Volatility and Dividends - Volatility Modelling with Cash Dividends and Simple Credit Risk

Abstract

This article shows how to incorporate cash dividends and credit risk into equity derivatives pricing and risk management. In essence, we show that in an arbitrage-free model the stock price process upon default must have the form St = (F ⁄ t i Dt)Xt + Dt where X is a (local) martingale with X0 = 1, the curve F ⁄ is the \risky forward and D is the ∞oor imposed on the stock price process in the form of appropriately discounted future dividends. This has already been shown in [1]. We show that the method presented is the only such method which is consistent with the assumption of cash dividends and simple credit risk. We discuss the implications for implied volatility, no-arbitrage conditions and we derive a version of Dupire’s formula which handles cash dividend and credit risk properly. We discuss pricing and risk management of European options, PDE methods and in quite some detail variance swaps and related derivatives such as gamma swaps, conditional variance swaps and corridor variance swaps. Indeed, to the our best if our knowledge, this is the flrst article which shows the correct handling of cash dividends when pricing variance swaps.