coupon bond with $1 par and one year maturity is worth $0.95
today.
(a) (10) Using the approach discussed in class, construct a
two-step
binomial tree to approximate the stock
price dynamics, with each step being
1 year
. List the stock price at each node at one and two years.
(b) (5) Compute the risk-neutral probability of going up and going down at each step.
(c) (20) Based on the binomial tree, compute (i) the current value and (ii) the delta of an
European put
option on the stock with a maturity of
two years
and a strike price of
$110
.
(d) (10) Based on the binomial tree, compute (i) the current value and (ii) the delta of an
American
put
option on the stock with a maturity of
two years
and a strike price of
$110
.
2. Consider a European put option with the underlying security spot price being $100, strike pricing being $90,
and time to maturity being one year. We also know that
N
(
d
1
) =
0
.
75 and
N
(
d
2
) =
0
.
7, and we further assume
zero interest rates and zero dividends for this question.
(a) (10) Compute the Black-Scholes (i) value and (ii) delta of the European put option.
(b) (15) Compared to this put option, are the following put options more, less, or the same in terms of their
sensitivity to the underlying price movement? (i) a 25-delta 10-year put option, (ii) a 50-delta 1-year put
option, (iii) a 75-delta 1-year put option. [
3 answers, one for each contract.
]
3. You have a portfolio of options on the same stock, with a delta
-
200 million and vega 400 million.
(a) (5) If the stock price suddenly falls by one dollar while the volatility does not change, how much do you
expect your portfolio value to change?
(b) (5) If the stock price does not change but the volatility suddenly goes up by one percentage point (0
.
01,
or 1%), how much do you expect your portfolio value to change?
(c) (10) If you want to alter your risk exposure using (i) the underlying stock and (ii) a put option with a
delta of
-
0
.
5 and a vega of 20. How many of these two contracts do you need to long or short in order
to make your portfolio to delta and vega neutral?
4. You are trying to infer the return distribution of a currency at one month horizon. The one-month at-the-money
straddle is quoted at 20%. The one-month 25-delta risk-reversal is quoted at
-
10%. The one-month 25-delta
butterfly spread is quoted at 10%.
(a) (5) Compared to a normal distribution, does the currency return has fatter tails or thinner tails?
(b) (5) Compared to a normal distribution, is the currency return distribution negatively or positively skewed?
1
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Final — Version
A
FIN9797: Options Markets, Liuren Wu, Spring 2015
Please write all answers on blue book.
Assume zero interest rates (domestic or foreign) and zero dividends for all
questions, whenever applicable.
1. Simple answer questions
3 points for each question. Please highlight final answer. No partial credits.
1. Let
W
t
denote a standard Brownian motion, compute the mean (
) and volatility (standard deviation)
(
) of 15%
+
10%
(
W
T
-
W
t
)
, with
T
-
t
=
4.
2. Consider a one step tree for a stock with a current stock price of $100 that can go either up to $110 or down
to $80. The step size is 3 months.
(a) What is the risk-neutral probability of going up to $110? (
)
(b) What is the risk-neutral probability of going down to $80? (
)
(c) Consider a one-year put option with a strike of $100, what is the payoff of this option if the stock price
goes up to $110? (
) What is the payoff if the stock price goes to $80? (
)
(d) Based on the tree, compute the delta (
) of this one-year $100-strike put option. How much is
this option worth? (
)
3. Which one of the following options (all on the same stock) has the highest strike? (
)
(a) 1-year 20-delta put, (b) 1-year 30-delta call, (c) 1-year 50 delta call, (d) 1-year 60 delta put
4. Which one of the following options (all on the same stock) has the highest volatility exposure? (
)
(a) 1-year 20-delta put, (b) 1-year 30-delta call, (c) 1-year 50 delta call, (d) 2-year 50 delta put
5. Which one of the following options (all on the same stock) has the highest gamma (curvature)? (
)
(a) 1-year 20-delta put, (b) 1-year 30-delta call, (c) 1-year 50 delta call, (d) 2-year 50 delta put
6. An option portfolio has a total delta of 400, how many shares of the underlying stock you need to buy or sell
to neutralize the exposure? (
) (use negative shares for sell, and positive shares for buy)
7. An option portfolio is delta neutral but has a vega of 40. You can use the underlying stock and a call option
with a delta of 0.5 and a vega of 4 to hedge the portfolio. (Use negative shares for sell, positive shares for buy)
(a) How many shares of stock (
) and the call (
) you need to neutralize the portfolio?
(b) How many (minimum) shares of stock (
) and the call (
) you need to keep the delta
exposure below
±
0
.
5 and vega exposure below
±
4?
8. Consider an option portfolio with a delta of 50 and vega of -150.
(a) If the stock price goes up by $1, how much will the portfolio value change? (
)
(b) If the volatility goes up by 1%, how much will the portfolio value change? (
)
9. The current stock price is at $100. The one-year options implied volatilities at $80, $100, and $120 strikes are
30$, 20%, and 40%, respectively?
(a) Is the option-implied risk-neutral distribution skewed? (
)
(a) negatively skewed, (b) positively skewed, (c) symmetric, (d) cannot tell.
(b) Are the two tails of the return distribution fatter or thinner than a normal distribution? (
)
(a) fatter, (b) thinner, (c) same, (d) cannot tell
2. Essay questions
Partial credits possible.
10. Consider a stock with a current price (
S
t
) of $100 and a constant annualized return volatility (
σ
) of 30%.
(a) (10) Using the approach discussed in class, construct a
two-step
binomial tree to approximate the stock
price dynamics, with each step being
3 months
. List the stock price at each node.
(b) (5) Compute the risk-neutral probability of going up and going down at each step.
(c) (10) Based on the binomial tree, compute (i) the current value and (ii) the delta of an
American
put
option on the stock with a maturity of
6 months
and a strike price of
$60
.
11. The current stock price is $100. The 2-year 25-delta call option has a strike of $110.
N
(
d
2
)
for this option is
0
.
20.
(a) (10) Compute the (i) Black-Scholes value and (ii) delta of this call option
(b) (10) Compute the (i) Black-Scholes value and (ii) delta of the put option at the same strike and expiry.
(c) (5) Which of the two options (2-year $110-strike call and put) have a higher gamma?
2
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Final Exam
FIN9797: Options Markets, Liuren Wu
Assume a continuously compounding dollar interest rate of 5% for all maturities, whenever applicable.
1.
(15) Under the Black-Scholes model, the stock price
S
t
follows a geometric Brownian motion under
the risk-neutral measure,
dS
t
/
S
t
= (
r
-
q
)
dt
+
σ
dW
t
,
where
r
is the continuously compounded interest rate and
q
is the dividend yield. Assume that the stock
has an annualized dividend yield of 3% per year and an annualized volatility
σ
=
20%. Consider the
log return ln
(
S
T
/
S
t
)
with a horizon of three months (
T
-
t
=
3
/
12). What is the mean and variance of
the log return? What’s its distribution?
2.
(10) Let
W
t
denote a Brownian motion starting at zero. Compute the mean and volatility of the following
processes:
(a)
5%
+
20%
W
t
with
t
=
2.
(b)
2%
+
10%
(
W
T
-
W
t
)
with
t
=
1 and
T
=
3.
3.
(30) Consider a two-year 25-delta call option on dollar price of pound (pound is the foreign currency).
The current pound interest rate is also 5% (the same as the dollar rate). The current exchange rate is
$2.00 dollar. The strike of the option is
K
=
2
.
52, with
N
(
d
2
) =
0
.
17.
(a)
Compute the value of the call option based on the Black-Scholes formula.
(b)
Compute the value of a put option at the same strike and maturity.
(c)
Compute the delta of the call option and the put option.
4.
(20) Assume that you have an option portfolio with delta 10 and vega 200, and you want to alter your
delta and vega exposure using two liquid contracts. The first contract has a delta of 0.5 and vega of 2.
The second contract has delta of zero and vega of 6.
(a)
How many of these two contracts do you need in order to balance your portfolio to delta and vega
neutral?
(b)
Suppose you want to achieve delta neutral but are willing to have a vega exposure within
±
10.
What are the minimum number of contracts you need to get your portfolio within your target
exposure range?
5.
(15) The current spot price is $100, consider three call options at 80, 100, and 120, which one has the
highest delta exposure? Which one has the highest vega exposure? How do your answers change if the
three options are put options?
6.
(10) Consider a 25-delta put option on a stock. When the stock price goes up by one dollar, approxi-
mately how much will the put option price move? In what direction (up or down)?
1
7.
2
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Final Exam
FIN9797: Options Markets, Liuren Wu
Assume a continuously compounding dollar interest rate of 5% for all maturities, whenever applicable.
1.
(15) Under the Black-Scholes model, the stock price
S
t
follows a geometric Brownian motion under
the risk-neutral measure,
dS
t
/
S
t
= (
r
-
q
)
dt
+
σ
dW
t
,
where
r
is the continuously compounded interest rate and
q
is the dividend yield. Assume that the stock
has an annualized dividend yield of 3% per year and an annualized volatility
σ
=
20%. Consider the
log return ln
(
S
T
/
S
t
)
with a horizon of three months (
T
-
t
=
3
/
12). What is the mean and variance of
the log return? What’s its distribution?
2.
(10) Let
W
t
denote a Brownian motion starting at zero. Compute the mean and volatility of the following
processes:
(a)
5%
+
20%
W
t
with
t
=
2.
(b)
2%
+
10%
(
W
T
-
W
t
)
with
t
=
1 and
T
=
3.
3.
(30) Consider a two-year 25-delta call option on dollar price of pound (pound is the foreign currency).
The current pound interest rate is also 5% (the same as the dollar rate). The current exchange rate is
$2.00 dollar. The strike of the option is
K
=
2
.
52, with
N
(
d
2
) =
0
.
17.
(a)
Compute the value of the call option based on the Black-Scholes formula.
(b)
Compute the value of a put option at the same strike and maturity.
(c)
Compute the delta of the call option and the put option.
4.
(20) Assume that you have an option portfolio with delta 10 and vega 200, and you want to alter your
delta and vega exposure using two liquid contracts. The first contract has a delta of 0.5 and vega of 2.
The second contract has delta of zero and vega of 6.
(a)
How many of these two contracts do you need in order to balance your portfolio to delta and vega
neutral?
(b)
Suppose you want to achieve delta neutral but are willing to have a vega exposure within
±
10.
What are the minimum number of contracts you need to get your portfolio within your target
exposure range?
5.
(15) The current spot price is $100, consider three call options at 80, 100, and 120, which one has the
highest delta exposure? Which one has the highest vega exposure? How do your answers change if the
three options are put options?
6.
(10) Consider a 25-delta put option on a stock. When the stock price goes up by one dollar, approxi-
mately how much will the put option price move? In what direction (up or down)?
1
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7.
2
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Final Exam
FIN9797: Options Markets, Liuren Wu, Spring 2009
1. Consider stock with a current price (
S
t
) of $100 and a constant annualized return volatility (
σ
) of 20%. The
stock does not pay dividends. A risk-free zero-coupon bond with $1 par and one year maturity is worth $0.95
today.
(a) (10) Using the approach discussed in class, construct a
two-step
binomial tree to approximate the stock
price dynamics, with each step being
1 year
. List the stock price at each node at one and two years.
(b) (5) Compute the risk-neutral probability of going up and going down at each step.
(c) (20) Based on the binomial tree, compute (i) the current value and (ii) the delta of an
European put
option on the stock with a maturity of
two years
and a strike price of
$110
.
(d) (10) Based on the binomial tree, compute (i) the current value and (ii) the delta of an
American
put
option on the stock with a maturity of
two years
and a strike price of
$110
.
2. Consider a European put option with the underlying security spot price being $100, strike pricing being $90,
and time to maturity being one year. We also know that
N
(
d
1
) =
0
.
75 and
N
(
d
2
) =
0
.
7, and we further assume
zero interest rates and zero dividends for this question.
(a) (10) Compute the Black-Scholes (i) value and (ii) delta of the European put option.
(b) (15) Compared to this put option, are the following put options more, less, or the same in terms of their
sensitivity to the underlying price movement? (i) a 25-delta 10-year put option, (ii) a 50-delta 1-year put
option, (iii) a 75-delta 1-year put option. [
3 answers, one for each contract.
]
3. You have a portfolio of options on the same stock, with a delta
-
200 million and vega 400 million.
(a) (5) If the stock price suddenly falls by one dollar while the volatility does not change, how much do you
expect your portfolio value to change?
(b) (5) If the stock price does not change but the volatility suddenly goes up by one percentage point (0
.
01,
or 1%), how much do you expect your portfolio value to change?
(c) (10) If you want to alter your risk exposure using (i) the underlying stock and (ii) a put option with a
delta of
-
0
.
5 and a vega of 20. How many of these two contracts do you need to long or short in order
to make your portfolio to delta and vega neutral?
4. You are trying to infer the return distribution of a currency at one month horizon. The one-month at-the-money
straddle is quoted at 20%. The one-month 25-delta risk-reversal is quoted at
-
10%. The one-month 25-delta
butterfly spread is quoted at 10%.
(a) (5) Compared to a normal distribution, does the currency return has fatter tails or thinner tails?
(b) (5) Compared to a normal distribution, is the currency return distribution negatively or positively skewed?
1
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Final — Version
A
FIN9797: Options Markets, Liuren Wu, Spring 2015
Please write all answers on blue book.
Assume zero interest rates (domestic or foreign) and zero dividends for all
questions, whenever applicable.
1. Simple answer questions
3 points for each question. Please highlight final answer. No partial credits.
1. Let
W
t
denote a standard Brownian motion, compute the mean (
) and volatility (standard deviation)
(
) of 15%
+
10%
(
W
T
-
W
t
)
, with
T
-
t
=
4.
2. Consider a one step tree for a stock with a current stock price of $100 that can go either up to $110 or down
to $80. The step size is 3 months.
(a) What is the risk-neutral probability of going up to $110? (
)
(b) What is the risk-neutral probability of going down to $80? (
)
(c) Consider a one-year put option with a strike of $100, what is the payoff of this option if the stock price
goes up to $110? (
) What is the payoff if the stock price goes to $80? (
)
(d) Based on the tree, compute the delta (
) of this one-year $100-strike put option. How much is
this option worth? (
)
3. Which one of the following options (all on the same stock) has the highest strike? (
)
(a) 1-year 20-delta put, (b) 1-year 30-delta call, (c) 1-year 50 delta call, (d) 1-year 60 delta put
4. Which one of the following options (all on the same stock) has the highest volatility exposure? (
)
(a) 1-year 20-delta put, (b) 1-year 30-delta call, (c) 1-year 50 delta call, (d) 2-year 50 delta put
5. Which one of the following options (all on the same stock) has the highest gamma (curvature)? (
)
(a) 1-year 20-delta put, (b) 1-year 30-delta call, (c) 1-year 50 delta call, (d) 2-year 50 delta put
6. An option portfolio has a total delta of 400, how many shares of the underlying stock you need to buy or sell
to neutralize the exposure? (
) (use negative shares for sell, and positive shares for buy)
7. An option portfolio is delta neutral but has a vega of 40. You can use the underlying stock and a call option
with a delta of 0.5 and a vega of 4 to hedge the portfolio. (Use negative shares for sell, positive shares for buy)
(a) How many shares of stock (
) and the call (
) you need to neutralize the portfolio?
(b) How many (minimum) shares of stock (
) and the call (
) you need to keep the delta
exposure below
±
0
.
5 and vega exposure below
±
4?
8. Consider an option portfolio with a delta of 50 and vega of -150.
(a) If the stock price goes up by $1, how much will the portfolio value change? (
)
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(b) If the volatility goes up by 1%, how much will the portfolio value change? (
)
9. The current stock price is at $100. The one-year options implied volatilities at $80, $100, and $120 strikes are
30$, 20%, and 40%, respectively?
(a) Is the option-implied risk-neutral distribution skewed? (
)
(a) negatively skewed, (b) positively skewed, (c) symmetric, (d) cannot tell.
(b) Are the two tails of the return distribution fatter or thinner than a normal distribution? (
)
(a) fatter, (b) thinner, (c) same, (d) cannot tell
2. Essay questions
Partial credits possible.
10. Consider a stock with a current price (
S
t
) of $100 and a constant annualized return volatility (
σ
) of 30%.
(a) (10) Using the approach discussed in class, construct a
two-step
binomial tree to approximate the stock
price dynamics, with each step being
3 months
. List the stock price at each node.
(b) (5) Compute the risk-neutral probability of going up and going down at each step.
(c) (10) Based on the binomial tree, compute (i) the current value and (ii) the delta of an
American
put
option on the stock with a maturity of
6 months
and a strike price of
$60
.
11. The current stock price is $100. The 2-year 25-delta call option has a strike of $110.
N
(
d
2
)
for this option is
0
.
20.
(a) (10) Compute the (i) Black-Scholes value and (ii) delta of this call option
(b) (10) Compute the (i) Black-Scholes value and (ii) delta of the put option at the same strike and expiry.
(c) (5) Which of the two options (2-year $110-strike call and put) have a higher gamma?
2
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Final — Version
A
FIN9797: Options Markets, Liuren Wu, Fall 2017
Please write all answers on blue books. Points total: 137
1. Simple answer questions
3 points for each question. Please highlight final answer. No partial credits.
1. The Black-Scholes model assumes that the underlying security price follows a geometric Brownian motion:
dS
t
/
S
t
=
μdt
+
σ
dW
t
. Assume that the underlying security is AAPL, with the instantaneous expected return
μ
t
=
15% and the instantaneous volatility
σ
=
30%.
(a) Compute the mean (
) and standard deviation (
) of the 3-month log return on AAPL
(ln
S
T
/
S
t
with
T
-
t
=
1
/
4).
(b) Suppose you have $1,000 and decide to put $500 of the money in cash (which generates no return) and
the remaining $500 is AAPL (which follows the above process). Compute the mean (
) and the
standard deviation (
) of the 6-month log return on your portfolio, which includes both cash and
AAPL.
2. Suppose the prices of stocks A, B, C all follow the Black-Scholes model (geometric Brownian motion) but
with different mean (
μ
=
10%
,
20%
,
30%, respectively) and volatility (
σ
=
15%
,
25%
,
35%, respectively). The
current stock price levels are $100 for all three. Assume zero rates/dividends.
(a) Three-month at-the-money call options are traded on all three stocks. Which stock has the highest call
option price (
), which stock has the lowest call option price (
)?
(b) One-year at-the-money put options are traded on all three stocks. Which stock has the highest put option
price (
), which stock has the lowest put option price (
)?
(c) One-year $80-strike call options are written on the three stocks. Call on which stock has the highest
delta (
), call on which stock has the highest vega (
)?
3. Consider a one step tree for a stock with a current stock price of $50 that can go either up to $70 or down to
$40. The step size is 6 months. Continuously compounding interest rate is 5%. No dividends.
(a) What the present value of a 6-month zero-coupon bond that pays $1 at expiry? (
)
(b) What is the risk-neutral probability of going up to $70? (
)
(c) What is the risk-neutral probability of going down to $40? (
)
(d) Consider a 6-month call option with a strike of $60, what is the payoff of this option if the stock price
goes up to $70? (
) What is the payoff if the stock price goes to $40? (
)
(e) Suppose you want to replicate the payoff of the option with the 6-month bond and the stock. How many
shares of the stock (
) and the bond (
) do you need for the replication? What’s the
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cost of stock (
) and bond (
) in the replicating portfolio? What’s the total cost of the
replication (
)?
(f) Suppose you are long in the option and want to hedge the risk of the option with the stock, how many
shares of the stock (
) do you need for the hedge (negative for short, positive for long)?
(g) Suppose 6-month later, the stock price goes to $60 (different from tree assumption), what’s the payoff
of the option (
) and what’s the payoff of your replication in (e) (
)? If you long the
option and short the replication, how much do you make/lose (
)? (negative for loss)
(h) Suppose 6-month later, the stock price goes to $30 instead, what’s the payoff of the option (
)
and what’s the payoff of your replication in (e) (
)? If you long the option and short the repli-
cation, how much do you make/lose (
)?
4. If you think that the market views a currency’s return distribution as negatively skewed and fat-tailed, do
you expect the risk reversal quote on this currency to be positive or negative (
)? Do you expect the
butterfly quotes on this currency to be positive or negative (
)?
5. When the stock price goes up by one dollar, approximately how much will the following options on the stock
move: (i) 3-month 25-delta put (
) (ii) 2-year 25-delta call (
) (i) 1-year delta-neutral straddle
(
) (negative for down and positive for up)
6. An option portfolio has a delta of 100 and a vega of 50.
(a) If you can use the stock and a delta-neutral straddle with vega of 10 to hedge, how many shares of the
stock (
) and the straddle (
) you need to neutralize the portfolio’s delta and vega?
(b) If you only have access to a 50-delta call and a 50-delta put, both with vega of 5, how many shares of
the call (
) and the put (
) you need to neutralize the portfolio’s delta and vega?
7. The current stock price is $100. The 1-year 25-delta call option has
N
(
d
2
)
at 0
.
20 and its option value at 2
.
0,
infer the option’s strike price based on the Black-Scholes formula (
) . Assume 0 rates/dividends.
2. Essay questions
Partial credits possible.
8. Consider a stock with a current price (
S
t
) of $100 and a constant annualized return volatility (
σ
) of 20%.
Assume 0 rates/dividends.
(a) (10) Using the approach discussed in class, construct a
two-step
binomial tree to approximate the stock
price dynamics, with each step being
6 months
. List the stock price at each node.
(b) (6) Compute the risk-neutral probability of going up and going down at each step.
(c) (10) Based on the binomial tree, compute the value and the delta
at each node
of a European call option
on the stock with a maturity of
one-year
and a strike price of
$110
.
2
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Final — Version
A
FIN9797: Options Markets, Liuren Wu, Fall 2018
Please write all answers on blue books. Points total: 125
1. Consider the following 4 options on the same stock AAPL: (i) one-month at-the-money call, (ii) one-year
at-the-money put, (iii). one-month 25-delta call, and (iv) one-year 75-delta put.
(a) (3) If Apple stock price goes up, everything else held fixed, which one of the four options will go up the
most in value?
(b) (3) If Apple stock return volatility goes up, everything else held fixed, which one of the four options will
go up the most in value?
(c) (3) If Apple stock price goes up, everything else held fixed, which one of the four options will go down
the most in value?
(d) (3) Which of the four options has the highest gamma?
2. Consider a one step tree for a stock with a current stock price of $100 that can go either up to $120 or down
to $80 in 3 months. Assume zero interest rates and no dividends.
(a) (4) Consider a 3-month European put option with a strike of $110, compute its terminal payoff on the
tree.
(b) (4) Suppose you want to replicate the payoff of the option with the 3-month bond and the stock. Compute
the replicating portfolio weights (shares of stocks and bonds) and the replicating cost.
(c) (3) What’s the delta of the option?
(d) (4) Suppose the market maker makes the market on the put option at $20 (with zero bid-ask spread).
What trade do you propose to make to lock in a profit based on the tree?
(e) (3) Suppose 3-month later, the stock price goes to $110, what will be the terminal payoff from your
above trade?
(f) (3) What happens to your trade if the stock price goes to $130?
(g) (3) What happens to your trade if the stock price goes to $60?
(h) (3) In general, your trade can lose money when the stock moves more or less than suggested by the tree?
Why (be concise).
3. (5) The stock price on CBOE (Chicago Board of Options Exchange) is at $100. The 1-year 25-delta call
option on the stock has a strike (
K
) of 120, and
N
(
d
2
)
at 0
.
18. Assume 0 rates and dividends. Compute its
option value based on the Black-Scholes-Merton formula.
4. (5) The stock price on LVS (Las Vegas Sands) is at $55. The 2-year 25-delta put option on the stock has a
strike (
K
) of 40, and
N
(
d
2
)
at 0
.
20. Assume 0 rates and dividends. Compute its option value based on the
Black-Scholes-Merton formula.
5. The stock on Canada Goose (GOOS) is traded at $60. The return on the stock has an annualized volatility (
σ
)
of 75%. Ignore dividend, and assume that the riskfree interest rate is flat at 5% across maturities.
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(a) (12) Based on the approach on the lecture notes, construct a 2-step tree, with each step being 3 month
(
∆
t
=
1
/
4). List the stock price at each node of the tree.
(b) (6) Such a tree maintains the same risk-neutral probability of going up and down. Compute the risk-
neutral probability of going up (
p
u
) and going down (
p
d
) for each node.
(c) (6) Based on the above tree, compute (i) the terminal payoffs, (ii) the delta, and (iii) the current value of
a 3-month at-the-money-spot call option (
K
=
60).
(d) (6) Based on the above tree, compute (i) the terminal payoffs, (ii) the delta, and (iii) the current value of
a 3-month at-the-money-spot put option (
K
=
60).
(e) (10) Based on the above tree, compute (i) the value and (ii) the delta of a 6-month American put option
at
K
=
100.
6. An option portfolio (on the same stock) has a delta of
-
20 million and a vega of 500 million.
(a) (3) If the stock price goes up by $2, how much will approximately the portfolio value change?
(b) (3) If the stock return volatility goes up by 0.5%, how much approximately will the portfolio value
change?
(c) (10) Suppose you have a risk limit of
±
5 million delta and
±
10 million vega. If you can use the stock
and a 50-delta call option with a vega of 2 to manage the risk of the portfolio, how many shares of the
stock and the call option at the minimum you need to bring the portfolio within risk limits.
7. (10) Suppose the market is pricing in an imminent large market crash. If you plot the S&P 500 index 3-month
option implied volatility against the strike-forward ratio (
K
/
F
), do you expect the implied volatility curve to
be higher at low strike or high strike? Draw a schematic implied volatility plot against the strike-forward ratio
to illustrate.
8. (10) Suppose the market is expecting more extreme possibilities (of either direction) than implied by a normal
distribution for the return on the dollar-euro exchange rate, do you expect the implied volatilities on the dollar-
euro exchange rate to be higher at the money or out of the money. Draw a schematic implied volatility plot
against the strike-forward ratio to illustrate.
2
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The binomial tree model: a simple example of pricing financial derivatives
SO9: Financial Physics. Prof. C. J. Foot, Physics Dept, University of Oxford
Assumed background knowledge
This an extension of the ‘coin-toss’ market (shown in Fig. 2-2 of the book by Johnson
et
al.
)
in which the two possible outcomes may have unequal probabilities.
The Binomial
distribution is described in the second-year lectures on probability and the book
Concepts
in Thermal Physics
by Profs. Blundell and Blundell. (Sometimes a single step is called a
Bernoulli trial.)
This description of the binomial tree model is structured as an answer to the following
question (similar to one on the examination paper in 2011).
Question
Consider a binomial tree model for the stock price process
{
x
n
: 0
≤
n
≤
3
}
. Let
x
0
= 100
and let the price rise or fall by 10 % at each time-step. The interest rate is
r
= 5 %. The
contract we wish to price is a European put option with strike price 110 at time-step 3.
(a) Find the risk neutral probabilities for the tree.
(b) Find the initial value of the option.
(c) After time-step 2, the stock price has three possible values. For each of these three
cases, determine what trading strategy the writer of the option should follow to hedge
the option.
(d) If the price movements for the asset are up, up and down find the trading strategy
required to hedge the option.
Answer
(a) Probability in the binomial model
Denote the risk neutral probability as
p
for rising, and 1
-
p
for falling. In an arbitrage-
free market the increase in share values matches the (riskless) increase from interest. This
corresponds to the mathematical expression
px
0
(1 + 10%) + (1
-
p
)
x
0
(1
-
10%) =
x
0
(1 + 5%)
.
Or more generally, the price goes up by a factor
u
and down by
d
, with an interest rate
r
.
Therefore at every time-step
pux
n
+ (1
-
p
)
dx
n
= (1 +
r
)
x
n
hence
pu
+ (1
-
p
)
d
=
(1 +
r
)
p
=
1 +
r
-
d
u
-
d
=
1 + 0
.
05
-
0
.
9
0
.
2
= 0
.
75
[An interest rate r=0 gives
p
= 0
.
5 which takes us back to the simple ‘coin-toss’ market.]
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(b) Find the price of the option
There are 4 possible states of the market at time
n
= 3. The corresponding stock prices
and payoffs of the option are shown in the following figure.
(b) Find the price of the option
There are 4 possible states of the market at time
n
= 3. The corresponding stock prices
and payoffs of the option are shown in the following figure.
121
0
133
.
1
1
.
1
108
.
9
99
20
.
9
89
.
1
81
37
.
1
72
.
9
110
90
100
Here the numbers are stock prices (below)
and the option payoff (above).
The expectation value of the option payoff in this binomial model is
E
(payoff)
=
(
p
3
,
3
p
2
(1
-
p
)
,
3
p
(1
-
p
)
2
,
(1
-
p
)
3
)
0
1
.
1
20
.
9
37
.
1
=
27
64
,
27
64
,
9
64
,
1
64
0
1
.
1
20
.
9
37
.
1
=
27
×
1
.
1 + 9
×
20
.
9 + 37
.
1
64
= 3
.
98
This is
not
the value of the option
V
because we have to account for interest. An amount
V
at time step 0 becomes worth
V
(1+
r
)
3
at time step 3, so the value of the option is given
by
V
(1 +
r
)
3
=
E
(payoff), i.e.
V
=
E
(payoff)
(1 +
r
)
3
=
3
.
98
1
.
158
= 3
.
44
This gives the initial value of the option for this market model.
Similarly the value of the option at other nodes is found by working backwards from the
final payoff.
0
.
262
121
0
133
.
1
1
.
1
108
.
9
5
.
76
99
20
.
9
89
.
1
23
.
8
81
37
.
1
72
.
9
1
.
56
110
9
.
77
90
3
.
44
100
(c) Determine a suitable hedging strategy
The writer of the option will lose money if the stock price goes down so that the holder
receives a payoff—the losses or gains of the writer and holder are equal and opposite. Either,
or both, of these parties to the futures contract can hedge to eliminate the risk. In practice
in may be only the writer, e.g. a big financial institution such as an investment bank, that
has the wherewithal to implement dynamic hedging.
Therefore we consider the option
writer’s portfolio which consists of
φ
shares of stock and
ψ
units of risk-free asset. The
Figure 1: A Binomial tree model with 3 time-steps.
The expectation value of the option payoff in this binomial model is
E
(payoff)
=
(
p
3
,
3
p
2
(1
-
p
)
,
3
p
(1
-
p
)
2
,
(1
-
p
)
3
)
0
1
.
1
20
.
9
37
.
1
=
27
64
,
27
64
,
9
64
,
1
64
0
1
.
1
20
.
9
37
.
1
=
27
×
1
.
1 + 9
×
20
.
9 + 37
.
1
64
= 3
.
98
This is
not
the value of the option
V
because we have to account for interest. An amount
V
0
at time-step 0 becomes worth
V
0
(1 +
r
)
3
at time-step 3, so the value of the option is
given by
V
0
(1 +
r
)
3
=
E
(payoff), i.e.
V
0
=
E
(payoff)
(1 +
r
)
3
=
3
.
98
1
.
158
= 3
.
44
This gives the initial value of the option for this market model. Other ways of writing this
expectation value are given below.
Similarly the value of the option at other nodes can be found by working backwards from
the final payoff. (This is not asked for in the exam-style question above but it is useful for
the purposes of this lecture, and easily implemented on a spreadsheet).
(b) Find the price of the option
There are 4 possible states of the market at time
n
= 3. The corresponding stock prices
and payoffs of the option are shown in the following figure.
121
0
133
.
1
1
.
1
108
.
9
99
20
.
9
89
.
1
81
37
.
1
72
.
9
110
90
100
Here the numbers are stock prices (below)
and the option payoff (above).
The expectation value of the option payoff in this binomial model is
E
(payoff)
=
(
p
3
,
3
p
2
(1
-
p
)
,
3
p
(1
-
p
)
2
,
(1
-
p
)
3
)
0
1
.
1
20
.
9
37
.
1
=
27
64
,
27
64
,
9
64
,
1
64
0
1
.
1
20
.
9
37
.
1
=
27
×
1
.
1 + 9
×
20
.
9 + 37
.
1
64
= 3
.
98
This is
not
the value of the option
V
because we have to account for interest. An amount
V
at time step 0 becomes worth
V
(1+
r
)
3
at time step 3, so the value of the option is given
by
V
(1 +
r
)
3
=
E
(payoff), i.e.
V
=
E
(payoff)
(1 +
r
)
3
=
3
.
98
1
.
158
= 3
.
44
This gives the initial value of the option for this market model.
Similarly the value of the option at other nodes is found by working backwards from the
final payoff.
0
.
262
121
0
133
.
1
1
.
1
108
.
9
5
.
76
99
20
.
9
89
.
1
23
.
8
81
37
.
1
72
.
9
1
.
56
110
9
.
77
90
3
.
44
100
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(c) Determining a suitable hedging strategy for the option writer
The writer of the option will lose money if the stock price goes down so that the holder
receives a payoff—the losses or gains of the writer and holder are equal and opposite. Either,
or both, of these parties to the contract can hedge to eliminate the risk. In practice it may
be only the writer, e.g. a big financial institution such as an investment bank, that has the
wherewithal to implement dynamic hedging. Let us consider the option writer’s portfolio
which consists of
φ
shares of stock and
ψ
units of risk-free asset. The risk-free asset increases
by (1 +
r
) at each step.
At a given step
n
the market will be at a node in the binomial tree as in the following figure
At a given step
n
the market will be at a node in the binomial tree as in the following figure
a
b
Here
a, b
are the two possible payoffs of the derivative after the next step. We need to find
a portfolio (
φ, ψ
) that replicates the payoff (where the risk-free asset has a unit price of
B
n
at step
n
); this can achieved by satisfying the conditions
φ
n
ux
n
+
ψ
n
(1 +
r
)
B
n
=
a
φ
n
dx
n
+
ψ
n
(1 +
r
)
B
n
=
b
Solving this set of two simultaneous equations with two unknowns to gives
φ
n
x
n
=
a
-
b
u
-
d
,
and
ψ
n
B
n
=
bu
-
ad
u
-
d
×
1
1 +
r
Hence for the case where
u
= 1 + 10%,
d
= 1
-
10% and 1 +
r
= 1 + 5%
φ
n
x
n
=
5(
a
-
b
)
,
and
(1)
ψ
n
B
n
=
11
b
-
9
a
2
×
1
.
05
(2)
Note that is only the total amount held in riskless assets
ψ
n
B
n
that is important, not the
actual number
ψ
n
, e.g. this could be cash in a bank account with units of either pounds
sterling (GBP) or Euro. (We are ignoring any transaction or currency exchange charges.)
1
We now apply this to the path indicated in the following figure.
0
.
262
121
133
.
1
1
.
1
108
.
9
99
89
.
1
81
72
.
9
1
.
56
110
90
3
.
44
100
From the formulae above, we calculate the strategy
2
•
At time 0,
φ
0
=
1
.
56
-
9
.
77
100
×
5 =
-
0
.
4107
,
ψ
0
B
0
=
11
×
9
.
77
-
9
×
1
.
56
2
×
1
.
05
= 44
.
51
•
At time 1,
φ
1
=
0
.
262
-
5
.
76
110
×
5 =
-
0
.
2500
,
ψ
1
B
1
=
11
×
5
.
63
-
9
×
0
.
262
2
×
1
.
05
= 29
.
06
•
At time 2,
φ
2
=
-
1
.
1
121
×
5 =
-
0
.
0455
,
ψ
2
B
2
=
11
×
1
.
1
2
.
1
= 5
.
76
Here
a, b
are the two possible values of the derivative after the next step. A portfolio (
φ, ψ
)
that replicates the payoff can be found that satisfies the conditions
φ
n
ux
n
+
ψ
n
(1 +
r
)
B
n
=
a
φ
n
dx
n
+
ψ
n
(1 +
r
)
B
n
=
b
where the risk-free asset has a unit price of
B
n
at step
n
. Solving this set of two simultaneous
equations with two unknowns gives
φ
n
x
n
=
a
-
b
u
-
d
,
and
ψ
n
B
n
=
bu
-
ad
u
-
d
×
1
1 +
r
Hence for the case where
u
= 1 + 10%,
d
= 1
-
10% and 1 +
r
= 1 + 5%
φ
n
x
n
=
5(
a
-
b
)
,
and
(1)
ψ
n
B
n
=
11
b
-
9
a
2
×
1
.
05
(2)
Note that is only the total amount held in riskless assets
ψ
n
B
n
that is important, not the
actual number
ψ
n
, e.g. this could be cash in a bank account with units of either pounds
sterling (GBP) or Euro. (We are ignoring any transaction or currency exchange charges.)
1
We now apply this to the three possible prices of the binomial model after 2 time-steps,
viz.
81, 99 and 121. From the eqn (1) above, we calculate the number of shares
•
For
x
2
= 121,
φ
2
=
0
-
1
.
1
121
×
0
.
2
=
-
0
.
0455,
•
For
x
2
= 99
,
φ
2
=
1
.
1
-
20
.
9
99
×
0
.
2
=
-
1,
•
For
x
2
= 81
,
φ
2
=
20
.
9
-
37
.
1
81
×
0
.
2
=
-
1.
The values of
φ
2
=
-
1 for the two lowest prices correspond to the option writer hedging by
short selling 1 share (equivalent to owning
-
1 units of the stock). Short selling is mathe-
matically equivalent to a negative amount of the assets, i.e., the practice of selling assets,
1
If desired,
one could calculate definite values of
ψ
n
for a particular starting value of
B
0
,
e.g.
(
B
0
, B
1
, B
2
, B
3
) = (1
,
1
.
05
,
1
.
1025
,
1
.
1576).
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such as securities, that have been borrowed from a third party (e.g. a broker). The borrowed
assets need to be returned to the lender at a later date. The prices
x
2
= 81 and 99 are so far
below the strike price of 110 that there will certainly be a payoff (there is no chance of going
above the strike price after the third and final time-step). Thus the holder of the European
put option will definitely exercise the option at the expiry date. Correspondingly the writer
of the option will certainly have to pay the strike price to the holder for 1 unit of stock.
However the +1 shares of the stock that the writer receives ‘cancels’ the
-
1 units of stock
from short selling, in other words the option writer passes on the stock to the lender thus
returning what was borrowed. (Note that under conditions for which the option is certain
to be exercised we have the same situation as for a forward contract, whose discounted
pricing was described in previous lectures.)
Conversely, if the stock price were to be so high that there is no chance that the put option
will be exercised then the writer does not need to do any short selling (or anything else).
This is not quite true for
x
2
= 121 but the only possible payoff is small and hence
φ
2
is
close to zero.
A financial institution that sells an option for an initial value
V
, and then follows a hedging
strategy as the stock price varies does not lose or gain (risk-free portfolio). They can actu-
ally sell the option for slightly more than
V
, and make a (riskless) profit on the margin.
Some exercises for the reader
You should use a spread-sheet programme for some of the repetitive calculations, e.g. from
iii) onwards.
i) Redo parts b) and c) above for a European put option with strike price 108 at time-step 3.
(Answers are not given but obviously there are only small changes cf. a strike price of 110. )
ii) Redo part a) above for an interest rate
r
= 4%.
iii) Redo b) and c) above for an interest rate
r
= 4% and a European put option with strike
price 110, or 108, at time-step 3.
iv) Redo some, or all, of the above for a European call option.
v) Show that there is Put-call parity.
vi) Warning: this part is open-ended and has not been checked—you may need to adjust
some of the parameters. Set up a binomial tree model, on a spreadsheet, with about 12,
or more, time-steps. [Take the interest rate
r
as the input and calculate the risk neutral
probability
p
from it.] Compare the value of options that you calculate at various times with
the predictions of the Black-Scholes approach. (An Excel version of a B-S calculator is on
the course website.) These should agree in the limit of a large number of small time-steps
if you match the volatilities.
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(d) Determining a suitable hedging strategy for the option writer along a par-
ticular path
We now apply eqns (1) and (2) to the path indicated in the following figure.
φ
n
x
n
=
5(
a
-
b
)
,
and
(1)
ψ
n
B
n
=
11
b
-
9
a
2
×
1
.
05
(2)
Note that is only the total amount held in riskless assets
ψ
n
B
n
that is important, not the
actual number
ψ
n
, e.g. this could be cash in a bank account with units of either pounds
sterling (GBP) or Euro. (We are ignoring any transaction or currency exchange charges.)
1
We now apply this to the path indicated in the following figure.
0
.
262
121
133
.
1
1
.
1
108
.
9
99
89
.
1
81
72
.
9
1
.
56
110
90
3
.
44
100
From the formulae above, we calculate the strategy
2
•
At time 0,
φ
0
=
1
.
56
-
9
.
77
100
×
5 =
-
0
.
4107
,
ψ
0
B
0
=
11
×
9
.
77
-
9
×
1
.
56
2
×
1
.
05
= 44
.
51
•
At time 1,
φ
1
=
0
.
262
-
5
.
76
110
×
5 =
-
0
.
2500
,
ψ
1
B
1
=
11
×
5
.
63
-
9
×
0
.
262
2
×
1
.
05
= 29
.
06
•
At time 2,
φ
2
=
-
1
.
1
121
×
5 =
-
0
.
0455
,
ψ
2
B
2
=
11
×
1
.
1
2
.
1
= 5
.
76
1
If desired, one could calculate definite values of
ψ
n
for a particular starting value of
B
0
, e.g.
(
B
0
, B
1
, B
2
, B
3
) = (1
,
1
.
05
,
1
.
1025
,
1
.
1576).
2
Short selling is mathematically equivalent to buying a ‘negative’ amount of the assets; it is the practice
of selling assets, usually securities, that have been borrowed from a third party (e.g. a broker) with the
intention of buying identical assets back at a later date to return to the lender. This definition has been
adapted from Wikipedia.
From the formulae above, we calculate the strategy
•
At time 0,
φ
0
=
1
.
56
-
9
.
77
100
×
5 =
-
0
.
4107
,
ψ
0
B
0
=
11
×
9
.
77
-
9
×
1
.
56
2
×
1
.
05
= 44
.
51
•
At time 1,
φ
1
=
0
.
262
-
5
.
76
110
×
5 =
-
0
.
2500
,
ψ
1
B
1
=
11
×
5
.
63
-
9
×
0
.
262
2
×
1
.
05
= 29
.
06
•
At time 2,
φ
2
=
-
1
.
1
121
×
5 =
-
0
.
0455
,
ψ
2
B
2
=
11
×
1
.
1
2
.
1
= 5
.
76
These results can be summarised in a table, where Π
n
=
φ
n
x
n
+
ψ
n
B
n
-
V
is the total value
of the portfolio (of the option writer)
step,
n
=
0
1
2
3
x
n
100
110
121
108.9
V
n
3.44
1.56
0.262
1.1
φ
n
x
n
-
41
.
07
-
27
.
5
-
5
.
5
–
φ
n
-
0
.
4107
-
0
.
25
-
0
.
0455
–
ψ
n
B
n
44.51
29.06
5.76
–
φ
n
x
n
+
ψ
n
B
n
3.44
1.56
0.262
–
Π
n
0
0
0
–
Buying and selling shares as dictated by Eqn. (1) and the riskless asset according to Eqn.
(2) replicates the value of the option at each time-step, i.e. making Π
n
= 0 at each step
means that
V
=
φ
n
x
n
+
ψ
n
B
n
The concept of a replicating portfolio is important in financial mathematics.
Comparison with other Financial Mathematics texts
The compounding of interest at each time-step to give a factor of (1 +
r
)
3
after 3 steps
has been used to keep things simple in the binomial model.
Most texts use continuous
compounding, i.e., riskless assets increase by
e
3
r
after 3 time periods, and the exponential
form is mathematically more convenient.
The difference is small for
r
1.
For the
example in a binomial market with interest rate
r
= 5%, as used above, a calculation using
continuously compounded interest yields a risk neutral probability of
p
= 0
.
7564 cf. 0
.
75,
and an initial value for the European put option with a strike price of 110 of
V
= 3
.
28 cf.
3
.
44.
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The expectation value is usually written in a more complicated form; at time
t
the value
for a European put option with a strike price of
K
which expires at time
T
is given by
V
t
(
x
t
) =
e
-
r
(
T
-
t
)
E
[max(
K
-
x
T
,
0)
|
x
t
]
.
(3)
For a call option, with the same strike price and expiry date,
V
t
=
e
-
r
(
T
-
t
)
E
[max(
x
T
-
K,
0)
|
x
t
]
.
(4)
Or for continuous probability distribution functions, eqn 6.20 in the book of Johnson
et al
gives the initial value of an option in terms of the expectation value of the payoff at expiry,
V
0
[
x
0
, K, T
] =
h
V
T
[
x
T
, K
]
i
x
T
=
Z
∞
0
V
T
[
x
T
, K
]
p
[
x
T
|
x
0
]
dx
T
.
(5)
where
p
[
x
T
|
x
t
] is the conditional probability of a final price
x
T
for a starting price
x
0
at
t
= 0, and
K
is used for the strike price instead of
X
; N.B. the interest rate has been set
to zero for simplicity (at the beginning of section 6.4.2 of the book). For a European call
option this gives
V
t
=
Z
∞
K
(
x
T
-
K
)
p
[
x
T
|
x
t
]
dx
T
.
(6)
Risk Neutral Pricing
(N.B. the change of notation from
x
to
S
for the price of the asset.)
This note aims to clarify the crucially important difference between the risk-neutral proba-
bilities (such as those used in the binomial tree model) and real-world, or physical, proba-
bilities. In a risk-neutral world investors are assumed to require no extra return on average
for bearing risks. The valuation of an option, or other derivative, in this risk-neutral world
gives the correct price for the derivatives in all worlds (and avoids the need to know real-
world, or physical probabilities). This is closely related to the elimination of the real-world
drift in the price of the asset in the derivation of the Black-Scholes equation, and its re-
placement by a risk-neutral drift (which is not the actual expected drift but equals the cost
of financing the asset and its yield). To repeat this crucial point: this approach does not
try to predict or take expectations of real world events—this was the great insight of Black
and Scholes; others had already established similar diffusion equations but thought that
drift was determined by risk preferences.
Consider an asset currently priced at
S
whose price one period later can be
Su
with prob-
ability
q
, or
Sd
with probability 1
-
q
.
How should that asset be priced in a world of
risk-averse investors? Any risky asset is priced as the discounted value of its future expec-
tation because of the inherent risk. The expected future price of the asset is
qSu
+(1
-
q
)
Sd
,
and the current price can be written as
S
=
qSu
+ (1
-
q
)
Sd
1 +
k
.
(7)
where
k
is the risky discount factor consisting of the risk free rate
r
plus a risk premium.
It turns out that there values
p
and 1
-
p
that can be substituted for
q
and 1
-
q
which
permit us to change
k
to the risk-free rate
r
, in the absence of arbitrage. Consider a market
with no arbitrage opportunities in which there are only two assets: the stock and a risk-free
bond. There would be an arbitrage opportunity if one could borrow an amount equal to the
price of the stock at the risk-free rate, purchase the stock and guarantee earning at least
the risk-free rate one period later. This would arise if
u > d >
1 +
r
(where the factor
u
is
greater than
d
by definition). Also 1 +
r > u
cannot arise since then it would be possible
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to short the stock and use the proceeds to buy the bond to obtain a return of 1 +
r
-
u
.
Consequently, no arbitrage requires that
u >
1 +
r > d.
(8)
Given these inequalities, it is possible to find weights
p
and 1
-
p
consistent with
pu
+ (1
-
p
)
d
= 1 +
r
. Hence
S
=
pSu
+ (1
-
p
)
Sd
1 +
r
(9)
Comparison with eqn 7 shows that the price of an asset can be restated by changing the
probabilities and discounting at the risk-free rate.
The only information required is the
volatility (as represented by
u
and
d
) and the risk-free rate. This no-arbitrage argument
shows that it is possible to state the price of the asset in terms of risk-neutral probabilities
with discounting at the risk-free rate.
Calling
p
and 1
-
p
risk neutral probabilities is a
source of much confusion.
There is a definite relationship between the probability in the binomial tree model and the
actual probability. Define the stocks risk premium as
φ
= (
E
-
r
)
/σ
where
E
is the expected
return on the stock, defined as
E
=
qu
+ (1
-
q
)
d
-
1. The stocks variance is defined as
σ
2
=
q
(1
-
q
)(
u
-
d
)
2
. This is the volatility of a one period binomially distributed variable
that can go up to u or down to d (which is straightforward to calculate in the usual way).
Substituting these values into the risk premium and noting that
p
= (1 +
r
-
d
)
/
(
u
-
d
), we
obtain the result
p
=
q
-
φ
p
q
(1
-
q
)
(10)
Thus, the binomial probability is the actual probability minus the risk premium times the
square root term arising from the volatility of a binomial process. In short, knowing the
binomial probability
p
= (1+
r
-
d
)
/
(
u
-
d
) is sufficient without having to know (or estimate)
the risk aversion,
φ
, and the actual probability
q
.
Note that we have not considered options in this treatment. It is easily shown that a call
(or put) option can be replicated by positions in the asset and the risk-free bond; therefore
options do not change the nature of the market or the treatment given here, provided that
they are properly priced allowing no arbitrage. This point has been made using a simple
binomial framework but it can also be shown for continuous-time models, albeit with more
mathematical complexity, e.g., in the book by S.N. Neftci,
An Introduction to the Mathe-
matics of Financial Derivatives
. San Diego: Academic Press (1996) Chapters 1, 14, and
15. [This way of looking at risk-neutral probabilities borrows extensively from an (online)
teaching note by Prof. D.M. Chance, Louisiana University. See also the Hull’s book.]
Implied Volatility
The Black-Scholes solution gives the price of an option for a given strike price and volatility.
However, if the assumptions of the B-S model are thought not to be valid, or for other
reasons, the value of the option can differ from the B-S value. A common way to characterise
this is to quote the volatility (assumed constant in time) that would have to be put into
the analytic solutions of the B-S equation in order to match the observed value.
This is
the implied volatility.
As a crude example of this the following spreadsheet uses a one-
step binomial model to calculate the value of an option for various strike prices, and then
evaluates the volatility that would give the same value when input into the solutions of
the B-S equation. (A separate spreadsheet that implements the B-S solutions is available
on the course website, or at Wolfram alpha). Clearly a model with only two possible final
prices is not the same as assuming a Gaussian distribution, hence the implied volatility is
not constant. The plot of implied volatility against strike price exhibits a ‘frown’, whereas
other cases can lead to a ‘volatility smile’.
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Implied volatility
= volatility that gives the same price according
to the Black-Scholes formula.
One-step binomial tree model with the following parameters:
u= 1.16
d= 0.84
r= 0.01
for 1 step.
p= 0.5314
p= (exp[rT]-d)/(u-d) = (exp[r]-d)/(u-d)
Call/put prices calculated from the binomial model with starting price =$50.
Strike price
Call price
Put price
Implied Volatility
($)
($)
($)
%
42
8.42
0.00
44
7.37
0.93
58.8
46
6.31
1.86
66.6
48
5.26
2.78
69.5
50
4.21
3.71
69.2
52
3.16
4.64
66.1
54
2.10
5.57
60.0
56
1.05
6.50
49.0
58
0.00
7.42
From JC Hull, Options, Futures and other Derivatives, Sect. 19.8
CJ Foot, Financial Physics short option, Oxford Physics, 2012.
0.0
10.0
20.0
30.0
40.0
50.0
60.0
70.0
80.0
44
46
48
50
52
54
56
Impliedvola.lity(%)
Strike price ($)
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●
The final exam format will be the same as the midterm.
○
40 questions, 2.5 points each.
○
December 15, 6-8pm, 2hours
●
For multiple blank questions, follow the instructions on formatting (in terms of integer or
decimals).
●
There are questions where I give you multiple choices: (1) 54-delta optionxx, (2)straddle,
(3) xxx, (4)xx. Just answer in integer. Do not add parentheses, commas etc. 1
Summary topics
●
Binomial tree (replication, hedging, probability)-- one step, 2-step (how to build the tree)
●
BMS model:
○
some behavior of Brownian motion and geometric Brownian motion
○
BMS option pricing formula – know how to implement it in excel, and also know how to plug things in. Know how to use put-call parity, if needed, to find the value of put from the value of call, and vice versa.
○
Whether the option is in the money, out of the money, intrinsic value
●
Greeks
○
Relation between delta and strike
○
The delta of put and call at the same maturity and strike
○
Relation between delta and stock price exposure.
○
Relation between vega and strike(delta)/maturity, the meaning of vega exposure.
○
Relation between gamma and strike(delta)/maturity
○
Know how to make a portfolio delta and vega (or gamma) neutral
○
How to make a portfolio’s delta and vega (gamma) to be within a certain limit
●
Implied vol smile
○
the shape of the smile and the distribution shape (e.g., normal, positive or negatively skewed, fat tailed)
—
S=55, N(-d1)=0.25, N(d2)=0.2, K=40 → N(-d2)=1-N(d2)=0.8
put = - S N(-d1) + K *exp(-r*t) *N(-d2)
=-55*0.25+ 40*1*0.8
=18.25
Put delta = -0.25
Compute the call option value and delta at the same maturity and strike.
call delta= N(d1)= 1- N(-d1)=0.75
call = SN(d1) - K *exp(-r*t)*N(d2)
=55*0.75 - 40*1*0.2=33.25
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Which is out of money? put is out of money
Instrinsic value for put =0. For call is 33.25-18.25=15
(F-K)*exp(-r t) = c-p
Correction: The step is 2y, not 1y.
S=200
Su=220
Sd=180
r=4%
T-t=1y
K=200 call option, → Payoff will be max(0, S-K)
(i) = max(0, 220-200)=20
(iii) B + D*220 = 20; B + D*180 =0
=> Delta = (20-0)/(220-180)=0.50
B=-D*180=-0.5*180=-90
B=20-D*220=-90
Value = -90*exp(-0.04*2)+0.5*200=16.92
For American options, at each step, check the exercise value. Suppose it is a call option, the exercise value = max(0, S-K)=max(0,200-200)=0
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–
Out of money call and in the money put have high strike.
→ b, 30-delta call (The smaller is delta, the more out of the money).
Also remember, without dividends, 30-delta call and 70-delta put have the same strike.
An option portfolio has a delta of 100 and a vega of 50.
(a) If you can use the stock and a delta-neutral straddle with vega of 10 to hedge, how many shares of the stock ( ) and the straddle ( ) you need to neutralize the portfolio’s delta and vega?
(b) If you only have access to a 50-delta call and a 50-delta put, both with vega of 5, how many shares of the call ( ) and the put ( ) you need to neutralize the portfolio’s delta and vega?
(c ) If you have the stock and the delta-neutral straddle with vega of 10 to hedge, what’s the minimum number of shares you need to trade on each to reduce delta to be within (+/- 50), and reduce vega within (+/- 30)?
Vega first (option is expensive so we want to deal with options first). I want to reduce by 20. That means I need to short 20/10=2 of the straddle.
Delta: I need to reduce delta by 50, which I can do by shorting 50 of the stock. (-50) stock and (-2) on straddle.
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[d] If you have the stock and a 50-delta call with vega of 10 to hedge, what’s the minimum number of shares you need to trade on each to reduce delta to be within (+/- 50), and reduce vega
within (+/- 30)?
Vega first: Reduce vega by 20. We need to short -2 shares of the call.
But short 2 call generate delta of -2*0.5=-1. My original delta is 100. The new delta becomes 100-1=99. To reduce this within (+/-50), I can just short 49 shares of the stock.
(a)
stock has 0 vega, delta 1→ hedge delta. Short 100 shares (-100) of stock → delta is neutral.
straddle has 0 delta and vega of 10, → -50/10=-5 (-100) stock (-5) for straddle
b. call delta 0.5, vega = 5
put delta = -0.5, vega =5
(1)Delta: c*0.5+p*(-0.5)+100=0
Vega: c*5 + p*5 + 50=0 ⇒
(2) c*.5 + p*.5 + 5=0
(1)+(2): c+105=0 → c=-105
(1)-(2): -p +95=0 → p =95
r=5%, q=3%, sigma=20%. tau=¼
log(S_T/S_t) is normal with mean ((r-q)*tau- 0.5*sigma^2*tau), variance (sigma^2*tau)
mean= (5%-3%)*¼-(½)*.2^2*(¼)=0
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variance = 0.2^2*0.25=0.01
2(a), mean is 5%, vol is 20%sqrt(2)=28.28%
(b)
mean=2%, vol= 10%*sqrt(2) = 14.14%
(c)
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Lists of Topics for the Final Exam7 1.
Binomial tree: How to build
a multi-step binomial tree and how to price and hedge
options (European and American) based on the tree. a.
Example: Q1 from 2009 spring final, Q1b-c from 2008 fall final, Q1b-c from 2008 spring final 2.
Understand the basic meaning of the Black-Scholes model: Given a dynamics specification, understand its implication for distribution. [With concrete numbers] a.
Example: If /
t
t
t
dS
S
dt
dW
µ
σ
=
+
, what’s the distribution of ?
Normal with mean (
), variance of (
). b.
If , what is the distribution of ln
/
T
t
S
S
?
Normal with mean and variance c.
What’s the distribution of (
)
,
,
,
T
t
T
T
t
W
W a
bW
b W
W
σ
+
−
…?
d.
Q1a from 2008 fall final. Q1&2 from 2007 fall final 3.
Know how to use the Black-Scholes formula. a.
Compute call and put option prices based on the BS formula, and know that
(
)
(
)
1
.
N
x
N
x
−
= −
I’ll tell you the value of (
)
N
x
. b.
Put-call parity. Use put-call parity to solve for the value of a call, a put, the underlyng forward, the underlying spot,… given other information. c.
Example: Price a one-year 25-delta put option using the BSM formula. Also price the call option at the same strike. The current underlying stock price is $100. The stock pays no dividend and the interest rate is 0. The stock has a volatility of 20%. The option has an N(d2)=.70 and strike of $90. Answer: Since put delta is 25 and no dividend, N(-d1)=0.25 and N(d1)=1-
0.25=0.75. Since N(d2)=.7, N(-d2)=1-N(d2)=1-0.7=0.3. Since there is no dividend and rate is zero, forward F=S=100. The put option value is p=-SN(-d1)+KN(-d2)=-100*.25+90*0.3=-25+27=$2. The call option value is c=SN(d1)-KN(d2)=100*.75-90*0.7=$12. (Note that I did not write the r,q terms because the question assumes zero rates and zero dividends). d.
More examples: Q3 from 2007Fall, Q1d from 2008Spring, Q1e from 2008Fall,Q2a from 2009Spring 4.
Greeks and hedging: a.
Know how to achieve a certain exposure. Example: An investor has an option portfolio with delta 2, vega 200. He wants to achieve delta and vega neutral using the underlying stock and a straddle with zero delta and vega of 6. How much of the underlying and the straddle he needs to achieve his objective?
He needs to short 200/6 straddle and short 2 share of stock.
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Example: An investor has an option portfolio with delta 2, gamma 0.5. He wants to achieve delta and gamma neutral using the underlying stock and a call option with delta of 0.5 and gamma of 0.1. How much of the underlying and the option he needs to achieve his objective?
He needs to short 0.5/.1=5 of the call option to neutralize gamma. This reduces his delta to 2-0.5*5=-0.5. Hence, he needs to long another half share of the stock to neutralize the delta. Example: An investor has an option portfolio with delta 1.5, gamma 0.5. He wants to bring his delta within +/-0.5 and bring his gamma exposure to +/-0.1 using the underlying stock and a call option with delta of 0.5 and gamma of 0.1. How much of the underlying and the option he needs trade at a minimum to achieve his objective?
He needs to short (0.5-0.1)/.1=4 of the call option to reduce his gamma exposure to 0.1. This reduces his delta to 1.5-0.5*4=-0.5, which is within his delta target. So he does not need to take any additional position in the stock. Example: Q2 from 2008fall, Q3 from 2009 spring b.
Know the basic link between delta and moneyness, as well as the relative exposure of different contracts (what contracts have higher/lower delta, higher/lower vega, higher/lower gamma) Example: The current spot price is $100, consider three call options at 80, 100, and 120, which one has the highest delta exposure? Which one has the highest vega exposure?
at-the-money option (100-strike) has the highest vega. 80-strike call is in the money and has the highest delta. c.
Write down the delta, vega, gamma formulae for the Black-Scholes model. Example: Q2 from 2009Spring, Q1e from 2008 fall. Q5 from 2007 fall. d.
The meaning of delta, vega, gamma. Example: Consider an option portfolio on IBM has a delta of 1.5 million, and vega of -$6 million. •
If the IBM stock price goes up by $1 dollar, how much the option portfolio value will change?
The portfolio value will go up by 1.5 million. •
If the IBM stock return volatility increases by one percentage point, how much the option portfolio value will change?
The portfolio value will decline by 6*0.01=0.06million or 60thousand dollars. •
The portfolio is net long or short in IBM?
Long •
The portfolio is net long or short options?
Short 5.
The basic meaning of the implied volatility surface: Map the smile to a distribution. a.
I draw different implied vol smiles/smirks, you tell me their implication in terms of the underlying return risk-neutral distribution.
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b.
I gave you some description of the return distribution, you plot implied vol against moneyness/strike. c.
What is the appropriate way to measure moneyness? –remember standardization Example: Q4 from 2009 spring, Q3 from 2008 fall, 2008 spring.
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Forward and Futures: Summary and Exercises
Liuren Wu
Forward payoff:
If you long a forward on an asset with a delivery price K, and the underlying spot price
of the asset at expiry (time T), then the payoff you have from this investment is
(
S
T
-
K
)
. If you short this
forward contract, your payoff is
(
K
-
S
T
)
.
Examples:
•
Suppose you are long 1 million forward contracts on Google with a delivery price of $600 and an expiry
date of March 14th, 2007. Plot your payoff as a function of possible Google stock prices on March 14th,
2007 (time
T
). (Today is sometime 2006).
Answer: The payoff as a function of the stock price at time
T
is:
(
S
T
-
600
)
×
1 million.
400
450
500
550
600
650
700
750
800
-200
-150
-100
-50
0
50
100
150
200
Google price at time T, S
T
Forward Payoff, millions
•
Following the above example, suppose the Google stock price on 3/14 is $650, what will be your
payoff? If you are short 1 million of this contract instead, what will be your payoff?
Answer: If
S
T
=
650
,
the payoff for a long position is 650
-
600
=
50 million. The payoff of a short (1
million) position is
-
50 million.
Forward position value:
If you long a forward on an asset with a delivery price
K
and expiry date
T
.
The current forward price at the same expiry date is
F(t,T)
. The current riskfree interest rate (continuously
compounding) over the same maturity is r
(
t
,
T
)
. Then, the current (time-
t
) value of your forward position is
e
-
r
(
t
,
T
)(
T
-
t
)
(
F
(
t
,
T
)
-
K
)
. The value of a short forward position is the opposite:
-
e
-
r
(
t
,
T
)(
T
-
t
)
(
F
(
t
,
T
)
-
K
)
.
Examples:
•
Suppose you short one million forward contracts with a delivery price of $1.30. The current forward
price at the same expiry date is $1.60. Assume a continuously compounding riskfree rate of 5%, and a
maturity for the forward contract of two years.
Answer: Value of short 1 million contract with delivery price of $1.30 is (in millions):
-
e
-
r
(
t
,
T
)(
T
-
t
)
(
F
(
t
,
T
)
-
K
) =
-
e
-
0
.
05
*
2
(
1
.
60
-
1
.
30
) =
-
0
.
27145
.
Forward pricing
:
The forward price of a contract F
(
t
,
T
)
should be equal to the cost of buying the underlying
security and carrying over to the expiry date. The usual cost includes interest cost (as you are buying now
1
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instead at expiry) and storage cost (if it is costly to store).
There are also benefits to buy today, such as
dividend receipts from buying a stock today, and interest receipts from buying a foreign currency today. If
the forward price is higher than the cost of buy and carry, you can short the forward contract as the current
(high) price and long the replication portfolio (buy today and carry it over to expiry) to lock in a profit for
free. If the forward price is lower than the buy-carry cost, you can long the forward contract at the current
(low) forward price and short the replication (short sell the underlying security).
Examples:
•
Consider a forward contract on the IBM stock. Assume that the stock is currently traded at $100 per
share. IBM pays quarterly dividend of $1 per share in the next 5 years (no change). The continuously
compounding interest rate is flat at 5% (across all maturities). Compute the forward price of the IBM
stock at one-, and two-year maturities.
Answer: For IBM stock, the cost is interest cost, and the benefit is dividends. The forward prices at
one- and two-year maturities are, respectively,
F
(
t
,
t
+
1
)
=
100
e
0
.
05
×
1
-
parenleftBig
1
e
0
.
05
×
3
/
4
+
1
e
0
.
05
×
2
/
4
+
1
e
0
.
05
×
1
/
4
+
1
parenrightBig
=
101
.
05
.
F
(
t
,
t
+
2
)
=
100
e
0
.
05
×
2
-
parenleftBig
1
e
0
.
05
×
7
/
4
+
1
e
0
.
05
×
6
/
4
+
1
e
0
.
05
×
5
/
4
+
1
e
0
.
05
×
4
/
4
parenrightBig
-
parenleftBig
1
e
0
.
05
×
3
/
4
+
1
e
0
.
05
×
2
/
4
+
1
e
0
.
05
×
1
/
4
+
1
parenrightBig
=
102
.
16
.
•
Following the above example, if the market quote for the one-year forward contract is at $103.00 (with
zero bid-ask spread), what can you do to lock in a profit?
Answer: Since the market quote (103) is higher than the replication (buy & carry) cost (101.05), we
should sell high and buy low to lock in a profit. “Sell high” here means to go short on the forward
contract, “buy low” means to buy the stock and carry it over to the expiry date (one year later). Specif-
ically,
today
we (1) sign the forward contract to take the short position on 1 share of IBM stock, (2)
borrow $100 from a bank at an interest rate of 5%, and buy one share of IBM stock. [We should do
millions of shares of this, but let’s use one share as an example].
At expiry (one year later),
the short
forward contract expires, and we need to sell 1 share of IBM for $103 (the delivery price) to satisfy our
short-forward obligation. Since we bought one share of IBM one year ago, we can use that one share of
IBM to cover the selling position. Furthermore, we need to pay back the $100 loan, which will become
100
e
0
.
05
×
1
=
105
.
13 where 5.13 is the interest cost for your borrowing. Finally, buying the stock one
year ago also generates four $1 dividend payments at each quarter. If we save these payments in the
bank at the same 5% rate, you will receive
(
1
e
0
.
05
×
3
/
4
+
1
e
0
.
05
×
2
/
4
+
1
e
0
.
05
×
1
/
4
+
1
)
=
4
.
0761 in cash
at the expiry ($4 dividend plus interest). Take everything together, you use your IBM stock to satisfy
your forward obligation, hence you do not have any IBM exposure left–whether the IBM stock price
is at $400 or $40 does not make a difference to you. Consolidating your cash in- and out-flows leaves
you: 103
-
105
.
13
+
4
.
0761
=
1
.
9461
.
That is what you end up with from all the trading, which is not
bad given that it is completely riskfree (it does not depend on how IBM stock performs).
•
Consider another case where the market quote for the one-year forward contract is $99. How can you
trade to lock in a profit?
Answer: In this case, the forward quote is lower than the replication cost, so we would long the forward
contract and short the replication, i.e., short selling the IBM stock. By short selling 1 share of IBM
stock, we receive $100, which we save in the bank. At expiry, we pay $99 to receive 1 share of IBM
2
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stock to satisfy the forward obligation. Since we borrowed one share of IBM stock one year ago, we
return this IBM share to the original lender, and we pay the lender back all the dividend payments during
the past year (plus interests), which amounts to
(
1
e
0
.
05
×
3
/
4
+
1
e
0
.
05
×
2
/
4
+
1
e
0
.
05
×
1
/
4
+
1
)
=
4
.
0761
.
That is, by borrowing 1 share of IBM stock one year ago, we need to return the share plus all the
dividends we receive (with interests). The money $100 we saved one year ago becomes 100
e
0
.
05
×
1
=
105
.
13
.
In total, our cashflow is:
-
99
-
4
.
0761
+
105
.
13
=
2
.
0539
.
That’s what you are left with after
the exercise, and it does not depend on how IBM stock price moves since you do not have any IBM
stock exposure left (you bought a share and gave it back to the original lender, and hence have nothing
left on IBM).
•
Consider a two-year forward contract on the dollar price of pound. The current exchange rate is $1
.
90
per pound. The continuously compounding US dollar interest rate is flat at 4%, and the corresponding
pound rate is at 5%. What should be the forward price at the two-year maturity?
Answer:
F
(
t
,
T
) =
S
t
e
(
r
d
-
r
f
)
(
T
-
t
)
=
1
.
90
×
e
(
0
.
04
-
0
.
05
)
×
2
=
1
.
8624
.
Here, buying pound costs dollar.
Hence, dollar rate is the cost, pound rate is the benefit.
•
Following the above example, if the market quote for the two-year forward is $1
.
89
,
what you can do
to lock in a profit?
Answer: The forward price is higher (1.89) than the replication cost (1.86). Hence, I would short the
forward today and buy the pound. Suppose I short the forward for one pound, I would need to buy
1
e
-
0
.
05
×
2
=
0
.
90484 pound today and save this pound in the bank to earn a 5% pound interest rate.
0
.
90484 grows at 5% for two years becomes 1: 0
.
90484
e
0
.
05
×
2
=
1
.
0
.
To buy 0
.
90484 pound, I need to
spend 0
.
90484
×
1
.
90
=
1
.
7192 dollars as each pound is worth $1
.
90 dollar now. I would borrow this
money ($1
.
7192) from the bank and pay dollar interest rate (4%) on this debt.
At expiry, my pound savings grow to one pound, with which I can satisfy my short forward obligation
and receive the delivery price of $1
.
89. My borrowed dollar (debt) $1
.
7192 grows to $1
.
7192
e
0
.
04
×
2
=
1
.
8624 based on the 4% dollar rate. My net cashflow is: $1
.
89
-
1
.
8624
=
0
.
0276, which is a small
profit that I lock in with no currency exposure.
The key in this example is that we do not need to buy one pound to have one pound at expiry because
pound can grow over time with its interest rate.
•
Consider an opposite example where the two-year forward market quote is $1
.
84
.
What can you do to
lock in a profit?
Answer: Since the market quote (1.84) is lower than the replication cost (1.86), I would go long on the
forward and short sell the pound. Long the forward implies that I would receive one pound at expiry.
Hence, I can borrow some pound today knowing that I would have 1 pound to pay back at expiry.
But I cannot borrow one pound as the debt will grow. I can only borrow 1
e
-
0
.
05
×
2
=
0
.
90484 pound
as this debt will grow to 1 pound at expiry with the 5% rate on pound. With the borrowed pound, I
sell it to the spot market to make $1
.
90 per pound and hence 0
.
90484
×
1
.
90
=
1
.
7192 dollars total. I
save these dollars in the bank and make 4% dollar rate on them. At expiry, the dollar saving grows to
$1
.
7192
e
0
.
04
×
2
=
1
.
8624. My pound debt grows to one pound. To satisfy the long forward obligation, I
spend $1
.
84 (the delivery price) to buy one pound and use this pound to pay back my pound debt. What
is left is $1
.
8624 I receive for my dollar savings and $1
.
84 I pay for the long forward delivery. My net
profit is $1
.
8624
-
1
.
84
=
0
.
0224, a small but nevertheless sure profit regardless of how pound or dollar
moves.
3
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Questions on Greeks
1.
Consider a call option with a strike of $70 on a stock with a current stock price of $50. Is the delta of this option closer to (i) 25, (ii) 50, or (iii) 75?
2.
Suppose we have two call options both with a strike of $70 on the same stock with a spot of $50. Option A
has a maturity of 1 month, option B has a maturity of 1 year. Which option has a higher delta?
3.
Suppose now volatility doubles (but the spot is still $50), should the delta of the above options increase or
decrease?
4.
How do your answers change for questions 1,2,3 if the option is a put?
Answers:
1.
Consider a call option with a strike of $70 on a stock with a current stock price of $50. Is the delta of this option closer to (i) 25, (ii) 50, or (iii) 75?
Answer
: This is an out-of-the-money option, hence delta should be smaller, possibly 25 (i).
For a put option
, high strike is in the money. Hence, it should be closer to 75. (negative of course).
2.
Suppose we have two call options both with a strike of $70 on the same stock with a spot of $50. Option A
has a maturity of 1 month, option B has a maturity of 1 year. Which option has a higher delta?
Answer
: Both are out-of-the-money options. 1-month
is more out of the money than 1-year (because it is more difficult to get to 70 from 50 within 1 month than within 1 year). Both options have low delta (lower than 50), the more out of money one has lower
delta. Hence, the 1-month option has lower delat. 1-yr
option has higher delta.
For put options
, the delta are higher than 50. More out of money will move away fro 50 and hence become higher. So 1-month has higher delta (in absolute term). 1-year has a delta closer to 50 and hence is lower in absolute magnitude.
3.
Suppose now volatility doubles (but the spot is still $50), should the delta of the above options increase or
decrease?
Answer: Volatility is higher, it becomes easier to reach 70. The delta should become closer to 50. Hence, the delta of the call will increase.
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For put options
, the delta is higher than 50 because it
is in the money. Moving closer to 50 means that the delta of the put will decline in absolute magnitude.
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Problem 9.9 Suppose that a European call option to buy a share for $100.00 costs $5.00 and is held until maturity. Under what circumstances will the holder of the option make a profit? Under what circumstances will the option be exercised? Draw a diagram illustrating how the profit from a long position in the option depends on the stock price at maturity of the option.
Ignoring the time value of money, the holder of the option will make a profit if the stock price at maturity of the option is greater than $105. This is because the payoff to the holder of the option is, in these circumstances, greater than the $5 paid for the option. The option will be exercised if the stock price at maturity is greater than $100. Note that if the stock price is between $100 and $105 the option is exercised, but the holder of the option takes a loss overall. The profit from a long position is as shown in Figure S9.1. Figure S9.1
Profit from long position in Problem 9.9 Problem 9.10 Suppose that a European put option to sell a share for $60 costs $8 and is held until maturity. Under what circumstances will the seller of the option (the party with the short position) make a profit? Under what circumstances will the option be exercised? Draw a diagram illustrating how the profit from a short position in the option depends on the stock price at maturity of the option.
Ignoring the time value of money, the seller of the option will make a profit if the stock price at maturity is greater than $52.00. This is because the cost to the seller of the option is in these circumstances less than the price received for the option. The option will be exercised if the stock price at maturity is less than $60.00. Note that if the stock price is between $52.00 and $60.00 the seller of the option makes a profit even though the option is exercised. The profit from the short position is as shown in Figure S9.2.
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Figure S9.2
Profit from short position in Problem 9.10 Problem 9.12 A trader buys a call option with a strike price of $45 and a put option with a strike price of $40. Both options have the same maturity. The call costs $3 and the put costs $4. Draw a diagram showing the variation of the trader’s profit with the asset price.
Figure S9.4 shows the variation of the trader’s position with the asset price. We can divide the alternative asset prices into three ranges: a)
When the asset price less than $40, the put option provides a payoff of 40
T
S
−
and the call option provides no payoff. The options cost $7 and so the total profit is 33
T
S
−
. b)
When the asset price is between $40 and $45, neither option provides a payoff. There is a net loss of $7. c)
When the asset price greater than $45, the call option provides a payoff of 45
T
S
−
and the put option provides no payoff. Taking into account the $7 cost of the options, the total profit is 52
T
S
−
. The trader makes a profit (ignoring the time value of money) if the stock price is less than $33 or greater than $52. This type of trading strategy is known as a strangle and is discussed in Chapter 11.
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Figure S9.4
Profit from trading strategy in Problem 9.12 Problem 10.10 What is a lower bound for the price of a two-month European put option on a non-dividend-
paying stock when the stock price is $58, the strike price is $65, and the risk-free interest rate is 5% per annum?
The lower bound is 0 05 2 12
65
58
6 46
e
$
− .
× /
−
=
.
Problem 10.11 A four-month European call option on a dividend-paying stock is currently selling for $5. The stock price is $64, the strike price is $60, and a dividend of $0.80 is expected in one month. The risk-free interest rate is 12% per annum for all maturities. What opportunities are there for an arbitrageur?
The present value of the strike price is 0 12 4 12
60
57 65
e
$
− .
× /
=
. The present value of the dividend is 0 12 1 12
0 80
0 79
e
− .
× /
.
=
.
. Because 5
64
57 65
0 79
<
−
.
−
.
the condition in equation (10.8) is violated. An arbitrageur should buy the option and short the stock. This generates 64
5
59
$
−
=
. The arbitrageur invests $0.79 of this at 12% for one month to pay the dividend of $0.80 in one month. The remaining $58.21 is invested for four months at 12%. Regardless of what happens a profit will materialize. If the stock price declines below $60 in four months, the arbitrageur loses the $5 spent on the option but gains on the short position. The arbitrageur shorts when the stock price is $64, has to pay dividends with a present value of $0.79, and closes out the short position when the stock price is $60 or less. Because $57.65 is the present value of $60, the short position generates at least 64
57 65
0 79
5 56
$
−
.
−
.
=
.
in present value terms. The present value of the arbitrageur’s gain is therefore at least 5 56
5 00
0 56
$
.
− .
=
.
.
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If the stock price is above $60 at the expiration of the option, the option is exercised. The arbitrageur buys the stock for $60 in four months and closes out the short position. The present value of the $60 paid for the stock is $57.65 and as before the dividend has a present value of $0.79. The gain from the short position and the exercise of the option is therefore exactly equal to 64
57 65
0 79
5 56
$
−
.
−
.
=
.
. The arbitrageur’s gain in present value terms is exactly equal to
5 56
5 00
0 56
$
.
− .
=
.
. Problem 10.12 A one-month European put option on a non-dividend-paying stock is currently selling for 2 50
$
.
. The stock price is $47, the strike price is $50, and the risk-free interest rate is 6% per annum. What opportunities are there for an arbitrageur?
In this case the present value of the strike price is 0 06 1 12
50
49 75
e
− .
× /
=
.
. Because 2 5
49 75
47 00
.
<
.
−
.
the condition in equation (10.5) is violated. An arbitrageur should borrow $49.50 at 6% for one month, buy the stock, and buy the put option. This generates a profit in all circumstances. If the stock price is above $50 in one month, the option expires worthless, but the stock can be sold for at least $50. A sum of $50 received in one month has a present value of $49.75 today. The strategy therefore generates profit with a present value of at least $0.25. If the stock price is below $50 in one month the put option is exercised and the stock owned is sold for exactly $50 (or $49.75 in present value terms). The trading strategy therefore generates a profit of exactly $0.25 in present value terms. Problem 11.10 Suppose that put options on a stock with strike prices $30 and $35 cost $4 and $7, respectively. How can the options be used to create (a) a bull spread and (b) a bear spread? Construct a table that shows the profit and payoff for both spreads.
A bull spread is created by buying the $30 put and selling the $35 put. This strategy gives rise to an initial cash inflow of $3. The outcome is as follows: Stock Price Payoff Profit 35
T
S
≥
0 3 30
35
T
S
≤
<
35
T
S
−
32
T
S
−
30
T
S
<
5
−
2
−
A bear spread is created by selling the $30 put and buying the $35 put. This strategy costs $3 initially. The outcome is as follows: Stock Price Payoff Profit 35
T
S
≥
0 3
−
30
35
T
S
≤
<
35
T
S
−
32
T
S
−
30
T
S
<
5
2
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Problem 11.12 A call with a strike price of $60 costs $6. A put with the same strike price and expiration date costs $4. Construct a table that shows the profit from a straddle. For what range of stock prices would the straddle lead to a loss?
A straddle is created by buying both the call and the put. This strategy costs $10. The profit/loss is shown in the following table: Stock Price Payoff Profit 60
T
S
>
60
T
S
−
70
T
S
−
60
T
S
≤
60
T
S
−
50
T
S
−
This shows that the straddle will lead to a loss if the final stock price is between $50 and $70. Problem 12.9 A stock price is currently $50. It is known that at the end of two months it will be either $53 or $48. The risk-free interest rate is 10% per annum with continuous compounding. What is the value of a two-month European call option with a strikeprice of $49? Use no-arbitrage arguments.
At the end of two months the value of the option will be either $4 (if the stock price is $53) or $0 (if the stock price is $48). Consider a portfolio consisting of: shares
1
option
+∆
:
−
:
The value of the portfolio is either 48
∆
or 53
4
∆ −
in two months. If 48
53
4
∆ =
∆ −
i.e., 0 8
∆ =
.
the value of the portfolio is certain to be 38.4. For this value of ∆
the portfolio is therefore riskless. The current value of the portfolio is: 0 8
50
f
. ×
−
where f
is the value of the option. Since the portfolio must earn the risk-free rate of interest 0 10 2 12
(0 8
50
)
38 4
f e
.
× /
. ×
−
=
.
i.e., 2 23
f
=
.
The value of the option is therefore $2.23. This can also be calculated directly from equations (12.2) and (12.3). 1 06
u
=
.
, 0 96
d
=
.
so that 0 10 2 12
0 96
0 5681
1 06
0 96
e
p
.
× /
−
.
=
=
.
.
−
.
and 0 10 2 12
0 5681
4
2 23
f
e
− .
× /
=
× .
×
=
.
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Problem 12.10 A stock price is currently $80. It is known that at the end of four months it will be either $75 or $85. The risk-free interest rate is 5% per annum with continuous compounding. What is the value of a four-month European put option with a strike price of $80? Use no-arbitrage arguments.
At the end of four months the value of the option will be either $5 (if the stock price is $75) or $0 (if the stock price is $85). Consider a portfolio consisting of: shares
1
option
−∆
:
+
:
(Note: The delta, ∆
of a put option is negative. We have constructed the portfolio so that it is +1 option and −∆
shares rather than 1
−
option and +∆
shares so that the initial investment is positive.) The value of the portfolio is either 85
−
∆
or 75
5
−
∆ +
in four months. If 85
75
5
−
∆ =−
∆ +
i.e., 0 5
∆ = − .
the value of the portfolio is certain to be 42.5. For this value of ∆
the portfolio is therefore riskless. The current value of the portfolio is: 0 5
80
f
. ×
+
where f
is the value of the option. Since the portfolio is riskless 0 05 4 12
(0 5
80
)
42 5
f e
.
× /
. ×
+
=
.
i.e., 1 80
f
=
.
The value of the option is therefore $1.80. This can also be calculated directly from equations (12.2) and (12.3). 1 0625
u
=
.
, 0 9375
d
=
.
so that 0 05 4 12
0 9375
0 6345
1 0625
0 9375
e
p
.
× /
−
.
=
=
.
.
−
.
1
0 3655
p
−
= .
and 0 05 4 12
0 3655
5
1 80
f
e
− .
× /
=
× .
×
= .
Problem 13.9 A stock price has an expected return of 16% and a volatility of 35%. The current price is $38. a)
What is the probability that a European call option on the stock with an exercise price of $40 and a maturity date in six months will be exercised? b)
What is the probability that a European put option on the stock with the same exercise price and maturity will be exercised? a)
The required probability is the probability of the stock price being above $40 in six months time. Suppose that the stock price in six months is T
S
2
0 35
ln
(ln38
(0 16
)0 5 0 35
0 5)
2
T
S
.
+
.
−
. , .
.
ϕ
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i.e., ln
(3 687 0 247)
T
S
.
, .
ϕ
Since ln 40
3 689
=
.
, the required probability is 3 689
3 687
1
1
(0 008)
0 247
N
N
.
− .
−
=−
.
.
From normal distribution tables N(0.008) = 0.5032 so that the required probability is 0.4968. b)
In this case the required probability is the probability of the stock price being less than $40 in six months time. It is 1
0 4968
0 5032
−
.
=
.
Problem 13.14 What is the price of a European put option on a non-dividend-paying stock when the stock price is $69, the strike price is $70, the risk-free interest rate is 5% per annum, the volatility is 35% per annum, and the time to maturity is six months?
In this case 0
69
S
=
, 70
K
=
, 0 05
r
=
.
, 0 35
=
.
σ
and 0 5
T
=
.
. 2
1
2
1
ln(69 70)
(0 05
0 35
2)
0 5
0 1666
0 35
0 5
0 35
0 5
0 0809
d
d
d
/
+
.
+
.
/
× .
=
=
.
.
.
=
−
.
.
=− .
The price of the European put is 0 05 0 5
70
(0 0809)
69
( 0 1666)
e
N
N
− .
× .
.
−
− .
0 025
70
0 5323
69
0 4338
e
− .
=
× .
−
×
6 40
=
.
or $6.40. Problem 15.9 A foreign currency is currently worth $1.50. The domestic and foreign risk-free interest rates are 5% and 9%, respectively. Calculate a lower bound for the value of a six-month call option on the currency with a strike price of $1.40 if it is (a) European and (b) American.
Lower bound for European option is 0 09 0 5
0 05 0 5
0
1 5
1 4
0 069
f
r T
rT
S e
Ke
e
e
−
−
− .
× .
− .
× .
−
=
.
− .
=
.
Lower bound for American option is 0
0 10
S
K
−
= .
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Problem 15.10 Consider a stock index currently standing at 250. The dividend yield on the index is 4% per annum, and the risk-free rate is 6% per annum. A three-month European call option on the index with a strike price of 245 is currently worth $10. What is the value of a three-month put option on the index with a strike price of 245?
In this case 0
250
S
=
, 0 04
q
=
.
, 0 06
r
=
.
, 0 25
T
=
.
, 245
K
=
, and 10
c
=
. Using put–call parity 0
rT
qT
c
Ke
p
S e
−
−
+
=
+
or 0
rT
qT
p
c
Ke
S e
−
−
=
+
−
Substituting: 0 25 0 06
0 25 0 04
10
245
250
3 84
p
e
e
− .
× .
− .
× .
=
−
=
.
The put price is 3.84. Problem 16.8 Suppose you buy a put option contract on October gold futures with a strike price of $900 per ounce. Each contract is for the delivery of 100 ounces. What happens if you exercise when the October futures price is $880?
An amount (900
880)
100
2 000
$
−
×
= ,
is added to your margin account and you acquire a short futures position obligating you to sell 100 ounces of gold in October. This position is marked to market in the usual way until you choose to close it out. Problem 16.9 Suppose you sell a call option contract on April live cattle futures with a strike price of 90 cents per pound. Each contract is for the delivery of 40,000 pounds. What happens if the contract is exercised when the futures price is 95 cents?
In this case an amount (0 95
0 90)
40 000
2 000
$
.
−
.
×
,
=
,
is subtracted from your margin account and you acquire a short position in a live cattle futures contract to sell 40,000 pounds of cattle in April. This position is marked to market in the usual way until you choose to close it out.
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- 2. Suppose you have the following information concerning a particular options.Stock price, S = RM 21Exercise price, K = RM 20Interest rate, r = 0.08Maturity, T = 180 days = 0.5Standard deviation, = 0.5a. What is correct of the call options using Black-Scholes model? b. Compute the put options price using Black-Scholes model. 3Suppose a European put options has a price higher than that dictated by the putcall parity.a. Outline the appropriate arbitrage strategy and graphically prove that the arbitrage is riskless.Note: Use the call and put options prices you have computed in the previous question 2 above.b. Name the options/stock strategy used to proof the put-call parity. c. What would be the extent of your profit in (a) depend on?arrow_forwardes Suppose you observe the following situation: Security Pete Corp. Repete Co. Beta 1.75 1.44 Pete Corp. Repete Co. What is the risk-free rate? (Do not round intermediate calculations. Round the final answer to 3 decimal places.) Expected Return 0.185 0.158 Risk-free rate Assume these securities are correctly priced. Based on the CAPM, what is the expected return on the market? (Do not round intermediate calculations. Round the final answers to 2 decimal places.) % Expected Return on Market %arrow_forwardHelp sirr mearrow_forward
- You are given the following information on some company's stock, as well as the risk- free asset. Use it to calculate the price of the call option written on that stock, as well as the price of the put option. (HINT: You should use the Black-Scholes formula!) (Do not round intermediate calculations and round your final answers to 2 decimal places, e.g., 32.16.) Today's stock $72 price Exercise price = $70 Risk-free rate = deviation of Option maturity = 4 months Standard annual stock returns = Call price Put price 4.3% per year, compounded continuously = 61% per yeararrow_forwardSuppose you observe the following situation: Security Pete Corporation Repete Company Beta 1.70 1.39 a. Expected return on market b. Risk-free rate Expected Return .180 .153 a. Assume these securities are correctly priced. Based on the CAPM, what is the expected return on the market? (Do not round intermediate calculations and enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.) b. What is the risk-free rate? (Do not round intermediate calculations and enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.) % %arrow_forwardSuppose you observe the following situation: Security Beta Expected Return Pete Corp. 1.70 0.180 Repete Col 1.39 0.153 What is the risk-free rate? (Do not round intermediate calculations. Round the final answer to 3 decimal places) Risk-free rate % Assume these securities are correctly priced. Based on the CAPM, what is the expected return on the market? (Do not round intermediate calculations. Round the final answers to 2 decimal places.) Expected Return on Market Pete Corp. Repete Co.%arrow_forward
- Consider the following multifactor (APT) model of security returns for a particular stock. Factor Risk Premium 8% 9 7 Factor Inflation Industrial production Oil prices Factor Beta 1.1 0.6 0.3 a. If T-bills currently offer a 7% yield, find the expected rate of return on this stock if the market views the stock as fairly priced. (Do not round intermediate calculations. Round your answer to 1 decimal place.) X Answer is complete but not entirely correct. Expected rate of return 19.0 X %arrow_forwardSuppose that many stocks are traded in the market and that it is possible to borrow at the risk-free rate, r4. The characteristics of two of the stocks are as follows: Correlation= Rate of return Stock Expected Return 7% 14% Required: a. Calculate the expected rate of return on this risk-free portfolio? (Hint: Can a particular stock portfolio be formed to create a "synthetic" risk-free asset?) (Round your answer to 2 decimal places.) O Yes O No Standard Deviation 30% 70% b. Could the equilibrium rybe greater than rate of return?arrow_forwardYou are given the following information on some company's stock, as well as the risk- free asset. Use it to calculate the price of the call option written on that stock, as well as the price of the put option. (HINT: You should use the Black-Scholes formula!) (Do not round intermediate calculations and round your final answers to 2 decimal places, e.g., 32.16.) Today's stock = $74 price Exercise price = $70 Risk-free rate = Option maturity = 4 months Standard deviation of annual stock returns 4.4% per year, compounded continuously Call price Put price = 62% per yeararrow_forward
- Suppose that many stocks are traded in the market and that it is possible to borrow at the risk-free rate, rf. The characteristics of two of the stocks are as follows: Correlation = -1 Stock Rate of return B O Yes ● No Expected Return Required: a. Calculate the expected rate of return on this risk-free portfolio? (Hint: Can a particular stock portfolio be formed to create a "synthetic" risk-free asset?) (Round your answer to 2 decimal places.) % 6% 12% Standard Deviation 25% 75% b. Could the equilibrium rf be greater than rate of return?arrow_forwardIn this problem we assume the stock price S(t) follows Geometric Brownian Motion described by the following stochastic differential equation: dS = µSdt + o Sdw, where dw is the standard Wiener process and u = 0.13 and o = current stock price is $100 and the stock pays no dividends. 0.20 are constants. The Consider an at-the-money European call option on this stock with 1 year to expiration. What is the most likely value of the option at expiration? Please round your numerical answer to 2 decimal places.arrow_forwardUse the Black-Scholes formula to find the value of a call option based on the following inputs. Note: Do not round intermediate calculations. Round your final answer to 2 decimal places. Stock price Exercise price Interest rate Dividend yield Time to expiration Standard deviation of stock's returns Call value $ 51 $ 64 0.068 0.04 0.50 0.265arrow_forward
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