By Kerry Back
"Deals with pricing and hedging monetary derivatives.… Computational equipment are brought and the textual content includes the Excel VBA exercises equivalent to the formulation and approaches defined within the publication. this is often useful given that machine simulation may also help readers comprehend the theory….The book…succeeds in proposing intuitively complicated spinoff 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
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 . = S Y 2 We can apply our previous example to compute the dynamics of Y when Y is the numeraire. This shows that the drift of dY /Y is (r + σy2 ) dt.
Let Y (t) denote its value at each date t ≤ T . We seek to ﬁnd Y (0). 17) tells us that Y (0) = num(0)E num Y (T ) num(T ) , 22 1 Asset Pricing Basics for any numeraire with price num(t). We want to choose the numeraire to simplify the calculation of the expectation. The expectation only involves the states of the world in which S(T ) ≥ K, because Y (T ) = 0 when S(T ) < K. 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).
The numbers d1 and d2 are diﬀerent, and hence these are diﬀerent probabilities, even though they are both probabilities of the option ﬁnishing in the money (S(T ) ≥ K). They are diﬀerent probabilities because they are computed under diﬀerent numeraires. A Remark It seems worthwhile here to step back a bit from the calculations and try to oﬀer some perspectives on the methods developed in this chapter. The change of numeraire technique probably seems mysterious. Even though one may agree that it works after following the steps in the chapter, there is probably a lingering question about why it works.
A course in derivative securities : introduction to theory and computation by Kerry Back