Assignment questions. Help.
1 You are given that the fair price to pay at time t for a derivative paying X at time 7 is style ve = e"(-EQ [X|F ], where Q is the risk-neutral probability measure and F, is the filtration with respect to the underlying process. The price movements of a non-dividend-paying share are governed by the stochastic differential equation dS, = S, (rdt + adB, ) , where B, is standard Brownian motion under the risk-neutral probability measure. (i) Solve the above stochastic differential equation. [4] Determine the probability distribution of Sy | S, . [2] (iii) Hence show that the fair price to pay at time t for a forward on this share, with forward price K and time to expiry 7 -t , is: Vi = S, -Ke (T-t) [4] [Total 10] .2 The process S, is defined by S, = Sped ho )+GM , where We is standard Brownian motion under Lyle a probability measure , and & and of are constants. (i) (a) State the name given to the process S, . (b) Give two real-world quantities that are commonly modelled using such a process. [3] (Wi) (a) State what is meant by 'equivalent probability measures'. (b) State how the Cameron-Martin-Girsanov Theorem could be applied here if we wished to work with a process of the form S, =Self-So Ittow , where W, is standard Brownian motion and / is the risk-free rate of interest. [3] (iii) (a) Determine the stochastic differential equation for d5, in terms of dt and dw,. (b) State the drift of the process in (ili)(a) and comment on your answer. [5] [Total 11]3 The random variable S, is defined by S, = Spell-*o )to2 , where Zy - N(0,T) under a yle probability measure Q. The random variable X is defined by X=max(S, -K,0) , where K is a positive constant. (1) Show that: EglX |Fo] = Spend(d2 +avT )-KQ(d2) where da = log(Sp/K) + (r- X202 )T [7] OVT Hint: You may wish to use the lognormal integral formulae given on page 18 of the Tables. (ii) Explain carefully the relevance of this result in financial mathematics. [3] [Total 10] 4 (0) State the general risk-neutral pricing formula for the price of a derivative at time t in Myle terms of the derivative payoff Xy at the maturity date 7 and the constant risk-free force of interest r . [1] Assume that the price of a share, which pays a constant force of dividend yield q, follows geometric Brownian motion. (ii) (a) Derive the formula for the price at time t of Derivative 1, which pays one at time T provided the share price at that time is less than K . (b) Derive the formula for the price at time t of Derivative 2, which pays the share price at time 7 provided the share price at that time is less than K. (c) Hence derive the formula for the price of a European put option with strike price K. [12] Hint for (ii)(a) and (ii)(b): You may wish to use the lognormal integral formulae given on page 18 of the Tables. (iii) An exotic forward contract provides a payoff equal to the value of the square of the share price at the maturity date 7 in return for a payment equal to the square of the forward price. Derive the formula for the value of this contract at time t. [3] [Total 16]
