Monte Carlo Methods

    1 Background 1.1 Stock Options The trader has calibrated a specialised risk neutral process for some underlying stock price. Given the current stock is S0, market prices indicate the risk-neutral distribution of the stock price at time t is given by: St ∼ N(f(S0, t), v2 (S0, t)) (1) for some calibrated functions f and v 2 . If (1) describes the risk neutral distribution, then the formula to value a call option V with payoff g(S) at time t = T is given by V (S0, 0) = e −rT E Q[g(ST )]. (2) Then to carry out a Monte Carlo valuation of an option, we may use samples from a standard random normal distribution φ ∼ N(0, 1) (3) to write the equation ST = f(S0, T) + v(S0, T)φ. (4) Equation (5) then generates a single random path, from which we can value a payoff V i = e −rT g f(S0, T) + v(S0, T)φ i . (5) If n simulations are performed, then (as described in the notes) we merely average out the V i to yield an approximation for the value of the portfolio, i.e. V (S0, t = 0) = Pi=n i=1 V i n (6) Note that if we write z = X − f(S0, T) v(S0, T) , then a Put Option V with a terminal condition V (S, T) = max(X − S, 0) written on the asset S has the analytic solution V (S0, t = 0) = XN(z) + v(S0, T) 1 √ 2π e −z 2/2 − f(S0, T)N(z) e −rT , where N(z) is the standard cummulative normal distribution. 1 Created in Master PDF Editor 1.2 Path Dependent Options Now assume that the stochastic process follow a standard Geometric Brownian motion governed by dS = (µ − D0)Sdt + σSdW. Then the following options will depend on S(tk) which are the share prices at K + 1 equally spaced sampling times t0, t1, ..., tK with t0 = 0 and tK = T (unlike part (a), the computation cannot proceed from t = 0 to t = T in one step). Full details are given the the lecture notes - but the important point to note is that S i tk = S i tk−1 exp[(r − D0 − 1 2 σ 2 )(tk − tk−1) + σ p tk − tk−1φi ] to estimate the underlying asset values at each time, where each of the K increments dWk involves drawing φk from a Normal distribution. Asian Option Assume that a discretely sampled Asian option has a payoff depending on the discretely sampled average given by A = 1 K X K k=1 S(tk). Then we can write V (S, A, t = T) = f(S, A), where f is the payoff function depending the type of option. There are different classes of Asian option, resulting in different payoff conditions. In this coursework we look at simple European style call or put options. A fixed strike call option will have the payoff f(S, A) = max(A − X, 0) where X is the strike price and a floating strike call option would be f(S, A) = max(S − A, 0). where A is sometimes calles the average strike price. A fixed strike put option will have the payoff f(S, A) = max(X − A, 0) where X is the strike price and a floating strike put option would be f(S, A) = max(A − S, 0). where A is the strike price. Lookback Option The discretely sampled Lookback option has a payoff depending on the discretely sampled maximum or minimum given by A = max k S(tk), or A = min k S(tk). Then we can write V (S, A, t = T) = f(S, A),