Computational Methods for Pricing Structured Products

[points] 1. Consider the Cheuk-Vorst algorithm for pricing floating strike lookback put options. The truncated binomial tree for the process is constructed for Y t = S max t S t , Y t ≥ 1. Let V n j denote the numerical approximation to V t = V t /S t at the (n, j) th node of the binomial tree for Y t , where Y t = u j , j ≥ 0. (i) Explain how to derive the binomial formula V n j = e −r∆t [p V n+1 j−1 u + (1 − p) V n+1 j+1 d], j ≥ 1. [3] (ii) How to modify the above formula when j = 0? [2] (iii) How to determine the terminal condition in the binomial calculations? [1] (iv) How to modify the above binomial formula in order to incorporate the American early exercise feature? [2] 2. Explain how to construct the forward shooting grid algorithm for pricing the European style fixed strike lookback call option under (i) continuous monitoring, (ii) discrete monitoring. In particular, explain how to construct the grid function that models the lookback feature. [5] 3. (a) Explain why the stock price process and its underlying volatility process are negatively correlated in general. [2] (b) We assume the volatility to be function of time (no dependence on the stock price). Recall the formula: ∫ t t * σ 2 (u) du = σ 2 imp (t * , t)(t − t *), we would like to obtain the term structure of volatility σ(t) given the implied volatility σ 2 imp (t * , t) at discrete time points. Suppose we assume σ(t) to be piecewise constant over (t i−1 , t i), where σ(t) = σ i , t i−1 < t < t σ i = √ (t i − t *)σ 2 imp (t * , t i) − (t i−1 − t *)σ 2 imp (t * , t i−1) t i − t i−1 , t i−1 < t < t i. [4] (c) What is the difference between " implied volatility " and " local volatility " ? [2] (d) The usual binomial tree is generated based on constant volatility in the underlying Geometric Brownian motion of the stock price process. The implied binomial tree constructed by the Derman-Kani algorithm uses quite different approach.