Continuous-time lognormal model

[Harashima Senju]

Are the following results true

• In a real world we have: \(ln\frac{S_t}{S_s} \sim N( \mu(t-s), \sigma^2(t-s))\)
• In a real neutral world : \(ln\frac{S_t}{S_s} \sim N( (r-0.5\sigma^2)(t-s), \sigma^2(t-s))\)

 

[John Potter]

They look a bit mixed up.

In the real world

When we start with the SDE dSt = St {mu.dt + sigma dZt}, we get the solution...

ln(St/S0) ~ N[ (mu - 0.5sigma^2)t, sigma^2 t]

Because mu = 0.5 sigma^2 is just one parameter, we may as well call it "theta", say. Then someone else doesn't like "theta", they like "mu", so they use "mu" and stick it in the cts-time logN model. So, it's the SAME model but NOT the same "mu".

In the risk-neutral world

Swapping from P to Q is the same as swapping the "mu" for an "r". This must be done on the "mu" in the dSt = St {mu.dt + sigma dZt} presentation, where ln(St/S0) ~ N[ (mu - 0.5sigma^2)t, sigma^2 t].

It's the CMG theorem that allows us to do this swap...

dZ~t = dZt + gamma. dt
gamma = (mu - r) / sigma (market price of risk)

Good luck!
John