Separating Simulated Underlying Volatility from Option Pricing Volatility

# Monte Carlo Simulation of price processes


# Monte Carlo Simulation of price processes












before I ask my question I want to illustrate what I think I know about Monte Carlo simulation: say I want to simulate the price paths of one European Call Option with fixed strike and maturity in the Black Scholes model. This can be done in two steps:

- Create paths of geometrian Brownian motions with parameters $\theta$ (in this example $\theta$ would contain the volatility $\sigma$ and the risk-free rate $r$).

- Along each path use the Black-Scholes formula with parameters $\theta$ to get the corresponding call option price path.

In this setting I use the same set of parameters $\theta$ to simulate the paths and to price the option.

I recently stumbled upon notes of a former colleague who used different sets of parameters, e.g. he would

- Create paths of geometrian Brownian motions with parameters $\theta_\text{simul}$.

- Along each path use the Black-Scholes formula with (different) parameters $\theta_\text{pricing}$ to get the corresponding call option price path.

In particular he used different volatilities for the simulation part and the pricing part. Also the $\theta_\text{simul}$ could contain a non-zero drift. In one particular example he would even generate paths from a completly different process (say Heston) and then use Black-Scholes in the second part anyway.

Is this common market practice? Or how could one justify such an approach?

## Answer by AlRacoon (score 2)

https://quant.stackexchange.com/a/38240

As you simulate the underlying, you will have a path of possible valuations that are moving around. However, the strike of your options and the maturity of your options remain constant. The volatility you are using in your simulation of the underlying is the realized/historical underlying volatility, while the volatility to price options is the market or implied volatility. Also, there is a skew and term structure of volatility. In other words, the moneyness and the expiry of the option will drive the implied vol to use to attain the market value of the options in question.

So your colleague was probably adjusting the vol by adding the skew component depending on where the strike was relative to the simulated underlying value. He was probably also adjusting this vol to account for the option being one period closer to expiry with each step of the Monte Carlo simulation. The Monte Carlo simulation you describe is only creating a simulated path of the underlying. The vol parameter needs to be adjusted depending on where the strike is relative to the simulation and expiry. He also may have used stochastic volatility as in the Heston model.

Here is an example of the volatility surface (moneyness or strike vs time to maturity) of the S&P 500 index. As you can see, all the options are not priced using the same or constant volatility.

The top row has the moneyness in %, with the strikes below that row.

The left column has the expiry of the options.