By Kerry Back
This booklet goals at a center flooring among the introductory books on by-product securities and people who supply complex mathematical remedies. it's written for mathematically able scholars who've no longer inevitably had past publicity to likelihood thought, stochastic calculus, or computing device programming. It presents derivations of pricing and hedging formulation (using the probabilistic switch of numeraire method) for normal strategies, alternate strategies, recommendations on forwards and futures, quanto recommendations, unique recommendations, caps, flooring and swaptions, in addition to VBA code enforcing the formulation. It additionally includes an creation to Monte Carlo, binomial versions, and finite-difference methods.
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Extra info for A course in derivative securities: introduction to theory and computation
For example, in the Black-Scholes model, the most important assumption is that the volatility of the underlying asset price is constant. We will occasionally need to compute the volatilities of products or ratios of random processes. These computations follow directly from Itˆ o’s formula. Suppose dY dX = µx dt + σx dBx = µy dt + σy dBy , and X Y where Bx and By are Brownian motions with correlation ρ, and µx , µy , σx , σy , and ρ may be quite general random processes. 15) gives us dZ = (µx + µy + ρσx σy ) dt + σx dBx + σy dBy .
Given information at time t, the logarithm of S(u) for u > t is normally distributed with mean (u − t)(µ − σ 2 /2) and variance (u − t)σ 2 . Because S is the exponential of its logarithm, S can never be negative. For this reason, a geometric Brownian motion is a better model for stock prices than is a Brownian motion. 23) is d log S(t) = 1 µ − σ2 2 dt + σ dB(t) . 24) We conclude: The equation dS = µ dt + σ dB S is equivalent to the equation d log S(t) = 1 µ − σ2 2 dt + σ dB(t) . 23). 24) implies that the change in the logarithm of S is ∆ log S = 1 µ − σ 2 ∆t + σ ∆B .
In these states of the world, the value of the digital is already a constant K, so we should take the numeraire to have a constant value at T , so that the ratio Y (T )/num(T ) will be constant in the states in which S(T ) ≥ K. This means that we should take the numeraire to be the risk-free asset. For this numeraire, the pricing formula is Y (0) = e−rT E R [Y (T )] = e−rT K × probR S(T ) ≥ K , so we need to compute the risk-neutral probability that S(T ) ≥ K. We will do this by using the fact that S(t)/R(t) = e−rt S(t) is a martingale under the risk-neutral probability measure.