# Method for using Historical Simulation method on an Instrument priced using Monte Carlo

I was speaking to a very esteemed professional in Financial Risk and he mentioned that he always prefers to use Historical Simulation as the method for his VaR even if he prices his Exotic derivatives (such as Barrier Options) by using the Monte Carlo simulation method.

I asked how this would be done, re-running the simulation for everyday for the past year (if the in-sample period is one-year for example) and getting the payoff that way?

He said no, actually what he does - using an AutoCallable Swap on three indices as an example - is:

1. Calculate the estimated Price of the instrument using the Monte Carlo method: generating many simulated paths of the underlying indices and getting the mean of the simulated payoffs and present valuing to today
2. Getting the delta of the instrument which each index by varying them before re-pricing and seeing what the change in the price is
3. Getting the one-year time series of each index, multiplying each index by their delta, and adding them together and using that as the implied one-year time series of the index

I can't seem to find any reference to this method and I haven't been able to contact him since to ask him directly. Does this method have a name or a source that I could look up?

I am not sure what you mean by your second point, but to my knowledge, computing historical Value at Risk on a derivative is a full valuation exercise where you use historical data to simulate changes to your risk factors, then reprice your portfolio under these new factors. For a derivative, it requires back-casting the price of the underlying to reprice the instrument for every point in your in-sample period. This is because you are trying to evaluate the risk of your instrument today (with its specific moneyness and time to expiry), but it might've been deep in the money or OTM 50 days ago.

For example, for an OTM option on some underlying A with one month to expiry:

1. Obtain the price of your instrument today using your Monte Carlo engine
2. Observe the input changes to your option on your sample period (underlying, risk-free rate, dividend yield, time to expiry, etc...) and reprice your instrument at each step. For example, if the current price of A is \\$1000, and the return was -10% yesterday, you would use the back-cast price of $$1000 \times 0.9$$.
3. The final step would be to compute your return time series as a difference between the price of your instrument today, and the prices you computed in step 2. This time series is the one used for VaR.

Most of what I've described I've found here a while ago. I hope this is helpful.

• Hi Arida, thank you for your reply. This was not a valuation exercise; the person I was talking has had a very long and successful career in Fund Risk, and was clear that this was not the case, so I take his word on that. He said for all of his OTC derivatives he does it this way: price the instrument, get the delta of the each underlying with respect to the price, and use the sum of the underlying's historical time series', weighted by their deltas. It is a technique I can't find any reference to, but it sounds quite useful, hence my question. Sep 23 at 12:36

I have been able to get a hold of the man again and he clarified for me.

Suppose you are running the Monte Carlo Simulation 10,000 times. You are not going to feasibly be able to re-price the asset with this method for every day in the pass 255 business days (assuming a daily VaR with 1-year in-sample). Also, the risk associated with the asset now is what you care about, whereas 6 months ago for example the theta decay of an option with be very different to what it is now.

So what you do is recall that the Total Derivative is mathematically the best first order approximation of a function at that point (or you can think of it as a mutli-variable Taylor Series expansion).

If $$P(x_1, \ldots, x_k)$$ is the Monte Carlo simulated price of the asset, which depends on the parameters $$x_i(t)$$, all of which depend on time, then we have: $$$$P = P(x_1, \ldots, x_k)\bigg|_{t=0} + \sum \frac{\partial P}{\partial x_i}\bigg|_{t=0}x_i + (\text{higher order terms})$$$$

Since we are looking at VaR which is the amount of loss, we can neglect the first constant term. We also neglect the higher order terms, because:

1. If it's daily VaR then the squared differences will be small, and
2. Monte Carlo already has a large "error-bar" already which would mean that higher order partial derivatives wouldn't be very accurate.

This makes for a very computationally cheap and easily applicable Historical Simulation VaR model. You would get the time series of the underlying parameters, multiply them by the partial derivative of the Monte Carlo price with respect to this parameter, then add them all together to get a time series for the asset!

Whether it's robust or not however is another matter.