1. Investment Analysis and Portfolio Management
Lecture 11
Gareth Myles

2. Introduction Asset prices follow a random walk
They can go up and down but cannot be predicted
This follows from the efficient markets hypothesis
Past information is fully incorporated into share prices
Market prices respond immediately to new information

3. Price Process To model this process let S be the asset price
Then dS/S is the proportional change in asset price
Assume:
A predictable growth component mdt in time interval dt
A random change sdX where dX is drawn from a mean-zero normal distribution with variance dt 
s is the volatility

4. Price Process Taken together the proportional price change satisfies
dS/S = sdX + mdt
This is a stochastic differential equation
It cannot be solved to give a deterministic solution but determines a random solution
If s = 0 then dS/S = mdt so that
S = S0em(t – t0)

5. Price Process Taking the expectation of the price process
E[dS] = E[sSdX] + E[mSdt]
= mSdt
The expected change is given by the deterministic component
A higher m implies faster growth in S
A larger s implies more variability in S

6. Ito’s Lemma
This is the rule for differentiating functions involving random variables
Take a function f(S)
Taylor’s series gives
For a non-random variable it would be claimed dS2 is small so

7. Ito’s Lemma But for the random price process dS2 = [sSdX + EmSdt]2
= s 2S 2dX 2 + 2smS 2dXdt + m 2S 2dt 2
Now dX = O(√dt) (i.e. as dt → 0, dX/dt → 1) so to leading order
dS2 = s 2S 2dX 2
But dX 2 → dt so
dS2 → s 2S 2dt

8. Ito’s Lemma
This gives
More generally if f = f (S, t) then

9. Ito’s Lemma This expression depends on:
dX which is random
dt which is deterministic
The random component prevents the partial differential equation from being solved
With the binomial model the risk was hedged away
A portfolio of option and (short) underlying
The same can be done in this case

10. Ito’s Lemma
To eliminate dX at each instant let
g = f - DS
Then