- Risky Assets 63
Proposition 3.5
Given that the stock priceS(n) has become known at timen, the risk-neutral
conditional expectation ofS(n+ 1) will be
E∗(S(n+1)|S(n)) =S(n)(1 +r).Proof
Suppose thatS(n)=xafterntime steps. Then
E∗(S(n+1)|S(n)=x)=p∗x(1 +u)+(1−p∗)x(1 +d)becauseS(n+ 1) takes the valuex(1 +u) with probabilityp∗andx(1 +d)
with probability 1−p∗.Butp∗(1 +u)+(1−p∗)(1 +d)=1+rby (3.4), which
implies that
E∗(S(n+1)|S(n)=x)=x(1 +r)
for any possible valuexofS(n), completing the proof.
Dividing both sides of the equality in Proposition 3.5 by (1 +r)n+1,we
obtain the following important result for thediscounted stock pricesS ̃(n)=
S(n)(1+r)−n.
Corollary 3.6 (Martingale Property)
For anyn=0, 1 , 2 ,...
E∗(S ̃(n+1)|S(n)) =S ̃(n).We say that the discounted stock pricesS ̃(n)formamartingaleunder the
risk-neutral probabilityp∗. The probabilityp∗itself is also referred to as the
martingale probability.
Exercise 3.19
Letr=0.2. Find the risk-neutral conditional expectation ofS(3) given
thatS(2) = 110 dollars.3.3 Other Models
This section may be skipped at first reading because the main ideas to follow
later do not depend on the models presented here.