By Kerry Back
"Deals with pricing and hedging monetary derivatives.… Computational equipment are brought and the textual content comprises the Excel VBA workouts reminiscent of the formulation and approaches defined within the ebook. this can be precious considering the fact that desktop simulation may help readers comprehend the theory….The book…succeeds in featuring intuitively complex by-product modelling… it offers an invaluable bridge among introductory books and the extra complex literature." --MATHEMATICAL REVIEWS
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Extra info for A course in derivative securities : introduction to theory and computation
9b) where Bx and By can be diﬀerent Brownian motions. The relation between the two Brownian motions is determined by their covariance or correlation. Given dates t < u, we know that both changes Bx (u) − Bx (t) and By (u) − By (t) are normally distributed with mean 0 and variance equal to u − t. There will exist a (possibly random) process ρ such that the covariance of these two normally distributed random variables, given the information at date t, is u ρ(s) ds . Et t The process ρ is called the correlation coeﬃcient of the two Brownian motions, because when it is constant the correlation of the changes Bx (u) − Bx (t) and By (u) − By (t) is u ρ ds (u − t)ρ covariance =√ t √ =ρ.
The derivative of the normal distribution function N is the normal density function n deﬁned as 2 1 n(d) = √ e−d /2 . 8) which simpliﬁes the calculations for the Black-Scholes call option pricing formula. , ∆t. One may have noticed also that the symbol for vega is a little diﬀerent from the others; this reﬂects the fact that vega is not actually a Greek letter. 5 Delta Hedging 55 ∂d1 + qe−qT S N(d1 ) ∂T ∂d2 − re−rT K N(d2 ) + e−rT K n(d2 ) ∂T ∂d1 ∂d2 − = e−qT S n(d1 ) ∂T ∂T Θ = −e−qT S n(d1 ) + qe−qT S N(d1 ) − re−rT K N(d2 ) σ = −e−qT S n(d1 ) √ + qe−qT S N(d1 ) − re−rT K N(d2 ) , 2 T ∂d ∂d2 1 − e−rT K n(d2 ) V = e−qT S n(d1 ) ∂σ ∂σ ∂d ∂d 1 2 − = e−qT S n(d1 ) ∂σ ∂σ √ = e−qT S n(d1 ) T , ∂d1 ∂d2 − e−rT K n(d2 ) + T e−rT K N(d2 ) ρ = e−qT S n(d1 ) ∂r ∂r ∂d2 ∂d1 − + T e−rT K N(d2 ) = e−qT S n(d1 ) ∂r ∂r = T e−rT K N(d2 ) .
If Z = eX , then (dX)2 dZ = dX + . 17) Logarithms. If Z = log X, then dZ = 1 dX − X 2 dX X 2 . 18) 38 2 Continuous-Time Models Compounding/Discounting. Let t Y (t) = exp q(s) ds 0 for some (possibly random) process q and deﬁne Z = XY for any Itˆ o process X. The usual calculus gives us dY (t) = q(t)Y (t) dt, and the product rule above implies dX dZ = q dt + . 19) Z X This is the same as in the usual calculus. 7 Reinvesting Dividends Frequently, we will assume that the asset underlying a derivative security pays a “constant dividend yield,” which we will denote by q.