By Kerry Back
"Deals with pricing and hedging monetary derivatives.… Computational equipment are brought and the textual content includes the Excel VBA exercises resembling the formulation and methods defined within the publication. this can be worthy due to the fact machine simulation will help readers comprehend the theory….The book…succeeds in providing intuitively complex spinoff modelling… it offers an invaluable bridge among introductory books and the extra complex literature." --MATHEMATICAL REVIEWS
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Additional info for A course in derivative securities : introduction to theory and computation
5 Introduction to Option Pricing 21 S(T ) Y (T ) Y (t) =0 . 15) for expectations using S as the numeraire that we can write this as Y (T ) Y (t) =0. , E 1A EtS Y (t) Y (T ) − =0, S(T ) S(t) or, equivalently EtS Y (t) Y (T ) = . 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.
3 Puts and Calls A European call option pays S(T )−K at date T if S(T ) > K and 0 otherwise. Again letting 1 if S(T ) > K , x= 0 otherwise , the payoﬀ of the call can be written as xS(T ) − xK. This is equivalent to one share digital minus K digitals, with the digitals paying in the event that S(T ) > K. The share digital is worth e−qT S(0) N(d1 ) at date 0 and each digital is worth e−rT √ N(d2 ). 4) for d1 and d2 imply d2 = d1 − σ T . 4) and d2 = d1 − σ T . 5) A European put option pays K − S(T ) at date T if S(T ) < K and 0 otherwise.
9), then dZ = 4 ∂g ∂g 1 ∂2g ∂g 1 ∂2g 2 dt + dX + dY + (dX) + (dY )2 ∂t ∂x ∂y 2 ∂x2 2 ∂y 2 ∂2g (dX)(dY ) . 14) to be valid. Note also that we are using a short-hand notation here. The partial derivatives of g will generally depend on t, X(t) and Y (t) just as g does. 6 Examples of Itˆ o’s Formula The following are the applications of Itˆ o’s formula that will be used most frequently in the book. They follow from the boxed formula at the end of the previous section by taking g(x, y) = xy or g(x, y) = y/x or g(x) = ex or g(x) = log x.
A course in derivative securities : introduction to theory and computation by Kerry Back