# Are the sin, cos, tan functions used in some financial calculations?

I ask because those functions are on the TI BA II Plus financial calculator.

• The question is weird because it comes from a funky reason. But I'm fine having somebody giving an example of trigonometric functions in quantitative finance. – SRKX May 9 '14 at 10:38
• sinh and cosh are used in some formulations of the Heston model of stochastic volatility, but you're not going to be doing those on a calculator. – experquisite May 9 '14 at 15:53
• @SRKX Now that I think about it, I don't think this reason is funky at all. Who else but the non-mathematically inclined would ask such? – BCLC May 11 '14 at 9:18

One can use the Karhunen–Loève expansion to approximate a trajectory of a Wiener Process, which can be used to model the evolvement of returns in time. (http://en.wikipedia.org/wiki/Karhunen%E2%80%93Lo%C3%A8ve_theorem#The_Wiener_process)

Though the Karhunen–Loève expansion has theoretical advantages to other variants to generate a trajectory of a Wiener Process, many users will use different methods because on computers evaluation of trignometric is very expensive in terms of calculation time.

Fourier methods use sine and cosine functions, and are used in calculating option prices, VaR, time series analysis etc. It is an alternative process for doing many things in finance. Some links Fourier Methods in trading on StackExchange and Wiki

You can use $\sin$ or $\cos$ to model seasonality. If all you have is a calculator it might be the most practical way.

When you do Monte Carlo simulation and would like to draw sample from the normal distribution $\mathcal{N}(\mu,\sigma^2)$, you may use Box-Muller transform and come up with formulas using $\sin$ and $\cos$.

Trigonometric functions show up in econometric models for business cycles. For example: the average length of a cycle of an AR(2) process is

$k = \frac{2 \pi}{\cos^{-1}( \phi_1/ (2 \sqrt{-\phi_2}))}$

For an AR(2) model given by $r_t = \phi_0 + \phi_1 r_{t-1} + \phi_2 r_{t-2} + a_t$

with complex roots, $\phi_1^2 + 4\phi_2 <0$

Solving some heat/diffusion equations under certain conditions needs trigonometric functions.

Black-Scholes reduces to a heat/diffusion equation by a change of variables.

• But the initial conditions in Black Scholes lead to the normal distribution as a solution (i.e. not trig functions)! – vonjd May 11 '14 at 11:33

Trigonometric functions are WAVE phenomena. As such, they are best used to model so-called periodic functions, that is, functions with cycles of a fixed period in length. That's why they are good for modelling, seasonal, annual, "blue moon" (once every two and half years), or other functions with set "periods."