By Kerry Back
This e-book goals at a center floor among the introductory books on by-product securities and people who supply complicated mathematical remedies. it truly is written for mathematically able scholars who've no longer unavoidably had earlier publicity to chance conception, stochastic calculus, or desktop programming. It presents derivations of pricing and hedging formulation (using the probabilistic swap of numeraire strategy) for normal innovations, trade suggestions, ideas on forwards and futures, quanto ideas, unique ideas, caps, flooring and swaptions, in addition to VBA code enforcing the formulation. It additionally comprises 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
Therefore, the drift of dS/S must be (r − q) dt under the risk-neutral measure. Because the change of measure does not aﬀect the volatility, this implies: dS = (r − q) dt + σs dBs∗ , S where Bs∗ is a Brownian motion under the risk-neutral measure. Underlying as the Numeraire When V is the numeraire, the process Z(t) deﬁned as t exp 0 r(s) ds R(t) Z(t) = = V (t) V (t) is a martingale. 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.
Then the dividend received at date t is qS(t)X(t) dt, which can be used to purchase qX(t) dt new shares. This implies that dX(t) = qX(t) dt, or dX(t)/dt = qX(t), and it is easy to check (and very well known) that this equation is solved by X(t) = eqt X(0). In our case, with X(0) = 1, we have X(t) = eqt . The dollar value of the trading strategy just described will be X(t)S(t) = eqt S(t). Denote this by V (t). This is the value of a non-dividend-paying portfolio, because all dividends are reinvested.
5 Introduction to Option Pricing A complete development of derivative pricing requires the continuous-time mathematics to be covered in the next chapter. However, we can present the basic ideas using the tools already developed. Consider the problem of pricing a European call option. Let T denote the maturity of the option and K its strike price, and let S denote the price of the underlying. We will assume for now that the underlying does not pay dividends, but we will make no assumptions about the distribution of its price S(T ) at the maturity of the option.
A course in derivative securities: introduction to theory and computation by Kerry Back