Local Volatility Model for Callable Quanto Option Valuation

A model is presented for computing the price, in the domestic currency, of European standard call and put options on an underlying foreign equity (stock or index)

Local Volatility Model for Callable Quanto Option

We review a model for computing the price, in the domestic currency, of European standard call and put options on an underlying foreign equity (stock or index) with tenor of up to 7 years. The function implements a local volatility based pricing method.

The payoff in the domestic currency of a European option at maturity T is given by

where

The equity price process satisfies a risk-neutral stochastic differential equation (SDE) when where are no dividend payments. Let St denote the equity price at time t. We assume that the process satisfies a SDE of the form under the domestic risk-neutral probability measure:

where

We note that the volatility σ depends only on time and on the instantaneous value of the state variable S, but does not explicitly depend on W.

The quanto-adjustment q is calculated by

where

Note that we also implement a second quanto-adjustment technique that is of the form

where w_k is a weight.

Consider the calibration of the local volatility function based on market option prices or, equivalently, market Black’s implied volatilities. If there exists a smooth surface of either option price or implied volatility as a function of option strike and maturity, then this surface uniquely determines the local volatility function.

Moreover, an explicit expressions for local volatility is provided. For example, if C denotes the price of a call option on a non-dividend paying stock, with a constant risk-free interest rate, then

Local volatility model can also be applied to value callable exotic notes. Callable option is more volatile as callable events make the remaining part of the trade potentially be cancelled as a result of a trigger condition or an exercise option. You can find a good illustration on the topic of callable notes at https://finpricing.com/lib/EqCallable.html

In general, local volatility model is very useful tool for pricing equity derivatives.