I need to compute the value at risk of a given portfolio as an exercise for a class at university but I have trouble understanding how correlated assets affect the price of the portfolio. Could you point out what I am doing wrong? This is my reasoning so far:
Let $X_t$ and $Y_t$ be stock prices following a geometric brownian motion (i.e. $dS_t = S_t(\mu dt+\sigma dW_t)$)
where the $(W_i)_t$ are correlated Brownian motions with a factor of $\rho$.
Let say that I hold a portfolio with the following derivatives:
- A call option on $X$ with strike $K_1$ and maturity $1$ (year).
- A digital put option on $Y$ with strike $K_2$ and maturity $1$.
Giving an initial price to such a portfolio is quite straightforward to me:
However if I would like to evaluate the price at some later point before maturity (for example to compute the Value at Risk) then I would follow the following steps:
- Simulate an important amount of scenarios up to the date I want to re-evaluate my portfolio.
- Compute the new price of the portfolio based on the outcome for all these scenarios.
- (Then I could compute the value at Risk by looking at the distribution of my profits (laterPrice minus initialPrice) for example).
I implemented the results on Matlab but when I play with the values of $\rho$ it does not impact the new price in any way. I think the correlation should at least impact the risk of the portfolio and therefore the later price. Is my reasoning wrong?
Does anyone have an idea on where I could make a mistake? Or simply redirect me to some material explaining on how to price portfolios that contain products that are correlated?
Thanks a lot !!