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Non-risk neutral Black Scholes

how would you derive a BS, taking an PDE approach, but assuming that we are operating in a non-risk neutral world, thus allowing the possiblity of arbitrage?
 
I believe that you would add on some higher risk premium, p, everywhere you see the risk free rate, r. The risk compensation variable p would depend on some sort of risk aversion, or how much people want to be compensated for bearing volatility in the future. This is difficult to estimate.

However, I believe the prices of options would end up the same at the end of the day, as long as you still allow traders to borrow and lend, and trade options, freely. Risk neutral is just a way of looking at the same world, not a different world altogether -- it changes the expected drift, not the volatility (Girsanov), and options are priced on the vol, which is the same in all worlds (or views of the same world), not priced on the drift, which cancels out between the stock and the options that depend on that stock, and hence doesn't really matter for the options themselves, which is why you can change it with impugnity. It's just a trick to make the math easier.

Breaking no arbitrage is really tricky, because it implies that the same object can have 2 prices at the same time, or that there are ways to create new risk free rates above the stated risk free rates, e.g. with linear combinations of risky assets. Then the system can have several risk free rates to the same maturity, which is odd. Breaking academic arbitrage relies on someone else making mistakes, essentially. It's not impossible, but it's unlikely to last.
 

bob

Faculty (Undercover)
how would you derive a BS, taking an PDE approach, but assuming that we are operating in a non-risk neutral world, thus allowing the possiblity of arbitrage?

In concept, it can be done using the Feynman-Kac theorem with discounting.

Using an underlying that obeys the SDE:
(dS = \mu S dt + \sigma S dW)

We seek the pricing formula:
(V(S,t) = E^{S,t}[e^{-r(T-t)}g(S(T))])

As long as our payoff function g is well-behaved under this probability measure, Feynman-Kac guarantees that:
(V_{t} + \mu S V_{S} + \frac{1}{2} \sigma^2 S^2 V_{SS} = rV)
With the terminal condition
(V(S,T) = g(S))

Shreve's Stochastic Calculus for Finance II has a discussion of this theorem and a few proofs on pp.268-272, if you want some background.

Obviously, if you're dealing with European options with friendly payoffs, you'd much rather find this V from the distribution of the terminal price, not the PDE. But I suppose if you're pricing American puts, say, under this assumption, then the PDE might come in handy.
 

dstefan

Baruch MFE Director
Bob, do you have any idea how to derive BS from CAPM ? Thanks

this is the original Cox-Rubinstein derivation, starting from CAPM in continuous time. A few details are in Wilmott's book Frequently Asked Questions in Quantitative Finance, on page 267.
 
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