- Risky Assets 67
Introducing a sequence of independent random variablesξ(n), each with
two values
ξ(n)={
+
√
τ with probability 1/2,
−√
τ with probability 1/2,we can write the logarithmic return as
k(n)=mτ+σξ(n).Exercise 3.23
Find the expectation and variance ofξ(n)andk(n).Exercise 3.24
WriteS(1) andS(2) in terms ofm,σ,τ,ξ(1) andξ(2).Next, we introduce an important sequence of random variablesw(n), called
asymmetric random walk, such that
w(n)=ξ(1) +ξ(2) +···+ξ(n),andw(0) = 0. Clearly,ξ(n)=w(n)−w(n−1).Because of the last equality,
theξ(n) are referred to as the increments ofw(n).
From now on we shall often writeS(t)andw(t) instead ofS(n)andw(n)
fort=τn,wheren=1, 2 ,....
Proposition 3.7
The stock price at timet=τnis given by
S(t)=S(0) exp(mt+σw(t)).Proof
By (3.2)
S(t)=S(nτ)=S(nτ−τ)ek(n)
=S(nτ− 2 τ)ek(n−1)+k(n)
=···=S(0)ek(1)+···+k(n)
=S(0)emnτ+σ(ξ(1)+···+ξ(n))
=S(0)emt+σw(t),as required.