Black-Scholes Model and Risk Neutral Pricing

Transcription

1 Inroducion echniques Exercises in Financial Mahemaics Lis 3 UiO-SK45 Soluions Hins Auumn 5 eacher: S Oriz-Laorre Black-Scholes Model Risk Neural Pricing See Benh s book: Exercise 44, page 37 See Benh s book: page 3 3 See Benh s book: Exercise 48, pages See Benh s book: Exercises 49 4, pages a Consider he process Z = logs aking ino accoun ha x logx = /x, xx logx = /x ds = S µd + S σdw, ds = S σ d, we can apply Iô s formula o ge ha Z = logs + µsds + aking he exponenial we obain he resul S = exp Z = S exp µsds + b Consider he process M = exp µs rs dw s σs σsdw s σsdw s σ sds σ sds µs rs ds, [, ] σs Noe ha, as min [, ] σ > σ >, we have ha µr σ L [, ], by exercise 4 e in Lis, M is a maringale under P Alernaively, we can use Novikov s heorem because [ E exp = exp µ r d] σ µ r d <, σ due o he fac ha µ, r σ are deerminisic µr σ L [, ] herefore, according o Girsanov s heorem, we can define a probabiliy measure Q by considering he following Radon-Nikodym derivaive wih respec o P dq dp = M = exp µ r dw σ µ r d σ Las updaed: November 5, 5

2 Noe ha, as M >, Q is acually equivalen o P In addiion, also by Girsanov s heorem, we have ha he process W = µs rs ds + W, σs is a Brownian moion under Q Rewriing he dynamics of S in erms of W we ge ha ds = S µd + S σdw = S µd + S σ d W µ r d σ = S rd + S σd W Moreover, B = exp rdds which yields B = exp rdds db = rb d Hence, by he inegraion by pars formula, we have d S = db S = B ds + S db = B + = {}} { db ds {S rd + S σd W } + S { rb d } = S σd W As W is a BM under Q, we ge ha S = S exp σsd W s σ sds, is a maringale under Q Alernaively, one can prove ha S σ belongs o L a, c By assumpion, we have ha Ṽφ is a maringale under Q V φ = max, S K Hence, exp rsds V φ = Ṽφ = E Q [Ṽ φ F ] ] = E Q [max, S K exp rsds F, which yields V φ = E Q [max, S K exp ] rsds F Using he expression for S in secion a rewriing i in erms of W we can wrie S = S exp rsds + σsd W s σ sds, or, alernaively, S exp rsds = S exp σsd W s σ sds Plugging his expression in equaion we obain V φ = E Q [max, S exp σsd W s σ sds K exp rsds F ] Las updaed: November 5, 5

3 Recall he following general propery of he condiional expecaion Le X be a G- measurable rom variable Y be a rom variable independen of G, hen for any Borel measurable funcion Ψ such ha E[ ΨX, Y ] < we have ha E [ΨX, Y G] = E [Ψx, Y ] x=x Applying his propery o G = F, X = S, Y = exp σsd W s Ψx, y = max, x exp y σ sds we ge V φ = F, S where F, x = E Q [max, x exp σsd W s K exp σ sds K exp rsds, rsds Noe ha F W = F W σsd W s is independen of F because W has independen incremens under Q d In order o find an explici expression for F, x, noe ha under Q Define σ, = σ sds, r, = σsd W s N, rsds hen, ] F, x = E Q [max, x exp σsd W s σ, Ke r, ] σ sds ] = e r, E Q [max, exp σsd W s + logx + r, σ, K = e r, E Q [max, Z K], where logz N logx + r, σ,, σ,, under Q Hence, we can use he Black-Scholes formulae o obain F, x = xφd Ke r, Φd, wih d = log x K + r, + σ,, σ, d = log x K + r, σ, σ, 6 By he risk neural pricing formula we ge, as F = {Ω, }, ha n /n π H = e r E Q max S i K,, where Q is he unique risk-neural probabiliy measure in he Black-Scholes model ha is, he probabiliy measure given by dq dp = exp u r u r σ W σ 3 Las updaed: November 5, 5

4 Recall ha S can be wrien as S = S exp σ W + r σ, where W is a Brownian moion under Q Moreover, we have he recursion S i = S exp σ W i + r σ i = S exp σ W i + r σ i exp σ Wi W i + r σ i i = S i exp W i + r σ i, where W i W i W i i i i Ieraing his recursion we ge S i = S i k= n n S i = S k= exp W k + i exp W k + aking logarihm we obain n /n Z log S i = logs + n = logs + n n i + W i + n Obviously, under Q, Z has normal disribuion wih Var [Z] = Var E[Z] = logs + n [ n r σ k, r σ k i k= r σ { W k + } r σ k n n i + k r σ n n i + k ] n i + W i = n n n i + i, where o compue he variance we have used ha W i are independen of each oher Hence, we have reduced he problem o compue π H = e r E [ max e Z K, ], where Z N logs + n r σ n n i + k, n n i + i A formula for his expecaion is given in he soluion of exercise 33 in Lis 7 hese coningen claims are examples of he so called packages, which are linear combinaions of simpler opions posiions in cash Le C, S ; K denoe he arbirage free price a ime of a call opion wih srike K exercise ime 4 Las updaed: November 5, 5

5 a he payoff H = min maxs, K, K wih K > K > can be rewrien as H = K + max, S K max, S K Using he risk-neural pricing formula we ge ha π H = e r E Q [H F ] = e r E Q [K + max, S K max, S K F ] = e r K + e r E Q [max, S K F ] e r E Q [max, S K F ] = e r K + C, S ; K C, S ; K b he payoff H = maxs, S e r K, wih K > can be rewrien as H = max, S S e r + S e r K Using he risk-neural pricing formula we ge ha π H = e r E Q [H F ] 8 See See Benh s book: pages 79-8 = e r E Q [max, S S e r + S e r K F ] = e r E Q [max, S S e r F ] + e r { S e r K } = C, S ; S e r + S e r e r K 5 Las updaed: November 5, 5