Matlab coding problem (Black-Scholes Option Pricing content)

[spenghali]

So I have made this code to price a european call, hedge, and record the hedging error at each time step. I then take the mean and variance of the hedging error which is suppose to go to zero as dt goes to zero. Here's what I have, my question will follow:

clear;

%Parameters

M = 200;
dt = 0.5;
T = repmat(365/4:-dt:0,M,1);
N = size(T,2);
K(1:M,1:N) = 50;
r(1:M,1:N) = 0.01;
sigma(1:M,1:N) = 0.25;

toss = randn(M,N-1);
S = cumsum([K(:,1), toss],2); %Stock Paths

V = blsprice(S, K, r, T, sigma); %BSM Price
delta = blsdelta(S, K, r, T, sigma); %BSM delta
X = V-(delta.*S); %Portfolio value

mean = mean(mean(X)) %Sample mean
var = var(var(X)) %Sample varianceSo my problem is that when I decrease dt to make the time steps smaller, some of the underlyings (stock prices, S) become negative and then the 'blsprice' function cannot compute the price of the option because it is expecting a non-negative entry. Any suggestions on how to solve this? The model works fine for dt = 0.5, I just cannot make it smaller. Thanks for any input.

 

[mmwave]

Hello,

Thank you for sharing your code and explaining your problem. It seems like you have a good understanding of the concepts behind pricing a European call and hedging it. Based on the information provided, here are a few suggestions that may help you solve your problem:

1. Check your input parameters: It is important to make sure that the input parameters for your BSM functions (blsprice and blsdelta) are appropriate for the stock prices and time steps you are using. For example, if you are using a very small time step (dt), you may need to adjust your volatility (sigma) to compensate for the smaller time intervals.

2. Use a different pricing model: If your BSM model is not able to handle negative stock prices, you may want to consider using a different pricing model that can handle such scenarios. For example, you could try using the binomial or trinomial tree models, which can handle negative stock prices.

3. Implement boundary conditions: Another approach could be to implement boundary conditions to prevent your stock prices from becoming negative. This could involve setting a minimum stock price or using a reflective boundary condition, where negative stock prices are reflected back to positive values.

4. Consider using a different time step: If none of the above solutions work, you may need to reconsider using a smaller time step. Sometimes, a larger time step may be more appropriate for certain scenarios, and it may still provide accurate results.

I hope these suggestions are helpful and that you are able to find a solution to your problem. Keep in mind that there is no one-size-fits-all solution, and it may require some experimentation and fine-tuning to get the results you are looking for. Good luck!