MATH3075/3975 Financial Mathematics
Tutorial 11: Solutions
Exercise 1 We consider the Black-Scholes model M = (B,S) with the initial stock price S0 = 9,
the continuously compounded interest rate r = 0.01 per annum and the stock price volatility σ = 0.1
per annum. Recall that dBt = rBt dt with B0 = 1 (equivalently, B(t, T ) = e
−r(T−t) and
dSt = St
(
r dt+ σ dWt
)
, S0 > 0,
where W is a standard Brownian motion under the martingale measure P̃.
(a) Using the Black-Scholes call option pricing formula
C0 = S0N
(
d+(S0, T )
)
−Ke−rTN
(
d−(S0, T )
)
we compute the price C0 of the European call option with strike price K = 10 and maturity
T = 5 years. We find that
d+(S0, T ) = −0.13578, d−(S0, T ) = −0.35938
and thus C0 = 0.59285.
(b) Using the Black-Scholes put option pricing formula
P0 = Ke
−rTN
(
− d−(S0, T )
)
− S0N
(
− d+(S0, T )
)
we find that the price P0 = 1.10514
(c) The put-call parity relationship holds since
C0 − P0 = 0.59285− 1.10514 = −0.51229 = 9− 10e−0.05 = S0 −Ke−rT .
(d) We now recompute the prices of call and put options for modified maturities T = 5 months
and T = 5 days.
– We note that 5 months is equivalent to T = 0.416667 and thus
d+(S0, T ) = −1.53541, d−(S0, T ) = −1.59996.
Hence C0 = 0.015315 and P0 = 0.973735.
– We note that 5 days is equivalent to T = 0.013699 and thus
d+(S0, T ) = −8.98455, d−(S0, T ) = −8.99615.
Hence C0 = 1.49E − 21 and P0 = 0.99863.
(e) The call option (respectively, put option) price decreases to zero (respectively, increases to
K−S0 = 1) when the time to maturity tends to zero. This is related to the fact that S0 < K
and thus for short maturities it is unlikely (respectively, very likely) that the call option
(respectively, put option) will be exercised at expiration.
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Exercise 2 Assume that the stock price S is governed under the martingale measure P̃ by the
Black-Scholes stochastic differential equation
dSt = St
(
r dt+ σ dWt
)
where σ > 0 is a constant volatility and r is a constant short-term interest rate. Let 0 < L < K
be real numbers. We consider a path-independent contingent claim with the payoff X at maturity
date T > 0 given as
X = min
(
|ST −K|, L
)
.
(a) It is easy to sketch the profile of the payoff X as the function of the stock price ST . The
decomposition of X in terms of the payoffs of standard call and put options reads
X = L− CT (K − L) + 2CT (K)− CT (K + L).
Note that other decompositions are possible.
(b) The arbitrage price πt(X) satisfies, for every t ∈ [0, T ],
πt(X) = Le
−r(T−t) − Ct(K − L) + 2Ct(K)− Ct(K + L).
(c) We will now find the limits of the arbitrage price limL→0 π0(X) and limL→∞ π0(X). We
observe the payoff X increases when L increases. Hence the price π0(X) is also an increasing
function of L. Moreover,
lim
L→0
π0(X) = −C0(K) + 2C0(K)− C0(K) = 0.
By analysing the payoff X when L tends to infinity (obviously, we no longer assume here that
the inequality L < K holds since K is fixed and L tends to infinity), we obtain
lim
L→∞
min
(
|ST −K|, L
)
= |ST −K| = (K − ST )+ + (ST −K)+ = PT (K) + CT (K)
and thus
lim
L→∞
π0(X) = P0(K) + C0(K).
(d) To find the limit limσ→∞ π0(X), we observe that
lim
σ→∞
d+(S0, T ) =∞, lim
σ→∞
d−(S0, T ) = −∞,
so that
lim
σ→∞
N
(
d+(S0, T )
)
= 1, lim
σ→∞
N
(
d−(S0, T )
)
= 0.
Hence the price of the call option satisfies, for all strikes K ∈ R+,
lim
σ→∞
C0(K) = S0.
This in turn implies that limσ→∞ π0(X) = Le
−rT = π0(L).
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Exercise 3 We denote by v the Black-Scholes call option pricing, that is, the function v : R+ ×
[0, T ]→ R such that Ct = v(St, t) for all t ∈ [0, T ].
(a) We need to show that, for every s ∈ R+,
lim
t→T
v(s, t) = (s−K)+
For this purpose, we observe that d+(s,K) and d−(s,K) tend to ∞ (respectively, −∞) when
t → T and s > K (respectively, s < K). Consequently, N(d+(s,K)) and N(d−(s,K)) tend
to 1 (respectively, 0) when t→ T and s > K (respectively, s < K). This in turn implies that
v(s, T ) tends to either s − K or 0 depending on whether s > K or s < K. The case when
s = K is also easy to analyse and to check that limt→T v(s, t) = 0 when s = K.
(b) (MATH3975) Observe that v(s, t) = c(s, T − t) where the function c is such that Ct =
c(St, T − t). Our goal is to check that the pricing function of the European call option satisfies
the Black-Scholes partial differential equation (PDE)
∂v
∂t
+
1
2
σ2s2
∂2v
∂s2
+ rs
∂v
∂s
− rv = 0, ∀ (s, t) ∈ (0,∞)× (0, T ), (1)
with the terminal condition v(s, T ) = (s−K)+. Equivalently, the function c satisfies
−
∂c
∂t
+
1
2
σ2s2
∂2c
∂s2
+ rs
∂c
∂s
− rc = 0, ∀ (s, t) ∈ (0,∞)× (0, T ),
with the initial condition c(s, 0) = (s−K)+. From the Black-Scholes theorem, we know that
v is given by the following expression
v(s, t) = sN(d+(s, T − t))−Ke−r(T−t)N(d−(s, T − t)). (2)
Straightforward computations show that the partial derivatives are:
vs(s, t) = N(d+(s, T − t)),
vss(s, t) =
n(d+(s, T − t))
σs
√
T − t
,
vt(s, t) = −
σs
2
√
T − t
n(d+(s, T − t))−Kre−r(T−t)N(d−(s, T − t))
where n(x) is the density function of the standard normal distribution. Hence
−
sσ
2
√
T − t
n(d+(s, T − t))−Kre−r(T−t)N(d−(s, T − t))
+
1
2
σ2s2
n(d+(s, T − t))
sσ
√
T − t
+ rsN(d+(s, T − t))− rv(s, t) = 0
where we have also used the equality (2).
It is worth noting that the pricing function w(s, t) = p(s, T − t) for the put option also
satisfies the Black-Scholes PDE but with the terminal condition w(s, T ) = (K − s)+. This
can be checked either by computing directly the partial derivatives or by combining already
established PDE (1) with the put-call parity relationship, which reads
v(s, t)− w(s, t) = s−Ke−r(T−t).
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Exercise 4 (MATH3975) We consider the stock price process S given by the Black and Scholes
model.
(a) We will first show that Ŝt = e
−rtSt is a martingale with respect to the filtration F = (Ft)t≥0
generated by the stock price process S. We observe that this filtration is also generated by
W . Using the properties of the conditional expectation, we obtain, for all s ≤ t,
EP̃
(
Ŝt
∣∣Fs) = EP̃ (Ŝs eσ(Wt−Ws− 12σ2(t−s)) ∣∣Fs)
= Ŝs e
− 1
2
σ2(t−s) EP̃
(
eσ(Wt−Ws) | Fs
)
= Ŝs e
− 1
2
σ2(t−s) EP̃
(
eσ(Wt−Ws) | Fs
)
= Ŝs e
− 1
2
σ2(t−s) EP̃
(
eσ(Wt−Ws)
)
where in the last equality we used the independence of increments of the Wiener process.
Recall also that Wt −Ws =
√
t− sZ where Z ∼ N(0, 1), and thus
EP̃
(
Ŝt | Fs
)
= Ŝs e
− 1
2
σ2(t−s) EP̃
(
eσ
√
t−sZ).
It is known (and easy to check by integration) that if Z ∼ N(0, 1) then for any real number a
EP̃
(
eaZ
)
= e
1
2
a2 . (3)
By setting a = σ
√
t− s, we obtain
EP̃
(
Ŝt | Ŝu, u ≤ s
)
= Ŝs e
− 1
2
σ2(t−s) e
1
2
σ2(t−s) = Ŝs,
which shows that Ŝ is a martingale under P̃.
(b) To compute the expectation EP̃(St), we observe that
EP̃(St) = e
rt EP̃(Ŝt) = e
rt EP̃(Ŝ0) = e
rtŜ0 = e
rtS0.
To compute the variance Var P̃(St), we recall that
Var P̃(St) = EP̃(S
2
t )−
[
EP̃(St)
]2
where in turn
EP̃(S
2
t ) = S
2
0e
2rt EP̃
[
e2σWt−σ
2t
]
= S20e
2rteσ
2t EP̃
[
e2σWt−
1
2
(2σ
√
t)2
]
= S20e
2rteσ
2t EP̃
[
eaZ−
1
2
a2
]
where we denote a = 2σ
√
t and Z ∼ N(0, 1). Since (see (3))
EP̃
[
eaZ−
1
2
a2
]
= 1
we conclude that
EP̃(S
2
t ) = S
2
0e
2rteσ
2t
and thus
Var P̃(St) = S
2
0e
2rt
(
eσ
2t − 1
)
.
4