In this post, I augment a constant coefficient geometric Brownian motion process by a single jump whose time of occurrence is known. The random variable representing the jump size follows a normal mixture distribution. To get a feeling for the impact of predictable jumps on option prices, we inspect the shape of a few model implied probability densities and volatility smiles.
We consider a simple and tractable model for an asset price process that is subject to a large jump with a known time of occurrence. These situations often arise around scheduled news releases such as quarterly earnings announcements, monetary policy decisions or political elections. This post is the first in a series on this topic. I provide the general pricing setup and derive the characteristic function of the corresponding logarithmic return process including drift adjustment. Future posts then make specific choices for the jump size distribution and provide examples for the corresponding implied volatility smiles.
In the last post, I provided a brief introduction to forward mode automatic differentiation with CppAD. In this post, I propose to use automatic differentiation for the computation of cumulants of option pricing models based on characteristic functions. This is useful, for example, when pricing European vanilla options using the Fang and Oosterlee (2008) COS method. Here, the first four cumulants are used to determine the integration range.
As shown in Schoutens et al. (2004), different dynamics for the underlying asset price are able to achieve a similarly good calibration to the market prices of European plain vanilla options. Consequently, these models are able to generate very similar marginal distributions for all future points in time. However, their path-behaviour differs thus often yielding significantly different prices for exotic options. In this post, we provide some intuition for the sign of the price differences between local and stochastic volatility models for barrier and American binary options. These arguments are not novel and we refer to e.g. Baker et al. (2004) for a more in-depth and rigorous discussion.
In the presence of an implied volatility smile, it is not clear which volatility to use when pricing barrier options in the Black-Scholes framework. A simple, although theoretically not justified, tweak is an approach related to outside barrier options. For these contracts, the payoff is determined by one asset while the barrier trigger is linked to another.
Jim Gatheral’s book “The Volatility Surface – A Pratitioner’s Guide” (Wiley, 2006) provides an excellent treatment on volatility modelling. Most of the proofs and derivations are only outlined in the book and it is left to the reader to do the intermediate steps. The attached document provides the missing steps in the derivation of the local volatility as a function of the implied volatility on pages 11 – 13 in the book.
I recently had to derive the pricing formula for an American digital call option (or American binary/one touch call option). Although the final solution can be readily found in the internet, I was unable to find a full derivation of both, the first passage time of a drifted Brownian motion and the option value.