Stress Testing Down-and-In Puts with Spot and Volatility Shifts

# Volatility model for pricing of Down-In Put options


# Volatility model for pricing of Down-In Put options












What is the best volatility model to price Down-In Puts (DIP) under several stress scenarios (e.g downward movement of underlying price by -5%, -10%, -15%)?

Please note that I got the market prices for all vanilla put options on the underlying assets and I have calculated the implied volatilities. Also, I use the formulae in the following link to price DIP.

https://www.asc.tuwien.ac.at/~juengel/simulations/fincalculator/doc/Barrier.pdf

Thanks

## Answer by AKdemy (score 2, accepted)

https://quant.stackexchange.com/a/74126

I would suggest to simply use the spot shift and add a few implied vol (IVOL) shifts. All else will be a massive task, with little to no benefit in my opinion. Explanation below:

Simple scenarios in most professional setups will be isolated shifts keeping all else equal. Since you already have implied vol (IVOL) for all options, you could plug in the IVOL associated with the shift. Ideally, you create a vol surface and interpolate to the exact value, but an actual shift of, for example 15%, will most certainly have such an impact on IVOL that the current surface will anyhow not be accurate. Also, for a move of 15% to occur, you would realistically have some time shift as well.

Using something like Local Vol (LV) will not improve your results because it just uses the existing vol surface (vanilla prices) in any case. Also, you would not be able to use your simple pricing formula anymore. Generally, IVOL comes from the correlation between the spot process and the vol process. As a result, where the (instantaneous) correlation is nonzero, you will have a skew. You do not have this in LV though, because the entire skew comes from the input surface. If you incorporate a time shift, the forward skew will be under pronounced in LV, since option prices' skew tends to flatten with extended maturities.

Theoretically, Stochastic Vol (SV) like Heston allows for this. It would be best to use term Heston which has several periods. Ideally, a Stochastic Local Vol (SLV) model which mixes LV with SV and calibrates to existing barrier options should be used.

Getting from your current setup to a SLV model is akin to going from running a simple OLS model $ \widehat{\mathbf{\beta}}=\mathbf{X^\prime X}^{-1}\mathbf{X^\prime y} $ to writing an entire Statsmodels package. You can have a look at this answer to get a basic idea of what complexity this involves.

Generally, there are two types of scenarios:

- implicit scenarios: simple custom scenarios where you change parameters of a deal (shift IVOL, spot etc.)

- explicit scenarios, also called predictive scenarios that are regression based and take indices like S&P, Eurostoxx, FTSE and fetch historical data to check for correlation etc.

All available option pricing models have one thing in common; they use current data only and scenarios are based on implicit scenarios. You could look at historical data and see how an actual 15% shift affected the parameters (e.g., what happened when Lehman Brothers collapsed, or during COVID and so forth - this approach is called historical scenarios, but it really is just a subtype of implicit scenarios).

If you want an explicit scenario, you need to get estimates of the likely impact on parameters based on statistical methods. In theory, this should be more reliable than simple implicit scenarios. In reality though, no market crisis will be the same. For example, a spot shift in your underlying could be because there is bad news in your underlying only, or because the entire economy is plunging.

Since you do not look at stress of a portfolio of various underlyings and products, I honestly think the simplest method of keeping your simple model and just shifting spot will be sufficient. If you want to get a bit fancy, plug in the IVOL associated with the shift, and maybe even assume an additional upward shift in IVOL based on historical estimates.