The easy part in financial mathematics, as far as I’m concerned, is the ‘mathematics’ part while the ‘financial’ part is obscure. This leads to many books displaying both bad thinking and dirty mathematics. I still think that financial mathematics is the least clean domain of maths (ever) but one gets used to it. In this post I wish to explain the way I see it and present the most clean derivation of the Black-Scholes I managed to construct over time. The clean part in this is the stochastic Ito calculus which I will not derive here however. The lesser part is the semantics, starting with the no-arbitrage concept.

No arbitrage is the same as the ‘no free lunch’ idea. The danger with free lunches is the danger of creating imbalances in a domain (finance or other). It means there is a flow in one direction which is not counter-balance. Nature in general abhors anything not balanced but in the man-made world it took some catastrophes (read: financial crises) to grasp it; imbalance does not work in the long term. With options pricing and the derivatives market it’s the story that if prices are not balanced well there are opportunities for people (and systems) to benefit in a way which creates in the long run dead-locks. More concretely, one should not be able to pick up some money somewhere, pay somewhere else with profit to repay the loan and still make profit in the end. Mathematically this would amount to something like

$latex P(V_t)>0 \wedge V_0=0$

where V is some time-dependent quantity, say a portfolio. It says, with initial investment zero you have always a positive (profit), statistically speaking. The statistics is important here; the evolution is not deterministic but induced by some underlying stochastic process. In nature this would amount to saying that you start off with some closed system at zero energy and end up with some non-zero energy in the future. That’s not OK, there is no magic and financial markets discovered that it doesn’t work either.

Say you have some financial portfolio V and some dynamics S(V) the no-arbitrage tells you that

S(V) == V + interest

If this wasn’t the case you could either borrow money to invest or invest to pay back borrowed money (plus interest). More precise

$latex S(V_t) = V_t\;e^{rt}$

where the exponential is the accumulated interest (rate r). This equation is in ‘small’ form

$latex dV_t – rV_t\,dt = 0 $

At this point one has to make some simplifying assumptions. The interest is not time dependent, the portfolio consists of options and a fixed amount of securities, the so-called cash account is ignored. It’s at this point that things become murky in various expositions. They become obscure either because the mathematical level is low or because the odd mathematical details have to be explained by means of concepts which are themselves flawed.

The portfolio

$latex V_t = F_t – \Delta S$

is considered as an option packet F together with a fixed amount of securities $latex \Delta$, where the security $latex S$ is a Wiener (stochastic) process of the form

$latex dS_t = \mu S dt + \sigma dW_t.$

Because the amount of securities is kept constant, we have

$latex dV_t = dF_t – \Delta dS_t$

and Ito’s rule then gives

$latex dV_t = F_t dt + (F_S – \Delta) dS + \frac{1}{2} F_{SS} \sigma^2 dt$

A lot of books then tell you that defining

$latex \Delta = F_S$

is simplifying the solution. Of course, this does not define $latex \Delta$, it’s a first order constraint on $latex V_t$! So, effectively, we have this **constraint** together with

$latex dV_t = (F_t + \frac{1}{2} \sigma^2 F_{SS}) dt$

and upon inserting this in the no-arbitrage constraint you get

$latex F_t + \frac{1}{2} \sigma^2 F_{SS} -r F + rSF_S =0.$

This is then the Black-Scholes equation and needs further constraints in the shape of boundary and start/end conditions. Note that the constraint on $latex \Delta$ is absorbed into the final equation and thus does not need to be supplied separately.

I posted a question on StackExchange to highlight the issue, according to me, with how books derive or formulate the derivation of Black-Scholes.

### References

- An undergraduate introduction to financial mathematics by J.Buchanan, chapter 7. A very nice and clean intro to financial maths without being hard on the theorems and rigour.
- The mathematics of derivatives by R.Navin, chapter 5. A pleasant style of presenting the material, quite enjoyable.
- Stochastic differential equations by B.Oksendal. Solid and classic intro to the maths of stochastics.