By Kerry Back
"Deals with pricing and hedging monetary derivatives.… Computational equipment are brought and the textual content includes the Excel VBA workouts akin to the formulation and techniques defined within the e-book. this can be invaluable due to the fact that desktop simulation can assist readers comprehend the theory….The book…succeeds in featuring intuitively complex by-product modelling… it presents an invaluable bridge among introductory books and the extra complicated literature." --MATHEMATICAL REVIEWS
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Extra info for A course in derivative securities : introduction to theory and computation
Then the payoﬀ of the share digital at date T is xS(T ). Let Y (t) denote the value of this claim for 0 ≤ t ≤ T . We have Y (T ) = xS(T ) and we want to ﬁnd Y (0). From Sect. 7, we know that V (t) = eqt S(t) is the price of a non-dividendpaying portfolio. 17), using V as the numeraire, we have Y (0) = S(0)E V Y (T ) eqT S(T ) = e−qT S(0)E V [x] . As in the previous section, E V [x] = probV (x = 1), so we need to compute this probability of the option ﬁnishing in the money. We follow the same steps as in the previous section.
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 .
The reason that there are many arbitrage-free values for a call (or put) is that a call cannot be replicated in a trinomial model using the stock and risk-free asset; we can say equivalently that there is no “delta hedge” for a call option. Recall that we ﬁrst found the value of a call in the binomial model by ﬁnding the replicating portfolio and calculating its cost. A similar analysis is impossible in the trinomial model. To see this, consider a portfolio of a dollars invested in the risk free asset and b dollars invested in the stock.
A course in derivative securities : introduction to theory and computation by Kerry Back