Chapter 6
Martingale Approach to Pricing and
Hedging
In this chapter we present the probabilistic martingale approach method to
the pricing and hedging of options. In particular, this allows one to compute
option prices as the expectations of the discounted option payoffs, and to
determine the associated hedging portfolios.
6.1 Martingale Property of the Itô Integral
Recall (Definition 5.5) that an integrable process (Xt )t∈R+ is said to be a
martingale with respect to the filtration (Ft )t∈R+ if
IE Xt | Fs = Xs ,
0 6 s 6 t.
Examples of martingales
1. Given F ∈ L2 (Ω) a square-integrable random variable and (Ft )t∈R+ a
filtration, the process (Xt )t∈R+ defined by Xt := IE[F | Ft ] is an (Ft )t∈R+ martingale under P, as follows from the tower property:
IE Xt | Fs = IE IE[F | Ft ] | Fs = IE[F | Fs ] = Xs , 0 6 s 6 t, (6.1)
by the tower property (17.37).
2. Any integrable stochastic process (Xt )t∈R+ with centered and independent increments is a martingale with respect to its own filtration (Ft )t∈R+ :
IE[Xt | Fs ] = IE[Xt − Xs + Xs | Fs ]
= IE[Xt − Xs | Fs ] + IE[Xs | Fs ]
= IE[Xt − Xs ] + Xs
= Xs ,
0 6 s 6 t.
173
(6.2)
N. Privault
In particular, the standard Brownian motion (Bt )t∈R+ is a martingale
because it has centered and independent increments. This fact can also
be recovered from Proposition 6.1 since Bt can be written as
wt
Bt =
dBs ,
t ∈ R+ .
0
3. The driftless geometric Brownian motion
Xt := X0 e σBt −σ
2
t/2
is a martingale. Indeed, we have
2
IE[Xt | Fs ] = IE X0 e σBt −σ t/2 | Fs
2
= X0 e −σ t/2 IE e σBt | Fs
2
= X0 e −σ t/2 IE e σ(Bt −Bs )+σBs | Fs
2
= X0 e −σ t/2+σBs IE e σ(Bt −Bs ) | Fs
2
= X0 e −σ t/2+σBs IE e σ(Bt −Bs )
= X0 e −σ
2
t/2+σBs
= X0 e σBs −σ
= Xs ,
2
eσ
2
(t−s)/2
s/2
0 6 s 6 t.
This fact can also be recovered from Proposition 6.1 since Xt satisfies the
equation
dXt = σXt dBt ,
i.e. it can be written as the Brownian stochastic integral
wt
t ∈ R+ ,
Xt = X0 + σ Xu dBu ,
0
cf. Proposition 6.1 below. The discounted asset price
Xt = X0 e (µ−r)t+σBt −σ
2
t/2
is a martingale when µ = r. The case µ 6= r will be treated in Section 6.3
using risk-neutral measures and the Girsanov theorem, cf. (6.9) below.
4. The discounted value
Vet = e −rt Vt
of a self-financing portfolio is given by
wt
Vet = Ve0 +
ξu dXu ,
0
174
This version: June 12, 2017
http://www.ntu.edu.sg/home/nprivault/indext.html
t ∈ R+ ,
"
Martingale Approach to Pricing and Hedging
cf. Lemma 5.2 is a martingale when µ = r by Proposition 6.1 because
wt
Vet = Ve0 + σ ξu Xu dBu ,
t ∈ R+ ,
0
since
dXt = Xt ((µ − r)dt + σdBt ) = σXt dBt .
Since the Black-Scholes theory is in fact valid for any value of the parameter µ we will look forward to including the case µ 6= r in the sequel.
The following result shows that the indefinite Itô integral is a martingale with
respect to the Brownian filtration (Ft )t∈R+ . It is the continuous-time analog
of the discrete-time Proposition 2.6.
r
t
Proposition 6.1. The indefinite stochastic integral
u dBs
of a
0 s
t∈R+
square-integrable adapted process u ∈ L2ad (Ω × R+ ) is a martingale, i.e.:
w
w
t
s
IE
uτ dBτ Fs =
uτ dBτ ,
0 6 s 6 t.
(6.3)
0
0
Proof. The statement is first proved in case u is a simple predictable process,
and then extended to the general case, cf. e.g. Proposition 2.5.7 in [Pri09].
For example, for u of the form us := F 1[a,b] (s) with F an Fa -measurable
random variable and t ∈ [a, b], we have
i
i
hw ∞
hw ∞
IE
us dBs Ft = IE
F 1[a,b] (s)dBs Ft
0
0
= IE F (Bb − Ba ) Ft
= F IE Bb − Ba Ft
= F (Bt − Ba )
wt
=
us dBs
a
wt
=
us dBs ,
0
a 6 t 6 b.
On the other hand, when t ∈ [0, a] we have
wt
0
us dBs = 0,
hence we need to check that
i
i
hw ∞
hw ∞
IE
us dBs Ft = IE
F 1[a,b] (s)dBs Ft
0
0
= IE F (Bb − Ba ) | Ft
= IE IE F (Bb − Ba ) | Fa Ft
= IE F IE Bb − Ba | Fa Ft
"
175
This version: June 12, 2017
http://www.ntu.edu.sg/home/nprivault/indext.html
N. Privault
= 0,
0 6 t 6 a,
where we used the tower property (6.1) of conditional expectations and the
fact that (Bt )t∈R+ is a martingale:
IE Bb − Ba | Fa = IE Bb | Fa − Ba = Ba − Ba = 0.
The extension from simple processes to square-integrable processes in L2ad (Ω×
R+ ) can be proved as in Proposition 4.9. Indeed, given (u(n) )n∈N be a sequence of simple predictable processes converging to u in L2 (Ω × [0, T ]) cf.
Lemma 1.1 of [IW89], pages 22 and 46, by Fatou’s Lemma 17.1 and the
continuity of the conditional expectation on L2 we have:
"
#
i2
hw ∞
wt
IE
us dMs − IE
us dMs Ft
0
0
6 lim inf IE
n→∞
"
w
t
0
u(n)
s dMs
− IE
hw ∞
0
i2
us dMs Ft
#
w∞
i2 IE
u(n)
us dMs Ft
s dMs −
n→∞
0
0
w
2 w∞
∞
6 lim IE IE
u(n)
dM
−
u
dM
Ft
s
s
s
s
n→∞
0
0
w
2
∞
= lim IE
(u(n)
s − us )dMs
n→∞
0
hw ∞
i
2
= lim IE
|u(n)
s − us | ds
= lim IE
n→∞
hw ∞
0
= 0,
where we used the Itô isometry (4.14).
rt
In particular, 0 us dBs is Ft -measurable,
Relation (6.3) applied with t = 0 recovers
centered random variable:
i
hw ∞
hw ∞
us dBs
us dBs = IE
IE
0
0
t ∈ R+ , and since F0 = {∅, Ω},
the fact that the Itô integral is a
i w0
us dBs = 0.
F0 =
0
6.2 Risk-neutral Measures
Recall that by definition, a risk-neutral measure is a probability measure P∗
under which the discounted asset price process
(Xt )t∈R+ := ( e −rt St )t∈R+
176
This version: June 12, 2017
http://www.ntu.edu.sg/home/nprivault/indext.html
"
Martingale Approach to Pricing and Hedging
is a martingale, cf. Proposition 5.6. Note that when µ = r, the process
(Xt )t∈R+ is a martingale under P∗ = P, which is a risk-neutral measure.
In this section we address the construction of a risk-neutral measure in the
general case µ 6= r using the Girsanov theorem. Note that the relation
dXt = Xt ((µ − r)dt + σdBt )
can be rewritten as
dXt = σXt dB̃t ,
where (B̃t )t∈R+ is a drifted Brownian motion given by
B̃t :=
µ−r
t + Bt ,
σ
t ∈ R+ .
Therefore the search for a risk-neutral measure can be replaced by the search
for a probability measure P∗ under which (B̃t )t∈R+ is a standard Brownian
motion.
Let us come back to the informal interpretation of Brownian motion via its
infinitesimal increments:
√
∆Bt = ± dt,
with
√ √ 1
P ∆Bt = + dt = P ∆Bt = − dt = .
2
14
12
10
8
~
Bt
6
4
2
0
-2
-4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
2.0
t
Fig. 6.1: Drifted Brownian path.
Clearly, given ν ∈ R, the drifted process B̃t := νt+Bt is no longer a standard
Brownian motion because it is not centered:
IE[νt + Bt ] = νt + IE[Bt ] = νt 6= 0,
"
177
This version: June 12, 2017
http://www.ntu.edu.sg/home/nprivault/indext.html
N. Privault
cf. Figure 6.1. This identity can be formulated in terms of infinitesimal increments as
IE[νdt + dBt ] =
√
√
1
1
(νdt + dt) + (νdt − dt) = νdt 6= 0.
2
2
In order to make νt+Bt a centered process (i.e. a standard Brownian motion,
since νt + Bt conserves all the other properties (i)-(iii) in the definition of
Brownian motion, one may change the probabilities of ups and downs, which
have been fixed so far equal to 1/2.
That is, the problem is now to find two numbers p∗ , q ∗ ∈ [0, 1] such that

√
√
 p∗ (νdt + dt) + q ∗ (νdt − dt) = 0

p∗ + q ∗ = 1.
The solution to this problem is given by
p∗ =
√
1
(1 − ν dt)
2
and q ∗ =
√
1
(1 + ν dt).
2
Coming back to Brownian √
motion considered as a discrete random walk with
independent increments ± ∆t, we try to construct a new probability measure denoted P∗ , under which the drifted process B̃t := νt + Bt will be a
standard Brownian motion. This probability measure will be defined through
its density
√
√
dP∗
P∗ (∆Bt1 = 1 ∆t, . . . , ∆BtN = N ∆t)
√
√
:=
dP
P(∆Bt1 = 1 ∆t, . . . , ∆BtN = N ∆t)
√
√
P∗ (∆Bt1 = 1 ∆t) · · · P∗ (∆BtN = N ∆t)
√
√
=
P(∆Bt1 = 1 ∆t) · · · P(∆BtN = N ∆t)
√
√
1
=
P∗ (∆Bt1 = 1 ∆t) · · · P∗ (∆BtN = N ∆t),
(1/2)N
1 , . . . , N ∈ {−1, 1}, with respect to the historical probability measure P,
obtained by taking the product of the above probabilities divided by the reference probability 1/2N corresponding to the symmetric random walk.
Interpreting N = T /∆t as an (infinitely large) number of discrete time steps
and under the identification {0 < t < T } ' {t1 , t2 , . . . , tN }, this density can
be rewritten as
Y 1 1 √ 1
dP∗
'
∓ ν ∆t
dP
(1/2)N
2 2
0<t<T
where 2
N
becomes a normalization factor. Using the expansion
178
This version: June 12, 2017
http://www.ntu.edu.sg/home/nprivault/indext.html
"
Martingale Approach to Pricing and Hedging
√ √
1 √
log 1 + ν ∆t = ν ∆t − (ν ∆t)2 + o(∆t),
2
for small values of ∆t, this density can be informally shown to converge as
follows as N tends to infinity, i.e. as the time step ∆t = T /N tends to zero:
Y Y 1 1 √ √ 1 ∓ ν ∆t
2N
∓ ν ∆t =
2 2
0<t<T
0<t<T
!
Y √ = exp log
1 ∓ ν ∆t
0<t<T
X
= exp
√ log 1 ∓ ν ∆t
!
0<t<T
√
√
1 X
∓ ∆t −
(∓ν ∆t)2
2
0<t<T
0<t<T
!
X √
ν2 X
∆t
= exp −ν
± ∆t −
2
0<t<T
0<t<T
!
X
ν2 X
= exp −ν
∆Bt −
∆t
2
0<t<T
0<t<T
ν2
= exp −νBT − T ,
2
based on the identifications
X √
BT '
± ∆t
!
X
' exp ν
and T '
0<t<T
X
∆t.
0<t<T
6.3 Change of Measure and the Girsanov Theorem
In this section we restate the Girsanov theorem in a more rigorous way, using
changes of probability measures. Recall that, given Q a probability measure
on Ω, the notation
dQ
=F
dP
means that the probability measure Q has a density F with respect to P,
where F : Ω −→ R is a nonnegative random variable such that IE[F ] = 1.
We also write
dQ = F dP,
which is equivalent to stating that
w
w
IEQ [G] =
G(ω)dQ(ω) =
G(ω)F (ω)dP(ω) = IE [F G] ,
Ω
"
Ω
179
This version: June 12, 2017
http://www.ntu.edu.sg/home/nprivault/indext.html
N. Privault
where G is an integrable random variable. In addition we say that Q is equivalent to P when F > 0 with P-probability one.
Recall that here, Ω = C0 ([0, T ]) is the Wiener space and ω ∈ Ω is a
continuous function on [0, T ] starting at 0 in t = 0. Consider the probability
Q defined by
ν2
dQ(ω) = exp −νBT − T dP(ω).
2
Then the process νt + Bt is a standard (centered) Brownian motion under
Q.
For example, the fact that νT + BT has a standard (centered) Gaussian law
under Q can be recovered as follows:
w
IEQ [f (νT + BT )] =
f (νT + BT )dQ
Ω
w
1
=
f (νT + BT ) exp −νBT − ν 2 T dP
Ω
2
w∞
2
1 2
dx
=
f (νT + x) exp −νx − ν T e −x /(2T ) √
−∞
2
2πT
w∞
−(νT +x)2 /(2T ) dy
√
=
f (νT + x) e
−∞
2πT
w∞
−y 2 /(2T ) dy
√
=
f (y) e
,
−∞
2πT
i.e.
IEQ [f (νT + BT )] =
w
Ω
f (νT + BT )dQ =
w
Ω
f (BT )dP = IEP [f (BT )]. (6.4)
The Girsanov theorem can actually be extended to shifts by adapted processes as follows, cf. e.g. Theorem III-42, page 141 of [Pro04]. Section 15.6
will cover the extension of the Girsanov theorem to jump processes.
Theorem 6.2. Let (ψt )t∈[0,T ] be an adapted process satisfying the Novikov
integrability condition
w
1 T
IE exp
|ψt |2 dt
< ∞,
(6.5)
2 0
and let Q denote the probability measure defined by
w
T
dQ
1wT 2
= exp −
ψs dBs −
ψs ds .
0
dP
2 0
Then
B̃t := Bt +
180
wt
0
ψs ds,
0 6 t 6 T,
This version: June 12, 2017
http://www.ntu.edu.sg/home/nprivault/indext.html
"
Martingale Approach to Pricing and Hedging
is a standard Brownian motion under Q.
The Girsanov Theorem allows us to extend (6.4) as
w
w·
T
1wT
IE[F ] = IE F B· + ψs ds exp −
ψs dBs −
(ψs )2 ds , (6.6)
0
0
2 0
for all random variables F ∈ L1 (Ω), see also Exercise 6.15.
When applied to
ψt :=
µ−r
,
σ
the Girsanov theorem shows that
B̃t :=
µ−r
t + Bt ,
σ
0 6 t 6 T,
(6.7)
is a standard Brownian motion under the probability measure P∗ defined by
µ−r
(µ − r)2
dP∗
= exp −
BT −
T .
(6.8)
2
dP
σ
2σ
Hence by Proposition 6.1 the discounted price process given by
dXt = (µ − r)Xt dt + σXt dBt = σXt dB̃t ,
t ∈ R+ ,
(6.9)
is a martingale under P∗ , hence P∗ is a risk-neutral measure. We obviously
have P = P∗ when µ = r.
6.4 Pricing by the Martingale Method
In this section we give the expression of the Black-Scholes price using expectations of discounted payoffs.
Recall that from the first fundamental theorem of mathematical finance,
a continuous market is without arbitrage opportunities if there exists (at
least) a risk-neutral probability measure P∗ under which the discounted price
process
Xt := e −rt St ,
t ∈ R+ ,
is a martingale under P∗ . In addition, when the risk-neutral measure is
unique, the market is said to be complete.
In case the price process (St )t∈R+ satisfies the equation
dSt
= µdt + σdBt ,
St
"
t ∈ R+ ,
S0 > 0
181
This version: June 12, 2017
http://www.ntu.edu.sg/home/nprivault/indext.html
N. Privault
we have
St = S0 e σBt −σ
2
t/2+µt
,
and Xt = S0 e (µ−r)t+σBt −σ
2
t/2
,
t ∈ R+ ,
hence from Section 6.2 the discounted price process is a martingale under the
probability measure P∗ defined by (6.8), and P∗ is a martingale measure.
We have
dXt = (µ − r)Xt dt + σXt dBt = σXt dB̃t ,
t ∈ R+ ,
(6.10)
hence by Lemma 5.2 the discounted value Vet of a self-financing portfolio can
be written as
wt
Vet = Ve0 +
ξu dXu
0
wt
eu ,
= Ve0 + σ ξu Xu dB
t ∈ R+ ,
0
and by Proposition 6.1 it becomes a martingale under P∗ .
As in Chapter 3, the value Vt at time t of a self-financing portfolio strategy
(ξt )t∈[0,T ] hedging an attainable claim C will be called an arbitrage price of
the claim C at time t and denoted by πt (C), t ∈ [0, T ].
Theorem 6.3. Let (ξt , ηt )t∈[0,T ] be a portfolio strategy with price
Vt = ηt At + ξt St ,
t ∈ [0, T ],
and let C be a contingent claim, such that
(i) (ξt , ηt )t∈[0,T ] is a self-financing portfolio, and
(ii) (ξt , ηt )t∈[0,T ] hedges the claim C, i.e. we have VT = C.
Then the arbitrage price of the claim C is given by
Vt = e −r(T −t) IE∗ [C | Ft ],
0 6 t 6 T,
(6.11)
where IE∗ denotes expectation under the risk-neutral measure P∗ .
Proof. Since the portfolio strategy (ξt , ηt )t∈R+ is self-financing, by Lemma 5.2
and (6.10) we have
Vet = Ve0 + σ
wt
0
ξu Xu dB̃u = V0 +
wt
0
ξu dXu ,
t ∈ R+ ,
which is a martingale under P∗ from Proposition 6.1, hence
Vet = IE∗ VeT Ft
182
This version: June 12, 2017
http://www.ntu.edu.sg/home/nprivault/indext.html
"
Martingale Approach to Pricing and Hedging
= e −rT IE∗ [VT | Ft ]
= e −rT IE∗ [C | Ft ],
which implies
Vt = e rt Vet = e −r(T −t) IE∗ [C | Ft ].
When the process (St )t∈R+ has the Markov property, the value
Vt = e −r(T −t) IE∗ [φ(ST ) | Ft ] = C(t, St ),
0 6 t 6 T,
of the portfolio at time t ∈ [0, T ] can be written from (6.11) as a function
C(t, St ) of t and St .
This is the case in particular when (St )t∈R+ is a geometric Brownian motion, in which case by Proposition 5.11 the function C(t, x) solves the BlackScholes PDE

∂C
∂C
1
∂2C

 rC(t, x) =
(t, x) + rx
(t, x) + x2 σ 2 2 (t, x)
∂t
∂x
2
∂x


C(T, x) = φ(x), x > 0.
In the case of European options with payoff function φ(x) = (x − K)+ we recover the Black-Scholes formula (5.17), cf. Proposition 5.16, by a probabilistic
argument.
Proposition 6.4. The price at time t of a European call option with strike
price K and maturity T is given by
C(t, St ) = St Φ dT+ (t) − K e −r(T −t) Φ d− (T − t) ,
0 6 t 6 T, (6.12)
with
d+ (T − t) :=
log(x/K) + (r + σ 2 /2)(T − t)
√
,
σ T −t
d− (T − t) :=
log(x/K) + (r − σ 2 /2)(T − t)
√
.
σ T −t
Proof. The proof of Proposition 6.4 is a consequence of (6.11) and Lemma 6.5
below. Using the relation
ST = St e r(T −t)+σ(B̃T −B̃t )−σ
2
(T −t)/2
,
0 6 t 6 T,
by Theorem 6.3 the price of the portfolio hedging C is given by
Vt = e −r(T −t) IE∗ [C | Ft ]
= e −r(T −t) IE∗ (ST − K)+ | Ft
"
183
This version: June 12, 2017
http://www.ntu.edu.sg/home/nprivault/indext.html
N. Privault
2
= e −r(T −t) IE∗ (St e r(T −t)+σ(B̃T −B̃t )−σ (T −t)/2 − K)+ | Ft
2
= e −r(T −t) IE∗ (x e r(T −t)+σ(B̃T −B̃t )−σ (T −t)/2 − K)+ x=St
= e −r(T −t) IE∗ ( e m(x)+X − K)+ x=S ,
0 6 t 6 T,
t
where
m(x) = r(T − t) − σ 2 (T − t)/2 + log x
and X = σ(B̃T − B̃t ) is a centered Gaussian random variable with variance
Var [X] = Var σ(B̃T − B̃t ) = σ 2 Var B̃T − B̃t = σ 2 (T − t)
under P∗ . Hence by Lemma 6.5 below we have
C(t, St ) = Vt
= e −r(T −t) IE∗ ( e m(x)+X − K)+ x=S
t
= e −r(T −t) e m(St )+σ
2
(T −t)/2
Φ(v + (m(St ) − log K)/v)
−K e −r(T −t) Φ((m(St ) − log K)/v)
= St Φ(v + (m(St ) − log K)/v) − K e −r(T −t) Φ((m(St ) − log K)/v)
= St Φ dT+ (t) − K e −r(T −t) Φ d− (T − t) ,
0 6 t 6 T.
Relation (6.12) can also be written as
e −r(T −t) IE∗ (ST − K)+ | Ft = e −r(T −t) IE∗ (ST − K)+ | St
(6.13)
2
log(x/K) + (r + σ /2)(T − t)
√
= St Φ
σ T −t
log(x/K) + (r − σ 2 /2)(T − t)
√
−K e −r(T −t) Φ
,
0 6 t 6 T.
σ T −t
Lemma 6.5. Let X be a centered Gaussian random variable with variance
v 2 . We have
2
IE ( e m+X − K)+ = e m+v /2 Φ(v + (m − log K)/v) − KΦ((m − log K)/v).
Proof. We have
2
2
1 w ∞ m+x
IE ( e m+X − K)+ = √
(e
− K)+ e −x /(2v ) dx
2πv 2 −∞
2
2
1 w∞
( e m+x − K) e −x /(2v ) dx
= √
2πv 2 −m+log K
2
2
2
2
em w ∞
K w∞
= √
e x−x /(2v ) dx − √
e −x /(2v ) dx
2πv 2 −m+log K
2πv 2 −m+log K
184
This version: June 12, 2017
http://www.ntu.edu.sg/home/nprivault/indext.html
"
Martingale Approach to Pricing and Hedging
2
2
2
2
2
e m+v /2 w ∞
K w∞
= √
e −(v −x) /(2v ) dx − √
e −x /2 dx
2
−m+log
K
(−m+log
K)/v
2π
2πv
2
2
2
e m+v /2 w ∞
= √
e −x /(2v ) dx − KΦ((m − log K)/v)
2πv 2 −v2 −m+log K
= e m+v
2
/2
Φ(v + (m − log K)/v) − KΦ((m − log K)/v).
Call-put parity
Let
P (t, St ) := e −r(T −t) IE∗ (K − ST )+ | Ft
denote the price of the put option with strike price K and maturity T .
Proposition 6.6. We have the relation
C(t, St ) − P (t, St ) = St − e −r(T −t) K.
(6.14)
Proof. From Theorem 6.3 we have
C(t, St ) − P (t, St )
= e −r(T −t) IE∗ (ST − K)+ | Ft − e −r(T −t) IE∗ (K − ST )+ | Ft
= e −r(T −t) IE∗ (ST − K)+ − (K − ST )+ | Ft
= e −r(T −t) IE∗ ST − K | Ft
= St − e −r(T −t) K.
Relation (6.14) is called the put-call parity, and it shows that the put option
price is given by
P (t, St ) = C(t, St ) − St + e −r(T −t) K
= St Φ dT+ (t) + e −r(T −t) K − St − e −r(T −t) KΦ d− (T − t)
= −St (1 − Φ dT+ (t) ) + e −r(T −t) K(1 − Φ d− (T − t) )
= −St Φ − dT+ (t) + e −r(T −t) KΦ − d− (T − t) .
6.5 Hedging Strategies
The martingale method can be used to recover the Black-Scholes PDE of
Proposition 5.11 by using the fact that the discounted price Ṽt = e −rt g(t, St )
of a self-financing hedging portfolio is a martingale by Lemma 5.2 and Proposition 6.1, and by noting that from e.g. Corollary II-1, page 72 of [Pro04], all
"
185
This version: June 12, 2017
http://www.ntu.edu.sg/home/nprivault/indext.html
N. Privault
terms in dt should vanish in the expression of
d( e −rt g(t, St )) = −r e −rt g(t, St )dt + e −rt dg(t, St )
as in the proof of e.g. Proposition 5.11.
In the next Proposition 6.7 we compute a self-financing hedging strategy leading to an arbitrary square-integrable random claim C admitting a stochastic
integral representation formula of the form
C = IE∗ [C] +
wT
0
(6.15)
ζt dB̃t ,
where (ζt )t∈[0,t] is a square-integrable adapted process. Consequently, the
mathematical problem of finding the stochastic integral decomposition (6.15)
of a given random variable has important applications in finance. For example
we have
wT
BT2 = T + 2
Bt dBt ,
0
and
BT3 = 3
wT
0
(T − t + Bt2 )dBt ,
cf. Exercise 4.4.
Recall that the risky asset follows the equation
dSt
= µdt + σdBt ,
St
t ∈ R+ ,
S0 > 0,
and the discounted asset price satisfies
dXt = σXt dB̃t ,
t ∈ R+ ,
X0 = S0 > 0,
where (B̃t )t∈R+ is a standard Brownian motion under the risk-neutral probability measure P∗ .
The following proposition applies to arbitrary square-integrable payoff
functions, i.e. it covers exotic and path-dependent options.
Proposition 6.7. Consider a random payoff C ∈ L2 (Ω) such that (6.15)
holds, and let
e −r(T −t)
ζt ,
σSt
−r(T −t)
e
IE∗ C | Ft − ξt St
ηt =
,
At
(6.16)
ξt =
0 6 t 6 T.
Then the portfolio (ξt , ηt )t∈[0,T ] is self-financing, and letting
186
This version: June 12, 2017
http://www.ntu.edu.sg/home/nprivault/indext.html
(6.17)
"
Martingale Approach to Pricing and Hedging
Vt = ηt At + ξt St ,
0 6 t 6 T,
(6.18)
we have
Vt = e −r(T −t) IE∗ [C | Ft ],
0 6 t 6 T.
(6.19)
In particular we have
VT = C,
(6.20)
i.e. the portfolio (ξt , ηt )t∈[0,T ] yields a hedging strategy leading to C, starting
from the initial value
V0 = e −rT IE∗ [C].
Proof. Relation (6.19) follows from (6.17) and (6.18), and it implies
V0 = e −rT IE∗ [C] = η0 A0 + ξ0 S0
at t = 0, and (6.20) at t = T . It remains to show that the portfolio strategy
(ξt , ηt )t∈[0,T ] is self-financing. By (6.15) and Proposition 6.1 we have
Vt = ηt At + ξt St
= e −r(T −t) IE∗ [C | Ft ]
wT
= e −r(T −t) IE∗ IE∗ [C] +
ζu dB̃u Ft
0
wt
= e −r(T −t) IE∗ [C] +
ζu dB̃u
0
wt
= e rt V0 + e −r(T −t)
ζu dB̃u
0
wt
= e rt V0 + σ ξu Su e r(t−u) dB̃u
0
wt
= e rt V0 + e rt
ξu dXu ,
0 6 t 6 T,
0
which shows that the discounted portfolio value Vet = e −rt Vt satisfies
wt
Vet = V0 +
ξu dXu ,
0 6 t 6 T,
0
and this implies that (ξt , ηt )t∈[0,T ] is self-financing by Lemma 5.2.
The above proposition shows that there always exists a hedging strategy
starting from
V0 = IE∗ [C] e −rT .
In addition, since there exists a hedging strategy leading to
VeT = e −rT C,
then (Vet )t∈[0,T ] is necessarily a martingale with
"
187
This version: June 12, 2017
http://www.ntu.edu.sg/home/nprivault/indext.html
N. Privault
Vet = IE∗ VeT Ft = e −rT IE∗ C | Ft ,
and initial value
0 6 t 6 T,
Ve0 = IE∗ VeT = e −rT IE∗ [C].
In practice, the hedging problem can now be reduced to the computation of
the process (ζt )t∈[0,T ] appearing in (6.15). This computation, called the Delta
hedging, can be performed by application of the Itô formula and the Markov
property, see e.g. [Pro01].
Definition 6.8. The Markov semi-group (Pt )06t6T associated to (St )t∈[0,T ]
is the mapping Pt defined on functions f ∈ Cb2 (R) as
Pt f (x) := IE∗ [f (St ) | S0 = x],
t ∈ R+ .
By the Markov property and time homogeneity of (St )t∈[0,T ] we also have
Pt f (Su ) := IE∗ f (St+u ) | Fu = IE∗ f (St+u ) | Su , t, u ∈ R+ ,
and the semi-group (Pt )06t6T satisfies the relation
Pt Ps = Ps+t ,
s, t ∈ R+ .
In addition, the process (PT −t f (St ))t∈[0,T ] is an Ft -martingale, i.e.:
IE∗ PT −t f (St ) | Fu = IE∗ IE∗ [f (ST ) | Ft ] | Fu
= IE∗ f (ST ) | Fu
= PT −u f (Su ),
(6.21)
0 6 u 6 t 6 T , and we have
St
Pt−u f (x) = IE∗ f (St ) | Su = x = IE∗ f x
,
Su
0 6 u 6 t. (6.22)
The next lemma allows us to compute the process (ζt )t∈[0,T ] in case the payoff
C is of the form C = φ(ST ) for some function φ. In case C ∈ L2 (Ω) is the
payoff of an exotic option, the process (ζt )t∈[0,T ] can be computed using the
Malliavin gradient on the Wiener space, cf. [NØP09], [Pri09].
Lemma 6.9. Let φ ∈ Cb2 (Rn ). The stochastic integral decomposition
wT
φ(ST ) = IE∗ φ(ST ) +
ζt dB̃t
0
(6.23)
is given by
ζt = σSt
∂
(PT −t φ)(St ),
∂x
0 6 t 6 T.
(6.24)
Proof. Since PT −t φ is in C 2 (R), we can apply the Itô formula to the process
188
This version: June 12, 2017
http://www.ntu.edu.sg/home/nprivault/indext.html
"
Martingale Approach to Pricing and Hedging
t 7−→ PT −t φ(St ) = IE∗ φ(ST ) | Ft ,
which is a martingale from the tower property (6.1) of conditional expectations as in (6.21). From the fact that the finite variation term in the Itô
formula vanishes when (PT −t φ(St ))t∈[0,T ] is a martingale, (see e.g. Corollary II-6-1 page 72 of [Pro04]), we obtain:
PT −t φ(St ) = PT φ(S0 ) + σ
wt
0
Su
∂
(PT −u φ)(Su )dB̃u ,
∂x
0 6 t 6 T, (6.25)
with PT φ(S0 ) = IE∗ φ(ST ) . Letting t = T , we obtain (6.24) by uniqueness
of the stochastic integral decomposition (6.23) of C = φ(ST ).
By (6.22) we also have
∂ ∗
IE φ(ST ) | St = x x=St
∂x
∂ ∗
= σSt
IE φ(xST /St ) x=S ,
0 6 t 6 T,
t
∂x
ζt = σSt
hence
1 −r(T −t)
e
ζt
σSt
∂
= e −r(T −t)
IE∗ φ(xST /St ) x=St ,
∂x
ξt =
(6.26)
0 6 t 6 T,
which recovers the formula (5.13) for the Delta of a vanilla option. As a consequence we have ξt > 0 and there is no short selling when the payoff function
φ is nondecreasing.
In the case of European options, the process ζ can be computed via the next
proposition.
Proposition 6.10. Assume that C = (ST − K)+ . Then for 0 6 t 6 T we
have
ST
S
ζt = σSt IE∗
1[K,∞) x T
.
(6.27)
St
St
x=St
Proof. This result follows from Lemma 6.9 and the relation
x
) ,
PT −t f (x) = IE∗ f (St,T
after approximation of x 7−→ φ(x) = (x − K)+ with C 2 functions.
By evaluating the expectation (6.27) in Proposition 6.10 we can recover the
formula (5.20) in Proposition 5.13 for the Delta of a European call option in
the Black-Scholes model.
"
189
This version: June 12, 2017
http://www.ntu.edu.sg/home/nprivault/indext.html
N. Privault
Namely, Proposition 6.11 shows that the Black-Scholes self-financing hedging
strategy is to hold a (possibly fractional) quantity
log(St /K) + (r + σ 2 /2)(T − t)
√
ξt = Φ dT+ (t) = Φ
>0
(6.28)
σ T −t
of the risky asset, and to borrow a quantity
log(St /K) + (r − σt2 /2)(T − t)
√
60
− ηt = K e −rT Φ
σ T −t
(6.29)
of the riskless (savings) account, cf. also Corollary 12.14 in Chapter 12.
In the next proposition we provide another proof of the result of Proposition 5.13.
Proposition 6.11. The Delta of a European call option with payoff function
f (x) = (x − K)+ is given by
log(St /K) + (r + σ 2 /2)(T − t)
√
ξt = Φ dT+ (t) = Φ
,
0 6 t 6 T.
σ T −t
Proof. By Propositions 6.7 and 6.10 we have
1 −r(T −t)
e
ζt
σSt
S
S
T
1[K,∞) x T
= e −r(T −t) IE∗
St
St
x=St
ξt =
= e −r(T −t)
h
i
2
2
× IE∗ e σ(B̃T −B̃t )−σ (T −t)/2+r(T −t) 1[K,∞) (x e σ(B̃T −B̃t )−σ (T −t)/2+r(T −t) )
x=St
w∞
1
σy−σ 2 (T −t)/2−y 2 /(2(T −t))
e
dy
=p
2π(T − t) σ(T −t)/2−r(T −t)/σ+σ−1 log(K/St )
w∞
2
1
e −(y−σ(T −t)) /(2(T −t)) dy
=p
√
2π(T − t) −d− (T −t)/ T −t
2
1 w∞
=√
e −(y−σ(T −t)) /2 dy
2π −d− (T −t)
2
1 w∞
=√
e −y /2 dy
2π −dT+ (t)
1 w dT+ (t) −y2 /2
=√
e
dy
2π −∞
T
= Φ d+ (t) .
190
This version: June 12, 2017
http://www.ntu.edu.sg/home/nprivault/indext.html
"
Martingale Approach to Pricing and Hedging
As noted above, the result of Proposition 6.11 recovers (5.21) which is obtained by a direct differentiation of the Black-Scholes function as in (5.13) or
(6.26).
Exercises
2
/2)t
Exercise 6.1 Given the price process (St )t∈R+ defined as St := S0 e σBt +(r−σ
,
t ∈ R+ , price the option with payoff function φ(ST ) by writing e −rT IE φ(ST )
as an integral.
Exercise 6.2 Consider an asset price (St )t∈R+ which is a martingale under
the risk-neutral measure P∗ in a market with interest rate r = 0, and let
φ(x) = (x − K)+ be the (convex) European call payoff function.
Show that, for any two maturities T1 < T2 and p, q ∈ [0, 1] such that
p + q = 1, the price of the average option with payoff φ(pST1 + qST2 ) is upper
bounded by the price of the European call option with maturity T2 , i.e. show
that
IE∗ φ(pST1 + qST2 ) 6 IE∗ φ(ST2 ) .
Hint 1: For φ a convex function we have φ(px + qy) 6 pφ(x) + qφ(y) for any
x, y ∈ R and p, q ∈ [0, 1] such that p + q = 1.
Hint 2: Any convex function φ(St ) of a martingale St is a submartingale.
Exercise 6.3
Consider an underlying asset price process (St )t∈R+ .
a) Show that the price at time t of a European call option with strike price
K and maturity T is lower bounded by (St − K e −r(T −t) )+ , i.e.
e −r(T −t) IE∗ (ST − K)+ | Ft > (St − K e −r(T −t) )+ , 0 6 t 6 T.
b) Show that the price at time t of a European put option with strike price
K and maturity T is lower bounded by K e −r(T −t) − St , i.e.
e −r(T −t) IE∗ (K − ST )+ | Ft > (K e −r(T −t) − St )+ , 0 6 t 6 T.
Exercise 6.4 The following two graphs describe the payoff functions φ of
bull spread and bear spread options with payoff φ(SN ) on an underlying SN
at maturity time N .
"
191
This version: June 12, 2017
http://www.ntu.edu.sg/home/nprivault/indext.html
N. Privault
100
100
80
80
60
60
40
40
20
20
0
0
50
K1
100
x
150
K2
(a) Bull spread payoff.
200
0
0
50
K1
100
x
150
K2
200
(b) Bear spread payoff.
Fig. 6.2: Payoff functions of bull spread, bear spread, and collar options.
Show that in each of the cases (a)-(b)-(c), the corresponding option can be
realized by purchasing and/or short selling standard European call and put
options with strike prices to be specified.
Hints: An option with payoff φ(SN ) is priced (1+r)−N IE∗ φ(SN ) at time 0.
The payoff of the European call (resp. put) option with strike price K is
(SN − K)+ , resp. (K − SN )+ .
Exercise 6.5 Forward contracts revisited. Consider a risky asset whose price
2
St is given by St = S0 e σBt +rt−σ t/2 , t ∈ R+ , where (Bt )t∈R+ is a standard
Brownian motion. Consider a forward contract with maturity T and payoff
ST − κ.
a) Compute the price Ct of this claim at any time t ∈ [0, T ].
b) Compute a hedging strategy for the option with payoff ST − κ.
Exercise 6.6 Option pricing with dividends (Exercise 5.1 continued).
Consider an underlying asset price process (St )t∈R+ modeled under the riskneutral measure as
dSt = (r − δ)St dt + σSt dBt ,
where (Bt )t∈R+ is a standard Brownian motion and δ > 0 is a continuoustime dividend rate. Compute the price at time t ∈ [0, T ] of the European call
option in a market with dividend rate δ by the martingale method.
Exercise 6.7 Forward start options [Rub91]. A forward start European call
option is an option whose holder receives at time T1 (e.g. your birthday) the
value of a standard European call option at the money and with maturity
T2 > T1 . Compute the price
192
This version: June 12, 2017
http://www.ntu.edu.sg/home/nprivault/indext.html
"
Martingale Approach to Pricing and Hedging
e −r(T1 −t) IE∗ e −r(T2 −T1 ) IE∗ (ST2 − ST1 )+ | FT1 | Ft
at time t ∈ [0, T1 ], of the forward start European call option using the BlackScholes formula.
Exercise 6.8 Power option. (Exercise 5.2 continued). Consider the price
process (St )t∈[0,T ] given by
dSt
= rdt + σdBt
St
and a riskless asset of value At = A0 e rt , t ∈ [0, T ], with r > 0. In this
problem, (ηt , ξt )t∈[0,T ] denotes a portfolio strategy with value
Vt = ηt At + ξt St ,
0 6 t 6 T.
a) Compute the arbitrage price
C(t, St ) = e −r(T −t) IE∗ |ST |2 | Ft ,
at time t ∈ [0, T ], of the power option with payoff |ST |2 .
b) Compute a self-financing portfolio strategy (ηt , ξt )t∈[0,T ] hedging the claim
|ST |2 .
Exercise 6.9
(Exercise 5.7 continued).
a) Solve the stochastic differential equation
dSt = αSt dt + σdBt
in terms of α, σ > 0, and the initial condition S0 .
b) For which value αM of α is the discounted price process S̃t = e −rt St ,
0 6 t 6 T , a martingale under P?
c) For each value of α, build a probability measure Pα under which the
discounted price process S̃t = e −rt St , 0 6 t 6 T , is a martingale.
d) Compute the arbitrage price
C(t, St ) = e −r(T −t) IEα exp(ST ) | Ft
at time t ∈ [0, T ] of the contingent claim with payoff exp(ST ), and recover
the result of Exercise 5.7.
e) Explicitly compute the portfolio strategy (ηt , ξt )t∈[0,T ] that hedges the
contingent claim exp(ST ).
f) Check that this strategy is self-financing.
"
193
This version: June 12, 2017
http://www.ntu.edu.sg/home/nprivault/indext.html
N. Privault
Exercise 6.10 Let (Bt )t∈R+ be a standard Brownian motion generating a
filtration (Ft )t∈R+ . Recall that for f ∈ C 2 (R+ ×R), Itô’s formula for (Bt )t∈R+
reads
w t ∂f
f (t, Bt ) = f (0, B0 ) +
(s, Bs )ds
0 ∂s
w t ∂f
1 w t ∂2f
+
(s, Bs )ds.
(s, Bs )dBs +
0 ∂x
2 0 ∂x2
2
a) Let r ∈ R, σ > 0, f (x, t) = e rt+σx−σ t/2 , and St = f (t, Bt ). Compute
df (t, Bt ) by Itô’s formula, and show that St solves the stochastic differential equation
dSt = rSt dt + σSt dBt ,
where r > 0 and σ > 0.
b) Show that
2
IE e σBT | Ft = e σBt +σ (T −t)/2 ,
0 6 t 6 T.
Hint: Use the independence of increments in the decomposition
BT = (BT − Bt ) + (Bt − B0 )
2 2
and the Laplace transform IE e αX = e α η /2 when X ' N (0, η 2 ).
c) Show that the process (St )t∈R+ satisfies
IE ST | Ft = e r(T −t) St ,
0 6 t 6 T.
d) Let C = ST − K denote the payoff of a forward contract with exercise
price K and maturity T . Compute the discounted expected payoff
Vt := e −r(T −t) IE C | Ft .
e) Find a self-financing portfolio strategy (ξt , ηt )t∈R+ such that
Vt = ξt St + ηt At ,
0 6 t 6 T,
where At = A0 e rt is the price of a riskless asset with interest rate r > 0.
Show that it recovers the result of Exercise 5.4-(c).
f) Show that the portfolio (ξt , ηt )t∈[0,T ] found in Question (e) hedges the
payoff C = ST − K at time T , i.e. show that VT = C.
Exercise 6.11
Binary options. Consider a price process (St )t∈R+ given by
dSt
= rdt + σdBt ,
St
S0 = 1,
194
This version: June 12, 2017
http://www.ntu.edu.sg/home/nprivault/indext.html
"
Martingale Approach to Pricing and Hedging
under the risk-neutral measure P∗ . A binary (or digital) call, resp. put, option
is a contract with maturity T , strike price K, and payoff


 $1 if ST > K,
 $1 if ST 6 K,
Cd :=
resp. Pd :=


0 if ST < K,
0 if ST > K.
Recall that the prices πt (Cd ) and πt (Pd ) at time t of the binary call and put
options are given by the discounted expected payoffs
πt (Cd ) = e −r(T −t) IE Cd | Ft and πt (Pd ) = e −r(T −t) IE Pd | Ft .
(6.30)
a) Show that the payoffs Cd and Pd can be rewritten as
Cd = 1[K,∞) (ST )
and
Pd = 1[0,K] (ST ).
b) Using Relation (6.30), Question (a), and the relation
IE 1[K,∞) (ST ) | St = x = P∗ (ST > K | St = x),
show that the price πt (Cd ) is given by
πt (Cd ) = Cd (t, St ),
where Cd (t, x) is the function defined by
Cd (t, x) := e −r(T −t) P∗ (ST > K | St = x).
c) Using the results of Exercise 4.17-(c) and of Question (b), show that the
price πt (Cd ) of the binary call option is given by
(r − σ 2 /2)(T − t) + log(x/K)
√
Cd (t, x) = e −r(T −t) Φ
σ T −t
= e −r(T −t) Φ d− (T − t) ,
where
d− (T − t) =
(r − σ 2 /2)(T − t) + log(St /K)
√
.
σ T −t
d) Assume that the binary option holder is entitled to receive a “return
amount” α ∈ [0, 1] in case the underlying ends out of the money at maturity. Compute price at time t ∈ [0, T ] of this modified contract.
e) Using Relation (6.30) and Question (a), prove the call-put parity relation
πt (Cd ) + πt (Pd ) = e −r(T −t) ,
0 6 t 6 T.
(6.31)
If needed, you may use the fact that P∗ (ST = K) = 0.
"
195
This version: June 12, 2017
http://www.ntu.edu.sg/home/nprivault/indext.html
N. Privault
f) Using the results of Questions (e) and (c), show that the price πt (Pd ) of
the binary put option is given by
πt (Pd ) = e −r(T −t) Φ − d− (T − t) .
g) Using the result of Question (c), compute the Delta
ξt :=
∂Cd
(t, St )
∂x
of the binary call option. Does the Black-Scholes hedging strategy of such
a call option involve short selling? Why?
h) Using the result of Question (f), compute the Delta
ξt :=
∂Pd
(t, St )
∂x
of the binary put option. Does the Black-Scholes hedging strategy of such
a put option involve short selling? Why?
Exercise 6.12 Computation of Greeks. Consider an underlying asset whose
price (St )t∈R+ is given by a stochastic differential equation of the form
dSt = rSt dt + σ(St )dWt ,
where σ(x) is a Lipschitz coefficient, and an
and price
h
C(x, T ) = e −rT IE φ(ST )
option with payoff function φ
i
S0 = x ,
where φ(x) is a twice continuously differentiable (C 2 ) function, with S0 = x.
Using the Itô formula, show that the sensitivity
ThetaT =
i
∂ −rT h
e
IE φ(ST ) S0 = x
∂T
of the option price with respect to maturity T can be expressed as
h
i
h
i
ThetaT = −r e −rT IE φ(ST ) S0 = x + r e −rT IE St φ0 (ST ) S0 = x
h
i
1
+ e −rT IE φ00 (ST )σ 2 (ST ) S0 = x .
2
Problem 6.13 Chooser options. In this problem we denote by C(t, St , K, T ),
resp. P (t, St , K, T ), the price at time t of a European call, resp. put, option
with strike price K and maturity T , on an underlying asset priced under the
2
risk-neutral measure as St = S0 e σBt +rt−σ t/2 , t ∈ R+ .
196
This version: June 12, 2017
http://www.ntu.edu.sg/home/nprivault/indext.html
"
Martingale Approach to Pricing and Hedging
a) Prove the call-put parity formula
C(t, St , K, T ) − P (t, St , K, T ) = St − K e −r(T −t) ,
0 6 t 6 T. (6.32)
b) Consider an option contract with maturity T , which entitles its holder to
receive at time T the value of a European put option with strike price K
and maturity U > T .
Write down the price this contract at time t ∈ [0, T ] using a conditional
expectation under the risk-neutral measure P∗ .
c) Consider now an option contract with maturity T , which entitles its holder
to receive at time T either the value of a European call option or a European put option whichever is higher. The European call and put options
have same strike price K and same maturity U > T .
Show that at maturity T , the payoff of this contract can be written as
P (T, ST , K, U ) + max 0, ST − K e −r(U −T ) .
Hint: Use the call-put parity formula (6.32).
d) Price the contract of Question (c) at any time t ∈ [0, T ] using the put and
call pricing functions C(t, x, K, T ) and P (t, x, K, U ).
e) Using the Black-Scholes formula, compute the self-financing hedging strategy (ξt , ηt )t∈[0,T ] with portfolio value
Vt = ξt St + ηt e rt ,
0 6 t 6 T,
for the option contract of Question (c).
f) Consider now an option contract with maturity T , which entitles its holder
to receive at time T the value of either a European call or a European put
option, whichever is lower. The two options have same strike price K and
same maturity U > T .
Show that the payoff of this contract at maturity T can be written as
C(T, ST , K, U ) − max 0, ST − K e −r(U −T ) .
g) Price the contract of Question (f) at any time t ∈ [0, T ].
h) Using the Black-Scholes formula, compute the self-financing hedging strategy (ξt , ηt )t∈[0,T ] with portfolio price
Vt = ξt St + ηt e rt ,
0 6 t 6 T,
for the option contract of Question (f).
i) Give the price and hedging strategy of the contract that yields the sum
of the payoffs of Questions (c) and (f).
"
197
This version: June 12, 2017
http://www.ntu.edu.sg/home/nprivault/indext.html
N. Privault
j) What happens when U = T ? Give the payoffs of the contracts of Questions (c), (f) and (i).
Problem 6.14
St = S0 e
Consider a risky asset priced
σBt +µt−σ 2 t/2
,
i.e.
dSt = µSt dt + σSt dBt ,
t ∈ R+ ,
a riskless asset priced At = A0 e rt , and a self-financing portfolio (ηt , ξt )t∈R+
with value
Vt := ηt At + ξt St ,
t ∈ R+ .
a) Using the portfolio self-financing condition dVt = ηt dAt +ξt dSt , show that
we have
wT
wT
VT = Vt +
(rVs + (µ − r)ξs Ss )ds + σ
ξs Ss dBs .
t
b)
t
Show that under the risk-neutral measure P the portfolio value Vt
satisfies the Backward Stochastic Differential Equation (BSDE)
∗
∗
Vt = VT −
wT
t
rVs ds −
wT
t
πs dB̃s ,
(6.33)
where πt := σξt St is the risky amount invested on the asset St , multiplied
by σ, and (B̃t )t∈R+ is a standard Brownian motion under P∗ . Show that
under the risk-neutral measure P∗ , the discounted portfolio value Ṽt :=
e −rt Vt can be rewritten as
wT
ṼT = Ṽ0 +
e −rs πs dB̃s .
(6.34)
0
c) Express dv(t, St ) by the Itô formula, where v(t, x) is a C 2 function of t
and x.
d) Consider now a more general BSDE of the form
Vt = VT −
wT
t
f (s, Ss , Vs , πs )ds −
wT
t
πs dBs ,
(6.35)
with terminal condition VT = g(ST ). By matching (6.35) to the Itô formula of Question c, find the PDE satisfied by the function v(t, x) defined
as Vt = v(t, St ).
µ−r
z, the PDE of Question d recovers
e) Show that when f (t, x, v, z) = rv+
σ
the standard Black-Scholes PDE.
µ−r
f) Assuming again f (t, x, v, z) = rv +
z and taking the terminal conσ
dition
∗
The Girsanov Theorem 7.3 states that B̃t := Bt + (µ − r)t/σ is a standard Brownian
motion under P∗ .
198
This version: June 12, 2017
http://www.ntu.edu.sg/home/nprivault/indext.html
"
Martingale Approach to Pricing and Hedging
VT = (S0 e σBT +(µ−σ
g)
h)
i)
j)
2
/2)T
− K)+ ,
give the process (πt )t∈[0,T ] appearing in the stochastic integral represen2
tation (6.34) of the discounted claim e −rT (S0 e σBT +(µ−σ /2)T − K)+ .∗
From now on we assume that short selling is penalized† at a rate γ > 0,
i.e. γSt |ξt |dt is subtracted from the portfolio value change dVt whenever
ξt < 0 over the time period [t, t + dt]. Rewrite the self-financing condition
using (ξt )− := − min(ξt , 0).
Find the BSDE of the form (6.35) satisfied by (Vt )t∈R+ , and the corresponding function f (t, x, v, z).
Under the above penalty on short selling, find the PDE satisfied by the
function u(t, x) when the portfolio value Vt is given as Vt = u(t, St ).
Assume that one can borrow only at a rate R which is higher‡ than the
risk-free rate r > 0, i.e. we have
dVt = Rηt At dt + ξt dSt
when ηt < 0, and
dVt = rηt At dt + ξt dSt
when ηt > 0. Find the PDE satisfied by the function u(t, x) when the
portfolio value Vt is given as Vt = u(t, St ).
k) Assume that the portfolio differential reads
dVt = ηt dAt + ξt dSt − dUt ,
where (Ut )t∈R+ is an increasing process. Show that the corresponding
portfolio strategy (ξt )t∈R+ is superhedging the claim VT = C.
Exercise 6.15 Girsanov theorem. Assume that the Novikov integrability condition (6.5) is not satisfied. How does this modify the statement (6.6) of the
Girsanov Theorem 6.2?
Problem 6.16 Log options.
a) Consider a market model made of a risky asset with price (St )t∈R+ as in
Exercise 4.21-(d) and a riskless asset with price At = $1 × e rt and riskless
interest rate r = σ 2 /2. From the answer to Exercise 4.21-(b), show that
the arbitrage price
General Black-Scholes knowledge can be used for this question.
SGX started to penalize naked short sales with an interim measure in September 2008.
‡
Regular savings account usually pays r=0.05% per year. Effective Interest Rates (EIR)
for borrowing could be as high as R=20.61% per year.
∗
†
"
199
This version: June 12, 2017
http://www.ntu.edu.sg/home/nprivault/indext.html
N. Privault
Vt = e −r(T −t) IE (log ST )+ | Ft
at time t ∈ [0, T ] of a log call option with payoff (log ST )+ is equal to
r
T − t −Bt2 /(2(T −t))
Bt
Vt = σ e −r(T −t)
e
+ σ e −r(T −t) Bt Φ √
.
2π
T −t
b) Show that Vt can be written as
Vt = g(T − t, St ),
where g(τ, x) = e −rτ f (τ, log x), and
r
τ −y2 /(2σ2 τ )
y
√
f (τ, y) = σ
e
+ yΦ
.
2π
σ τ
c) Figure 6.3 represents the graph of (τ, x) 7→ g(τ, x), with r = 0.05 = 5%
per year and σ = 0.1. Assume that the current underlying price is $1 and
there remains 700 days to maturity. What is the price of the option?
Price
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
700
600
500
400
T-t
300
200
100
0
0
0.5
1.5
1
2
St
Fig. 6.3: Option price as a function of the underlying and of time to maturity.
∂g
(T − t, St ) of St at
∂x
+
time t in a portfolio hedging the payoff (log ST ) is equal to
log S
1
√ t
ξt = e −r(T −t) Φ
,
0 6 t 6 T.
St
σ T −t
d) Show∗ that the (possibly fractional) quantity ξt =
∂g
e) Figure 6.4 represents the graph of (τ, x) 7→ ∂x
(τ, x). Assuming that the
current underlying price is $1 and that there remains 700 days to maturity,
how much of the risky asset should you hold in your portfolio in order to
hedge one log option?
∗
Recall the chain rule of derivation
∂
1 ∂f
f (τ, log x) =
(τ, y)|y=log x .
∂x
x ∂y
200
This version: June 12, 2017
http://www.ntu.edu.sg/home/nprivault/indext.html
"
Martingale Approach to Pricing and Hedging
Delta
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0 2
1.8
1.6
1.4
1.2
St
1
0.8
0.6
0.4
0.2
700
600
500
400
300
200
100
0
T-t
Fig. 6.4: Delta as a function of the underlying and of time to maturity.
f) Based on the framework and answers of Questions (c) and (e), should you
borrow or lend the riskless asset At = $1 × e rt , and for what amount?
∂2g
g) Show that the Gamma of the portfolio, defined as Γt =
(T − t, St ),
∂x2
equals
!
2
2
1
log S
1
p
√ t
Γt = e −r(T −t) 2
e −(log St ) /(2σ (T −t)) − Φ
,
St σ 2π(T − t)
σ T −t
0 6 t < T.
h) Figure 6.5 represents the graph of Gamma. Assume that there remains
60 days to maturity and that St , currently at $1, is expected to increase.
Should you buy or (short) sell the underlying asset in order to hedge the
option?
Gamma
1
0.8
0.6
0.4
0.2
0
-0.20.2
0.4
0.6
0.8
St
1
1.2
1.4
1.6
1.8
180200
140160
T-t
100120
60 80
Fig. 6.5: Gamma as a function of the underlying and of time to maturity.
i) Let now σ = 1. Show that the function f (τ, y) of Question (b) solves the
heat equation
"
201
This version: June 12, 2017
http://www.ntu.edu.sg/home/nprivault/indext.html
N. Privault

1 ∂2f
∂f



(τ, y)
(τ, y) =
∂τ
2 ∂y 2



f (0, y) = (y)+ .
Problem 6.17 Log options with a given strike price.
a) Consider a market model made of a risky asset with price (St )t∈R+ as in
Exercise 4.17, a riskless asset with price At = $1 × e rt , riskless interest
rate r = σ 2 /2 and S0 = 1. From the answer to Exercise A.4-(b), show
that the arbitrage price
Vt = e −r(T −t) IE∗ (K − log ST )+ | Ft
at time t ∈ [0, T ] of a log call option with strike price K and payoff
(K − log ST )+ is equal to
r
K/σ − Bt
T − t −(Bt −K/σ)2 /(2(T −t)) −r(T −t)
√
Vt = σ e −r(T −t)
e
+e
(K−σBt )Φ
.
2π
T −t
b) Show that Vt can be written as
Vt = g(T − t, St ),
where g(τ, x) = e −rτ f (τ, log x), and
r
τ −(K−y)2 /(2σ2 τ )
K −y
√
e
+ (K − y)Φ
.
f (τ, y) = σ
2π
σ τ
c) Figure 6.6 represents the graph of (τ, x) 7→ g(τ, x), with r = 0.125 per
year and σ = 0.5. Assume that the current underlying price is $3, that
K = 1, and that there remains 700 days to maturity. What is the price of
the option?
Price
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
2.2
2.4
2.6 2.8
St
3
3.2
3.4
3.6
3.8
0
100
200
300
400
500
600
700
T-t
Fig. 6.6: Option price as a function of the underlying and of time to maturity.
202
This version: June 12, 2017
http://www.ntu.edu.sg/home/nprivault/indext.html
"
Martingale Approach to Pricing and Hedging
∂g
(T − t, St ) of St at time t in a portfolio
∂x
hedging the payoff (K − log ST )+ is equal to
1
K − log St
√
ξt = − e −r(T −t) Φ
,
0 6 t 6 T.
St
σ T −t
d) Show∗ that the quantity ξt =
∂g
e) Figure 6.7 represents the graph of (τ, x) 7→ ∂x
(τ, x). Assuming that the
current underlying price is $3 and that there remains 700 days to maturity,
how much of the risky asset should you hold in your portfolio in order to
hedge one log option?
Delta
0
-0.05
-0.1
-0.15
-0.2
-0.25
-0.3
-0.35
-0.4
-0.45
-0.5
4
3.8
3.6
3.4
3.2
St
3
2.8
2.6
2.4
2.2
700
600
500
400
300
200
100
0
T-t
Fig. 6.7: Delta as a function of the underlying and of time to maturity.
f) Based on the framework and answers of Questions (c) and (e), should you
borrow or lend the riskless asset At = $1 × e rt , and for what amount?
∂2g
g) Show that the Gamma of the portfolio, defined as Γt =
(T − t, St ),
∂x2
equals
Γt = e
−r(T −t)
1
St2
1
σ
p
2π(T − t)
e
−(K−log St )2 /(2σ 2 (T −t))
+Φ
K − log St
√
σ T −t
0 6 t 6 T.
h) Figure 6.8 represents the graph of Gamma. Assume that there remains
10 days to maturity and that St , currently at $3, is expected to increase.
Should you buy or (short) sell the underlying asset in order to hedge the
option?
∗
Recall the chain rule of derivation
"
∂
1 ∂f
f (τ, log x) =
(τ, y)|y=log x .
∂x
x ∂y
203
This version: June 12, 2017
http://www.ntu.edu.sg/home/nprivault/indext.html
!
,
N. Privault
Gamma
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
010
20
30
T-t
40
50
60
70
80
90
100
3.8
3.6
3.4
3.2
3
2.6
2.8
St
2.4
2.2
Fig. 6.8: Gamma as a function of the underlying and of time to maturity.
i) Show that the function f (τ, y) of Question (b) solves the heat equation

∂f
σ2 ∂ 2 f



(τ, y) =
(τ, y)
∂τ
2 ∂y 2



f (0, y) = (K − y)+ .
204
This version: June 12, 2017
http://www.ntu.edu.sg/home/nprivault/indext.html
"