1.
a. Consider a two-step standard binomial model with a riskless bond with initial price e −2r∆T and a single risky asset S with initial price S0 = 1.1, with two equal time steps of length ∆T = 0.5. Assume that at each time step, S is multiplied by u = 1.04 with probability p, or multiplied by d = .95 with probability 1 − p and the bond price is multiplied by e r∆T , and the interest r = 0.02. Calculate the risk-neutral probability q of S going up at each time step. Price an American put option with strike K = 1. [40%]
b. Consider a discrete-time market with a single stock which evolves under the physical probability measure P as Sn = Sn−1e µ + σZn for 0 ≤ n ≤ N, and a riskless bond which evolves as Bn = e rn, where Zn is a sequence of independent and identically distributed random variables with Zn = −1, 0 or 1 with equal probability. Does this market admit arbitrage? Is the market complete? (just answer true or false, proofs not required). [30%]
c. Let C(u, v) be a general copula on [0, 1] × [0, 1]. Explain how to use C(.) to simulate two correlated uniform random variables U1 and U2 on [a, b]. You may use that a uniform random variable on [a, b] has density 1 /b−a . [30%]
2.
a. Let (Wt)t≥0 denote a standard Brownian motion on some probability space (Ω, F, P) with filtration Ft satisfying the usual conditions. Which of these statements is true
i. d/dtf(Wt) = f 0 (Wt) for any differentiable function f
ii. W2 t is an F W t -martingale
iii. W can go to ±∞ in finite time iv. E(W3 t ) = 0 [40%]
b. Write down an expression for E(f(Wt)|Ws = x) for any measurable function f for 0 ≤ s ≤ t. Does this expression simplify when s = t? [30%] c. Let dXt = σ(Xt)dWt . Derive the SDE satisfied by Yt = g(Xt), where g(x) = 1/σ(u) du (you do not need to invert g explicitly). [30%]
3.
Let (Wt)t≥0 denote a standard Brownian motion on some probability space (Ω, F, P) with filtration Ft . Consider the Black-Scholes model dSt = St(µdt + σdWt) for a Stock price process S.
a. Write down the Black-Scholes PDE and boundary condition for the price of an option which pays 1 at time T if S hits a barrier level B > S0 at any time t ∈ [T2, T] for 0 < T2 < T, and pays zero otherwise. [30%]
b. Write down an expression for the payoff at time T of the option in part a) in terms of the minumum or maximum of the stock price over a certain time interval. [30%]
c. State whether each of these statements are true or false (no explanation is required).
i. St tends to +∞ under Q as t → ∞ if r − σ 2 > 0;
ii. St can tend to infinity as t → ∞ under Q but not P;
iii. 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.
iv. The stock price can go to zero with non-zero probability before time 1 [40%]
4.
Consider a jump-diffusion model where the log stock price Xt = µt + σWt + PNt i=1(ξi − 1), where Wt is standard Brownian motion, the ξi ’s are i.i.d Exp(1) random variables and Nt is a Poisson process with N0 = 0 and arrival rate λ > 0, and W, the ξi ’s and Nt are all independent of each other.
a. Write down an integral expression for V (p) = log E(e pX1 ). Explain how to compute E(e −qτ + a ) when σ = 0 for q > 0, where τ + a = min{t : Xt ≥ a} for a > 0. You do not need to explicitly evaluate the expression for V (p). [40%]
b. Assume V 0 (0) > 0. What can we say about P(τa < ∞)? (Hint: draw a picture, and you may assume that V (p) → ∞ as p → ∞ without proof). [30%]
c. Let (Nt)t≥0 be a Poisson process. Write down an integral expression for Var(T1 + ...Tn), where Ti denotes the i’th jump time of N. [30%]
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