Abstract
This article provides a brief overview of the construction of mathematical finance and suggests some questions of philosophy of science raised by the study of this field.
Remarks and comments on this paper are welcome and may be addressesd to the author at: ghislaine.idabouk@gmail.com.
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Notes
- 1.
- 2.
I consider here, as is standard among finance academics, that “Modern Financial Theory” emerged after the publication of Harry Markowitz’s paper (Markowitz 1952). This boundary date with previous financial theory might be questioned. Yet, it is beyond the scope of this paper, which addresses mathematical finance after 1973, to do so.
- 3.
- 4.
In this article, the analysis is restricted to a limited corpus of academic articles identified in Sections 13.4 and 13.5, which I consider as a first, coherent building block of mathematical finance. The latter has then evolved into several directions to extend the Black and Scholes model (stochastic volatility, jump-diffusion processes) and to address issues of asset pricing in incomplete markets. The analysis of these extensions from a philosophy of science standpoint is left for future research.
- 5.
Fama’s PhD dissertation was published in the January 1965 issue of the Journal of Business (see Fama 1965).
- 6.
See Belze and Spieser (2005, p 198).
- 7.
See Belze and Spieser (2005, p 241).
- 8.
A portfolio is a combination of financial instruments. The weight of each asset in the portfolio is given in proportion of the total value of the portfolio.
- 9.
- 10.
This can formally be written as \(\forall i,E\left ({r}_{i}\right ) = {r}_{f} + {\beta }_{i}\left (E\left ({r}_{M}\right ) - {r}_{f}\right )\), where r i is the return of the i-th asset, r M is the return of the market portfolio, r f is the risk-free rate and \(\forall i,{\beta }_{i} = \frac{\mathrm{cov}\left ({r}_{i},{r}_{M}\right )} {V \left ({r}_{M}\right )}\).
- 11.
Modigliani and Miller are not the inventors of the notion of arbitrage. It is already present in the works of Irving Fisher. Yet Modigliani and Miller’s paper contributed to spreading the use of arbitrage reasoning among financial economists.
- 12.
Efficiency here is informational efficiency.
- 13.
- 14.
Mehrling claims though that the early reasons of Black’s interest in option pricing are unknown.
- 15.
A stochastic process is a collection of random variables.
- 16.
The Brownian motion is a particular stochastic process. It starts at 0 at time 0, has independent and stationary increments (if we consider that the filtration is the natural one), and the increments follow a centered normal distribution with a variance proportional to the time interval.
- 17.
Although Black acknowledges Merton’s contribution for the arbitrage approach.
- 18.
Black and Scholes do not justify their intuition. They seem to be reasoning as in traditional differential calculus, which in this case is a reasoning error. Yet, the intuition was correct and it can be properly proved using stochastic differential calculus.
- 19.
Black and Scholes reason in terms of null Beta (CAPM influence throughout).
- 20.
I used Black and Scholes’ notations here. w 1 is the partial derivative of the option price with respect to the stock price.
- 21.
Value here simply means number of units multiplied by price.
- 22.
Here, Black and Scholes implicitly use the assumption that the portfolio is “self-financing”, a notion which will be introduced later in Mathematical Finance. Otherwise their computation would be wrong.
- 23.
An arbitrage opportunity is the possibility to make a profit with no cost and no risk.
- 24.
Merton and Scholes were awarded the Alfred Nobel Memorial Prize the same year for their “new method to determine the value of derivatives” (Black, who died in 1995, could not receive it as it is not awarded posthumously).
- 25.
Initially, “rational option pricing” for Merton is simply such that no asset is dominant nor dominated. Obviously this definition being too generic, he then had to restrict it to be able to get a price for options.
- 26.
Anyways Black and Scholes were neither the first ones to use the Brownian motion (and not even the geometric Brownian motion) nor the first ones to use continuous time modeling in Financial Economics.
- 27.
Mostly Brownian motion in this first phase, though not exclusively. The study here is restricted to papers with Brownian motion models, see also footnote 5.
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Idabouk, G. (2010). Randomness, Financial Markets and the Brownian Motion: A Reflection on the Role of Mathematics in Their Interaction with Financial Theory After 1973. In: Suárez, M., Dorato, M., Rédei, M. (eds) EPSA Philosophical Issues in the Sciences. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-3252-2_13
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