Martingale process

In the academic literature, the bond price given by equation (3.15) evolves as a martingale process under the risk-neutral probability measure P. This process is the province of advanced fixed-income mathematics and lies outside the scope of this book. An introduction, however, is presented in chapter 4, which can be supplemented by the readings listed in the References section. 

The stochastic process d% (z) is driftless and thereupon a local martingale, if the deterministic function A t,z) solves the following ODE... 

Therefore, a stochastic process that is a martingale has no observable trend. The price process described by Equation (2.9) is not a martingale unless the drift component fi is equal to zero otherwise, a trend will be observed. A process that is observed to trend upwards is known as a submartingale, while a process that on average declines over time is known as a supermartingale. 

What is the significance of this Here we take it as given that because price processes can be described as equivalent martingale measures (which we do not go into here) they enable the practitioner to construct a risk-free hedge of a market instmment. By enabling a no-arbitrage portfolio to be described, a mathematical model can be set up and solved, including risk-free valuation models. 

The background and mathematics to martingales can be found in Harrison and Kreps (1979) and Harrison and Pliska (1981) as well as Baxter and Rennie (1996). For a description of how, given that price processes are martingales, we are able to price derivative instruments, see James and Webber (2000, Chapter 3). 

Brownian motion is very similar to a Wiener process, which is why it is common to see the terms used interchangeably. Note that the properties of a Wiener process require that it be a martingale, while no such constraint is required for a Brownian process. A mathematical property known as the Levy s theorem allows us to consider any Wiener process Z, with respect to an information set Ft as a Brownian motion Z, with respect to the same information set. 

A key element of the description of a stochastic process is a specification of the level of informatimi oti the behaviour of prices that is available to an observer at each point in time. As with the martingale property, a calculation of the expected future values of a price process requires information on current prices. Generally, fmancial valuation models require data on both the current and the historical security prices, but investors are only able to deal on the basis of current known information and do not have access to future information. In a stochastic model, this concept is captured via the process known as filtration. 

A default-free zero-coupon bond can be defined in terms of its current value imder an initial probability measure, which is the Wiener process that describes the forward rate dynamics, and its price or present value under this probability measure. This leads us to the HJM model, in that we are required to determine what is termed a change in probability measure , such that the dynamics of the zero-coupon bond price are transformed into a martingale. This is carried out using Ito s lemma and a transformatiOTi of the differential equation of the bmid price process. It can then be shown that in order to prevent arbitrage, there would have to be a relationship between drift rate of the forward rate and its volatility coefficient. 

Harrison, M., Pliska, S., 1981. Martingales and stochastic integrals in the theory of continuous trading. Stochastic Process. Appl. 11, 215-260. 

The stochastic process for default-free spot rates and default process are independent under martingale measure Q This last assumption implies that the default process is uncorrelated with default-free spot interest rates. 

Rogers, L., Wdliams, D. Diffusions, Markov Processes, and Martingales, 2nd edn. Cambridge University Press, Cambridge (2000). ISBN 978-0521775946... 

Most option pricing models use one of two methodologies, both of which are based on essentially identical assumptions. The first method, used in the Black-Scholes model, resolves the asset-price model s partial differential equation corresponding to the expected payoff of the option. The second is the martingale method, first introduced in Harrison and Kreps (1979) and Harrison and Pliska (1981). This derives the price of an asset at time 0 from its discounted expected future payoffs assuming risk-neutral probability. A third methodology assumes lognormal distribution of asset returns but follows the two-step binomial process described in chapter 11. 

Harrison, J., and S. Pliska. 1981. Martingales and Stochastic Integrals in the Theory of Continuous Trading. Stochastic Processes and Their Applications 11, 216-260. 

The integral with respect to the Wiener process is taken using the Ito approach that has an important advantage over the Stratonovich approach, namely the integral in this case will be a martingale. Definitions and... 

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