Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
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I. Basic math.
II. Pricing and Hedging.
III. Explicit techniques.
IV. Data Analysis.
V. Implementation tools.
1. Finite differences.
2. Gauss-Hermite Integration.
3. Asymptotic expansions.
4. Monte-Carlo.
A. Generation of random samples.
B. Acceleration of convergence.
C. Longstaff-Schwartz technique.
D. Calculation of sensitivities.
a. Pathwise differentiation.
b. Calculation of sensitivities for Monte-Carlo with optimal control.
5. Convex Analysis.
VI. Basic Math II.
VII. Implementation tools II.
VIII. Bibliography
Notation. Index. Contents.

Calculation of sensitivities for Monte-Carlo with optimal control.

e are interested in evaluation of MATH when MATH where the $\tau$ is the optimal stopping strategy. The sup is taken over all functional forms of $\tau$ . In the section on backward induction ( Backward induction ) and Bellman equation ( Bellman equation section ) we saw that the $\tau$ is a function of the state variable and the final condition, MATH . Here $s,y$ represent the time and the process state variables. The $T$ is the final time and $h$ is the payoff. In particular, the optimal stopping rule does not depend on the initial condition. Since we recover the $\tau$ when evaluating the MATH itself we simply use the $\tau$ that we already have.

For valuation of Vega (or similar sensitivity) we need a different argument but arrive to the same result. Assuming that the sup is attained on some $\tau^{\ast}$ , we have

MATH (Optimal stopping)
for any variation $\delta\tau$ . Hence, when evaluating Vega, MATH where we abuse the notation slightly: the MATH is the derivative with respect to any direction in $\delta\tau$ -space or a derivative with respect to any parameterization of $\tau$ . In any case, by the ( Optimal stopping ), MATH . Hence, again we may assume that the optimal stopping rule does not change.

The rest of the calculation may follow the section ( Pathwise differentiation ).

Notation. Index. Contents.

Copyright 2007