Consider a Russian option with value function V(S, J) in the case where thereare continuous dividends paid on the underlying asset S at rate D. Define thevariable J for this option. A similarity reduction of the form W(n) = V/J withn S/J leads to110²² W +(r− D)nW' rW = 0,which must be solved subject to-W(1) W'(1) = 0, W(no) = 1, W'(no) = 0,where n = ŋo is the optimal exercise boundary in similarity coordinates. Thecurrent value of the underlying is S = 3. What is the current value of theRussian option assuming that σ = 0.3, r = 0.1 and D = 0.05? Present yourresults to a minimum of 4 decimal places.

1. Understand the Problem SetupWe are given a Russian option whose value V(S, J) depends on the…

Answered: Consider a Russian option with value…

Essentials Of Investments

11th Edition

ISBN:9781260013924

Author:Bodie, Zvi, Kane, Alex, MARCUS, Alan J.

Publisher:Bodie, Zvi, Kane, Alex, MARCUS, Alan J.

Chapter1: Investments: Background And Issues

Section: Chapter Questions

Problem 1PS

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The Black-Scholes equation for the value of an option V(S,t), where S is the value of the underlying asset and t is time, can be transformed into the diffusion equation for a new variable u(x,7) using S = Ee™, t = T − 27/0², k = 2r/o² and - (k-1) x (k+1)² + T V=Eu(x,T) exp 2 4 where r is the risk-free interest rate, σ is the volatility, E is the strike price, and T is the expiry date. The diffusion equation has the solution u(x,T): = 1 TT or L 2√Tπ 1 uo (s) exp 4T : (x - s)²) ds. Determine, showing all steps, the function uo(s) for a European asset- or-nothing call with the pay-off A(S) = SH(S – E). -
We believe that the single factor model can predict any individual asset’s realized rate of return well. Both Portfolio A and Portfolio B are well-diversified: ri = E(ri) + βiF + Ei, where E(ei) = 0 and Cov(F, i) = 0 A B β 1.2 0.8 E(r) 0.1 0.08 (1) What is the rate of return of the risk-free asset? (2) What is the expected rate of return of the well-diversified portfolio C with βC = 1.6, which also exists in the market? (3) A fund constructs a well-diversified portfolio D. Studies show that βD = 0.6. The expected rate of return of D is 0.06. Is there an arbitrage opportunity? If so, construct a trading strategy to earn profits with no risk. If not, why?
Consider a two-period binomial market model for an underlying asset with price process S; at each time t. Branching fromt to t+1 results in St+1 and St+1 = dS; in case of a downward price move. You are asked to input values So = 2, u = 2, d = 1/2, r = 0.1, and 8 = 0 to price a European put option with strike price 3. uSt in case of an upward price move, a. Write down the evolution of the underlying stock and the payoff of the option along the tree. b. Determine the structure of the replicating portfolio at times 0 and 1 in every node. c. Determine the price of the option at time 0. d. Explain how the price computed at point (c) would change if the option were of American type.
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Consider the Black-Scholes equation for the option price V(S, t) if the underlying does not pay out dividends. (i) Show that V(S,t) = AS, where S is the underlying price, t is time, and A is a constant, is a solution to the Black-Scholes equation. (ii) For the solution V(S,t) = AS, define the value A (Delta) used for hedging.
a) Let VK(t, T) be the value of a forward contract on an asset with delivery price K, VK(t, T) = (F(t, T) − K)e −r(T −t) . a) Verify that VK(T, T) equals the payout of a forward contract with delivery price K. For an asset that pays no income, substitute the expression for its forward price into the above equation and give an intuitive explanation for the resulting expression. b) Suppose at time t0 you go short a forward contract on an asset that pays no income with maturity T (and with delivery price equal to the forward price). At time t, t0 < t < T, suppose both the price of the asset and interest rates are unchanged. How much money have you made or lost? This is sometimes called the carry of the trade.
Suppose that C is the price of a European call option to purchase a security whose present price is S.Show that if C > S then there is an opportunity for arbitrage (i.e. riskless profit). You may assume theinterest rate is r = 0 so that present value calculations are unnecessary.
The CAPM states that the expected (required) return on an asset is : E(Ri)=Rf+βi[E(RM)−Rf] where the term in square brackets is the risk-premium earned by the market portfolio. Therefore, the beta of the market portfolio (βM) must be equal to __________ . A) zero B) 0.5 C) 1.0 D) an unknown estimate
Use the put-call parity relationship to demonstrate that an at-the-money call option on a nondividend-paying stock must cost more than an at-the-money put option. Show that the prices of the put and call will be equal if S0 = (1 + r)T..
Question 1. Suppose t ≤ T1 ≤ T2 ≤ T3, where t is the current time, and ∆ > 0. Recall that Z(T1, T2) is the price at time T1 of a ZCB with maturity T2 and F(T1, T2, T3) is the forward price at time T1 for a forward contract with maturity T2 on a ZCB with maturity T3. a) For each of the pairs of A and B in the table, choose the most appropriate relationship out of ≥, ≤, = , ?, where ? means the relationship is indeterminate. Give brief reasoning. A ≥, ≤, = , ? B (i) Z(t, T1) 1 (ii) Z(T1, T1) 1 (iii) Z(t, T2) Z(t, T3) (iv) Z(T1, T2) Z(T1, T3) (v) Z(T1, T3) Z(T2, T3) (vi) Z(T1, T1 + ∆) Z(T2, T2 + ∆) (vii) F(t, T1, T2) F(t, T1, T3) (viii) F(t, T1, T3) F(t, T2, T3) (ix) limT→∞ Z(t, T) 0 Hint: Remember that at current time t, F(t, ·, ·) is known but Z(T, ·) is a random variable. b) What can you say about interest rates between T1 and T2 if i) Z(t, T1) = Z(t, T2)? ii) Z(t, T1) > 0 and Z(t, T2) = 0?
4. Valuation of a Derivative Consider a derivative on a stock with the time to expiration T and the following payoff: 0 K₁ 0 if ST K₁. What is the present value of the derivative? Provide an analytic expression of the price using N(), the cumulative probability distribution function of a standard normal random variable.
Let rf be the risk free rate of interest. E[r e ] be the expected return of some risky asset. Suppose that this risky asset pays out in states when the aggregate endowment is particularly low. There are three possibilities: ( a) E[r e ] > rf (b) E[r e ] = rf (c) E[r e ] < rf Which case applies to E[r e ] and why?

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