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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:

$\text{price}= e^{-rT}E_\mathbb{Q}(\text{payoff})$.

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:

  1. Simulate an important amount of scenarios up to the date I want to re-evaluate my portfolio.
  2. Compute the new price of the portfolio based on the outcome for all these scenarios.
  3. (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 !!

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  • $\begingroup$ Are you finding a future price which is close to the current price? $\endgroup$ – Daneel Olivaw Apr 2 at 12:08
  • $\begingroup$ yes ! and over a short time horizon (~0.1 year). $\endgroup$ – Ulysse Apr 2 at 12:32
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    $\begingroup$ Some remarks: 1) your portfolio price is a conditional expectation $E^\mathbb{Q}_t[\cdot]$, thus normally you need to simulate up to $t$, then again simulate up to maturity $T$ (for each simulation up to $t$: nested simulations) to get the future price at $t$; 2) these two simulations are not performed under the same measure: it seems you are interested in real-world VaR and you have some estimation or desired value of drift $\mu$, thus you simulate up to $t$ with drift $\mu$, but the nested simulation from $t$ to $T$ is for pricing purposes thus the drift must be the rate $r$. $\endgroup$ – Daneel Olivaw Apr 2 at 12:34
  • $\begingroup$ What drift are you using for the simulation? How are you "computing the new price" at $t$? $\endgroup$ – Daneel Olivaw Apr 2 at 12:35
  • $\begingroup$ Ok, thanks a lot it is much more clear now. I use direct integration to price the portfolio since otherwise I need indeed to have nested simulations which takes too much memory. I only simulate up to $t$ in the 'real world' measure (with proper $\mu$ and $\sigma$ and then price using $\mu=r$. So indeed the new price depends on the path the stocks took up to time $t$ and therefore the correlation has an impact on these paths. Thank you very much for your help. your comments were very helpful! $\endgroup$ – Ulysse Apr 2 at 23:06
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The correlation certainly has an impact on the price of your portfolio (of two options). If you simulate the prices at time $t < T$ then you get samples prices $X_t$ and $Y_t$ and the return between time $0$ and time $t$ reflects the correlations.

This means that if $\rho$ is positive then the $X_t-X_0$ and $Y_t-Y_0$ are likely to have the same sign. Then at time $t$ you revaluate your positions with the new price $X_t$ and $Y_t$ as "starting points", some assumptions on the volatilities and reduced time to maturity $T-t$.

Maybe you have a bug in the code?

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  • $\begingroup$ Ok, this makes much more sense now. Nevertheless the correlation has an impact on the price of my portfolio only at later times, correct? The initial price is still independent of the correlation. It is only after simulating the prices at time $t<T$ that the price of the portfolio will be dependent of the correlation factor? Thank you very much for your help! $\endgroup$ – Ulysse Apr 2 at 23:19
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    $\begingroup$ I would say: At time $0$ the spot prices are what they are. They enter the evaluation of your derivative where each one of them just has one underlying. When you assess the risk of your portfolio (real world measure) in the sense of spot prices in the future then it matters. $\endgroup$ – Ric Apr 3 at 9:22
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The correlation will impact the random numbers generated for the simulation. Use Cholesky Decomposition on the original correlation matrix to recalculate what the correlated random normal numbers will be and use those in the simulated path(s). If you're using Matlab or another canned scripting language, they usually have the function pre-coded.

In matlab: R = chol(A) factorizes symmetric positive definite matrix A into an upper triangular R that satisfies A = R'*R. If A is nonsymmetric , then chol treats the matrix as symmetric and uses only the diagonal and upper triangle of A.

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