# Local volatility pricer

I am testing a local volatility pricer by comparing its results under two settings:

1. Pricing a 5yr ATM call option with a flat volatility of $0.194$
2. Pricing the call option with the typically shaped equity vol surface: in particular, the minimum implied vol figure is $0.195$ (longest maturity / ATM strike) and the maximum implied vol is $0.245$ (shortest maturity / lowest strike)

By intuition, using a implied vol surface where all the individual volatility points are above $0.194$ should deliver a higher call price than using a flat volatility. However, the local vol pricer that I am testing (black-box) delivers lower option values when the input is the [$0.194$ to $0.245$] vol surface than when I use a single $0.194$ flat figure.

Is that enough evidence to conclude that the local vol implementation is flawed? Or should I distrust my intuition?

## 1 Answer

You can view the price of an option as the cost to dynamically replicate it. The more volatility, the more costs you will have trading the underlying to keep your delta equal to 0 (I'm assuming you sold the option, hence a negative gamma position). So, if at any spot, any date your local vol is above 0.194, rebalancing the portfolio will be constantly more expensive than doing the same job with a constant vol equal to 0.194. So the option needs to be more expensive.

• I did not say that the local vol points are above 0.194. The local volality is unknown to me (hence the black box). What is above 0.194 are all the implied vol points that were used as an input. – sets Nov 14 '14 at 16:06
• If the local vol model fits the market smile properly, it will give implied vol equal to the input. Here all the implied 5y vol are above 0.194 but the local vol model prices a 5y call at a lower price than a constant 0.194 vol Black-Scholes model. So by definition, the local vol model gives an implied vol that does not match the input. A possible explanation: the implied vol is >0.194 for all given strikes, but because of some overshoot in the interpolation, the implied vol of your specific strike happen to be <0.194 ? Or the model implementaion is wrong. – Istopopoki Nov 18 '14 at 18:39
• if you price an option whose strike is part of your known inputs, and observe the same behavior, you will know that for sure the model does not fit the input. But it may not come from a problem in the model if the input you gave it is incoherent. – Istopopoki Nov 18 '14 at 18:43