Historical Value At Risk on option portfolio

Question

I am new to Value at Risk subject in fact everything related to quant. Can any body validate the Value at Risk Model on the option price ? I am using a below explained approach .

our portfolio contains an Out-the-money call on Corn sept delivery month with one month until expiry. Based on Historical VaR, and assuming we are interested in the 1-day 95% VaR, you might be tempted to gather the returns on the option over the past 100 days (or 200, or 500, …), sort them, and select the 5th worst return. This would be wrong. The problem is that we are trying to evaluate the risk of an out-the-money call with one month to expiry, but 100 days ago the option had one month plus 100 days to expiry and may have been far in-the-money or out-of-the-money.What we need to do in this case is to go back every day for the past 100 day and determine not what the returns of our option were, but what the returns of our option would have been. We do this by going back over the past 100 days, observing the change in the inputs to our option price (underlying price, implied volatility, dividend yield, risk-free rate, and time to expiry), and then calculating what the price of our option would have been. For example, if the current price of Corn is 1,077.29 and the return on Corn 100 days ago was –0.76%, then we would use an underlying price of $1,069.10, $1,077.29 x (1 – 0.76%) = $1,069.10.

To price our option. We refer to this approach as back casting, and to the resulting prices as backcast prices.

The next step in backcasting with options is to take the backcast price of the option, and calculate a backcast return. For example, if the current price of our Corn option is 10 and the backcast price calculated ( Using Black Scholes, Black 76 , Binomial Tree ) in the previous step was $11, then our backcast return would be 10, (11 – 10)/10 = 10 percent. Notice that our starting price for calculating the backcast return is the current price of the option. If we are evaluating the risk of the option on 12/12/2013 and our 100 day backcast window runs from 7/23/2013 - 12/11/2013, then the first back cast return would use the 12/12/2013 and the backcast price on 7/23/2013. The second backcast return would use the 12/12/2013 and the backcast price on 7/24/2013. We repeat this process for each day in our backcast window (100 days, 200 days, 500 days, …).

The final step in backcasting with options is the same as it would be for any security: we sort the backcast returns and then select the return corresponding to our VaR (e.g. the 5th worst of 100 for 95% VaR). In summary, for Commodity options the historical method is a four step process: 1. Calculate backcast values of the underline using Spot Price *( 1+ daily return). 2. Calculate the backcast price of the option on each of the backcast dates(Using Binomial , Black Scholes, Black 76) . 3. Calculate the backcast returns for the option. (Maket Observed Return - Calcualte option price for the back cast Date)/ Maket Observed Return 4. Sort the backcast returns, and determine the VaR.

Assumption

While revaluing the option time to maturity in the option is constant from the simulation day to time to maturity . .

Would really appreciate if any one can help to validate the model ?

Answer 1

Your method does make sense to me, I do the same. The historical simulation is known to be a full evaluation approach: you simulate changes in market conditions by applying the same changes happened in the past to your risk factors, then you compute your portfolio value under the new market conditions.

Since Value at Risk (VaR) is the maximum loss with a given confidence level over a specified time frame (like 1 day or 10 business days), the time to maturity of the option should be adjusted for this time period: if you compute the 1 day VaR, then the time to maturity of the option is 1 day less when you compute the under new market conditions.

According to me, you tend to focus too much on time: the idea of historical simulation is to "borrow" data about changes in market conditions from history and simulate them today to see how your portfolio value changes. So you take the data and forget about time.

I give you a reference where I studied the method: Risk Management and Shareholders' Value in Banking, chapter 7. I think it is very clear and thorough.

Two notes about the comments:

• the delta-normal approach refers usually to the variance-covariance method (there is a sensitivity coefficient in the VaR formula that links VaR to the risk factor change in value in a linear relationship). The same book will offer you some clarification about it (chapter 5).

• About implied volatility, I believe using it is a good thing in case you evaluate with the usual Black-Scholes formula: it allows you to consider the skew/smile. However, in case of a portfolio of options, you would have a very simplified scenario: each option has its own implied volatility, but you have the same historical change to apply to all of them. This is quite unrealistic, since each volatility would change differently in value (maybe same direction, but almost surely different percentage change).