- Discrete Time Market Models 89
for anyj=1,...,m,anyi=1,...,kand anyn=0, 1 , 2 ,...,whereE∗(·|S(n))
denotes the conditional expectation with respect to probabilityP∗computed
once the stock priceS(n) becomes known at timen.
Example 4.6
We shall use the same scenariosω 1 ,ω 2 ,ω 3 ,ω 4 , stock pricesS(0),S(1),S(2)
and money market pricesA(0),A(1),A(2) as in Example 4.5. In addition, we
consider a European call option giving the holder the right (but no obligation)
to buy the stock for the strike price ofX= 85 dollars at time 2.
In this situation we need to consider an extended model with three assets,
the stock, the money market, and the option, with unit pricesS(n),A(n),CE(n),
respectively, whereCE(n) is the market price of the option at timen=0, 1 ,2.
The time 2 option price is determined by the strike price and the stock
price,
CE(2) = max{S(2)−X, 0 }.
The pricesCE(0) andCE(1) can be found using the Fundamental Theorem
of Asset Pricing. (Which explains the name of the theorem!) According to the
theorem, there is a probabilityP∗such that the discounted stock and option
pricesS ̃(n)=S(n)/A(n)andC ̃E(n)=CE(n)/A(n) are martingales, or else
an arbitrage opportunity would exist. However, there is only one probability
P∗turningS ̃(n) into a martingale, namely that found in Example 4.5. As a
result,C ̃E(n) must be a martingale with respect to the same probabilityP∗.
This gives
CE(1) =A(1)
A(2)E∗(C
E(2)|S(1)) and CE(0) =A(0)
A(1)E∗(CE(1)).
The values ofP∗for each scenario found in Example 4.5 can now be used to
computeCE(1) and thenCE(0). For example,
CE(1,ω 1 )=CE(1,ω 2 )=A(1)
A(2)
P∗(ω 1 )CE(2,ω 1 )+P∗(ω 2 )CE(2,ω 2 )
P∗(ω 1 )+P∗(ω 2 )=^110
12119
30 ×27 +19
60 ×^21
19
30 +19
60∼= 22. 73
dollars. Proceeding in a similar way, we obtain
Scenario CE(0) CE(1) CE(2)
ω 1 19. 79 22. 73 27. 00
ω 2 19. 79 22. 73 21. 00
ω 3 19. 79 3. 64 5. 00
ω 4 19. 79 3. 64 0. 00