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Monte Carlo VaR for a Forward and European Put Portfolio

Article Quant Q&A · Author: Kyle
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Summary

The document explores how to estimate value at risk for a portfolio containing forwards and European puts. It proposes calculating each instrument’s profit or loss at a simulated maturity price, adding those component outcomes, and taking a tail quantile from many simulations. It considers generating forward prices and converting them to spot prices before repricing the put with Black–Scholes, while questioning which volatility, strike, and contract assumptions are appropriate. A static historical volatility is raised as one possible input, not established as a recommended choice.

The post provides a tentative payoff-based setup, but its option-pricing equations contain apparent notation or formula issues, and it does not present a finished VaR method or empirical results. In particular, it leaves unspecified the risk horizon, current portfolio marks, option maturity relative to that horizon, market inputs, and dependence between price and volatility changes. The discussion points toward modeling the joint portfolio profit-and-loss distribution; it offers no evidence that separate simulations or separate VaR figures can be safely combined.

Key ideas

  • Portfolio VaR should reflect the combined profit and loss of the forward and put positions under shared market scenarios.
  • The proposed workflow simulates underlying prices, values each instrument, and forms a distribution of portfolio outcomes.
  • The post considers historical volatility as an input but does not justify a static estimate or specify a calibration method.
  • The risk horizon and the option’s remaining life need to be distinguished from contract maturity when constructing scenarios.
  • The document gives no completed model, validation, or empirical VaR result, and some displayed pricing notation appears problematic.

Tags

Risk managementOptionsFuturesDerivatives pricingStatistics
Full text
# How is VaR calculated for forward contracts accounting for European put options?


# How is VaR calculated for forward contracts accounting for European put options?












My initial idea is to create profit and loss using an equation like this: \begin{align} P\&L = & \text{European Put P&L} + \text{Forward P&L}\\ P\&L = & [(K-S_T)^+ - \text{Premium}] \cdot{} \text{N Put Contracts} + \\ & [(S_T - K) - \text{Contract Price}] \cdot \text{N Forward Contracts}\\ \end{align} Where $S_T$ is the underlying spot at maturity, $K$ is the strike price of the option or agreed-upon delivery price for the forward, $T$ is the time to maturity, and $S_T$ is derived from simulated forward prices from Monte Carlo Simulations and converted to spot via the BSM.

After obtaining a select forward price from an MC Simulation, we follow the BSM model. The forward value of the stock is:

$$F(t_0,T')=\mathbb{E}[S(T')]=e^{r(T'-t_0)}S_0$$

Where $T'$ is the contract maturity. We get the Stock value $S_0$ as a function of the forward value and the discount factor:

$$S_0=F(t_0,T')e^{-r(T'-t_0)}$$

Supposing that $t_0=0$, we get: $S_0=F(t_0,T')e^{-r T'}$.

The Black-Scholes formula based on the underlying stock is:

$$\text{P} = Ke^{-rT}N(-d_2) - S_0 N(-d_2)$$

With:

$$d_2=\frac{ln \left( \frac{S_0}{K} \right)+rT+0.5\sigma^2T}{\sigma \sqrt{T}}$$

Now, we substitute $F(t_0,T')e^{-r T'}$ for $S_0$ in the above to get:

\begin{align} \text{Put Premium } = & Ke^{-rT}N(-d_2) - F_e^{-rT'-T} \cdot N(-d_2)\\ d_2 = & \frac{\ln\left(\frac{Fe^{-rT'}}{K}\right)+(r-\sigma{}^2/2)T}{\sigma{}\sqrt{T}} \end{align}

After calculating P&L for thousands of simulations, I can calculate VaR.

So, I have historical forward prices. Theoretically, I should be able to simulate forward prices, get their spot and European put prices, and calculate profit and loss, then just do a simulated historical VaR calculation. However, I'm not sure about the parameters that I need to pick for the BSM model—volatility, strike prices, number of put contracts, etc. I was thinking about going forward with the volatility as a static historical annual volatility.

I'm not sure if I'm completely wrong or just overthinking it. Maybe I can somehow simulate European put options separately, the forward prices separately, and somehow combine the VaRs. But I'm not sure what assumptions to simulatenously make for both of those processes if I did that either.

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.