By Kerry Back
"Deals with pricing and hedging monetary derivatives.… Computational tools are brought and the textual content comprises the Excel VBA exercises similar to the formulation and techniques defined within the publication. this is often worthwhile seeing that laptop simulation may also help readers comprehend the theory….The book…succeeds in offering intuitively complicated by-product modelling… it presents 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
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 ) . We can calculate the Greeks of a European put option from the call option Greeks and put-call parity: Put Price = Call Price + e−rT K − e−qT S(0) .
10) 0 Furthermore, it can be shown that the sum of products satisﬁes N T ∆X(ti ) × ∆Y (ti ) → i=1 σx (t)σy (t)ρ(t) dt . 11) as N T ∆X(ti ) × ∆Y (ti ) = lim N →∞ (dX)(dY ) 0 i=1 T = (µx dt + σx dBx )(µy dt + σy dBy ) 0 T = σx (t)σy (t)ρ(t) dt . 9). In this case, Itˆo’s formula is4 T Z(T ) = Z(0) + 0 + ∂g dt + ∂t T 1 2 0 T + 0 T 0 ∂g dX(t) + ∂x T 2 1 ∂ g (dX(t))2 + ∂x2 2 0 T 0 2 ∂g dY (t) ∂y ∂ g (dY (t))2 ∂y 2 ∂2g (dX(t))(dY (t)) . 4d) to compute (dX(t))2 = σx2 (t) dt , (dY (t))2 = σy2 (t) dt , (dX(t))(dY (t)) = σx (t)σy (t)ρ(t) dt .
39) The instantaneous variance of dZ/Z is therefore dZ Z 2 = (σy dBy − σx dBx )2 = (σx2 + σy2 − 2ρσx σy ) dt . This implies: The volatility of Y /X is σx2 + σy2 − 2ρσx σy . 40) Further Discussion To understand why taking the square root of (dZ/Z)2 (dropping the dt) gives the volatility, consider for example the product case Z = XY . 38). 37) as dZ = (µx + µy + ρσx σy ) dt + σ dB . 42) From the discussion in Sect. 3, we know that B is a continuous martingale. We can compute its quadratic variation from 2 σx dBx + σs dBs σ 2 2 (σ + σs + 2ρσx σs ) dt = x , σ2 = dt .
A course in derivative securities : introduction to theory and computation by Kerry Back