By Kerry Back

ISBN-10: 3540253734

ISBN-13: 9783540253730

ISBN-10: 3540279008

ISBN-13: 9783540279006

"Deals with pricing and hedging monetary derivatives.… Computational equipment are brought and the textual content comprises the Excel VBA workouts akin to the formulation and systems defined within the booklet. this can be priceless for the reason that machine simulation can assist readers comprehend the theory….The book…succeeds in providing intuitively complicated spinoff modelling… it presents 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**

**Sample text**

1 for the binomial model. For a call option, such a portfolio consists of delta shares of the underlying asset and a short call option, or a short position of delta shares of the underlying and a long call option. These portfolios have no instantaneous exposure to the price of the underlying. To create a perfect hedge, the portfolio must be adjusted continuously, because the delta changes when the price of the underlying changes and when time passes. In practice, any hedge will therefore be imperfect, even if the assumptions of the model are satisﬁed.

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 . 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.

In the states of the world in which S(T ) ≥ K, the value of the share digital is S(T ). , the expected payoﬀ of a gamble that pays $1 when a fair die rolls a 6 is 1/6). This suggests we should use the stock as the numeraire, because then we will have S(T ) Y (T ) = =1 num(T ) S(T ) when S(T ) ≥ K, implying that E num Y (T ) = probS S(T ) ≥ K , num(T ) where probS denotes the probability using S as the numeraire. This implies that the value of the share digital is S(0) × probS S(T ) ≥ K . The remaining question is obviously how to compute the probability.

### A course in derivative securities : introduction to theory and computation by Kerry Back

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