Simple VaR Question

hilss · Post by **hilss** » Wed Sep 05, 2007 5:35 pm

Hello,

My question is about how to compute VaR (specifically when options are involved). If I have 3 assets with an expected return of 10%, 15% and 20% and a risk of 10%, 20% and 30%, and a correlation matrix, I can compute VaR (given the time and the confidence interval).

But say I have a portfolio that contains a stock and an option on the same stock. I have the dynamics of the stock, how do I go about computing the VaR of such a protfolio? Should I simply do an MC simulation? once I get the results, plug them into the option pay-off function? is it that simple? Is there a corelation between the stock and the option? is the corrlation equal to the delta of the option?

Thanks,

hilss

MoreLiver · Post by **MoreLiver** » Wed Sep 05, 2007 5:56 pm

Options are nonlinear instruments, so variance-covariance approach is obviously not enough. Google for delta-gamma VaR, or consider MC.

Sure there is correlation between the stock and option, but as the stock price changes and option maturity slowly approaches, the correlation changes as well (option has so-called gamma, so delta is not constant). The instantaneous correlation is roughly equal to delta.

BTW, this belongs to the Basics imho.

Johnny · Post by **Johnny** » Wed Sep 05, 2007 5:58 pm

The correlation between stock and option is one if the only source of uncertainty is the stock price (e.g. Black Scholes world).

hilss · Post by **hilss** » Wed Sep 05, 2007 6:10 pm

Thanks for your response.

The correlation can't be 1 (even in a Black-Scholes world). But we can disregard the correlation right now if we have a protfolio that has the stock and the option on the stock. Because the option value is a function of the stock, interest rate and vol. If we need to consider any correlation, it must be the correlation of the stock with the vol (and the interest rate possibly).

I will google "delta-gamma VaR", but what is the exact technique if I choose to use MC. Can someone spell it out for me?

Thanks,

Hilss

Johnny · Post by **Johnny** » Wed Sep 05, 2007 6:45 pm

[i]"If we need to consider any correlation, it must be the correlation of the stock with the vol" ...[/i] [i]"The correlation can't be 1 (even in a Black-Scholes world)"[/i]

In the Black Scholes world the volatility is constant (or at most deterministic) and has zero correlation with anything. You're talking through your, uh, hat. I'll spell that out for you. The correlation between asset and option is one; the delta is the delta, blah blah.

Anyway, your original question is straightforward. Use the covariance matrix to run some MC simulations of asset prices. Re-value the options at each asset price. Use all this to form a distribution of P&Ls and then use that to calculate VaR. It's slow doing it by MC this way, but ...

hilss · Post by **hilss** » Wed Sep 05, 2007 7:26 pm

Hi Johnny,

Conisder a way out-of-the-money option with little time left to expiry. If the underlying moves, how much will the option price move by? delta is zero, right? so the option price won't move (even with a constant vol). So the correlation is not 1 in this case (and is not 1 in most cases).

In any event, thanks for your input on the MC simulations. Is there a quick to derive the "risk" of an option given the dynamics of the underlying?

Thanks,

hilss

FDAXHunter · Post by **FDAXHunter** » Wed Sep 05, 2007 7:29 pm

LOL... you listen to him Johnny, you hear me? (Sorry couldn't resist, too phunny Smiley )

hilss · Post by **hilss** » Wed Sep 05, 2007 8:03 pm

FDAX,

You may appear a bit more intelligent if you remained quiet.

hilss

FDAXHunter · Post by **FDAXHunter** » Wed Sep 05, 2007 8:08 pm

Smiley You are right. Shoot first, ask questions later. I'm all for that.

Now... along the same lines of thinking... are we blending or what?

Johnny · Post by **Johnny** » Wed Sep 05, 2007 8:21 pm

Oh man ...