# Interpolating on the BS parameters and injecting in the BS formula vs interpolating directly on option prices

Let's consider a simple European call option. In practice, the way the Black-Scholes formula is used to price it is by injecting all of the parameters and paying special attention to the volatility and the dividends where their implied values are used. This gives then, by definition, the market price of the corresponding call option.

Now, suppose one wants to price a call option with maturity $T$ and strike $K$ that isn't found in the market (and hence has no corresponding implied volatility or dividends). One could either interpolate between the prices of the options with the closest maturities and strikes, or one could interpolate between the implied volatilities and dividends corresponding to the closest maturities and strikes and then inject into the BS formula.

Which one is the best approach ?

• How would you "interpolate" between prices given the nonlinear relationships? The only practical way is by using the BS formula and interpolate the INPUTS to the B-S formula. As you yourself said, this is how the BS formula is used in practice. – Alex C Mar 12 '16 at 20:40
• I could make the same remark about the interpolation on the inputs. If we suppose that the implied volatility is a function of $K$ and time-to-maturity $\tau$ then there is no reason to believe that interpolating between known values of the implied volatility will yield a good approximation because it is nonlinear in $K$ and $\tau$, except if we have many close data points. But then again, this could also be said about interpolating directly between the prices. – BS. Mar 12 '16 at 20:52
• The solution then would be to interpolate the implied volatility using an appropriate model such as SVI (quantitativefundamentalist.blogspot.com/2011/01/…) and then plug into the BS formula. These models exist for a reason, why throw them out and start from scratch with prices. – Alex C Mar 13 '16 at 4:48

## 1 Answer

The best way is to interpolate in volatility space. The reason is because it is closer to the intrinsic pricing of the option, and it is less likely to produce an arbitrage. Like Alex C noted in the comment - prices are nonlinear function of inputs, and interpolating in them does not make sense. Inputs are "free", and interpolated value of inputs will likely be somewhere in the smoothed area.

Really simple example - consider 3 european options with consecutive equidistant strikes, with the same black-scholes IV. If you don't know the middle option's price, you should interpolate vols, and then price the middle option. If you were to interpolate prices, then your middle option will be the average of the other two, and will create a free butterfly, which will be an arbitrage.