Question

Consider a lookback call option which has payoff X=(max0≤t≤T​St​−K)+. Then, we can approximate max0≤t≤T​St​ by max1≤i≤m​Sti​​ Use the Monte Carlo method to approximate the value of this option. Create a Matlab code and the inputs should have S0​,r,σ,T,K,m,n where n are the number of samples. Choosing S0​=50,r=0.10,σ=0.2,T=1,K=50,m=1000, Create a table with the different approximation with the sample size: n= 10,102,103,104.

Answer 1

The question asks for a MATLAB code to simulate the price of a lookback call option using the Monte Carlo method, considering different sample sizes for the approximation and intends to analyze the distribution of the simulated option payoffs.

Step-by-step explanation:

The question is related to the valuation of a lookback call option using the Monte Carlo method. This method involves generating a large number of random sample paths for the underlying asset's price, then using these samples to estimate the option's payoff and price. In this particular case, we're approximating the maximum stock price over a period (denoted by max0≤t≤T St) which is part of the option's payoff function at expiration.

To achieve this, we will write a MATLAB code that simulates the evolution of the stock's price process under the geometric Brownian motion assumption. The code will create different approximations of the lookback option’s value for various sample sizes, such as n = 10, 100, 1000, and 10000 iterations. The inputs for our simulation will be the initial stock price (S0), the risk-free interest rate (r), the volatility (σ), the time to maturity (T), the strike price (K), the number of monitoring points (m), and the number of sample paths (n).

A histogram of the distribution of the lookback option payoffs can be constructed using the results to visualize the simulation. Moreover, we expect the distribution of the option payoffs to become smoother and more refined as the number of samples increases which aligns with the law of large numbers.