By Kerry Back
"Deals with pricing and hedging monetary derivatives.… Computational equipment are brought and the textual content comprises the Excel VBA exercises resembling the formulation and techniques defined within the ebook. this can be worthy considering that machine simulation may also help readers comprehend the theory….The book…succeeds in featuring intuitively complex spinoff modelling… it offers an invaluable bridge among introductory books and the extra complicated literature." --MATHEMATICAL REVIEWS
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Extra resources for A course in derivative securities : introduction to theory and computation
37) Z The instantaneous variance of dZ/Z is calculated, using the rules for products of diﬀerentials, as dZ Z 2 = (σx dBx + σy dBy )2 = (σx2 + σy2 + 2ρσx σy ) dt . As will be explained below, the volatility is the square root of the instantaneous variance (dropping the dt). This implies: The volatility of XY is σx2 + σy2 + 2ρσx σy . 16) gives us dZ = (µy − µx − ρσx σy + σx2 ) dt + σy dBy − σx dBx . 39) The instantaneous variance of dZ/Z is therefore dZ Z 2 = (σy dBy − σx dBx )2 = (σx2 + σy2 − 2ρσx σy ) dt .
So, we are led to a study of continuous martingales. An important fact is that any non-constant continuous martingale must have inﬁnite total variation! So, the normal functions with which we are familiar are left behind once we enter the study of continuous martingales. There remains perhaps the question of why we focus on Brownian motion within the world of continuous martingales. The answer here is that any continuous martingale is really just a transformation of a Brownian motion. 3 Itˆ o Processes 31 A continuous martingale is a Brownian motion if and only if its quadratic variation over each interval [0, T ] equals T .
We omit here technical assumptions regarding the existence of expectations. 5 Introduction to Option Pricing 21 S(T ) Y (T ) Y (t) =0 . 15) for expectations using S as the numeraire that we can write this as Y (T ) Y (t) =0. , E 1A EtS Y (t) Y (T ) − =0, S(T ) S(t) or, equivalently EtS Y (t) Y (T ) = . 5 Introduction to Option Pricing A complete development of derivative pricing requires the continuous-time mathematics to be covered in the next chapter. However, we can present the basic ideas using the tools already developed.
A course in derivative securities : introduction to theory and computation by Kerry Back