Risk Free Rate & pricing of Options

[ChrisDerick]

Hey guys,

I'm currently reading Hull's book and I was wondering about an apparant inconsistency in my thoughts:

1. In the Black Scholes PDE, the only rate exist is risk free rate 'r'. The derivation from Ito Lemma is pretty straightforward (using the riskless portfolio assumption). The drift of underlying '\mu' is removed when we consider a riskless portfolio. My personal interpretation is that the option's value have little to do with the drift of underlying.

2. If we instead opt to use monte carlo method, we will have to simulate based on the GBM equation:
\(S_t = S_0 exp \large( \large( \mu -\frac{\sigma^2}{2} \right) t + \sigma W_t \right)\)

This however, does not assume \(\mu = r\) anywhere. In practice however, we usually take \(\mu = r\) to get obtain the result (which is equal to Black Scholes' formula).

My question is where did we assumed in case (2) that \(\mu = r\)? I'll be glad if anyone could point me to some reading about it.. Thanks.

 

[Keith Tan]

St=S0Exp(rdt). seems like u're trying to simulate rdt with a GBM. mu is mean. in a riskless world, the mean rate of return is r. to be more specific, mu can also be (r-q), where q is the dividend yield. so mu is open to interpretation.

 

[ChrisDerick]

Hi Keith,

Thanks for responding! I think I'm just a little confused... cleared up now. =)

 

[C S]

The thing to realize is there are two types of approaches, 1) PDE and 2) Martingale. When you do Monte Carlo, you are using the second approach. In the martingale approach, you use a change of measure that will change the drifts of your assets but not their volatilities. So if mu is the drift under the real world probability measure, then when you do the appropriate measure change, you get a new drift which is the riskless rate. This is what is required to make your discounted asset prices behave as a martingale.

 

[aaronhotchner]

Black-Scholes does not assume that there is no risk. It assumes that the drift of all assets is the risk-free rate, and options are priced in a risk-neutral world. I haven't looked at the proofs long enough to forget the actual details, but I'm fairly certain that the solution does not depend on mu, as opposed to concluding that mu = r.

 

[ChrisDerick]

Black-Scholes does not assume that there is no risk. It assumes that the drift of all assets is the risk-free rate, and options are priced in a risk-neutral world. I haven't looked at the proofs long enough to forget the actual details, but I'm fairly certain that the solution does not depend on mu, as opposed to concluding that mu = r.

Erg. at last I understand. (doink). Thanks!