1.
Let Φ(x) denote the standard cumulative Normal distribution function throughout this question.
a. Consider a random variable X on (0,∞) with cumulative distribution function FX(x) . Describe how we can simulate X using a standard U[0, 1] random variable. [30%]
b. Let Z ∼ N(0, 1) and let X = Z/a − b for a, b ∈ R with a 6= b. Calculate the following quantities in terms of Φ:
(i) the distribution function of X (ii) P(Z 4 ≥ x) for x ∈ R [40%]
c. Explain how to use a general copula C(u, v) on [0, 1]2 to simulate two correlated uniform random variables U1 and U2 on [−1, 1]. [30%]
2.
Let (Wt)t≥0 denote a standard Brownian motion throughout.
a. Compute the conditional distribution of aWt + b at time s ∈ [0, t] for a, b ∈ R. [40%]
b. Write down an integral expression for E(|Wt | 3 ) in terms of a probability density (you do not need to evaluate this integral explicitly). Write down a similar integral expression for E(W3 t ), and compute this integral explicitly. [30%]
c. Derive the SDE satisfied by Xt = W2 t + 1 in terms of Xt . Does E(XsXt) change if Wt is replaced by −Wt for s, t ∈ R? (explain your answer) [30%]
3.
Let (Wt)t≥0 denote a standard Brownian motion and recall the Black-Scholes model dSt = St(µdt + σdWt) for a stock price process S under the physical measure P.
a. Write down the conditional expectation and conditional variance of c log ST at time t ∈ [0, T] under the risk-neutral measure Q, for c ∈ R. [30%]
b. Compute the no-arbitrage price at time zero of a contract which pays aST + b at time T for a, b ∈ R. What is the vega of this contract at time t ∈ [0, T]? [30%]
c. State whether each of these statements are true or false (no explanation is required).
i. St tends to +∞ under P as t → ∞ if µ − σ 2 > 0;
ii. St can tend to infinity as t → ∞ under P but not Q;
iii. The stock price sample path is continuous under P;
iv. C(S, t) ≤ S, where C(S, t) denotes the price of a European call option at time t ∈ [0, T] under the Black-Scholes model with strike K and maturity T conditioned on St = S. [40%]
4.
Let (Wt)t≥0 denote a standard Brownian motion throughout this question. Consider the Black-Scholes model
dSt = St(µdt + σdWt)
for a stock price process S.
a. Let Mt = max0≤s≤tWs. Write down an integral expression for Var(Mt). [30%]
b. Consider a double One-Touch option which pays 1 at time T if S hits L < S0 or B > S0 before time T, and zero otherwise. Write down the boundary conditions satisfied by the price P(S, t) of this option at time t ∈ [0, T] given that St = S. What is the sign of the vega of this contract before and after S hits the barrier? [40%]
c. Write down a mathematical expression for the payoff of the option in part b) in terms of the running minimum and maximum processes of S. Draw a picture of P(S, 0) as a function of S. [30%]
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