Chapter 3
Pricing and Hedging in Discrete Time
We consider the pricing and hedging of options in a discrete-time financial
model with N + 1 time instants t = 0, 1, . . . , N . Vanilla options are treated
using backward induction, and exotic options with arbitrary payoff functions
are considered using the Clark-Ocone formula in discrete time.
3.1 Pricing of Contingent Claims
Let us consider an attainable contingent claim with (random) claim payoff
C > 0 and maturity N . Recall that by the Definition 2.10 of attainability
there exists a (self-financing) portfolio strategy (ξt )t=1,2,...,N that hedges the
claim C, in the sense that
ξ¯N · S̄N =
d
X
(k)
(k)
ξN S N = C
(3.1)
k=0
at time N . Clearly, if (3.1) holds, then investing the amount
V0 = ξ¯1 · S̄0 =
d
X
(k)
(k)
(3.2)
(k)
(k)
(3.3)
ξ1 S0
k=0
at time t = 0, resp.
Vt = ξ¯t · S̄t =
d
X
ξt St
k=0
at times t = 1, 2, . . . , N into a self-financing hedging portfolio will allow one
to hedge the option and to obtain the perfect replication (3.1) at time N .
Definition 3.1. The value (3.2)-(3.3) at time t of a self-financing portfolio
strategy (ξt )t=1,2,...,N 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, 1, . . . , N .
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N. Privault
Next we develop a second approach to the pricing of contingent claims,
based on conditional expectations and martingale arguments. We will need
the following lemma, in which Vet := Vt /(1 + r)t denotes the discounted portfolio value, t = 0, 1, . . . , N .
Relation (3.4) in the following lemma has a natural interpretation by saying
that when a portfolio is self-financing the value Vet of the (discounted) portfolio at time t is given by summing up the (discounted) profits and losses
registered over all time periods from time 0 to time t. Note that the use
the discounted asset price X̄t in (3.4) allows us to add up profits and losses
ξ¯t · (X̄t − X̄t−1 ) since they are expressed in units of currency “at time 0”.
Indeed, in general, $1 at time t = 0 cannot be added to $1 at time t = 1
without proper discounting.
Lemma 3.2. The following statements are equivalent:
(i) The portfolio strategy (ξ¯t )t=1,2,...,N is self-financing.
(ii) ξ¯t · X̄t = ξ¯t+1 · X̄t for all t = 1, 2, . . . , N − 1.
(iii) We have the discounted decomposition
Vet = Ve0 +
t
X
ξ¯k · (X̄k − X̄k−1 ),
t = 0, 1, . . . , N.
(3.4)
k=1
|
{z
}
sum of profits and losses
Proof. First, the self-financing condition (i)
ξ¯t−1 · S̄t−1 = ξ¯t · S̄t−1 ,
t = 1, 2, . . . , N,
is clearly equivalent to (ii) by division of both sides by (1 + r)t−1 .
Next, assuming that (ii) holds we have the telescoping identity
Vet = Ve0 +
= Ve0 +
= Ve0 +
= Ve0 +
t
X
k=1
t
X
k=1
t
X
k=1
t
X
(Vek − Vek−1 )
ξ¯k · X̄k − ξ¯k−1 · X̄k−1
ξ¯k · X̄k − ξ¯k · X̄k−1
ξ¯k · (X̄k − X̄k−1 ),
t = 1, 2, . . . , N.
k=1
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Pricing and Hedging in Discrete Time
Finally, assuming that (iii) holds we get
Vet − Vet−1 = ξ¯t · (X̄t − X̄t−1 ),
hence

 ξ¯t · X̄t − ξ¯t−1 · X̄t−1 = ξ¯t · (X̄t − X̄t−1 ),
¯
ξt−1 · X̄t−1 = ξ¯t · X̄t−1 ,
t = 1, 2, . . . , N.
In Relation (3.4), the term ξ¯t · (X̄t − X̄t−1 ) represents the profit and loss
Vet − Vet−1 = ξ¯t · (X̄t − X̄t−1 ),
of the self-financing portfolio strategy (ξ¯j )j=1,2,...,N over the time period
[t − 1, t], computed by multiplication of the portfolio allocation ξ¯t with the
change of price X̄t − X̄t−1 , t = 1, 2, . . . , N .
The sum (3.4) is also referred to as a discrete-time stochastic integral of
the portfolio strategy (ξ¯t )t=1,2,...,N with respect to the random process
(X̄t )t=0,1,...,N .
Remark 3.3. 1. As a consequence of the above Lemma 3.2, if a contingent
claim C with discounted payoff is attainable by a self-financing portfolio
strategy (ξ¯t )t=1,2,...,N then the discounted claim payoff
e :=
C
C
= ξ¯N · X̄N
(1 + r)N
rewrites as the sum of discounted profits and losses
e = ξ¯N · X̄N = VeN = Ve0 +
C
N
X
ξ¯t · (X̄t − X̄t−1 ).
(3.5)
t=1
2. By Proposition 2.8, (Xt )t=0,1,...,N is a martingale under the risk-neutral
measure P∗ , hence by Proposition 2.6 and Lemma 3.2, (Vet )t=0,1,...,N in
(3.4) is also martingale under P∗ , provided that (ξ¯t )t=1,2,...,N is a selffinancing and predictable process.
The above remarks will be used in the proof of Theorem 3.4.
Theorem 3.4. The arbitrage price πt (C) of an attainable contingent claim
C is given by
πt (C) =
1
IE∗ [C | Ft ],
(1 + r)N −t
t = 0, 1, . . . , N,
(3.6)
where P∗ denotes any risk-neutral probability measure.
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Proof. a) Short proof. Since the claim C is attainable, there exists a selfe = VeN .
financing portfolio strategy (ξt )t=1,2,...,N such that C = VN , i.e. C
In addition, by Lemma 3.2 and Remark 3.3 the process (Vet )t=0,1,...,N is a
martingale, hence we have
e | Ft ,
Vet = IE∗ VeN | Ft = IE∗ C
t = 0, 1, . . . , N,
(3.7)
which shows (3.8). To conclude, we note that by Definition 3.1 the arbitrage
price πt (C) of the claim at time t is equal to the value Vt of the self-financing
hedging C.
b) Long proof. For completeness, we include a self-contained, step by step
derivation of (3.7), as follows:
e Ft = IE∗ VeN Ft
IE∗ C
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#
N
X
= IE∗ Ve0 +
ξ¯k · (X̄k − X̄k−1 )Ft
k=1
N
X
= IE∗ Ve0 | Ft +
IE∗ ξ¯k · (X̄k − X̄k−1 )Ft
k=1
= Ve0 +
t
X
N
X
IE∗ ξ¯k · (X̄k − X̄k−1 ) | Ft +
IE∗ ξ¯k · (X̄k − X̄k−1 ) | Ft
k=1
= Ve0 +
t
X
k=t+1
ξ¯k · (X̄k − X̄k−1 ) +
k=1
= Vet +
N
X
N
X
IE∗ ξ¯k · (X̄k − X̄k−1 ) | Ft
k=t+1
IE∗ ξ¯k · (X̄k − X̄k−1 ) | Ft ,
k=t+1
where we used Relation (3.4) of Lemma 3.2. In order to obtain (3.8) we need
to show that
N
X
IE∗ ξ¯k · (X̄k − X̄k−1 ) | Ft = 0,
k=t+1
or
IE∗ ξ¯j · (X̄j − X̄j−1 ) | Ft = 0,
for all j = t + 1, . . . , N . Since 0 6 t 6 j − 1 we have Ft ⊂ Fj−1 , hence by the
“tower property” of conditional expectations we get
IE∗ ξ¯j · (X̄j − X̄j−1 ) | Ft = IE∗ IE∗ ξ¯j · (X̄j − X̄j−1 ) | Fj−1 | Ft ,
therefore it suffices to show that
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Pricing and Hedging in Discrete Time
IE∗ ξ¯j · (X̄j − X̄j−1 ) | Fj−1 = 0.
We note that the porfolio allocation ξ¯j over the time period [j − 1, j] is
predictable, i.e. it is decided at time j − 1, and it thus depends only on the
information Fj−1 known up to time j − 1, hence
IE∗ ξ¯j · (X̄j − X̄j−1 ) | Fj−1 = ξ¯j · IE∗ X̄j − X̄j−1 | Fj−1 .
Finally we note that
IE∗ X̄j − X̄j−1 | Fj−1 = IE∗ X̄j | Fj−1 − IE∗ X̄j−1 | Fj−1
= IE∗ X̄j | Fj−1 − X̄j−1
= 0,
j = 1, 2, . . . , N,
because (X̄t )t=0,1,...,N is a martingale under the risk-neutral measure P∗ , and
this concludes the proof of (3.7). Let
e=
C
C
(1 + r)N
denote the discounted payoff of the claim C. We will show that under any
risk-neutral measure P∗ the discounted value of any self-financing portfolio
hedging C is given by
e Ft ,
Vet = IE∗ C
t = 0, 1, . . . , N,
(3.8)
which shows that
Vt =
1
IE∗ [C | Ft ]
(1 + r)N −t
after multiplication of both sides by (1 + r)t . Next, we note that (3.8) follows
from the martingale transform result of Proposition 2.6.
Note that (3.6) admits an interpretation in an insurance framework, in which
πt (C) represents an insurance premium and C represents the random value
of an insurance claim made by a subscriber. In this context, the premium of
the insurance contract reads as the average of the values (3.6) of the random
claims after time discounting. In addition, the discounted price process
((1 + r)−t πt (C))t=0,1,...,N = Vet t=0,1,...,N
is a martingale under P∗ .
Remark 3.5. From Theorem 3.4, we can recover the fact that the selffinancing discounted portfolio process (Vet )t=0,1,...,N is a martingale under P∗
as in Remark 3.3, since
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N. Privault
e Ft
Vet = IE∗ C
h h i i
∗
e Ft+1 Ft
= IE IE∗ C
h
i
= IE∗ Vet+1 Ft ,
t = 0, 1, . . . , N − 1,
(3.9)
from the “tower property” (17.37) of conditional expectations, which also allows us to compute Vt by backward induction on t = 0, . . . , N − 1, starting
from VN = C.
In particular, at t = 0 we obtain the price of the contingent claim C at time
0:
1
e | F0 = IE∗ C
e =
IE∗ [C].
π0 (C) = IE∗ C
(1 + r)N
3.2 Pricing of Vanilla Options in the CRR Model
In this section we consider the pricing of contingent claims in the discretetime Cox-Ross-Rubinstein model, with d = 1. More precisely we are concerned with vanilla options whose payoffs depend on the terminal value of
the underlying asset, as opposed to exotic options whose payoffs may depend
on the whole path of the underlying asset price until expiration time.
Recall that the portfolio value process (Vt )t=0,1,...N and the discounted
portfolio value process respectively satisfy
Vt = ξ¯t · S̄t
and
Vet =
1
Vt = ξ¯t · X̄t ,
(1 + r)t
t = 0, 1, . . . , N.
Here we will be concerned with the pricing of vanilla options with payoffs of
the form
C = f (SN ),
e.g. f (x) = (x − K)+ in the case of a European call. Equivalently, the discounted claim
C
e=
C
(1 + r)N
e = fe(SN ) with fe(x) = f (x)/(1 + r)N . For example in the case of a
satisfies C
European call with strike price K we have
fe(x) =
1
+
(x − K) .
(1 + r)N
From Theorem 3.4, the discounted value of a portfolio hedging the attainable
e is given by
(discounted) claim C
h
i
Vet = IE∗ fe(SN ) | Ft ,
t = 0, 1, . . . , N,
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Pricing and Hedging in Discrete Time
under the risk-neutral measure P∗ . As a consequence, we have the following
proposition.
Proposition 3.6. The arbitrage price πt (C) at time t = 0, 1, . . . , N of the
contingent claim C = f (SN ) is given by
πt (C) =
1
IE∗ [f (SN ) | Ft ],
(1 + r)N −t
t = 0, 1, . . . , N.
(3.10)
In the next proposition we implement the calculation of (3.10).∗
Proposition 3.7. The price πt (C) of the contingent claim C = f (SN ) satisfies
πt (C) = v(t, St ),
t = 0, 1, . . . , N,
where the function v(t, x) is given by
 

N
Y
1
∗ 

v(t, x) =
(1
+
R
)
I
E
f
x
j
(1 + r)N −t
j=t+1
=
(3.11)
N
−t X
1
N −t
k
N −t−k
(p∗ )k (1 − p∗ )N −t−k f x (1 + b) (1 + a)
.
(1 + r)N −t
k
k=0
Proof. From the relations
SN = St
N
Y
(1 + Rj ),
j=t+1
and (3.10) we have, using Property (v) of the conditional expectation and
the independence of the returns {R1 , . . . , Rt } and {Rt+1 , . . . , RN },
1
IE∗ [f (SN ) | Ft ]
(1 + r)N −t
 


N
Y
1

∗ 

f
S
(1
+
R
)
=
I
E
Ft
t
j
(1 + r)N −t
j=t+1
 

N
Y
1
=
IE∗ f x
(1 + Rj )
.
N
−t
(1 + r)
j=t+1
πt (C) =
x=St
Next we note that the number of times Rj is equal to b for j ∈ {t + 1, . . . , N },
has a binomial distribution with parameter (N − t, p∗ ), where
p∗ =
∗
r−a
b−a
and
1 − p∗ =
b−r
,
b−a
(3.12)
Download the corresponding IPython notebook that can be run here.
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N. Privault
since the set of paths
from time t+1 to time N containing j times “(1+b)” has
cardinality Nj−t and each such path has the probability (p∗ )j (1 − p∗ )N −t−j ,
j = 0, . . . , N − t. Hence we have
1
IE∗ [f (SN ) | Ft ]
(1 + r)N −t
N
−t X
1
N −t
k
N −t−k
(p∗ )k (1 − p∗ )N −t−k f St (1 + b) (1 + a)
.
=
N
−t
(1 + r)
k
πt (C) =
k=0
In the above proof we have also shown that πt (C) is given by the conditional
expectation
πt (C) =
1
1
IE∗ [f (SN ) | Ft ] =
IE∗ [f (SN ) | St ]
(1 + r)N −t
(1 + r)N −t
given the value of St at time t = 0, 1, . . . , N , which is consistent with the
Markov property of (St )t=0,1,...,N . In particular, the price of the claim C is
written as the average (path integral) of the values of the contingent claim
over all possible paths starting from St .
The gearing at time t ∈ [0, T ] of the option with payoff C = h(ST ) is defined
as the ratio
St
St
Gt :=
=
,
t = 0, 1, . . . , N.
πt (C)
v(t, St )
The break-even price BEPt of the underlying is the value of S for which the
intrinsic option payoff h(S) equals the option price πt (C) at time t ∈ [0, T ].
For European call options it is given by
BEPt := K + πt (C) = K + v(t, St ),
t = 0, 1, . . . , N.
whereas for European put options it is given by
BEPt := K − πt (C) = K − v(t, St ),
t = 0, 1, . . . , N.
The option premium OPt can be defined as the variation required from the
underlying in order to reach the break-even price, i.e. we have
OPt :=
BEPt − St
K + v(t, St ) − St
,=
,
BEPt
K + v(t, St )
t = 0, 1, . . . , N,
for European call options, and
OPt :=
St − BEPt
St + v(t, St ) − K
=
,
BEPt
K − v(t, St )
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t = 0, 1, . . . , N,
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Pricing and Hedging in Discrete Time
for European put options. The term “premium” is sometimes also used to
denote the arbitrage price πt (C) of the option.
Pricing by Backward Induction
In the CRR model, the discounted portfolio price Vet can be computed by backward induction as in (3.9), using the martingale property of the discounted
price process (Vet )t=0,1,...,N under the risk-neutral measure P∗ . Namely, by
the “tower property” of conditional expectations, letting
ve(t, St ) :=
1
v(t, St ),
(1 + r)t
t = 0, 1, . . . , N,
we have
Vet = ve(t, St )
= IE∗ fe(SN ) Ft
= IE∗ IE∗ fe(SN ) Ft+1 | Ft
= IE∗ Vet+1 Ft
= IE∗ ve(t + 1, St+1 ) Ft
= ve (t + 1, (1 + a)St ) P∗ (Rt+1 = a) + ve (t + 1, (1 + b)St ) P∗ (Rt+1 = b)
= (1 − p∗ )e
v (t + 1, (1 + a)St ) + p∗ ve (t + 1, (1 + b)St ) ,
which shows that ve(t, x) satisfies the backward induction relation
ve(t, x) = (1 − p∗ )e
v (t + 1, x(1 + a)) + p∗ ve (t + 1, x(1 + b)) ,
(3.13)
while the terminal condition VeN = f˜(SN ) implies
ve(N, x) = fe(x),
x > 0.
3.3 Hedging of Contingent Claims
The basic idea of hedging is to allocate assets in a portfolio in order to protect oneself from a given risk. For example, a risk of increasing oil prices
can be hedged by buying oil-related stocks, whose value should be positively
correlated with the oil price. In this way, a loss connected to increasing oil
prices could be compensated by an increase in the value of the corresponding
portfolio.
In the setting of this chapter, hedging an attainable contingent claim C means
computing a self-financing portfolio strategy (ξ¯t )t=1,2,...,N such that
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N. Privault
ξ¯N · S̄N = C,
e
i.e. ξ¯N · X̄N = C.
(3.14)
Price, then hedge.
The portfolio allocation ξ¯N can be computed by first solving (3.14) for ξ¯N
from the payoff values C, based on the fact that the allocation ξ¯N depends
only on information up to time N − 1, by the predictability of (ξ¯k )16k6N .
If the self-financing portfolio value Vt is known, for example from (3.6), i.e.
Vt =
1
IE∗ [C | Ft ],
(1 + r)N −t
t = 0, 1, . . . , N,
(3.15)
we may similarly compute ξ¯t by solving ξ¯t · S̄t = Vt for all t = 1, 2, . . . , N − 1.
Hedge, then price.
If Vt is not known we can use backward induction to compute a self-financing
portfolio strategy. Starting from the values of ξ¯N obtained by solving
ξ¯N · S̄N = C,
we use the self-financing condition to solve for ξ¯N −1 , ξ¯N −2 , . . ., ξ¯4 , down to
ξ¯3 , ξ¯2 , and finally ξ¯1 .
In order to implement this algorithm we can use the N − 1 self-financing
equations
ξ¯t · X̄t = ξ¯t+1 · X̄t ,
t = 1, 2, . . . , N − 1,
(3.16)
allowing us in principle to compute the portfolio strategy (ξ¯t )t=1,2,...,N .
Based on the values of ξ¯N we can solve
ξ¯N −1 · S̄N −1 = ξ¯N · S̄N −1
for ξ¯N −1 , then
ξ¯N −2 · S̄N −2 = ξ¯N −1 · S̄N −2
for ξ¯N −2 , and successively ξ¯2 down to ξ¯1 . In Section 3.4 the backward induction (3.16) will be implemented in the CRR model, see the proof of Proposition 3.8, and Exercises 3.8 and 3.10 for an application in a two-step model.
The discounted value Vet at time t of the portfolio claim can then be obtained
from
Ve0 = ξ¯1 · X̄0
and
Vet = ξ¯t · X̄t ,
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t = 1, 2, . . . , N.
(3.17)
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Pricing and Hedging in Discrete Time
In addition we have shown in the proof of Theorem 3.4 that the price πt (C)
of the claim C at time t coincides with the value Vt of any self-financing
portfolio hedging the claim C, i.e.
πt (C) = Vt ,
t = 0, 1, . . . , N,
as given by (3.17). Hence the price of the claim can be computed either algebraically by solving (3.14) and (3.16) using backward induction and then
using (3.17), or by a probabilistic method by a direct evaluation of the discounted expected value (3.15).
3.4 Hedging of Vanilla Options in the CRR model
In this section we implement the backward induction (3.16) of Section 3.3 for
the hedging of contingent claims in the discrete-time Cox-Ross-Rubinstein
model. Our aim is to compute a self-financing portfolio strategy hedging a
vanilla option with payoff of the form
C = h(SN ).
(0)
Since the discounted price Xt
(0)
Xt
(0)
of the riskless asset satisfies
(0)
= (1 + r)−t St
(0)
= π0 ,
(0)
(1)
we will write π0 in place of Xt . We also let Xt be denoted as Xt , t =
0, 1, . . . , N . In Propositions 3.8∗ and 3.10 we present two different approaches
(1)
to hedging and to the computation of the predictable process (ξt )t=1,2,...,N ,
which is also called the Delta.
Proposition 3.8. Price, then hedge. The self-financing replicating portfolio
(0) (1)
(ξt , ξt )t=1,2,...,N hedging the contingent claim C = h(SN ) is given by
v (t, (1 + b)St−1 ) − v (t, (1 + a)St−1 )
St−1 (b − a)
ve (t, (1 + b)St−1 ) − ve (t, (1 + a)St−1 )
=
,
Xt−1 (b − a)/(1 + r)
(1)
ξt (St−1 ) =
(3.18)
and
(0)
ξt (St−1 ) =
=
(1 + b)v (t, (1 + a)St−1 ) − (1 + a)v (t, (1 + b)St−1 )
(0)
(b − a)πt
(1 + b)e
v (t, (1 + a)St−1 ) − (1 + a)e
v (t, (1 + b)St−1 )
(0)
(b − a)π0
, (3.19)
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here.
∗
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t = 1, 2, . . . , N , where the function ve(t, x) = (1+r)−t v(t, x) is given by (3.11).
Proof. We first compute the self-financing hedging strategy (ξ¯t )t=1,2,...,N by
solving
ξ¯t · X̄t = Vet ,
t = 1, 2, . . . , N,
from which we deduce the two equations

1+a
(0)
(0)
(1)


 ξt (St−1 )π0 + ξt (St−1 ) 1 + r Xt−1 = ve (t, (1 + a)St−1 )



1+b

 ξt(0) (St−1 )π0(0) + ξt(1) (St−1 )
Xt−1 = ve (t, (1 + b)St−1 ) ,
1+r
which can be solved as

(1 + b)e
v (t, (1 + a)St−1 ) − (1 + a)e
v (t, (1 + b)St−1 )
(0)


ξt (St−1 ) =

(0)


(b − a)π0


ve (t, (1 + b)St−1 ) − ve (t, (1 + a)St−1 )


 ξt(1) (St−1 ) =
,
(b − a)Xt−1 /(1 + r)
t = 1, 2, . . . , N , which only depends on St−1 , as expected. This is consistent
(1)
with the fact that ξt represents the (possibly fractional) quantity of the
risky asset to be present in the portfolio over the time period [t − 1, t] in
order to hedge the claim C at time N , and is decided at time t − 1.
The effective gearing at time t ∈ [0, T ] of the option with payoff C = h(ST )
is defined as the ratio
(1)
EGt := Gt ξt
St (1)
=
ξ
πt (C) t
=
St v (t, (1 + b)St−1 ) − v (t, (1 + a)St−1 )
,
St−1 v(t, St )(b − a)
t = 0, 1, . . . , N.
Note that if the function x 7−→ h(x) is nondecreasing then the function
x 7−→ ve(t, x) is also nondecreasing for all fixed t = 0, 1, . . . , N , hence the
(0) (1)
(1)
portfolio (ξt , ξt )t=1,2,...,N defined by (3.11) or (3.18) satisfies ξt > 0,
t = 1, 2, . . . , N and there is not short selling. This applies in particular to
(1)
European call options. Similarly we can show that ξt 6 0, t = 1, 2, . . . , N ,
i.e. short selling always occurs, when x 7−→ h(x) is a nonincreasing function,
as in the case of European put options.
Remark 3.9. We can check that the portfolio strategy (ξ¯t )t=1,2,...,N is selffinancing, as
(0)
(0)
(1)
(1)
ξ¯t+1 · X̄t = ξt+1 (St )π0 + ξt+1 (St )Xt
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(0) (1
= π0
+ b)e
v (t + 1, (1 + a)St ) − (1 + a)e
v (t + 1, (1 + b)St )
(0)
(b − a)π0
ve (t + 1, (1 + b)St ) − ve (t + 1, (1 + a)St )
+Xt
(b − a)Xt /(1 + r)
(1 + b)e
v (t + 1, (1 + a)St ) − (1 + a)e
v (t + 1, (1 + b)St )
=
(b − a)
ve (t + 1, (1 + b)St ) − ve (t + 1, (1 + a)St )
+
(b − a)/(1 + r)
r−a
b−a
=
ve (t + 1, (1 + b)St ) +
ve (t + 1, (1 + a)St )
b−a
b−a
∗
∗
= p ve (t + 1, (1 + b)St ) + q ve (t + 1, (1 + a)St )
= ve(t, St )
(0)
(0)
(1)
(1)
= ξt (St )π0 + ξt (St )Xt
= ξ¯t · X̄t ,
t = 0, 1, . . . , N − 1,
where we used (3.13) or the martingale property of the discounted price process (ṽ(t, St ))t=0,1,...,N , cf. Lemma 3.2.
(0) (0)
Aas a consequence of (3.19), the respective discounted amounts ξt π0
(1)
ξt Xt invested on the riskless and riskly assets are given by
(0)
ξt (St−1 ) =
(1 + b)e
v (t, (1 + a)St−1 ) − (1 + a)e
v (t, (1 + b)St−1 )
(b − a)
and
(1)
ξt (St−1 ) = Xt
(3.20)
ve (t, (1 + b)St−1 ) − ve (t, (1 + a)St−1 )
,
Xt−1 (b − a)/(1 + r)
t = 1, 2, . . . , N .
(0)
Regarding the quantity ξt
from the relation
of the riskless asset in the portfolio at time t,
(0) (0)
(1) (1)
Vet = ξ¯t · X̄t = ξt Xt + ξt Xt ,
t = 1, 2, . . . , N,
we also obtain
(0)
ξt
=
=
(1) (1)
Vet − ξt Xt
(0)
Xt
(1) (1)
Vet − ξt Xt
(0)
π0
(1)
=
"
(1)
ve(t, St ) − ξt Xt
(0)
π0
,
t = 1, 2, . . . , N.
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In the next proposition we compute the hedging strategy by backward induction, starting from
(1)
ξN (SN −1 ) =
h((1 + b)SN −1 ) − h((1 + a)SN −1 )
,
SN −1 (b − a)
and
(0)
ξN (SN −1 ) =
(1 + b)h((1 + a)SN −1 ) − (1 + a)h((1 + b)SN −1 )
(0)
(b − a)π0 (1 + r)N
,
that follow from (3.18) and (3.19) applied to the payoff function h(·).
Proposition 3.10. Hedge, then price. The self-financing replicating portfolio
(0) (1)
(ξt , ξt )t=1,2,...,N −1 hedging the contingent claim C = h(SN ) is given at
time N by
(1)
ξN (SN −1 ) =
h ((1 + b)SN −1 ) − kh ((1 + a)SN −1 )
,
SN −1 (b − a)
(3.21)
and
(0)
ξN (SN −1 ) =
(1 + b)h ((1 + a)SN −1 ) − (1 + a)h ((1 + b)SN −1 )
(0)
(b − a)πN
,
(3.22)
and then inductively by
(1)
(1)
ξt (St−1 ) =
(1)
(1 + b)ξt+1 ((1 + b)St−1 ) − (1 + a)ξt+1 ((1 + a)St−1 )
b−a
(0)
(0) ξt+1 ((1
+π0
(0)
+ b)St−1 ) − ξt+1 ((1 + a)St−1 )
,
(b − a)Xt−1 /(1 + r)
and
(1)
(0)
ξt (St−1 ) =
(1)
(1 + a)(1 + b)Xt−1 (ξt+1 ((1 + a)St−1 ) − ξt+1 ((1 + b)St−1 ))
(0)
(b − a)(1 + r)π0
+
(1 +
(0)
b)ξt+1 ((1
(0)
+ a)St−1 ) − (1 + a)ξt+1 ((1 + b)St−1 )
, (3.23)
b−a
t = 1, 2, . . . , N , and the pricing function ve(t, x) = (1 + r)−t v(t, x) is given by
(0) (0)
(1)
ve(t, St ) = π0 ξt (St−1 ) + Xt ξt (St−1 ),
t = 1, 2, . . . , N.
Proof. Relations (3.21)-(3.22) follow from (3.18)-(3.19) at time t = N . Next,
by the self-financing condition (3.16) we have
ξ¯t · X̄t = ξ¯t+1 · X̄t
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Pricing and Hedging in Discrete Time
i.e.

1+b
1+b
(0)
(0)
(1)
(0) (0)
(1)



 ξt π0 + ξt Xt−1 1 + r = ξt+1 ((1 + b)St−1 )π0 + ξt+1 ((1 + b)St−1 )Xt−1 1 + r


1+a
1+a
(0)
(0)
(1)

 ξt(0) π0(0) + ξt(1) Xt−1
= ξt+1 ((1 + a)St−1 )π0 + ξt+1 ((1 + a)St−1 )Xt−1
,
1+r
1+r
which can be solved as
(1)
(1)
(1 + b)ξt+1 ((1 + b)St−1 ) − (1 + a)ξt+1 ((1 + a)St−1 )
b−a
(1)
ξt (St−1 ) =
(0)
(0) ξt+1 ((1
+(1 + r)π0
(0)
+ b)St−1 ) − ξt+1 ((1 + a)St−1 )
,
(b − a)Xt−1
and
(1)
(1)
(1 + a)(1 + b)Xt−1 (ξt+1 ((1 + a)St−1 ) − ξt+1 ((1 + b)St−1 ))
(0)
ξt (St−1 ) =
(0)
+
(1 + b)ξt+1 ((1 +
(0)
(b − a)(1 + r)π0
(0)
a)St−1 ) − (1 + a)ξt+1 ((1
+ b)St−1 )
b−a
,
t = 1, 2, . . . , N − 1.
Remark 3.11. We can check that the corresponding discounted price process
(Ṽt )t=1,2,...,N = (ξ¯t · X̄t )t=1,2,...,N
is a martingale:
Ṽt = ξ¯t · X̄t
(0) (0)
(1)
= π0 ξt (St−1 ) + Xt ξt (St−1 )
(1)
=
(1)
(1 + a)(1 + b)Xt−1 (ξt+1 ((1 + a)St−1 ) − ξt+1 ((1 + b)St−1 ))
(b − a)(1 + r)
(0) (1
+π0
(0)
(0)
+ b)ξt+1 ((1 + a)St−1 ) − (1 + a)ξt+1 ((1 + b)St−1 )
(b − a)
(1)
+Xt
(1)
(1 + b)ξt+1 ((1 + b)St−1 ) − (1 + a)ξt+1 ((1 + a)St−1 )
b−a
(0)
(0) ξt+1 ((1
(0)
+ b)St−1 ) − ξt+1 ((1 + a)St−1 )
Xt−1 (b − a)
r − a (0) (0)
b − r (0) (0)
=
π ξ (St ) +
π ξ (St )
b − a 0 t+1
b − a 0 t+1
+(1 + r)Xt π0
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N. Privault
+
(r − a)(1 + b)
(b − r)(1 + a)
(1)
(1)
Xt ξt+1 (St ) +
Xt ξt+1 (St )
(b − a)(1 + r)
(b − a)(1 + r)
(0) (0)
(0) (0)
= p∗ π0 ξt+1 (St ) + q ∗ π0 ξt+1 (St )
1+a
1+b
(1)
(1)
Xt ξt+1 (St ) + q ∗
Xt ξt+1 (St )
+p∗
1+r
1+r
(0) (0)
(1)
= IE∗ π0 ξt+1 (St ) + Xt+1 ξt+1 (St ) | Ft
= IE∗ Ṽt+1 | Ft ,
t = 1, 2, . . . , N − 1.
3.5 Hedging of Exotic Options in the CRR Model
In this section we take p = p∗ given by (3.12) and we consider the hedging
of path-dependent options. Here we choose to use the finite difference gradient and the discrete Clark-Ocone formula of stochastic analysis, see also
[FS04], [LL96], [Pri08], Chapter 1 of [Pri09], [RdC01], or §15-1 of [Wil91].
See [NØP09] and Section 8.2 of [Pri09] for a similar approach in continuous
time. Given
ω = (ω1 , ω2 , . . . , ωN ) ∈ Ω = {−1, 1}N ,
and r = 1, 2, . . . , N , let
t
ω+
:= (ω1 , ω2 , . . . , ωt−1 , +1, ωt+1 , . . . , ωN )
and
t
ω−
:= (ω1 , ω2 , . . . , ωt−1 , −1, ωt+1 , . . . , ωN ).
We also assume that the return Rt (ω) is constructed as
t
Rt (ω+
)=b
and
t
Rt (ω−
) = a,
t = 1, 2, . . . , N,
ω ∈ Ω.
Definition 3.12. The operator Dt is defined on any random variable F by
t
t
Dt F (ω) = F (ω+
) − F (ω−
),
t = 1, 2, . . . , N.
(3.24)
Let the centered and normalized return Yt be defined by

b−r


= q,
ωt = +1,

Rt − r  b − a
=
t = 1, 2, . . . , N.
Yt :=

b−a

a−r


= −p, ωt = −1,
b−a
Note that under the risk-neutral measure P∗ we have
Rt − r
IE∗ [Yt ] = IE∗
b−a
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Pricing and Hedging in Discrete Time
a−r ∗
b−r ∗
P (Rt = a) +
P (Rt = b)
b−a
b−a
a−r
b−r
b−r
r−a
=
×
+
×
b−a b−a b−a b−a
= 0,
=
and
Var [Yt ] = pq 2 + qp2 = pq,
t = 1, 2, . . . , N.
In addition the discounted asset price increment reads
Xt − Xt−1 = Xt−1
1 + Rt
− Xt−1
1+r
1
Xt−1 (Rt − r)
1+r
b−a
=
Yt Xt−1 ,
t = 1, 2, . . . , N.
1+r
=
We also have
Dt Yt =
b−r
r−a
+
= 1,
b−a b−a
t = 1, 2, . . . , N,
and
Dk SN = S0 (1 + b)
N
Y
(1 + Rt ) − S0 (1 + a)
N
Y
(1 + Rt )
t=1
t6=k
t=1
t6=k
= S0 (b − a)
N
Y
(1 + Rt )
t=1
t6=k
= S0
=
N
b−a Y
(1 + Rt )
1 + Rk t=1
b−a
SN ,
1 + Rk
k = 1, 2, . . . , N.
The following stochastic integral decomposition formula for the functionals
of the binomial process is known as the Clark-Ocone formula in discrete time,
cf. e.g. [Pri09], Proposition 1.7.1.
Proposition 3.13. For any square-integrable random variables F on Ω we
have
∞
X
F = IE∗ [F ] +
IE∗ [Dk F |Fk−1 ]Yk .
(3.25)
k=1
The Clark-Ocone formula has the following consequence.
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N. Privault
Corollary 3.14. Assume that (Mk )k∈N is a square-integrable Ft -martingale.
Then we have
MN = IE∗ [MN ] +
N
X
Yk Dk Mk ,
N > 0.
k=1
Proof. We have
MN = IE∗ [MN ] +
= IE∗ [MN ] +
= IE∗ [MN ] +
∞
X
k=1
∞
X
k=1
∞
X
IE∗ [Dk MN |Fk−1 ]Yk
Dk IE∗ [MN |Fk ]Yk
Yk Dk Mk
k=1
= IE∗ [MN ] +
N
X
Yk Dk Mk .
k=1
In addition to the Clark-Ocone formula we also state a discrete-time analog of
Itô’s change of variable formula, which can be useful for option hedging. The
next result extends Proposition 1.13.1 of [Pri09] by removing the unnecessary
martingale requirement on (Mt )n∈N .
Proposition 3.15. Let (Zn )n∈N be an Fn -adapted process and let f : R ×
N −→ R be a given function. We have
f (Zt , t) = f (Z0 , 0) +
t
X
Dk f (Zk , k)Yk
k=1
+
t
X
(IE∗ [f (Zk , k)|Fk−1 ] − f (Zk−1 , k − 1)) .
(3.26)
k=1
Proof. First, we note that the process
t 7−→ f (Zt , t) −
t
X
(IE∗ [f (Zk , k)|Fk−1 ] − f (Zk−1 , k − 1))
k=1
is a martingale under P∗ . Indeed we have
"
#
t
X
∗
∗
IE f (Zt , t) −
(IE [f (Zk , k)|Fk−1 ] − f (Zk−1 , k − 1)) Ft−1
k=1
= IE∗ [f (Zt , t)|Ft−1 ]
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Pricing and Hedging in Discrete Time
−
t
X
(IE∗ [IE∗ [f (Zk , k)|Fk−1 ]|Ft−1 ] − IE∗ [IE∗ [f (Zk−1 , k − 1)|Fk−1 ]|Ft−1 ])
k=1
= IE∗ [f (Zt , t)|Ft−1 ] −
t
X
(IE∗ [f (Zk , k)|Fk−1 ] − f (Zk−1 , k − 1))
k=1
= f (Zt−1 , t − 1) −
t−1
X
(IE∗ [f (Zk , k)|Fk−1 ] − f (Zk−1 , k − 1)) ,
t > 1.
k=1
Note that if (Zt )t∈N is a martingale in L2 (Ω) with respect to (Ft )t∈N and
written as
t
X
uk Yk ,
t ∈ N,
Zt = Z0 +
k=1
where (ut )t∈N is a predictable process locally in L2 (Ω×N), (i.e. u(·)1[0,N ] (·) ∈
L2 (Ω × N) for all N > 0), then we have
Dt f (Zt , t) = f (Zt−1 + qut , t) − f (Zt−1 − put , t) ,
(3.27)
t = 1, 2, . . . , N . On the other hand, the term
IE[f (Zt , t) − f (Zt−1 , t − 1)|Ft−1 ]
is analog to the finite variation part in the continuous-time Itô formula, and
can be written as
pf (Zt−1 + qut , t) + qf (Zt−1 − put , t) − f (Zt−1 , t − 1) .
Naturally, if (f (Zt , t))t∈N is a martingale we recover the decomposition
f (Zt , t) = f (Z0 , 0)
t
X
+
(f (Zk−1 + quk , k) − f (Zk−1 − puk , k)) Yk
k=1
= f (Z0 , 0) +
t
X
Yk Dk f (Zk , k).
(3.28)
k=1
This identity follows from Corollary 3.14 as well as from Proposition 3.13. In
this case the Clark-Ocone formula (3.25) and the change of variable formula
(3.28) both coincide and we have in particular
Dk f (Zk , k) = IE[Dk f (ZN , N )|Fk−1 ],
k = 1, 2, . . . , N . For example this recovers the martingale representation
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N. Privault
Xt = S0 +
t
X
Yk Dk Xk
k=1
= S0 +
t
b−a X
Xk−1 Yk
1+r
k=1
= S0 +
= S0 +
t
X
k=1
t
X
Xk−1
Rk − r
1+r
(Xk − Xk−1 ),
k=1
of the discounted asset price.
Our goal is to hedge an arbitrary claim C on Ω, i.e. given an FN measurable random variable C we search for a portfolio (ξt , ηt )t=1,2,...,N such
that the equality
C = VN = ηN AN + ξN SN
(3.29)
holds, where AN = A0 (1 + r)N denotes the value of the riskless asset at time
N ∈ N.
The next proposition is the main result of this section, and provides a
solution to the hedging problem under the constraint (3.29).
Proposition 3.16. Given a contingent claim C, let
ξt =
(1 + r)−(N −t) ∗
IE [Dt C|Ft−1 ],
St−1 (b − a)
and
ηt =
t = 1, 2, . . . , N,
1 (1 + r)−(N −t) IE∗ [C|Ft ] − ξt St ,
At
(3.30)
(3.31)
t = 1, 2, . . . , N . Then the portfolio (ξt , ηt )t=1,2,...,N is self financing and satisfies
Vt = ηt At + ξt St = (1 + r)−(N −t) IE∗ [C|Ft ],
t = 1, 2, . . . , N,
in particular we have VN = C, hence (ξt , ηt )t=1,2,...,N is a hedging strategy
leading to C.
Proof. Let (ξt )t=1,2,...,N be defined by (3.30), and consider the process
(ηt )t=0,1,...,N defined by
η0 = (1 + r)−N
IE∗ [C]
S0
and ηt+1 = ηt −
(ξt+1 − ξt )St
,
At
t = 0, 1, . . . , N − 1. Then (ξt , ηt )t=1,2,...,N satisfies the self-financing condition
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Pricing and Hedging in Discrete Time
At (ηt+1 − ηt ) + St (ξt+1 − ξt ) = 0,
t = 1, 2, . . . , N − 1.
Let now
V0 :=
1
IE∗ [C],
(1 + r)N
and
Vet =
Vt := ηt At + ξt St ,
Vt
(1 + r)t
t = 1, 2, . . . , N,
t = 0, 1, . . . , N.
Since (ξt , ηt )t=1,2,...,N is self-financing, by Lemma 3.2 we have
Vet = Ve0 + (b − a)
t
X
k=1
1
Yk ξk Sk−1 ,
(1 + r)k
(3.32)
t = 1, 2, . . . , N . On the other hand, from the Clark-Ocone formula (3.25) and
the definition of (ξt )t=1,2,...,N we have
1
IE∗ [C|Ft ]
(1 + r)N
=
"
#
N
X
1
∗
∗
∗
I
E
I
E
[C]
+
Y
I
E
[D
C|F
]
F
k
k
k−1
t
(1 + r)N
k=0
t
X
1
1
IE∗ [C] +
IE∗ [Dk C|Fk−1 ]Yk
=
(1 + r)N
(1 + r)N
k=0
=
1
IE∗ [C] + (b − a)
(1 + r)N
t
X
k=0
1
ξk Sk−1 Yk
(1 + r)k
= Vet
from (3.32). Hence
Vet =
and
1
IE∗ [C|Ft ],
(1 + r)N
t = 0, 1, . . . , N,
Vt = (1 + r)−(N −t) IE∗ [C|Ft ],
t = 0, 1, . . . , N.
(3.33)
In particular, (3.33) shows that we have VN = C. To conclude the proof we
note that from the relation Vt = ηt At + ξt St , t = 1, 2, . . . , N , the process
(ηt )t=1,2,...,N coincides with (ηt )t=1,2,...,N defined by (3.31).
From Proposition 3.7, when C = f (SN ), the price πt (C) of the contingent
claim C = f (SN ) is given by
πt (C) = v(t, St ),
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N. Privault
where the function v(t, x) is given by
 

N
Y
1
1
∗
∗ 
v(t, St ) =
IE [C|Ft ] =
IE f x
(1 + Rj )
(1 + r)N −t
(1 + r)N −t
j=t+1
x=St
Note that in this case we have C = v(N, SN ), IE[C] = v(0, M0 ), and the
e = C/(1 + r)N = ve(N, SN ) satisfies
discounted claim payoff C
N
X
e = IE C
e +
Yt IE Dt ve(N, SN )|Ft−1
C
t=1
N
X
e +
= IE C
Yt Dt ve(t, St )
t=1
N
X
e +
= IE C
t=1
1
Yt Dt v(t, St )
(1 + r)t
N
X
e +
= IE C
Yt Dt IE [e
v (N, SN )|Ft ]
t=1
e +
= IE C
N
X
1
Yt Dt IE[C|Ft ],
(1 + r)N t=1
hence we have
IE[Dt v(N, SN )|Ft−1 ] = (1 + r)N −t Dt v(t, St ),
t = 1, 2, . . . , N,
and by Proposition 3.16 the hedging strategy for C = f (SN ) is given by
(1 + r)−(N −t)
IE[Dt v(N, SN )|Ft−1 ]
St−1 (b − a)
1
=
Dt v(t, St )
St−1 (b − a)
1
=
(v (t, St−1 (1 + b)) − v (t, St−1 (1 + a)))
St−1 (b − a)
1
=
(e
v (t, St−1 (1 + b)) − ve (t, St−1 (1 + a))) ,
Xt−1 (b − a)/(1 + r)
ξt =
t = 1, 2, . . . , N , which recovers Proposition 3.8 as a particular case. Note that
ξt is nonnegative (i.e. there is no short selling) when f is a non decreasing
function, because a < b. This is in particular true in the case of a European
call option for which we have f (x) = (x − K)+ .
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Pricing and Hedging in Discrete Time
3.6 Convergence of the CRR Model
In this section we consider the convergence of the discrete-time model to the
continuous-time Black Scholes model.
Continuous compounding - riskless asset
Consider the subdivision
T 2T
(N − 1)T
0, ,
,...,
,T
N N
N
of the time interval [0, T ] into N time steps. Note that
lim (1 + r)N = ∞,
N →∞
when r > 0, thus we need to renormalize r so that the interest rate on each
time interval becomes rN , with limN →∞ rN = 0. It turns out that the correct
renormalization is
T
rN := r ,
N
so that for T > 0,
lim (1 + rN )N = lim
N →∞
N →∞
1+r
T
N
N
= e rT .
(3.34)
Hence the price At of the riskless asset is given by
At = A0 e rt ,
t ∈ R+ ,
(3.35)
which solves the differential equation
dAt
= rAt ,
dt
A0 = 1,
also written as
dAt = rAt dt,
or
t ∈ R+ ,
dAt
= rdt.
At
(3.36)
(3.37)
Rewriting the last equation as
At+dt − At
= rdt,
At
with dAt = At+dt − At , we find the return (At+dt − At )/At of the riskless
asset equals rdt on the small time interval [t, t + dt]. Equivalently, one says
that r is the instantaneous interest rate per unit of time.
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The same equation rewrites in integral form as
AT − A0 =
wT
0
dAt = r
wT
0
At dt.
Continuous compounding - risky asset
The Galton board simulation of Figure 3.1 shows the convergence of the
binomial random walk to a Gaussian distribution in large time.
Fig. 3.1: Galton board simulation.∗
Figure 3.2 pictures a real-life Galton board.
Fig. 3.2: A real-life Galton board at Jurong Point # 03-01.
∗
The animation works in Acrobat Reader on the entire pdf file.
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In the CRR model we need to replace the standard Galton board by its
multiplicative version which converges to a lognormal distribution of the form
f (x) =
2
2
1
√
e −(−µ+log x) /(2σ T ) ,
xσ 2πT
x > 0,
as shown in Figure 3.3.
Fig. 3.3: Multiplicative Galton board simulation.∗
We need to apply a similar renormalization to the coefficients a and b of the
CRR model. Let σ > 0 denote a positive parameter called the volatility and
let aN , bN be defined from
√
1 + aN
= e −σ T /N
1 + rN
and
√
1 + bN
= e σ T /N ,
1 + rN
and
bN = (1 + rN ) e σ
i.e.
aN = (1 + rN ) e −σ
√
T /N
−1
√
T /N
− 1.
(N )
Rk
Consider the random return
∈ {aN , bN } and the price process defined
as
t
Y
(N )
(N )
St = S0
(1 + Rk ),
t = 1, 2, . . . , N.
k=1
Note that the risk-neutral probabilities are given by
√
bN − rN
e σ T /N − 1
∗
p
P (Rt = aN ) =
=
,
bN − aN
2 sinh σ 2 T /N
t = 1, 2, . . . , N,
and
∗
The animation works in Acrobat Reader on the entire pdf file.
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P∗ (Rt = bN ) =
√
rN − aN
1 − e −σ T /N
p
=
,
bN − aN
2 sinh σ 2 T /N
t = 1, 2, . . . , N,
which both converge to 1/2 as N goes to infinity.
Continuous-time limit
We have the following convergence result.
Proposition 3.17. Let f be a continuous and bounded function on R. The
(N ) price at time t = 0 of a contingent claim with payoff C = f SN
converges
as follows:
h
i
h
i
2
1
(N ) lim
IE∗ f SN
= e −rT IE f (S0 e σX+rT −σ T /2 ) (3.38)
N
N →∞ (1 + rT /N )
where X ' N (0, T ) is a centered Gaussian random variable with variance
T > 0.
Proof. This result is consequence of the weak convergence in distribution
(N )
of the sequence (SN )N >1 to a lognormal distribution, cf. Theorem 5.53 of
[FS04]. The convergence of the discount factor follows directly from (3.34).
Note that the expectation (3.38) can be written as a Gaussian integral:
2
h
w∞
√
i
e −x /2
2
2
dx,
e −rT IE f S0 e σX+rT −σ T /2 = e −rT
f S0 e σ T x+rT −σ T /2 √
−∞
2π
cf. also Lemma 6.5 in Chapter 6, hence we have
lim
N →∞
2
h
i
w∞
√
e −x /2
2
1
(N ) IE∗ f SN
= e −rT
f S0 e σ T x+rT −σ T /2 √
dx.
−∞
(1 + rT /N )N
2π
It is a remarkable fact that in case f (x) = (x−K)+ , i.e. when C = (ST −K)+
is the payoff of a European call option with strike price K, the above integral
can be computed according to the Black-Scholes formula:
h
i
2
e −rT IE (S0 e σX+rT −σ T /2 − K)+ = S0 Φ(d+ ) − K e −rT Φ(d− ),
where
d− =
and
(r − σ 2 /2)T + log(S0 /K)
√
,
σ T
2
1 wx
Φ(x) := √
e −y /2 dy,
2π −∞
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√
d+ = d− + σ T ,
x ∈ R,
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is the Gaussian cumulative distribution function.
The Black-Scholes formula will be derived explicitly in the subsequent
chapters using both the PDE and probabilistic method, cf. Propositions 5.16
and 6.4. It can be considered as a building block for the pricing of financial
derivatives, and its importance is not restricted to the pricing of options on
stocks. Indeed, the complexity of the interest rate models makes it in general
difficult to obtain closed form expressions, and in many situations one has
to rely on the Black-Scholes framework in order to find pricing formulas, for
example in the case of interest rate derivatives as in the Black caplet formula
of the BGM model, cf. Proposition 14.4 in Section 14.3.
Our aim later on will be to price and hedge options directly in continuoustime using stochastic calculus, instead of applying the limit procedure described in the previous section. In addition to the construction of the riskless
asset price (At )t∈R+ via (3.35) and (3.36) we now need to construct a mathematical model for the price of the risly asset in continuous time.
The return of the risky asset St over the time period [t, d + dt] will be
defined as
dSt
= µdt + σdBt ,
St
where in comparison with (3.37), we add a “small” Gaussian random perturbation σdBt which accounts for market volatility. Here, the Brownian
increment dBt is multiplied by the volatility parameter σ > 0. In the next
Chapter 4 we will turn to the formal definition of the stochastic process
(Bt )t∈R+ which will be used for the modeling of risky assets in continuous
time.
Exercises
Exercise 3.1 (Exercise 2.3 continued) Consider a two-period trinomial market model (St )t=0,1,2 with r = 0 and three return rates Rt = −1, 0, 1. Taking
S0 = 1, price the European put option with strike price K = 1 and maturity
N = 2 at times t = 0 and t = 1.
Exercise 3.2 Consider a two-period binomial market model (St )t=0,1,2 with
two return rates a = 0, b = 1 and S0 = 1, and with the riskless account
At = (1 + r)t where r = 0.5. Price and hedge the vanilla option whose payoff
C at time 2 is given by
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
3 if S2 = 4,





C = 1 if S2 = 2,





3 if S2 = 1.
Exercise 3.3 In a two-period trinomial market model (St )t=0,1,2 with interest
rate r = 0 and three return rates Rt = −0.5, 0, 1, we consider a down-an-out
barrier call option with exercise date N = 2, strike price K and barrier B,
whose payoff C is given by

+

min St > B,

 SN − K if t=1,2,...,N
+
o=
C = SN − K 1n

min St > B

0
if min St 6 B.
t=1,2,...,N
t=1,2,...,N
a) Show that P∗ given by r∗ = P∗ (Rt = −0.5) := 1/2, q ∗ = P∗ (Rt = 0) :=
1/4, p∗ = P∗ (Rt = 1) := 1/4 is risk-neutral.
b) Taking S0 = 1, compute the possible values of the option payoff C with
strike price K = 1.5 and barrier B = 1, at maturity N = 2.
c) Price the down-an-out barrier call option with exercise date N = 2, strike
price K = 1.5 and barrier B = 1, at time t = 0 and t = 1.
Hint: Use the formula
Vt =
1
IE∗ [C | St ],
(1 + r)N −t
t = 0, 1, . . . , N,
where N denotes maturity time and C is the option payoff.
d) Is this market complete? Is every contingent claim attainable?
Exercise 3.4
Analysis of a binary option trading website.
a) Using Proposition 3.6, compute the price at time t = 0 of the binary call
option with payoff

 $1 if SN > K,
C = 1[K,∞) SN =

0 if SN < K,
b) Compute the potential net returns obtained by purchasing one binary call
option.
c) Compute the corresponding expected return.
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d) A website proposes to pay a return of 86% in case the binary options
matures “in the money”, i.e. when SN > K. Compute the corresponding
expected return. What do you conclude?
Exercise 3.5 Consider an asset price (Sn )n=0,1,...,N which is a martingale under the risk-neutral measure P∗ , with respect to the filtration (Fn )n=0,1,...,N .
Given the (convex) function φ(x) := (x − K)+ , show that the price of an
Asian option with payoff
S1 + · · · + SN
φ
N
and maturity N > 1 is always lower than the price of the corresponding
European call option, i.e. show that
S1 + · · · + SN
IE∗ φ
6 IE∗ [φ(SN )].
N
Hint: Use in this order:
(i) the convexity inequality φ(x1 /N + · · · + xN /N ) 6 φ(x1 )/N + · · · +
φ(xN )/N ,
(ii) the martingale property Sk = IE∗ [SN | Fk ], k = 1, 2, . . . , N .
(iii) the Jensen inequality
φ(IE∗ [SN | Fk ]) 6 IE∗ [φ(SN ) | Fk ],
k = 1, 2, . . . , N,
(iv) the tower property IE∗ [IE∗ [φ(SN ) | Fk ]] = IE∗ [φ(SN )] of conditional expectations, k = 1, 2, . . . , N .
Exercise 3.6
(Exercise 2.4 continued)
a) We consider a forward contract on SN with strike price K and payoff
C := SN − K.
Find a portfolio allocation (ηN , ξN ) with price VN = ηN πN + ξN SN at
time N , such that
VN = C,
(3.39)
by writing Condition (3.39) as a 2 × 2 system of equations.
b) Find a portfolio allocation (ηN −1 , ξN −1 ) with price VN −1 = ηN −1 πN −1 +
ξN −1 SN −1 at time N − 1, and verifying the self-financing condition
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VN −1 = ηN πN −1 + ξN SN −1 .
Next, at all times t = 1, 2, . . . , N − 1, find a portfolio allocation (ηt , ξt )
with price Vt = ηt πt +ξt St verifying (3.39) and the self-financing condition
Vt = ηt+1 πt + ξt+1 St ,
where ηt , resp. ξt , represents the quantity of the riskless, resp. risky, asset
in the portfolio over the time period [t − 1, t], t = 1, 2, . . . , N .
c) Compute the arbitrage price πt (C) = Vt of the forward contract C, at
time t = 0, 1, . . . , N .
d) Check that the arbitrage price πt (C) satisfies the relation
πt (C) =
1
IE∗ [C | Ft ],
(1 + r)N −t
t = 0, 1, . . . , N.
Exercise 3.7 Power option. Let (Sn )n∈N denote a binomial price process with
returns −50% and +50%, and let the riskless asset be priced as Ak = $1,
k ∈ N. We consider a power option with payoff C := (SN )2 , and a predictable
self-financing portfolio strategy (ξk , ηk )k=1,2,...,N with price
Vk = ξk Sk + ηk A0 ,
k = 1, 2, . . . , N.
a) Find the portfolio allocation (ξN , ηN ) that matches the payoff C = (SN )2
at time N , i.e. that satisfies
VN = (SN )2 .
Hint: We have ηN = −3(SN −1 )2 /4.
b) Compute the portfolio price under the risk-neutral measure p∗ = 1/2
VN −1 = IE∗ [C | FN −1 ].
c) Find the portfolio allocation (ηN −1 , ξN −1 ) at time N − 1 from the relation
VN −1 = ξN −1 SN −1 + ηN −1 A0 .
Hint: We have ηN −1 = −15(SN −2 )2 /16.
d) Check that the portfolio satisfies the self-financing condition
VN −1 = ξN −1 SN −1 + ηN −1 A0 = ξN SN −1 + ηN A0 .
Exercise 3.8 Consider the discrete-time Cox-Ross-Rubinstein model with
N + 1 time instants t = 0, 1, . . . , N . The price St0 of the riskless asset evolves
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as St0 = π 0 (1 + r)t , t = 0, 1, . . . , N . The return of the risky asset, defined as
Rt :=
St − St−1
,
St−1
t = 1, 2, . . . , N,
is random and allowed to take only two values a and b, with −1 < a < r < b.
The discounted asset price is given by Xt = St /(1 + r)t , t = 0, 1, . . . , N .
a) Show that this model admits a unique risk-neutral measure P∗ and explicitly compute P∗ (Rt = a) and P(Rt = b) for all t = 1, 2, . . . , N , with
a = 2%, b = 7%, r = 5%.
b) Does there exist arbitrage opportunities in this model? Explain why.
c) Is this market model complete? Explain why.
d) Consider a contingent claim with payoff∗
C = (SN )2 .
Compute the discounted arbitrage price Vet , t = 0, 1, . . . , N , of a selffinancing portfolio hedging the claim C, i.e. such that
(SN )2
.
VeN = C̃ =
(1 + r)N
e) Compute the portfolio strategy
(ξ¯t )t=1,2,...,N = (ξt0 , ξt1 )t=1,2,...,N
associated to Vet , i.e. such that
Vet = ξ¯t · X̄t = ξt0 Xt0 + ξt1 Xt1 ,
t = 1, 2, . . . , N.
f) Check that the above portfolio strategy is self-financing, i.e.
ξ¯t · S̄t = ξ¯t+1 · S̄t ,
t = 1, 2, . . . , N − 1.
Exercise 3.9 We consider the discrete-time Cox-Ross-Rubinstein model with
N + 1 time instants t = 0, 1, . . . , N .
The price πt of the riskless asset evolves as πt = π0 (1 + r)t , t = 0, 1, . . . , N .
The evolution of St−1 to St is given by

 (1 + b)St−1 if Rt = b,
St =

(1 + a)St−1 if Rt = a,
∗
This is the payoff of a power call option with strike price K = 0.
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with −1 < a < r < b. The return of the risky asset is defined as
Rt :=
St − St−1
,
St−1
t = 1, 2, . . . , N.
Let ξt , resp. ηt , denote the (possibly fractional) quantities of the risky, resp.
riskless, asset held over the time period [t − 1, t] in the portfolio with value
Vt = ξt St + ηt πt ,
t = 0, 1, . . . , N.
(3.40)
a) Show that
Vt = (1 + Rt )ξt St−1 + (1 + r)ηt πt−1 ,
t = 1, 2, . . . , N.
(3.41)
b) Show that under the probability P∗ defined by
P∗ (Rt = a | Ft−1 ) =
b−r
,
b−a
P∗ (Rt = b | Ft−1 ) =
r−a
,
b−a
where Ft−1 represents the information generated by {R1 , R2 , . . . , Rt−1 },
we have
IE∗ [Rt | Ft−1 ] = r.
c) Under the self-financing condition
Vt−1 = ξt St−1 + ηt πt−1
t = 1, 2, . . . , N,
(3.42)
recover the martingale property
Vt−1 =
1
IE∗ [Vt | Ft−1 ],
1+r
using the result of Question (a).
d) Let a = 5%, b = 25% and r = 15%. Assume that the price Vt at time t of
the portfolio is $3 if Rt = a and $8 if Rt = b, and compute the price Vt−1
of the portfolio at time t − 1.
Exercise 3.10 Consider a two-step binomial random asset model (Sk )k=0,1,2
with possible returns a = 0 and b = 200%, and a risk-free asset Ak = A0 (1 +
r)k , k = 0, 1, 2 with interest rate r = 100%, and S0 = A0 = 1, under the
risk-neutral measure p∗ = (r − a)/(b − a) = 1/2.
a) Draw a binomial tree for the possible values of (Sk )k=0,1,2 and compute
the values Vk of the hedging portfolio at times k = 0, 1, 2 of a European
call option on ST with strike price K = 8 and maturity T = 2.
Hint: Consider three cases when k = 2, and two cases when k = 1.
b) Compute the self-financing hedging portfolio (ξk , ηk )k=1,2 with price
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Vk = ξk Sk + ηk Ak = ξk+1 Sk + ηk+1 Ak ,
at k = 1, hedging the European call with strike price K = 8 and maturity
T = 2.
Hint: Consider two separate cases for k = 2 and one case for k = 1.
Exercise 3.11 Consider a two-step binomial random asset model (Sk )k=0,1,2
with possible returns a = −50% and b = 150%, and a risk-free asset Ak =
A0 (1 + r)k , k = 0, 1, 2 with interest rate r = 100%, and S0 = A0 = 1, under
the risk-neutral measure p∗ = (r − a)/(b − a) = 3/4.
a) Draw a binomial tree for the values of (Sk )k=0,1,2 .
b) Compute the values Vk at times k = 0, 1, 2 of the hedging portfolio of a
European put option with strike price K = 5/4 and maturity T = 2 on ST .
c) Compute the self-financing hedging portfolio (ξk , ηk )k=1,2 with price
Vk = ξk Sk + ηk Ak = ξk+1 Sk + ηk+1 Ak ,
at k = 1, hedging the European put option with strike price K = 5/4 and
maturity T = 2.
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