68 Mathematics for Finance
In order to pass to the continuous-time limit we use the approximationex≈1+x+1
2
x^2 ,accurate for small values ofx, to obtain
S(nτ+τ)
S(nτ)=ek(n+1)≈1+k(n+1)+1
2
k(n+1)^2.Then, we compute
k(n+1)^2 =(mτ+σξ(n+1))^2 =σ^2 τ+···,where the dots represent all terms with powers ofτhigher than 1, which will
be omitted because they are much smaller than the leading term wheneverτ
is small. Next,
S(nτ+τ)
S(nτ)≈1+mτ+σξ(n+1)+1
2
σ^2 τ=1+
(
m+1
2
σ^2)
τ+σξ(n+1),and so
S(nτ+τ)−S(nτ)≈(
m+1
2
σ^2)
S(nτ)τ+σS(nτ)ξ(n+1).Sinceξ(n+1) =w(nτ+τ)−w(nτ), we obtain an approximate equation
describing the dynamics of stock prices:
S(t+τ)−S(t)≈(
m+1
2
σ^2)
S(t)τ+σS(t)(w(t+τ)−w(t)), (3.8)wheret=nτ. The solutionS(t) of this approximate equation is given by the
same formula as in Proposition 3.7.
For anyN=1, 2 ,...we consider a binomial tree model with time step of
lengthτ=N^1 .LetSN(t) be the corresponding stock prices and letwN(t)be
the corresponding symmetric random walk with incrementsξN(t)=wN(t)−
wN(t−N^1 ), wheret=Nnis the time afternsteps.
Exercise 3.25
Compute the expectation and variance ofwN(t), wheret=Nn.