Special functions in option pricing

The functions behind the formulas

scipy.special collects the named functions that appear again and again in formulas. The most useful for finance is the normal cumulative distribution $\Phi(x)$, available as special.ndtr. It is tied to the error function by $\Phi(x)=\tfrac12\big(1+\operatorname{erf}(x/\sqrt2)\big)$. The Black–Scholes price of a call with spot $S$, strike $K$, rate $r$, volatility $\sigma$ and time $T$ uses it twice: $C=S\,\Phi(d_1)-K e^{-rT}\Phi(d_2)$, where $d_1=\dfrac{\ln(S/K)+(r+\sigma^2/2)T}{\sigma\sqrt T}$ and $d_2=d_1-\sigma\sqrt T$. The sensitivity of the price to volatility is called vega, and it equals $S\,\varphi(d_1)\sqrt T$, where $\varphi$ is the normal density. Vega is what a Newton solver needs to find an implied volatility.

The other group of special functions is about numbers that are too big or too small. A factorial grows so fast that $171!$ already overflows a normal decimal number in a computer, but $\ln n!$ is easy: special.gammaln(n+1). special.comb(n, k) counts the paths with $k$ up-moves in an $n$-step tree. special.logsumexp computes $\ln\sum e^{x_i}$ safely by pulling out the largest $x$, so that $\ln(e^{1000}+e^{1000})=1000+\ln2$ does not overflow. special.expit(x)=1/(1+e^{-x}) turns any score into a probability between 0 and 1, as in a logistic model. The rule is: work in logs when a number can be huge or tiny, and exponentiate only at the end.

• Name the function: ndtr, erf, expit, gammaln, comb and logsumexp replace pages of hand-written maths.
• Work in logs for extremes: use gammaln and logsumexp instead of a factorial or a sum of huge exponentials.
• Vega drives the solver: a Newton step for implied volatility is the price gap divided by vega.

Lab · what's graded

• The normal function from erf10 pts
• A Black–Scholes call20 pts
• Vega10 pts
• An implied-volatility step15 pts
• What each function is for15 pts
• The safe way to add10 pts
• Why compute factorials through logs20 pts