Incorporating Indirect Channels into Option Greeks

Abstract

I propose a theoretical paradigm of reduced-form total Greeks which capture the total sensitivities of option premia to input variables without the necessity of modifying the underlying pricing model, e.g., Black-Scholes-Merton (for single risk-free rate) or Garman-Kohlhagen (for two risk-free rates), to incorporate the indirect channels that are overlooked in the model.
Applying this paradigm to two distinct instances of indirect channels, I propose (i) a total delta and total gamma that models a quadratic form of the volatility smile in options across asset classes, and (ii) a total rho that incorporates covered interest rate parity in currency options. I show that volatility smiles of certain shapes can push delta of call (put) options beyond the conventionally accepted upper limit of +1 (0) and the lower limit of 0 (-1). Furthermore, I propose a unified model of total Greeks for currency options to simultaneously tackle both volatility smiles and interest rate parity.

Keywords: Options, Option Greeks, Volatility Smile, Interest Rate Parity

JEL Classification: G13