Stochastic Calculus for Finance II - some Solutions to Chapter VII Matthias hul Last Update: June 9, 25 Exercise 7 Black-Scholes-Merton Equation for the up-and-out Call) i) We have ii) We first compute t δ pmτ, s) τ t τ { { σ τ [ ln s + r ± 2 ) ]} σ2 τ } ln s 2στ τ + r ± 2 σ2 2σ τ [ ) 2στ ln + τ s 2τ δ ± τ, ) qed) s r ± 2 σ2 ) τ ] s δ pmτ, s) { s σ τ sσ τ [ ln s + r ± 2 ) ]} σ2 τ Consequently for s x/c, we get he author can be contacted via <<firstname>><<lastname>>@gmailcom and http://wwwmatthiasthulcom
x δ ± τ, x ) c s δ ±τ, s) s x xσ τ and for s x/c we have x δ ± τ, x ) c s δ ±τ, s) s x xσ qed) τ iii) Since N δ ± τ, s)) { } exp δ2 ±τ, s), 2π 2 we have N δ + τ, s)) N δ τ, s)) { exp δ 2 2 τ, s) δ+τ, 2 s) )} { } exp 2 δ τ, s) δ + τ, s)) δ τ, s) + δ + τ, s)) Here, δ τ, s) δ + τ, s) σ τ and δ τ, s) + δ + τ, s) 2 ln s + 2rτ σ τ hus, N δ + τ, s)) N δ τ, s)) exp { ln s + rτ)} e rτ s qed) 2
and e rτ N d τ, s)) sn d + τ, s)) qed) iv) his result is immediately obvious from the definition of δ ± τ, s) and has been used in iv) already We have δ + τ, s) δ τ, s) σ2 τ σ τ σ τ qed) v) Again, this result follows immediately from the definition of δ ± τ, s) We have vi) We have δ ± τ, s) δ ) ± τ, s ln s ln s σ τ 2 ln s σ qed) τ { y2 N y) } exp y 2π 2 y } exp { y2 2π 2 yn y) qed) vii) o be continued Exercise 73 Markov Property for Geometric Brownian Motion and its Maximum to Date) he crucial steps for the solution to this problem have been derived in Section 744 First, remember that } St) S) exp {σŵ t), 3
where Ŵ t) is a drifted Brownian motion he maximum-to-date process is given by where { Y t) S) exp σ ˆMt) }, First notice that ˆMt) max Ŵ u) u t S ) St) S ) St) { )} St) exp σ Ŵ ) Ŵ t), where St) is Ft)-measurable and the increment Ŵ ) Ŵ t) is independent of the σ-algebra Ft) since by Definition 33 Brownian motion has independent increments Note that adding a drift to the Brownian motion does not change this property Similarly, Y ) Y t) Y ) Y t) { )} Y t) exp σ ˆM ) ˆMt) { [ ] } + Y t) exp σ max t u Ŵ u) ˆMt) { [ Y t) exp σ Ŵ u) Ŵ t)) max t u { [ ) Y t) exp max Ŵ σ u) Ŵ t) ln t u Again, ln Y t)/st)) is Ft)-measurable while ) Bt, ) max Ŵ σ u) Ŵ t) t u is independent of the filtration Ft) hus, ) ] } + ˆMt) Ŵ t) )] } + Y t) St) [ E [fs ), Y )) Ft)] E h St), Y t), S ) ) Ft)] St), Bt, ), 4
where h St), Y t), At, ), Bt, )} f St)At, ), Y t) exp { [ Bt, ) ln By Lemma 234, there exists a function gst), Y t)) defined by )] }) + Y t) St) such that [ gx, y) E h x, y, S ) )] St), Bt, ) E [fs ), Y )) Ft)] gst), Y t)) qed) Exercise 74 Cross Variation of Geometric Brownian Motion and its Maximum to Date) We have m Y t j ) Y t j )) S t j ) S t j )) j m Y t j ) Y t j ) S t j ) S t j ) j max S t j) S t j ) j m m Y t j ) Y t j ) j max j m S t j) S t j ) Y ) Y )) In the limit as maximum step size goes to zero, we get lim max S t j) S t j ) Y ) Y )), Π j m since the stock price St) is a continuous function of time Consequently, lim Π m Y t j ) Y t j )) S t j ) S t j )) qed) j Exercise 77 Zero-Strike Asian Call) i) We can split the integral in a Ft)-measurable part and a part independent of Ft) to get 5
[ ] E Q Su)du Ft) [ t ] [ ] E Q Su)du Ft) + E Q Su)du Ft) t Su)du + t E Q [Su)] du t Using the martingale property of the discounted stock price we can replace E Q [Su)] with e rt u) St) which yields t t Su)du + St) e rt u) du t Su)du + St) e r ) r It follows that and e r E Q [ ] Su)du Ft) e r t Su)du + St) r e r ) e r e r vt, x, y) y + x r ii) he partial derivatives of vt, x, y) are given by t ry x) e r, x e r, r y e r Substituting these into the PDE in Equation 758) yields ry x) e r e r e r + x + x e r e r ry + x All terms cancel out and the assertion follows We further have e r vt,, y) y v, x, y) y 6
hese are just the boundary conditions in Equations 759) and 75) for K Note that both expressions are non-negative since y is an integral over the nonnegative random variable St) iii) By the proof of heorem 75) and Remark 752), the hedge ratio is given by the first derivative of the option price wrt to the spot price t) e r r his quantity is non-random, since it only depends on time but not on the current value of St) or its history iv) he differential of the discounted portfolio value is d e rt Xt) ) t)e rt σst)dw t) Using the process for t) computed above, we get d e rt Xt) ) e rt e r σst)dw t) r he differential of the option value is dvt, St), Y t)) t dt + x dst) + dy t) y e r e r ry t) St)) dt + dst) + r e r e r ry t) dt + dst) r e r rvt, St), Y t))dt + σst)dw t r e r By the product rule, we obtain the differential of the discounted option value St)dt d e rt vt, St), Y t)) ) re rt vt, St), Y t))dt + e rt dvt, St), Y t)) e rt e r r σst)dw t) 7
Since the differential of the discounted portfolio value and the discounted option value agree, it follows that if we start with a portfolio that has initial value X) v, S), Y )) and hold t) shares of the asset at each point in time, then the terminal payoffs agree almost surely 8