Optimization problem: solve for the size of negative price shock within a stochastic simulation

I'm running a simulation for stock prices over a 12 months period and each period there are two randomly generated variables that affect the price

1. Normally distributed stochastic change in price
2. negative shock - that only occur p% of the time

at the end of each month, Price is given by: P(t+1) = P(t)*randn(t) - Shock, where Shock = S if rand(0,1) < 0.1 and Shock = 0 otherwise.

I need to solve for S such that Average Price at the end of the 12 months is a fixed number (say 40). Is there a code that will help me calibrate the size of the shock (s) with 100,000 simulations over 12 months?

Below is an example of the problem described. Thanks!

Answer 1

[Roger Stafford]

It is not necessary to run a simulation to solve this problem. Because the mean value of 'randn' is zero, after each month the mean value of P(t)*randn(t) will always be zero also, no matter what the size of P(t) is. Hence the mean value of P(t) after each month is always the same, namely -S*(p/100) with p% (= p/100). This is independent of the number of months and depends only on the S occurrence in the last month. To get 40 with p% the necessary value for S is easy to calculate: S = -4000/p.

However, if you ask about the variance, as opposed to the mean value, that is a different matter. It would depend on the number of months.

[Linda]

Thank you for your reply. Yes, you are correct.

I was actually just looking for some help with coding (new to MATLAB) that combines simulation and solver. Thus, I simplified the original question. Originally the question was designed so that the final price is only considered when P3 falls below 0. We set Average P3 = the average of max(-P3,0), then to calculate S requires the simulation, since it is now dependent on variance of the random variable (assuming that the random variable is not normally distributed).

Hope this is clear. Thanks.