Mixing Black Scholes with SABR

Question

I am new to the whole concept of stochastic volatility so I am experimenting with option pricing. I think the concept is really difficult to understand / grasp.

I was wondering if the following approach is way of or an appropriate strategy:

At day 0 I want to price a European Call with the underlying asset $S$ option that expires at time $T$. I observe market data for European Call options on $S$ with different strikes. Then I do the following:

• Observe market data (IV for different strikes for an option on $S$) and fit SABR to it to find estimates for $(\alpha,\beta,\rho)$

• Now that I have the SABR parameters: For a given strike ($K1)$ I comute it's volatility: $\hat{\sigma}_{K1} = \sigma_{SABR}(K1;\alpha,\beta,\rho)$

• Then I price the option with strike $K1$ with the naive Black Scholes formula and set volatility to $\hat{\sigma}_{K1}$

Is this a descent aproach for option pricing? Will my Call price with strike $K1$ be close to what is "real" fair value

Answer 1

In your first step, you will want to calibrate $\alpha, \rho, \nu$, and probably not $\beta$. See the related question Calibrate a SABR model?

How close your option price is from the market price will depend on the fit quality. SABR has only a few parameters and does not necessarily match well equity or equity index options of short maturities. It is more appropriate for foreign exchange options or interest rate swaptions.