Question

Write a function that simulates the price of a discretely monitored down-and-out call option with maturity T and payoff (ST​−K)+⋅1{min(St1​​,St2​​,…,Stm​​)≥H}​ where m is given. The function should have input parameters S0​,r,σ,T,K,H,m,n with n the sample size. Report the estimate and standard error for the price of the down-and-out call option with S0​=50,r=0.10,σ=0.2,T=1,K=50,H=45,m=100,n=10,102,103,104.

Answer 1

To simulate the price of a discretely monitored down-and-out call option, use Monte Carlo simulation. Generate random stock price paths, calculate the minimum price among the monitored points, and calculate the option payoff based on the minimum price. Repeat this process for the given sample size and calculate the average payoff and standard error.

Step-by-step explanation:

To simulate the price of a discretely monitored down-and-out call option, we can use Monte Carlo simulation. Here's a step-by-step guide:

1. Generate random stock price paths using the given parameters S0, r, σ, and T.
2. For each stock price path, calculate the minimum price among the m monitored points.
3. If the minimum price is greater than or equal to H, calculate the option payoff (ST - K)+. Otherwise, the option payoff is 0.
4. Repeat steps 1-3 for n sample size.
5. Calculate the average payoff and the standard error using the sample values.

For the specific values S0 = 50, r = 0.10, σ = 0.2, T = 1, K = 50, H = 45, m = 100, and n = 10, 102, 103, 104, you can plug them into the function and run the simulation to obtain the estimates and standard errors.