Riemann Zeta Function and the Brownian Motion of Asset Prices: Number Theory meets Quantitative Finance
June 12, 2011
Is there a connection between the Riemann Zeta function and the Brownian motion of the asset prices?
Riemann Zeta function is perhaps the most beautiful formula in Number theory and certainly ranks at par with Euler’s formula as one of the most beautiful formulas in mathematics. The Riemann Zeta function is simply expressed as:
The function is defined .In the above, definition, is a complex number. In a series form the zeta function can be expressed as:
All this is fine. But what has all this got to with quantitative finance?
Let’s see what happens when we take the inverse of the Riemann zeta function
The numerator of the above series are the values of the Mobius function, , which is zero if is square free, 1 if has even number of prime factors and -1 if it has an odd number of prime factors. If we look at the values of the Mobius function the look like
This can be thought of as a random walk in which a person – who is quite drunk – or a financial asset such as a stock price of an FX pair stays where it is, i.e. 0, or takes one step forward, i.e. +1 or takes one step backward, i.e. -1.
As David Wells writes in his excellent book, Prime Numbers, this observation was first made by Von Sternbach in 1896, who also listed the first 150,000 values of the Mobius function and estimated that the probability that was non-zero was around which is roughly equal to 0.608 with +1 and -1 having approximately equal probabilities of occurrence.
Therefore, the random walk, i.e. a Weiner process that models the asset price, is hidden deep within the Riemann zeta function.
Let us look at some more evidence of this.
In 1997, in a seminal paper, Broadie, Glasserman and Kou, professors at Columbia University and the University of Michigan respectively, proposed a beautiful formula for the adjustment that needs to be made when we move from a continuous barrier to a discrete barrier, while valuing a certain kind of financial derivative called the barrier options. Barrier options are extremely popular amongst sell side traders in the banks and institutional investors and are embedded in numerous structured products sold to retail investors as well.
All closed form solution for barrier options (knock-outs or knock-ins) are priced assuming a continuous monitoring of the barrier, owing to the use of continuous time stochastic calculus in such mathematical modeling. However, in real life all option traders observe any barrier – a knock-out or a knock-in level of the asset, given a certain asset price path – on a discrete basis, such as daily monitoring, weekly monitoring, etc. Therefore, adjustment needs to be made to compensate for this fact.
Broadie, Glasserman and Kou showed that if is denotes monitoring points (number of days or number of months, etc.), is the (theoretical) continuously monitored barrier level and is the corresponding (practical) discrete barrier level then the two should be approximately related by this formula
In the above formula, the authors showed that the constant is related to the Riemann zeta function by
In the above formula, is the value of the Riemann zeta function around .
The plus and the minus sign in front of the constant represent an up barrier and a down barrier option respectively. This beautiful formula is used today by traders to make adjustments to the barrier level when valuing barrier options and the formula.
The function, around has a special significance. The Riemann hypothesis states that the zeta function has non-real and non-trivial zeros only on the critical line for which the real part of is 1/2 .
The Riemann zeta function and the Riemann hypothesis are both related to the Prime number theorem and the distribution of prime numbers. Prime numbers are used in many powerful mathematical algorithms to we generate random numbers. These random numbers are in turn used to simulate Brownian motion while modeling asset price paths for the valuation of financial derivatives.
- Prime Numbers by David Wells (John Wiley & Sons, 2005)
- A Continuity Correction for Discrete Barrier Options, Mark Broadie, Paul Glasserman (Columbia Univerity), Steven Kou (University of Michigan), Mathematical Finance, Vol 7, No. 4 (October 1997)
- On Pricing of Discrete Barrier Options, S.G.Kou, Columbia University, April 2001 / 2003.
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