The book by Cont and Tankov (2004) is an excellent introduction to jump processes in finance. The attached document lists some potential typos/inconsistencies in the notation of the 2004 printing that are neither included in the errata published under http://www.cmap.polytechnique.fr/~rama/Jumps/ nor in an updated PDF version of some of the book chapters.
Stochastic vs. Local Volatility for Barrier Options
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.
Ornstein-Uhlenbeck Stochastic Clocks
After introducing the general idea of stochastic clocks in the previous post, we now consider the case where the instantaneous rate of activity follows a non-Gaussian Ornstein-Uhlenbeck process. The major reference is the paper by Barndorff-Nielsen and Shephard (2001b), see also Barndorff-Nielsen and Shephard (2001a). In this post, we discuss the general stochastic setup that applies to all such models that are driven by an almost surely non-decreasing Lévy processes. For the moment, we leave the particular dynamics of this subordinator unspecified but we will provide examples in future posts.
General Stochastic Clocks
This is the first post in a series on asset price dynamics that are subject to a stochastic clock. The main idea is to not model the volatility as a stochastic process but instead make the rate at which time progresses randomly. In this post, we outline the basic idea and define the terminology.
Two-Volatility Barrier Options
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.
A Course in Financial Calculus – Some Solutions
In the below files are some solutions to the exercises in Alison Etheridge’s textbook “A Course in Financial Calculus” (Cambridge University Press, 2002). The files are grouped by chapter and I will add new solutions from time to time.
Stochastic Calculus for Finance I – Some Solutions
In the below files are some solutions to the exercises in Steven Shreve’s textbook “Stochastic Calculus for Finance I – The Binomial Asset Pricing Model” (Springer, 2004). The files are grouped by chapter.
Stochastic Calculus for Finance II – Some Solutions
In the below files are some solutions to the exercises in Steven Shreve’s textbook “Stochastic Calculus for Finance II – Continuous Time Models” (Springer, 2004). The files are grouped by chapter.
Local Volatility in Terms of Implied Volatility
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.
Presentation on the Cox-Ingersoll-Ross Model
This is a presentation on the Cox-Ingersoll-Ross (CIR) model economy, the equilibrium interest rate process and the pricing of derivative securities. I had to hold this presentation in the PhD class FINS5591 – “Continuous Time Finance” at UNSW in the first semester of 2010.