By Kerry Back
This e-book goals at a center flooring among the introductory books on by-product securities and people who offer complex mathematical remedies. it's written for mathematically able scholars who've now not inevitably had previous publicity to likelihood idea, stochastic calculus, or desktop programming. It offers derivations of pricing and hedging formulation (using the probabilistic switch of numeraire process) for normal techniques, alternate ideas, techniques on forwards and futures, quanto suggestions, unique ideas, caps, flooring and swaptions, in addition to VBA code enforcing the formulation. It additionally includes an advent to Monte Carlo, binomial types, and finite-difference methods.
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Additional resources for A course in derivative securities: introduction to theory and computation
A simple example is a linear function: X(t) = at for some constant a. Then, taking ti − ti−1 = ∆t = T /N for each i, the sum of squared changes is N N [∆X(ti )]2 = i=1 [a ∆t]2 = N a2 (∆t)2 = N a2 i=1 T N 2 = a2 T 2 →0 N 30 2 Continuous-Time Models as N → ∞. Essentially the same argument shows that the quadratic variation of any continuously diﬀerentiable function is zero, because such a function is approximately linear at each point. Thus, the jiggling of a Brownian motion, which leads to the nonzero quadratic variation, is quite unusual.
11) of the probability of any event A, it can be shown that the expectation of any random variable X using S as the numeraire is E Xφ(T ) S(T ) S(0) . 12) The use of the symbol S to denote the price of the numeraire may be confusing, because S is usually used to denote a stock price. 1) that is suﬃcient in the binomial model. ” In general the expectation (or mean) of a random variable is an intuitive concept, and an intuitive understanding will be suﬃcient for this book, so I will not give a formal deﬁnition.
In keeping with the discussion of Sect. 22) as stating that µ dt is the expected rate of change of S and σ 2 dt is the variance of the rate of change in an instant dt. ” The geometric Brownian motion will grow at the average rate of µ, in the sense that E[S(t)] = eµt S(0). 23) log S(t) = log S(0) + µ − σ 2 t + σB(t) . 2 This shows that log S(t) − log S(0) is a (µ − σ 2 /2, σ)–Brownian motion. 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 .
A course in derivative securities: introduction to theory and computation by Kerry Back