Does Black Scholes need to assume no arbitrage?

Question

Since Girsanov's theorem guarantees a risk neutral measure for Geometric Brownian motion, by the fundamental theorem of asset pricing there can be no arbitrage. So, why does the model assume no arbitrage? In particular, where do we use it when we arrive at the PDE?

2. Why are we sure that the risk neutral measure in this case is unique? One logic I've seen is arguing that trade-able assets are quite large in number and using just that to derive uniqueness. But that can be assumed true for any model and not all models have unique risk neutral measures.

Seeking clarification on flow of logic between arbitrage, risk neutrality and completeness assumptions/inferences. Thank you

Answer 1

This is a far subtler and deeper rabbit hole than appears. Harrisson, Kreps and Pliska’s papers explain the completeness, uniqueness and risk-neutral aspects.

With regards to the BS framework, it simply states a real-world dynamics in the form of a GBM. The no-arb condition is essential in that it dictates that the riskless portfolio of option and hedge must grow at the risk-free rate, and from there allows the derivation of the BS formula. Note that no assumption or statement needs to be made about the risk-neutral dynamics of the stock price.

Subsequent findings about the necessary dynamics under no-arb conditions and completeness of the market have then yielded the risk-neutral GBM and the fact that the option value is the discounted expectation of the payoff under the risk-neutral measure. Under that framework, one naturally recovers the BS formula from a different angle.

But that latter point is historically unrelated to the derivation of the original BS formula.

In other words, one approach uses the real-world dynamics and a no-arb argument to arrive at the BS formula.

The other uses the no-arb argument and market completeness to derive a necessary fundamental pricing approach (expected value) and the risk-neutral asset dynamics. From these the BS formula can also be obtained.

In my opinion the formula can almost be thought as a law of nature (under certain assumptions on nature). It exists conceptually independently of its discoverers. What BS have done is discover one way to derive it, and subsequent research uncovered another way.

Answer 2

If you use the original reasoning by Black and Scholes, then Absence of Arbitrage is explicit in the derivation, it causes the dynamically hedged portfolio to earn the risk free rate, no more no less. Otherwise there would be an arbitrage between the Risk Free Asset and the dynamically hedged portfolio, both of which are riskless.

If you use the FTAP and the Martingale Methods of derivative pricing (which were developed by Harrison and Pliska many years later) then it may seem that you don't need Absence of Arbitrage. But that is not true. The whole theory is based on the fact that absence of arbitrage is equivalent to the existence of the martingale measure.

One way or the other Absence of Arbitrage is in there somewhere in the proof.