Local Volatility and Option Prices Under a Higher Volatility Surface

Summary

The document considers whether a local volatility pricer is flawed when it values a five-year at-the-money call below the price produced by a flat volatility input. The tested surface has implied volatilities above the flat input across the stated range, prompting the expectation that the surface-based price should be higher.

The accepted explanation uses dynamic replication: an option’s price reflects the cost of rebalancing a hedge, and greater volatility can raise those costs for a short gamma position. It argues that if local volatility is higher at every relevant spot and time, replication should cost more than under the lower constant volatility. This is an intuition-based explanation rather than a diagnosis of the black-box implementation. It assumes the stated pointwise comparison holds throughout the paths and times relevant to pricing; the document provides no independent test or detailed model conditions to establish that assumption.

Key ideas

• Option value can be understood through the cost of dynamically replicating its payoff.
• Higher volatility can increase hedging costs for a short gamma position.
• A local volatility surface above a constant volatility at every relevant state is expected to imply a higher option price.
• The explanation does not independently verify the pricer or specify all assumptions needed for the comparison.

Tags

# Local volatility pricer


# Local volatility pricer












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

- Pricing a 5yr ATM call option with a flat volatility of $0.194$

- 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?

## Answer by Istopopoki (score 5, accepted)

https://quant.stackexchange.com/a/15428

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.