The field of finance was revolutionized in 1973, allegedly, when Fischer Black and Myron Scholes came up with the Black-Scholes partial differential equation to price call options. A call option is a right, but not obligation to buy a share with some pre-agreed price and time. The original article considers a hedging approach, where a long position in the stock and short position in the option is created to replicate allegedly a risk-free financial product. Using the basic results of stochastic calculus, matching coefficients, then one can show that the Black-Scholes PDE is the following:
Where C(S,t) is the value of the option, r is the risk-free rate, sigma is volatility and S is the value of the underlying. The boundary condition reflects the provisions of the derivative agreement, i.e the value of the derivative at terminal time T. Fundamentally, in the various derivations, the hedging portfolio is the following:
Where x is the amount held short of the the option in the portfolio. The pair (1,x) is called a trading strategy. In the Black-Scholes model, it is assumed that this strategy is self-financing, so that
This assumption treats the trading strategy essentially as constant. This assumption makes it easy to match the coefficients, when one demands that the replicating portfolio above yields the risk-free rate.
Now we can proceed, assuming that the underlying stock follows geometric Brownian motion with some drift and volatility, we have
and on the other hand, the expansion for the call option is using Ito’s lemma:
For the replicating portfolio, we get:
The rBdt term on the right side of the equation is no-arbitrage condition: the replicating portfolio must yield the risk free rate r of a bond B. The procedure then finds a value for the call option such that the portfolio is instantly risk-free.
This only however works if we were to choose x in such a way that:
The Wiener process disappears. Then
Substituting this into the equation above, one obtains the Black-Scholes PDE.
But this contradicts the assumption of a self-financing portfolio, as it was assumed that the delta of the option is effectively a constant. This is not the case, unless we restrict to linearity in S.
The self-financing assumption has been criticized for example in:
https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4461094
https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4942398
The authors claim indeed that the Black-Scholes model is inherently wrong and leads to inconsistencies. Self-financing seems to be self-contradictory with the Black-Scholes pricing. If you assume self-financing, you cannot get rid of the Wiener process. The Black-Scholes model gives a fair value for the option such, which indicates that the market is risk-neutral as a whole. You can check this by considering the process
and then evaluate the expectation
Using Feynman-Kac, the expectation obeys the Black-Scholes PDE.
Alternative
What if we start from scratch? Assuming that the investor in a derivative security is risk averse, the general price can be understood by some discounted payoff of the derivative. Then we should have
So that the fair value for the derivative contract is just the expectation using the physical probability measure P. Of course we do not know what is the proper discount rate k. It seems that we should have at least k>r.
In risk neutral pricing, one constructs an artificial measure, and discounts using risk-free rate and calculates the expectation using this risk-adjusted measure. This is in principle possible. The question then is, how to construct the risk-neutral measure.
Suppose that the discount rate is just the drift or the instantaneous return of the asset, then we have
This could be a good guess, as this discount rate makes the discounted geometric Brownian motion a martingale. Paul Samuelson once said, that in efficient markets, properly discounted price processes must be martingales (=fair game).
https://www.jstor.org/stable/3003046
Using Feynman-Kac directly, we get the modified Black-Scholes PDE:
This agrees with original Black-Scholes, if the investor is risk neutral, ie. all instantaneous returns of assets converge to the risk free rate. What is the correct model? One could basically of course estimate k from real data. I am pretty sure it is higher than the risk free rate.
See:
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