Sochasic Modelling in Finance - Soluions o shee 8 8.1 The price of a defaulable asse can be modeled as ds S = µ d + σ dw dn where µ, σ are consans, (W ) is a sandard Brownian moion and (N ) is a one jump process which akes he values N = { < τ 1 τ where τ exp(γ) (i.e. τ is an exponenially disribued random variable wih parameer γ), independen of he process S. (a) Find a process (λ ) such ha where (M ) is a maringale. ds = λ d M, M + dm (1) We can consider he process (N τ ) in he quesion as he Poisson process wih inensiy γ. A he sopping ime τ we have a jump in he sock price (S ) of size S τ. Afer his jump he sock price remains in zero. To see his wrie ds as ds = S (µ γ) d + σs dw S (dn γ d). (2) We can see ha he righ-hand side is a muliple of S so if S τ = hen ds τ = and hence ds = for all τ. The second wo erms are maringales so we se dm = σs dw S (dn γ d). Calculaing he angle-bracke process of (M ) is d M, M = S 2 d σw, σw + S 2 d N + γ, N + γ = S 2 d[σw, σw ] + S 2 d N, N = S 2 (σ 2 + γ) d. Equaing he drifs in (1) and (2) we see ha S (µ γ) d = λ S 2 (σ 2 + γ) d and hence λ = λ S = µ γ S (σ 2 + γ). where λ = (µ γ)/(σ 2 + γ) is consan used in he nex par of he exercise. 1
(b) Suppose ha a measure P is defined via he densiy process ( dp ) dp := L = E λ u dm u. Is his measure an equivalen maringale measure for (S )? Explain your answer. The measure P is defined using ( dp dp = E λσ dw u + ) λ d(n u γu). If he Brownian moion (W ) and he Poisson process (N ) are independen we may wrie (see page 59 of he lecure noes) E ( λσw + λ(n γu)) = E ( λσw ) E (λ (N γu)). Hence using he formula (ii) on page 62 of he lecure noes we find ha ) dp dp = exp ( λσw λ2 σ 2 2 e γλ + λ N ) s (1 ) = exp ( λσw λ2 σ 2 2 exp(ln(1 + λ)n γλ). For P P we require ha (L ) is a sricly posiive maringale. Hence P P if and only if λ > 1. (c) Is i possible o specify an equivalen maringale measure for (S )? If λ > 1 hen he measure P in par (b) is an equivalen maringale measure for (S ). For λ 1 i is possible o consruc a maringale measure for (S ) by noing ha ds = S dx S d(n γ) (3) where dx = (µ γ) d + σ dw. The second erm in (3) is a local-maringale so we only need find a maringale measure of dx o ensure ha (S ) is he localmaringale. Repeaing he procedure in (a) we have dx = λ d M, M + d M where M = σ dw and λ = (µ γ)/σ 2. So he measure, defined via ( ) dp µ γ dp = E σ W is a maringale measure for (X ) and hence for (S ). 2
8.2 Suppose ha (Ω, F, F, Q) is a filered probabiliy space supporing wo sandard Brownian moions (W ) and (B ) which are correlaed wih correlaion coefficien ρ [ 1, 1]. The discouned price of wo asses are modeled under he risk neural measure Q using ds S = σ(s ) dw ; d = γ( ) db, where x σ(x) and x γ(x) are deerminisic, measurable funcions. Suppose ha he payoff of a coningen claim is given by C = g(s T )P T where x g(x) is a measurable funcion. (a) The discouned price of his opion is V (, S, )e r = E Q [g(s T )P T e rt F ]. Apply he Iô formula o find dv (, S, ). Applying Inegraion by pars and hen he Iô formula o V (, S, )e r = E Q [g(s T )P T e rt F ] gives dv (, S, )e r = e r dv (, S, ) re r V (, S, )d = e r V (, S, )d + e r V p (, S, )d + e r V s (, S, )ds + 1 2 e r V pp (, S, )d[p, P ] + 1 2 e r V ss (, S, )d[s, S] + e r V sp (, S, )d[s, P ] re r V (, S, )d (b) Find a PDE solved by V (, s, p) and a wo dimensional process (ϕ, ψ ) such ha V (, S, )e r = V (, S, P ) + ψ u dp u. Since (V (, S, )e r ) is a maringale he drif erm, i.e. he d par, has o be. This yields e r V (, S, ) + 1 2 e r V pp (, S, )P 2 γ 2 ( ) + 1 2 e r V ss (, S, )S 2 σ 2 (S ) and hus we ge he PDE + e r V sp (, S, )S σ(s )γ( )ρ re r V (, S, ) = 3
V (, s, p)+ 1 2 V pp(, s, p)p 2 γ 2 (p)+ 1 2 V ss(, s, s)s 2 σ 2 (s)+v sp (, s, p)spσ(s)γ(p)ρ rv (, s, p) =, wih boundary condiion V (T, p, s) = g(s)p. Furhermore, from he Io formula we see we should se ϕ u = e ru V s (u, P u, S u ) and ψ u = e ru V p (u, P u, S u ) for all u [, T ]. (c) Suppose ha i is no possible o purchase ( ). Find c R, (φ ) and a localmaringale (L ) such ha and [S, L] = for all. V (, S, )e r = c + We can wrie, using he usual decomposiion rick, φ u ds u + L d = γ( )db = γ( )(ρdw + 1 ρ 2 dw ) where W is a BM independen of W. Using his: V (,, S )e r = V (, P, S ) + = V (, P, S ) + = V (, P, S ) + + = V (, P, S ) + + = V (, P, S ) + Hence, we should se φ = ϕ + ψ ρ γ( ) S σ(s ) ψ u P u γ(p u ) 1 ρ 2 dw u ϕ u ds u ψ u dp u ψ u P u γ(p u )db u ψ u P u γ(p u )ρ S uσ(s u ) S u σ(s u ) dw u + L for all [, T ] and ψ u P u γ(p u )ρdw u ψ u P u γ(p u )ρ ds u + L. S u σ(s u ) so ha L = ψ u P u γ(p u ) 1 ρ 2 dw u, 4
[ [L, S] = ψ u P u γ(p u ) 1 ρ 2 dwu, = (d) Suppose ha Q is defined via dq dq = E Show ha We have S u σ(s u )dw u ] ψ u P u γ(p u ) 1 ρ 2 S u σ(s u )d[w, W ] u =. ( ) γ(p s ) db s 1 V (, S, ) = E Q [g(s T )e r(t ) F ]. V (, S, )e r = E Q [g(s T )P T e rt F ] = E Q [g(s T )P T e rt F ], dividing boh sides by and using Bayes rule as usual yields indeed E Q [g(s T )e r(t ) F ]. (e) Find (M ) such ha E Q [(M ) 2 ] <, c R and (φ ) such ha V (, S, ) e r = c + φ s dm s. Explain why his decomposiion is no useful for hedging. We have V (, S, ) e r = E Q [g(s T )e rt F ], where he rhs defines a maringale independen of P. Seing u via we ge using Io u(, S )e r = E Q [g(s T )e rt F ] d(u(, S )e r ) = e r du(, S ) re r u(, S )d = e r u (, S )d + e r u s (, S )ds + 1 2 e r u ss (, S )d[s, S] re r u(, S )d = e (u r (, S ) ru(, S ) + 1 ) 2 u ss(, S )S 2 σ 2 (S ) d + e r u s (, S )ds. 5
Noe ha ds S = σ(s )dw = σ(s )ρdb + σ(s ) 1 ρ 2 db = σ(s )ργ( )d + σ(s )ργ( )d + σ(s )ρdb + σ(s ) 1 ρ 2 db ( = σ(s )ργ( )d + σ(s ) ρd B + ) 1 ρ 2 db = σ(s )ργ( )d + σ(s )d W, where he 4h equaliy makes use of he fac ha B given by is a BM under Q. So, B = γ(p S )ds + B d(u(, S )e r ) = e (u r (, S ) ru(, S ) + 1 ) 2 u ss(, S )S 2 σ 2 (S ) d e r u s (, S )σ(s )ργ( )d + e r σ(s )u s (, S )d W. Using ha he drif has o be (maringale), we ge in he end u(, S )e r = u(, S ) + We conclude herefore ha for all φ = e r σ(s )u s (, S ). e rs σ(s s )u s (s, S s )d W s. The reason ha his is no useful for hedging purposes is he fac ha we can acually buy unis of W, hence his expression does no yield a hedging sraegy. 6