Modern Computational Finance

CHAPTER 8Manual Adjoint Differentiation

In this first introductory chapter to AAD, we introduce adjoint differentiation (AD) through simple examples. We explain how AD may compute many derivative sensitivities in constant time and show how this can be achieved by coding AD by hand.

In the next chapter, we will generalize and formalize adjoint mathematics, and show how adjoint calculations can be executed automatically through autogenerated calculation graphs without the need to code any form of differentiation manually.

In Chapter 10, we will turn these ideas into a general-purpose AAD library in C++, and, in subsequent chapters, we will apply this AAD library to differentiate serial and parallel Monte-Carlo simulations in constant time.

8.1 INTRODUCTION TO ADJOINT DIFFERENTIATION

We start with a simple example where we differentiate a well-known analytic function with respect to all its inputs and introduce adjoint calculations as a means to compute all those derivatives from one another in constant time. From the example, we abstract and illustrate a general methodology for the manual computation of adjoints and the production of corresponding code.

Example: The Black and Scholes formula

The well known Black and Scholes formula, derived in 1973 in [22], gives the price of a call option in the absence of arbitrage, under the assumption that the volatility of the underlying asset is a known constant, as a function of a small number of observable variables and the constant volatility. ...