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By James A. Primbs
This publication offers the quickest and least difficult path to nearly all of the consequences and equations in by-product pricing, and provides the reader the instruments essential to expand those principles to new occasions that they might come upon. It does so by way of concentrating on a unmarried underlying precept that's effortless to know, after which it exhibits that this precept is the foremost to the vast majority of the implications in spinoff pricing. In that experience, it presents the "big photo" of spinoff pricing through concentrating on the underlying precept and never on mathematical technicalities. After studying this e-book, one is supplied with the instruments had to expand the recommendations to any new pricing state of affairs.
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Extra resources for A Factor Model Approach to Derivative Pricing
17) where Y is a lognormal random variable. Using Ito’s lemma with f (x) = ln(x) allows one to write the solution in closed form as π(t) x(t) = e (a− 21 b2 )t+bz(t) i=1 Yi x(0). 18) But since Y is lognormal, each jump Yi can be written as Yi = eZi where Zi is a normal random variable. This allows us to write the product of Yi ’s as π(t) π(t) e Zi = e Yi = i=1 π(t) i=1 Zi . 19) Stochastic Differential Equations 37 which, conditioned on the number of jumps, π(t), follows the lognormal distribution.
To do this with a Poisson driven differential equation, we would like the second term to have zero instantaneous mean. Hence, we will often “compensate” the Poisson process to give it zero mean. This is done by simply subtracting the instantaneous mean from the second term and adding it to the first. It looks like dx(t) = a(x(t− ), t) + b(x(t− ), t)E[Y ]α dt + b(x(t− ), t) [Y dπ(t) − E[Y ]αdt] . We can also compute the instantaneous variance as 2 E[(dx(t) − [a(x(t− ), t) + b(x(t− ), t)E[Y ]α] dt) |x(t− )] = E[b2 (x(t− ), t)(Y dπ(t) − E[Y ]αdt)2 |x(t− )] = b2 (x(t− ), t)V ar(Y dπ(t)) = b2 (x(t− ), t) E[Y 2 dπ(t)2] − E[Y ]2 E[dπ(t)]2 = b2 (x(t− ), t) E[Y 2 ]E[dπ(t)2 ] − E[Y ]2 α2 dt2 = b2 (x(t− ), t) E[Y 2 ](αdt + α2 dt2 ) − E[Y ]2 α2 dt2 = b2 (x(t− ), t)E[Y 2 ]αdt + O(dt2) where V ar(Y dπ(t)) was used to denote the variance of Y dπ(t), and we used the identity V ar(X) = E[X 2 ] − (E[X])2 .
Appealing to our notion of the differential of a stochastic process, we will interpret this equation as x(t + dt) − x(t) = a(x(t), t)dt + b(x(t), t)(z(t + dt) − z(t)). 28) Since z(t) has independent increments, and a(x(t), t) and b(x(t), t) are evaluated at time t, they are considered independent of the increment dz(t) = z(t+dt)−z(t). This is important! It allows us to do the following simple calculations of the instantaneous mean, variance, and standard deviation of an Ito stochastic differential equation.
A Factor Model Approach to Derivative Pricing by James A. Primbs