Itos Lemma in the Presence of Jumps
Since Ito's lemma is valid between jumps, the application of Ito's lemma to a function g S t , t of the process given by Equation 2.149 gives dg t Ito's lemma applied to the drift-diffusion part of S t g t - g t_ 2.150 Jump in g due to jump in S If the jump in S t happens, g t jumps by an amount Ag. This amount is drawn from a distribution g , which may depend on other variables such as g t_ , the jump magnitude of S t , and S t- . As before, we can write g t -g t_ h t dtJAgagVg . dag dJg t...
The Markov Property of Solutions of SDE
A process X t is said to satisfy the Markov property, or to be Markovian, if the random properties of the process at time s gt t, conditional on information at time t, only depend on the value of the process at time t. This means that the process does not have a memory of events before the observation time that will influence its stochastic properties beyond the observation time. In other words, the behavior of the process beyond the observation time does not depend on the trajectory that the...
Chapter 4
Sampling from the Joint Distribution of the Random Process 81 Generating Scenarios by Numerical Integration of the Stochastic Differential Equations 86 Brownian Bridge Construction 94 Generating Scenarios with Brownian Bridges 95 Joint Normals by the Choleski Decomposition Approach 100 The Concept of Discrepancy 109 Discrepancy and Convergence The Koksma-Hlawka Proper Use of Quasi-Random Sequences 110 HJM for Instantaneous Forwards 113 LIBOR Rate Scenarios 115 Principal Component Analysis to...
Applying the Girsanov Theorem to Importance Sampling European Call Option
To illustrate the Girsanov approach to importance sampling in financial pricing, consider the case of a European call option on a non-dividend paying stock with the risk neutral log-normal price process, where r is the risk-free rate, assumed to be constant. K denotes the strike, and T is the time to maturity. The value of the option is, then, V 0 E exp -rT S T - K 5.55 In order to solve this problem by simulation, we construct trajectories of the stock price process from t 0 to t T. Because...
Case Study Application of Control Variates to Discretely Sampled StepUp Barrier
This simple example shows some of the practical issues to be considered in applying control variates. The traditional approach to control variates has been to select a highly correlated financial instrument as a control variate, for which the price is known, preferably analytically. This approach is typically limited to academic examples the best-known one is the arithmetic Asian option computed with the geometric Asian as control variate . A much more practical approach is to select a control...
Multidimensional Ito Processes
As an example, consider a two-dimensional drift-diffusion process, with components X t and Y t In this case, dWX, dWY, ax, and aj are scalars, and the product dXdY is dXdY axaY cov dWxdWY We can also use vector notation for the Wiener part of these processes Notice that here we have incorporated the coefficients of the linear combination in Equation 2.58 into the definition of a. The covariance of dX and dY is We can have an even more compact notation for multidimensional Ito processes by...
Products of Infinitesimal Increments of Wiener Processes
In using stochastic calculus as a practical tool, the product of two infinitesimal increments of Wiener processes is not a stochastic quantity. From the derivation of the second variation of the Wiener process we found that var W t At - W t 2 - At 2 At2 2.24 The quantity W t At - W t 2 - At is a stochastic process whose variance vanishes like At2 as At 0. We also know that E W t At -W t 2 At, or equivalently, E W t At - W t 2-At 0.Thismeans that W t At - W t 2 - At tends to zero as At 0. We can...
Properties of the Ito Integral
These properties of the Ito integral are useful in getting solutions to stochastic differential equations and other applications. The Ito Integral Is a Martingale This means that for gt t, f'g Y , dW o ftg Y , dW We can see why this is the case by representing the integral as jSg Y , dW j Y , dW jSg Y , dW 2.43 Jo Jo Jt When represented as the limit of partial sums, the second integral on the right is the sum of terms of the form g Y , t W i i - W . Since Y is determined by the information...
Multidimensional Wiener Processes
We briefly discussed multidimensional Wiener processes earlier this chapter. A multidimensional Wiener process W is a column vector W1, W2, ,Wk t of correlated Wiener processes Wi. As we saw earlier, the product of any two components of dW is the covariance between those components We can build correlated Wiener processes by taking combinations of uncorrelated Wiener processes. If Z is a column vector of uncorrelated Wiener processes, Z1, Z2, , ZK T, we can get the multidimensional Wiener...
Multidimensional Ito Lemma
Present Value as an Expectation of Future Values 4 Fundamentals of Stochastic Calculus 9 Filtration and the Revelation of Information 10 Measurable Stochastic Process 12 First Variation of a Differentiable Function 15 First Variation of the Wiener Process 15 Second Variation of a Differentiable Function 15 Second Variation of the Wiener Process 16 Products of Infinitesimal Increments of Wiener Processes 16 Properties of the Ito Integral 19 Multidimensional Wiener Processes 22 Multidimensional...