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I have a European call option with current stock price $S_0$, strike $K$, risk-free rate $r$, volatility $\sigma$, and time to maturity $T$ years.

I assume that the stock price at time $t$, which is given by $S_t$, follows a geometric brownian motion.

I need to use the delta-normal valuation method to compute the 95% VaR over a horizon of 3 days for a long position on the call. How do I do this?

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When the stockprice follows a GBM, the arbitrage-free value of an EU call is given by the Black Scholes model:

\begin{align} C(S, t) &= N(d_1)S_0 - N(d_2) Ke^{-r(T - t)} \\ d_1 &= \frac{1}{\sigma\sqrt{T - t}}\left[\ln\left(\frac{S_0}{K}\right) + \left(r + \frac{\sigma^2}{2}\right)(T - t)\right] \\ d_2 &= d_1 - \sigma\sqrt{T - t} \end{align}

The stockprice is given by $$S_t=S_0e^{(r-\sigma^2/2)t+tW_t},\quad W_t\sim N(0,t)$$

$S_t$ is the only random term and log-normally distributed.

Its inverse function for the VaR quantile can only be calculated numerically as by MATLAB logninv function (assuming 250 trading days): $$VaR_\alpha^S=\text{logninv}\left(\alpha,\mu_S=(r-\sigma^2)/2,\sigma_S=3/250\right)$$

As we know, the call option delta is positive, such that its value will always fall and increase with the stock price. Hence the Option VaR follows as:

$$VaR_\alpha^C=C(S_0,0)-C(VaR_\alpha^S, 3/250)$$

which corresponds to the loss at the $\alpha$ quantile.

The option value also falls deterministically with decreasing time to maturity, as represented by the theta Greek: $$Theta(t)=\frac{\partial C}{\partial t}= -\frac{S_0 N'(d_1) \sigma}{2 \sqrt{T - t}} - rKe^{-r(T - t)}N(d_2)\, -\frac{S_0 N'(d_1) \sigma}{2 \sqrt{T - t}} + rKe^{-r(T - t)}N(-d_2)$$ That loss is however already included in calculating the difference $C(S_0,0)-C(VaR_\alpha^S, 3/250)$.

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    $\begingroup$ The question asks explicitly for Delta normal. This answer is not giving the Delta normal approximation. Also, and more importantly, VaR is calculated under the true probability measure and not the risk neutral: there $\mu_S$ should have the actual drift rather than the risk free rate. $\endgroup$ – Kiwiakos Jan 5 '15 at 0:55
  • $\begingroup$ @Kiwiakos The option price is calculated under a change of measure, so I was assuming to use the riskfree drift? Delta Normal means that returns are normally distributed, which is the case here since $W_t$ is normal. $\endgroup$ – emcor Jan 5 '15 at 9:23
  • $\begingroup$ I think @Kiwiakos has a point here, I'd also use real-world drift $\mu$ instead of $r$ here. I agree that the option price is calculated using drift $r$, but for risk purposes this is irrelevant I think. $\endgroup$ – SRKX Mar 2 '15 at 2:42
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You could simulate many (100000) 3 day price paths for the stock using the geometric brownian motion. Then for each simulated path, calculate the option value and store them. Then calculate the return difference for each of the calls and order them from smallest to largest. The 5% cutoff is your 3 day 95% VaR.

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    $\begingroup$ Would you simulate $S_t$ under its $P$ or $Q$ representation? $\endgroup$ – emcor Jan 3 '15 at 22:05
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    $\begingroup$ You would do it under $\mathbb{P}$. $\mathbb{Q}$ does not represent the actual distribution of the call option or the stock. It is essentially a pricing shortcut that, conveniently, also says something about the lack of arbitrage in the market. $\endgroup$ – user9403 Jan 6 '15 at 11:38

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