What does oceanography have in common with the stock market? A lot, actually.

Consider the iconic Black-Scholes equation that prices European call and put options. In my previous post, I outlined how the Black-Scholes equation can be transformed into an analogue of the heat equation. I also mentioned how the heat equation came up in my Master’s thesis, which focused on fluid dynamics and physical oceanography. In this post I’ll explain to you why financial mathematics = fluid dynamics.

**The Black-Scholes Equation**

The Black-Scholes equation is given by:

* S *is stock price,

*the price of the option,*

**V(S,t)***is the annualized risk-free rate, and*

**r***is volatility of the stock. It’s assumed that the underlying stock pays no dividends, and that*

**σ***and*

**r***are constants.*

**σ**Finance people interpret the Black-Scholes equation as follows:

- The first term is
*theta*, or the time decay of the option’s price. - The second term, the one with
, involves**σ***gamma*and volatility. Gamma is the convexity of a derivative’s value with respect to the underlying value. - The third term is a short position consisting of ∂V/∂S shares of the underlying.
- The final term is the riskless return from a long position in the derivative.

The all means that the equation states that over any infinitesimal time interval, the loss from theta and the gain from gamma offset each other. The result is a return at the riskless rate.

Now that’s all fine and dandy, but this interpretation doesn’t really help me. I’m so far behind I think “Greeks” means Greeks that helped contribute to the Black-Scholes model. (Disclaimer: now I know what this means!). What’s more useful for me is if I can compare the Black-Scholes equation to something I already know.

**The Navier-Stokes and Continuity Equations**

This is where the Navier-Stokes equation comes in. Let’s consider the governing equations for ocean flow, something that I understand very well.

Here * (x,y,z) *are the zonal (east-west), meridional (north-south), and vertical directions,

**u**is the velocity vector,

*is zonal (east-west) velocity,*

**u***is meridional (north-south) velocity,*

**v***is pressure,*

**p****is density, and**

*ρ**is viscosity in the horizontal and vertical directions.*

**K**For the purposes of this post, it’s not necessary to go over what these equations are exactly, but it’s useful to have a bit of an idea. The first equation is the continuity equation, while the remaining make up the Navier-Stokes (NS) equation. The NS equation formalizes Newton’s famous second law, *F = ma*, as *a = F/m*. The Navier-Stokes equation is written here in the zonal, meridional, and vertical directions. In the vertical, the ocean is assumed to be in hydrostatic balance, which means that the vertical pressure gradient force (left hand side) balances the Coriolis force (right hand side). The Coriolis force is due to the rotation of the Earth.

**A Derivation**

Let’s see what happens if we make some assumptions- mathematicians love assumptions! First, let’s consider just the * x*-direction (the second equation above). Let’s also compute and expand the total derivative term (the first term on the left hand side). This gives us:

Next let’s assume any derivatives in the * y* and

*directions are zero. Finally, let’s say viscosity*

**z***is independent of*

**K***. We then arrive at:*

**x****Navier-Stokes = Black-Scholes?**

We can then draw some comparisons between the Black-Scholes equation and the equation above. The left hand side is the Black-Scholes, and the right hand side is the equation for ocean flow.

What does this all mean? Well, it means we can use insights into fluid dynamics to draw conclusions on financial markets, at least in this specific instance. Let’s consider what the terms in the Black-Scholes equation mean, based on what the terms mean in the equation for ocean flow. This way we derive an understanding of financial mathematics through an understanding of fluid dynamics.

- Term 1: Rate of change of option price with respect to time ~ Rate of change of velocity with respect to time.
- Term 2: Volatility/turbulence term ~ Viscosity/turbulence term.
- Term 3: Advection and randomness ~ Advection of horizontal velocity. Advection in this case refers to the evolution of the option price, and randomness refers to the random walk assumption of the Black Scholes, that is, stock price evolves according to a random walk and cannot be predicted. We can say this term is the evolution of the option price due to the random walk.
- Term 4: Market forces ~ Pressure gradient and Coriolis force, i.e. forces established by the environment. The annualized risk-free rate,
, is assumed to be constant in the Black-Scholes model, so it’s like a predetermined market force.**r**

Overall, then the rate of change of the price of an option is due to volatility, randomness, and market forces. This is pretty obvious- what else could it be due to? Let me know in the comments!

**Summary of Comparisons**

It’s certainly useful to compare the two formulas to help understand the Black-Scholes equation. My fluid dynamics-based interpretation of the first two terms is accurate, but the interpretation of the third and fourth terms requires a bit more imagination.

I interpret the third term, riskless return on a short position, to be like advection and randomness. Look at what this third term is in the Black-Scholes: The rate of change of the option price with respect to the underlying stock price, multiplied by the stock price and annualized risk-free rate, * r*. Physically, this is like advection. Keeping in mind that the Black-Scholes assumes stock prices are a random walk, this interpretation makes sense

The fourth term, riskless return on a long position, is like a predetermined market force, according to my interpretation. The pressure gradient force and Coriolis force are in a sense predetermined, since pressure gradient force can arise due to an initial distribution of sea surface height, and the Coriolis force can be set to a constant value. In the Black-Scholes model, * r *is chosen by whoever is writing the model. This fourth term then represents the predetermined factors, which we can call market factors.

**Conclusion**

The parallels between fluid dynamics and financial mathematics are clear. Does this mean fluid dynamics = financial mathematics? Yes, and no. Yes, because both fields use partial differential equations to model various real world phenomena, allowing a transfer of knowledge and tools between the two domains. No, because domain knowledge. I doubt a quantitative analyst in a hedge fund would be able to define the Reynolds number, and I’m not sure that a physical oceanographer would care to discuss spread. However, I believe that if equipped with the proper mathematical toolkit, a fluid dynamics specialist could become a financial markets specialist, and vice versa.

*
Also published on Medium. *

Wonderful write up! Just a small suggestion, the 3D Navier-Stokes should also contain the conservation of momentum in the z-direction. A full 3D N-S description contains 4 equations: conservation of mass, and momentum in the three (likely Cartesian) directions.

Again, this suggestion is quite minor. While most fluidicists would easily extrapolate the 4th equation, it may be a beneficial add for financial marketers who are less familiar with Navier-Stokes.

Hi John, glad you liked the post! Nice to have a comment from someone with fluid dynamics expertise. That’s a good point, although the ocean’s hydrostatic balance leads to the z-direction equation you see in the post, I suppose I can’t necessarily apply it to financial markets, so I could be more clear there. Thanks very much!

Thanks for the post, I am doing my masters in CFD and I love it, its good to know that such knowledge can help in other fields.

You’re welcome, Saeed! CFD is definitely a great field. All the best with your studies!

It is indeed a nice write-up. The content has merit.

I have worked on Black-Scholes model, heat equation, and the Navier stokes model. Though, the nonlinear BSM is yet to be linked with these.

is there an equation for the

futures trading markets

sp etc.