Lecture 3
Valuation of Contingent Claims and Replicating Portfolios
Definition 1. A contingent claim is a random variable X representing the payoff of a
contract at time t  1 .
The question we are addressing is: What is a fair price the buyer of the contract must pay
at time t  0 to get the payoff X at time t  1 ?
Definition 2. A contingent claim X is attainable or marketable if there is a trading
strategy H such that the value of the portfolio at time t  1 under that strategy is X.
N
Recall that Vt  H 0 Bt   H n Sn (t ), t  0,1 . If such a portfolio exists, it is called a
n 1
replicating portfolio.
Theorem 1. Assume that X is an attainable contingent claim, and the market model is
free of arbitrage. Then the price of the contingent claim at time t  0, denoted X 0 , is
exactly V0 .
Proof.
The idea of the proof is to show that in any of the cases in which X 0  V0 , arbitrage
opportunities are created.
Let us assume X 0  V0 . Then, an investor could sell the contingent claim at t  0, for X 0 ,
then follow the strategy that replicates the claim, investing V0 . At t  1 the replicating
portfolio will have the same value as the contingent claim in all states of the system.
Therefore the investor has made a riskless profit X 0  V0 at t  0 .
If X 0  V0 , thus  X 0  V0 . An investor could purchase the contingent claim, follow the
strategy  H , thus collecting V0 , and making a profit V0  X 0 .
Therefore the price of the contingent claim at time t  0 must be V0 .
QED
Observation 1. One could argue that the replicating portfolio may not be unique, and if
H and H are replicating portfolios such that V1 =V1  X , could it be that V0  V0 ? The
answer to this question is negative because of the lack of arbitrage assumption, and
therefore there is a unique cost for the contingent claim.
The next Theorem gives a computational formula for the cost of an attainable contingent
claim without involving the replicating portfolio.
Theorem 2. Assume that X is an attainable contingent claim, and the market model is
free of arbitrage. Then, the cost of the contingent claim at time t  0 is
X
X ( )
X 0  EQ (
)
Q( ) ,
1  r  1  r
for any risk neutral probability measure Q.
Proof. Let H be a replicating portfolio for the claim. Then, by Theorem 1, the cost at time
N
t  0 is X 0  V0  H 0   H n S n (0) . If Q is a risk neutral probability measure, by
n 1
V
X
).
definition we have V0  EQ (V1* )  EQ ( 1 )  EQ (
1 r
1 r
QED
Example 1. Let d  2, N  1, r  1/10, S (0)  6, S (1, 1 )  11/ 2, S (1, 2 )  77 /10 , and
consider the claim X (1 )  3, X (2 )  8 . We want to check whether X is attainable.
Therefore, we have to solve the system
11
 11
 H 0 10  H1 2  3
85
25
, which has the solution H 0   , H1  .

11
11
 H 11  H 77  8
0
1
 10
10
We saw in Example 6, Lecture 2 that there exists a unique risk neutral probability
1
X
3 10 1 8 10 1 110
)
 
 
5.
measure, Q(1 )  Q(2 )  . Then, X 0  EQ (
2
1 r
11 2 11 2 11
Let us now have a look at options. Recall that for a call option on a security S, with
expiration date t  1 , and strike price K, the holder will exercise the option if S (1)  K .
Therefore, the payoff is X ( )  max{S (1,  )  K , 0} . If the contingent claim is attainable,
there exists a strategy H, such that
H 0 (1  r )  H1S (1,  )  X ( ) for all   .
Example 2. Consider the data in Example 1, with a call option on S with strike price
11
1
77
27
5 
.
K  5 . We have X (1 )   5  , X (2 ) 
2
2
10
10
11 1
 11
 H 0 10  H1 2  2
50
, which has the solution H 0   , H1  1 . Therefore the call is

11
 H 11  H 77  27
0
1
 10
10 10
X
1 10 1 27 10 1 16
)        . Notice that the time
attainable and its cost is X 0  EQ (
1 r
2 11 2 10 11 2 11
50
16
zero value of the portfolio is V0  H 0  H1S (0)    6  .
11
11
Let us see now what happens in a more complex model.
Example 3. Let d  2, N  1, r  1/10.
S (0)
S (1)
1
2
3
6
11/2 77/10 33/10
Let us see under what conditions is a claim X attainable? We need to solve the system:
11
 11
 H 0 10  H1 2  X (1 )

77
 11
 X (2 ), which has a solution if and only if 2 X (1 )  X (2 )  X (3 )  0
 H 0  H1
10
 10
33
 11
 H 0 10  H1 10  X (3 )

Therefore, in this market model, a contingent claim is attainable if and only if
2 X (1 )  X ( 2 )  X ( 3 )  0 .
Example 4. With the data from example 3, consider a put option with strike price
77
K
. The payoff to the holder of the put is: X ( )  max{K  S (1,  ), 0}, which gives
10
22
44
X (1 )  , X (2 )  0, X (3 ) 
, which is attainable. We compute the cost at time
10
10
zero of the put using Theorem 2. We have from Example 7, Lecture 2, that the market
1  4
model has infinitely many risk-neutral probability measures, of the form Q (1 ) 
2
1  2
Q (2 ) 
, Q(3 )   . Then
2
X
10 22 1  4 44
X 0  EQ (
)  ( 
  )  1 .
1 r
11 10
2
10
So from Example 3 we saw that the fact that the model does not have arbitrage
opportunities (we saw that there exists at least one risk neutral probability measure) does
not guarantee that every claim is attainable.
Moreover, as it could be seen from Example 4, even if the market model has infinitely
many risk-neutral probability measures, the cost of an attainable contingent claim is
independent of the measure. In fact, we have a stronger result which characterizes the
attainable contingent claims in a market model that free of arbitrage.
Theorem 3. Suppose the market model is free of arbitrage. Then, a contingent claim X is
X
) takes the same value for all risk neutral probability
attainable if and only if E Q (
1 r
measures.
Proof. We will only prove one implication, which is immediate. If a claim X is attainable,
V
X
then X 0  V0  EQ ( 1 )  EQ (
) , for some replicating portfolio and a risk-neutral
1 r
1 r
probability measure, Q. But since V0 does not depend on the risk-neutral probability
X
) does not depend on Q.
measure, it follows that E Q (
1 r
QED
Definition 3. The market model is complete if every contingent claim is attainable.
Otherwise, the model is said to be incomplete.
For complete markets we have a very interesting result, pointed out in the next theorem.
Theorem 4. The market model is complete if and only if there exists a unique risk neutral
probability measure.
Proof.
One implication follows from Theorem 3. If there exists a unique risk neutral probability
X
) takes a unique value, and by
measure, then for every contingent claim X, E Q (
1 r
Theorem 3, X must be attainable. Therefore the model is complete.
Assume now that the model is complete, and suppose there exist two distinct risk-neutral
probability measures Q1 , Q2 and 0 such that Q1 (0 )  Q2 (0 ) . Define then
B ( )  1  r , if   0
X ( )   1 0
0, otherwise
X is a contingent claim, and since the model is complete, X must be attainable. By
X
) independent of the risk-neutral probability measure Q. In
Theorem 3, X 0  EQ (
1 r
X
X
)E (
), which is equivalent to
particular we must have X 0  E (
Q1 1  r
Q2 1  r
Q1 (0 )  Q2 (0 ) , but this contradicts our assumption, and therefore there must be a
unique risk neutral probability measure.
QED
EXERCISES
1. For the data in Example 1 consider a call option on the risky security with strike price
K  6 . Determine whether the payoff of the option is attainable, and in case it is find
its value.
2. For the data in Example 1 consider a put option on the risky security with strike price
K  6 . Determine whether the payoff of the option is attainable, and in case it is find
its value.
3. Suppose the interest rate is r, and denote by c and p the cost of a call and put options,
respectively, on a security S, with the same exercise date and strike price, K. Show
that both derivatives are attainable, or neither is attainable. In case they are attainable,
K
use Theorem 2 to prove the put-call parity formula: c  p  S (0) 
, where S (0) is
1 r
the value of the stock at time zero.