chapter 19 Extension of the Theoretical Framework for Pricing Derivatives (厦门大学,郑振龙).ppt

Extension of theTheoretical Framework for Pricing Derivatives:Martingales and Measures Chapter 19 Derivatives Dependent on a Single Underlying Variable Forming a Riskless Portfolio Market Price of Risk (Page 500) This shows that (m ?r )/s is the same for all derivatives dependent only on the same underlying variable, q, and t. We refer to (m ?r )/s as the market price of risk for q and denote it by l Differential Equation for ?(Equation 19.10, page 501) Using Ito抯 lemma(10.14) to obtain expressions for m and s in terms of m and s. The equation m-ls=r becomes Risk-Neutral Valuation Compare the equation with (12.A4), it shows that q is like a stock price paying a dividend yield of r ?m + ls This analogy shows that we can value ?in a risk-neutral world providing the drift rate of q is reduced from m to m ?ls Note: When q is not the price of an investment asset, the risk-neutral valuation argument does not necessarily tell us anything about what would happen with q in a risk-neutral world . Extension of the Analysisto Several Underlying Variables(Equations 19.12 and 19.13, page 503) Traditional Risk-Neutral Valuation with Several Underlying Variables A derivative can always be valued as if the would is risk neutral, provided that the expected growth rate of each underlying variable is assumed to be mi-雐si rather than mi. The volatility of the variables and the coefficient of the correlation between variables are not changed. (CIR,1985). Derivatives Dependent on Commodity Prices (Page 506) For a commodity the futures price gives the expected value in the traditional risk-neutral world. Martingales (Page 507) A martingale is a stochastic process with zero drfit A martingale has the property that its expected future value equals its value today Alternative Worlds A Key Result (Page 509) Forward Risk Neutrality We refer to a world where the market price of risk is the volatility of g as a world that is forward