1. Historical simulation for the option example
2. Historical simulation
Let's keep things really simple and consider that the investor's portfolio contains a single European call option on an equity index (the S&P 500).
Let's also consider the loss and profit distribution over a one-day time horizon.
When the option is valued using Black Scholes, then changes to the index value, the implied volatility, and the interest rate over the time horizon will all lead to changes in the value of the option.
Thus the risk factors are the index value, the implied volatility, and the interest rate.
Arguably, the first two of these are the most important and exhibit the greatest fluctuations. So let's concentrate on changes to the index value and implied volatility and assume that the interest rate remains constant.
To apply the historical simulation method, a loss operator function for the option is needed.
This function will take as input the log-returns of the equity index and the implied volatility and give as output the resulting loss or profit on the value of the derivative position.
3. Estimating VaR and ES
As before, the loss operator function - lossop() - has been precoded for the portfolio, and you can simply use it in the exercise.
When you apply lossop() to the log-return time series for the S&P 500 index and the VIX volatility index, you get historically simulated losses for the derivative portfolio.
Exactly as in the case of the equity investment portfolio, you can then measure the risk by estimating VaR and expected shortfall.
Recall that a simple method of estimating VaR is to compute a sample quantile for the historically simulated data.
A simple method of estimating ES is to compute the average of the losses exceeding the VaR estimate.
4. Let's practice!
So, now it is time for you to implement historical simulation and estimate VaR and expected shortfall for the options portfolio.