By Kerry Back

"Deals with pricing and hedging monetary derivatives.… Computational equipment are brought and the textual content comprises the Excel VBA exercises akin to the formulation and systems defined within the ebook. this is often beneficial seeing that machine simulation may help readers comprehend the theory….The book…succeeds in providing 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 resources for A course in derivative securities : introduction to theory and computation**

**Sample text**

9 Numeraires and Probabilities dV dZ = r dt − + Z V dV V 2 = (r − q + σs2 ) dt − 43 dS . S Because the drift of dZ/Z must be zero, this implies that the drift of dS/S is (r − q + σs2 ) dt. 28) S where now Bs∗ denotes a Brownian motion when V (t) = eqt S(t) is the numeraire. Another Risky Asset as the Numeraire When Y is the numeraire, Z(t) deﬁned as Z(t) = V (t) Y (t) must be a martingale. Using again the rule for ratios, we have dV dY dZ dV dY dY = − − + Z V Y V Y Y dY dV − − ρσs σy dt + σy2 dt = V Y dY dS − + (q − ρσs σy dt + σy2 ) dt .

4). We can simulate a path of S by simulating the changes ∆ log S. 25) has a normal distribution with zero mean and √ variance equal to σ 2 ∆t. We simulate it as σ ∆t multiplied by a standard normal. 9 Numeraires and Probabilities When we change probability measures, we cannot expect a process B that was a Brownian motion to remain a Brownian motion. The expected change in a Brownian motion must always be zero, but when we change probabilities, the expected change of B is likely to become nonzero.

Then for any date u > t, given the information at time t, the random variable B(u) − B(t) is normally distributed with mean zero and variance equal to u − t. Unless stated otherwise, our convention will be that a Brownian motion starts at B(0) = 0. We can generate an approximate Brownian motion in Excel. To do so, we take a small time period ∆t and deﬁne the value at the end of the period to be the value of the Brownian motion at the beginning plus a normally distributed variable with mean 0 and variance ∆t.