Project Assignment -

Q1. Let B_t(T) be the cost at time t of a zero coupon bond with maturity T (in years). Assume B₀(1) and B₀(2) are known at time 0 but B₁(2) will not be known until time 1.

(a) Prove or disprove B₀(1)B₁(2) = B₀(2) under the absence of arbitrage.

(b) Show that if we know with certainty that m ≤ B₁(2) ≤ M, then we can conclude mB₀(1) ≤ B₀(2) ≤ MB₀(1).

Q2. The purpose of this question is to compute the price of an Up-and-Out Put option. An Up-and-Out Put option (UOP for short) with strike K and barrier level H has the same payoff at time T as a European put option: (K - S_T)₊, unless the stock went above the barrier level H during the life of the option, in which case, the holder receives nothing.

(a) In this question will use the following explicit formula for the probability of crossing the barrier:

This question uses this formula to improve the estimators in parts (c) and (d). A new estimator for the price is:

C^~_{N_MC} = e^-rT 1/N_{MC k=1}∑^N_MC (K - S_T^(k)) + (1 - q(0, S0, T, S_T^(k))).

Show that this estimator is unbiased. Give an estimate of the UOP option with a 95% confidence interval. Is it different from the estimate in part (c)?

(b) Construct an estimator for the UOP option with rebate which is better. Are we overestimating or underestimating the price of the option?

Q3. Price an Asian call option with payoff (S^- - K)₊, with the mean S^- = _i=1∑ⁿ S_iΔt/n computed over n dates spaced Δt = T/n time units apart. Assume

dS_t = rS_tdt + √V_t S_tdW_t⁽¹⁾

dV_t = ξV_tdW_t⁽²⁾,

where W_t⁽¹⁾ and W_t⁽²⁾ are standard Brownian motions with E[dW_t⁽¹⁾dW_t⁽²⁾] = ρdt. Take r = 0.05, S₀ = 50, T = 1, K = 50, √V₀ = 0.3, ρ = 0.5 and n = 32.

(a) Apply variance reduction technique to improve your answer in part (a). Reconstruct Table 5.4 in "Asymptotically optimal importance sampling and stratification for pricing path-dependent options" by Glasserman, Heidelberger, Shahabuddin, Mathematical Finance, 9 (2), 117-152, 1999, with the last three column replaced by the variance ratio (Variance by your method=Variance by standard Monte Carlo).