We consider a continuous time market, where the interest rate r = 0, and the risky...
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We consider a continuous time market, where the interest rate r = 0, and the risky asset S = (St)o<t<r follows the Black-Scholes model with initial value So = 1, drift μ and volatility o>0 (without any dividend), so that dS₁ = µS₂dt + o S₁dB₂. a) (5 points) A self-financing portfolio is given by (r, o), where a represents the initial wealth of the portfolio, and ot represents the number of risky asset in the portfolio at time t. Let (II)[0,7] be the wealth process of the portfolio, write down the dynamic of II in form of dladt + 3₁dB₁, for some (to be founded) process (a, 3). b) (5 points) There exists a unique risky-neutral probability Q, together with a Brownian motion B under the probability measure Q. Please give the expression of the process S, as a function of (t, Bº). c) (40 points) We consider a derivaitive option with payoff g(ST) = log(ST) at maturity T. - (5 points) Compute the value - Vo := E[log(ST)]. (25 points) Let v(t, x):= log(x) - 0²(T-t), compute ov, d₂v and ou, and check that u satisfies the equation dv(t, x) + - 0²2²8²_,v(t, x) = 0, (t, x) = [0,T) × (0,∞), with terminal condition v(T, x) = log(r) for all x € (0,0). - (10 points) Remember that S, is a function of (t, B₁), apply the Itô formula on v(t, St) to deduce that -1.² log(ST) = Vo+ odSt, where ot := St Finally, deduce the (no-arbitrage) price of the derivative option with payoff g(ST) = log(ST). We consider a continuous time market, where the interest rate r = 0, and the risky asset S = (St)o<t<r follows the Black-Scholes model with initial value So = 1, drift μ and volatility o>0 (without any dividend), so that dS₁ = µS₂dt + o S₁dB₂. a) (5 points) A self-financing portfolio is given by (r, o), where a represents the initial wealth of the portfolio, and ot represents the number of risky asset in the portfolio at time t. Let (II)[0,7] be the wealth process of the portfolio, write down the dynamic of II in form of dladt + 3₁dB₁, for some (to be founded) process (a, 3). b) (5 points) There exists a unique risky-neutral probability Q, together with a Brownian motion B under the probability measure Q. Please give the expression of the process S, as a function of (t, Bº). c) (40 points) We consider a derivaitive option with payoff g(ST) = log(ST) at maturity T. - (5 points) Compute the value - Vo := E[log(ST)]. (25 points) Let v(t, x):= log(x) - 0²(T-t), compute ov, d₂v and ou, and check that u satisfies the equation dv(t, x) + - 0²2²8²_,v(t, x) = 0, (t, x) = [0,T) × (0,∞), with terminal condition v(T, x) = log(r) for all x € (0,0). - (10 points) Remember that S, is a function of (t, B₁), apply the Itô formula on v(t, St) to deduce that -1.² log(ST) = Vo+ odSt, where ot := St Finally, deduce the (no-arbitrage) price of the derivative option with payoff g(ST) = log(ST).
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a 5 points The dynamic of the wealth process II is dII IIdt IdB b 5 points Und... View the full answer
Related Book For
Essentials of Organizational Behavior
ISBN: 978-0134523859
14th edition
Authors: Stephen P. Robbins, Timothy A. Judge
Posted Date:
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