Let the interest rate r and the volatility o > 0 be constant. Let 1 St = So exp t+ oWt be a geometric Brownian motion with mean rate of return u, where the initial stock price So is positive. Let K be a positive constant. Show that, for T > 0, E[e-r" (ST – K)*] = S,N(d4) – Ke-rTN(d_), where So log K 1 1 o² = +p σνΤ + and ry Nw) = L 1 N(y) e-z²/2 dz.
Let the interest rate r and the volatility o > 0 be constant. Let 1 St = So exp t+ oWt be a geometric Brownian motion with mean rate of return u, where the initial stock price So is positive. Let K be a positive constant. Show that, for T > 0, E[e-r" (ST – K)*] = S,N(d4) – Ke-rTN(d_), where So log K 1 1 o² = +p σνΤ + and ry Nw) = L 1 N(y) e-z²/2 dz.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Let the interest rate r and the volatility σ > 0 be constant. Let
St = S0exp((µ − σ2/2)t + σWt )
be a geometric Brownian motion with mean rate of return µ, where the initial stock price S 0 is positive. Let K be a positive constant. Show that, for T > 0, satisfy the following equation shown in the below picture:
Text Book: STOCHASTIC CALCULUS FOR FINANCE, Shreve vol. II
![Let the interest rate r and the volatility o > 0 be constant. Let
((---) + ow.)
1
St = So exp
be a geometric Brownian motion with mean rate of return u, where the initial stock price S is positive. Let K be a
positive constant. Show that, for T > 0,
E[e¬r"(ST – K)+] = S,N(d+) – Ke¬r"N(d_),
-rT
-
where
(2) (
So
+
K
1
1
d+
log
T
oVT
and
1
N(y) =
/2T
e-=2/2 dz.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F593264c5-a4d3-4fc3-9c04-775c0b231240%2F9614fd66-a016-4e6a-a419-d0b7936742a6%2Fl00euwl_processed.png&w=3840&q=75)
Transcribed Image Text:Let the interest rate r and the volatility o > 0 be constant. Let
((---) + ow.)
1
St = So exp
be a geometric Brownian motion with mean rate of return u, where the initial stock price S is positive. Let K be a
positive constant. Show that, for T > 0,
E[e¬r"(ST – K)+] = S,N(d+) – Ke¬r"N(d_),
-rT
-
where
(2) (
So
+
K
1
1
d+
log
T
oVT
and
1
N(y) =
/2T
e-=2/2 dz.
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