VaR of simulation of delta instruments deviates from the delta normal model

quantdaddy

New Member
Hi. I am trying to understand why the value at risk from my simulation deviates from the value at risk based on the delta-normal model in which the 1 % VaR can be calculated as -2.33 * sqrt(delta^2*sigma^2).

The instrument is a european put option deep in the money. S_0 = 100, K=10000, T=1/365, r=0.04, sigma = 0.2 which can be priced with the black scholes model. The delta is -1 which is obtained from the black scholes model.

From my simulation i get a 1 % VaR of 2.419 and from the delta normal model I get 0.465.

The simulation is based on a geometric brownian motion to generate scenarios.
S_t = S_0 exp((mu-sigma^2/2)*t + sigma * W_t)

Does anybody have an idea as to what goes wrong?
 

IntoDarkness

Active Member
it would be accurate if you add theta. i usually use delta + gamma + theta. but your gamma is probably 0 in this case.
 

quantdaddy

New Member
Thank you @IntoDarkness. But I think the problem is somewhere else. Here is my code if anybody has time to look it over (It should give the results by simply running it all).
 
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quantdaddy

New Member
I would greatly appreciate if someone could help me by answering whether there is something wrong with my code or I have understood the theory wrong.
 

Ken Abbott

Managing Director
A deep-in-the-money option should have a 1-day VaR that looks like that of the cash position, unless the vol is really high. I assume the .2 is annualized.

You don’t say what the units are. USD? Percentage? What is the position size?

In general, I don’t think people will respond to many lines of code posted.
 

quantdaddy

New Member
Hi Ken. Thanks for taking the time to respond. Yes, the .2 volatility is annualized. The position is a single option S_0 = K = 100.

I am unsure whether or not I can compare the VaR estimated from monte carlo simulation where the underlying stock follows a geometric brownian motion with the VaR calucalted from the delta-normal model.
 

Ken Abbott

Managing Director
Hi Ken. Thanks for taking the time to respond. Yes, the .2 volatility is annualized. The position is a single option S_0 = K = 100.

I am unsure whether or not I can compare the VaR estimated from monte carlo simulation where the underlying stock follows a geometric brownian motion with the VaR calucalted from the delta-normal model.
Both are parametric. For large n, the results should converge. (In fact, this is a favorite exam question of mine.)
 
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