# 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:

1. Create paths of geometrian Brownian motions with parameters $\theta$ (in this example $\theta$ would contain the volatility $\sigma$ and the risk-free rate $r$).
2. 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

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

2. 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?

• The second approach could be used for risk management purpose as a means of generating economic scenarios under the physical measure P (e.g. a Heston dynamics if you believe the markets historically behaved that way and will continue doing so) while the pricing of the contingent claim remains under Q. This could be done for instance to get a feel for the P&L you get by hedging under your model assumptions while the market behaves differently. So it boils down to the P vs Q view discussed in many other questions e.g. here quant.stackexchange.com/questions/9172/sde-simulation-p-or-q – Quantuple Feb 1 '18 at 8:10
• Thank you for your feedback. That's what I thought at first. But for one, this whole procedure is only done to get the price process (so no risk management purpose). Second, if even these two sets of parameters would reflect two measures P,Q, shouldn't then at least the volatilities agree. And it also wouldn't make sense to say P is Heston and Q is Black Scholes, or would it? – Cettt Feb 1 '18 at 8:15
• Hi Cett. It's a matter of point of view. In the second approach you're basically specifiying the 'observed/historical' dynamics of the market and using a different model to capture it's 'future behaviour'. I agree this is not consistent from a pure theoretical perspective of getting an option price (typically the P associated to the pricing Q will be different than the measure under which you've specified the market dynamics as you've said)... But this may still provide you some useful information. Indeed I don't see what 'generating' the price process can be useful for except for risk mngt – Quantuple Feb 1 '18 at 8:20
• Usually we do not generate the option price path (only the underlying path under a given Q) but rather observe the realised path ex post (at the close of each trading for instance). The second approach emulates that: since the market realised dynamics will usually differ from that of your model, you specify it exogenously but still conduct the pricing under your favourite model hence Q. This is the idea anyway. – Quantuple Feb 1 '18 at 8:21

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.

• The price path was always obtained using Black Scholes with one constant volatility $\sigma_\text{pricing}$ which was different from the volatility he used to generate paths $\sigma_\text{simul}$. After thinking about this problem some more and reading your answer I think that $\sigma_\text{pricing}$ was (somehow) obtained from the implied vol surface. – Cettt Feb 16 '18 at 15:56