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

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

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

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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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.

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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