Hello Siddhi
One way to get the new SDE is by using Ito's lemma.
The question gives us an SDE that tells us how the share price changes over an instant of time:
dSt = (μ - λσ)St dt + σSt dWt
This is a diffusion or Ito process.
Mapping this Ito process to the one given on P46 of the Yellow Tables (in the section on Ito), we have:
x = St
z = Wt
a = drift = (μ - λσ)St
b = volatility = σSt
We now need to determine a second SDE, this time for exp(-rt)St. We can use Ito's lemma, as per Page 46 of the Yellow Tables, with:
G(St,t) = exp(-rt)St.
dG/dSt = exp(-rt)
d^2G/dSt^2 = 0
dG/dt = -rSt exp(-rt)
Note these are partial derivatives so that, when we are differentiating with respect to St, we assume t is constant and vice-versa.
Ito's lemma gives:
dG(St,t) = [a dG/dSt + 0.5b^2 d^2G/dSt^2 + dG/dt] dt + b dG/dSt dWt
= [(μ - λσ)St exp(-rt) + 0.5 (σSt)^2 * 0 + (-r)St exp(-rt)] dt + σSt exp(-rt) dWt
= [(μ - r - λσ)St exp(-rt)]dt + σSt exp(-rt) dWt
We want the drift term, ie the "dt" term, to be zero as martingales are processes with zero drift.
Hence:
μ - r - λσ = 0
and λ = (μ - r)/σ
Is this OK Siddhi?
Anna
