Estimating Short-Horizon VaR for a Long European Call
Summary
The document considers a three-day, 95% value-at-risk estimate for a long European call when the underlying follows geometric Brownian motion. It describes valuing the option with Black–Scholes, deriving a lower-tail stock-price quantile from the lognormal distribution, and repricing the call at that quantile. The loss is the difference between the current option value and its value at the horizon and quantile; this repricing incorporates the passage of time as well as the stock-price move.
A second response proposes Monte Carlo simulation: generate many three-day stock paths, revalue the option at the horizon for each path, sort the resulting changes, and take the adverse tail cutoff. The material is illustrative rather than a complete implementation guide. It does not provide inputs for a numerical VaR, and its quantile and time conventions should be checked carefully in a practical calculation. The simulation approach also requires a specified pricing model and assumptions for the underlying dynamics.
Key ideas
- Under geometric Brownian motion, the underlying stock price has a lognormal distribution over the VaR horizon.
- The option can be repriced using Black–Scholes at the stock-price quantile and the horizon date.
- For a long call, a lower underlying price produces an adverse value change, while time decay is reflected in horizon repricing.
- Monte Carlo paths offer another way to estimate the tail loss by repricing the option across simulated outcomes.
- The discussion gives no numerical estimate and leaves implementation conventions to the practitioner.
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Full text
# How to compute the VaR for European Call, using the delta-normal method?
# How to compute the VaR for European Call, using the delta-normal method?
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?
## Answer by emcor (score -1, accepted)
https://quant.stackexchange.com/a/16075
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 $C(S_0,0)-C(VaR_\alpha^S, 3/250)$.
## Answer by QuantK (score 3)
https://quant.stackexchange.com/a/16074
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.