## Using RcppArmadillo to price European Put Options

Davis Vaughan and Dirk Eddelbuettel — written Feb 28, 2018 — source

### Introduction

In the quest for ever faster code, one generally begins exploring ways to integrate C++ with R using Rcpp. This post provides an example of multiple implementations of a European Put Option pricer. The implementations are done in pure R, pure Rcpp using some Rcpp sugar functions, and then in Rcpp using RcppArmadillo, which exposes the incredibly powerful linear algebra library, Armadillo.

Under the Black-Scholes model The value of a European put option has the closed form solution:

$V = K e^{-rt} N(-d_2) - S e^{-yt} N(-d_1)$

where

\begin{equation} \begin{aligned} V &= \text{Value of the option} \\ r &= \text{Risk free rate} \\ y &= \text{Dividend yield} \\ t &= \text{Time to expiry} \\ S &= \text{Current stock price} \\ K &= \text{Strike price} \\ N(.) &= \text{Normal CDF} \end{aligned} \end{equation}

and

\begin{equation} \begin{aligned} d_1 &= \frac{log(\frac{S}{K}) + (r - y + \frac{1}{2} \sigma^2)t}{\sigma \sqrt{t}} \\ d_2 &= d_1 - \sigma \sqrt{t}\\ \end{aligned} \end{equation}

Armed with the formulas, we can create the pricer using just R.

 5.52021

 5.52021 4.58142 3.68485 2.85517 2.11883 1.49793


Let’s see what we can do with Rcpp. Besides explicitely stating the types of the variables, not much has to change. We can even use the sugar function, Rcpp::pnorm(), to keep the syntax as close to R as possible. Note how we are being explicit about the symbols we import from the Rcpp namespace: the basic vector type, and well the (vectorized) ‘Rcpp Sugar’ calls log() and pnorm() calls. Similarly, we use sqrt() and exp() for the calls on an atomic double variables from the C++ Standard Library. With these declarations the code itself is essentially identical to the R code (apart of course from requiring both static types and trailing semicolons).

We can call this from R as well:

 5.52021

 5.52021 4.58142 3.68485 2.85517 2.11883 1.49793


Finally, let’s look at RcppArmadillo. Armadillo has a number of object types, including mat, colvec, and rowvec. Here, we just use colvec to represent a column vector of prices. By default in Armadillo, * represents matrix multiplication, and % is used for element wise multiplication. We need to make this change to element wise multiplication in 1 place, but otherwise the changes are just switching out the types and the sugar functions for Armadillo specific functions.

Note that the arma::normcdf() function is in the upcoming release of RcppArmadillo, which is 0.8.400.0.0 at the time of writing and still in CRAN’s incoming. It also requires the C++11 plugin.

Use from R:

        [,1]
[1,] 5.52021

        [,1]
[1,] 5.52021
[2,] 4.58142
[3,] 3.68485
[4,] 2.85517
[5,] 2.11883
[6,] 1.49793


Finally, we can run a speed test to see which comes out on top.

  test replications elapsed relative
2 Arma          100   6.409    1.000
3 Rcpp          100   7.917    1.235
1    R          100   9.091    1.418


Interestingly, Armadillo comes out on top here on this (multi-core) machine (as Armadillo uses OpenMP where available in newer versions). But the difference is slender, and there is certainly variation in repeated runs. And the nicest thing about all of this is that it shows off the “embarassment of riches” that we have in the R and C++ ecosystem for multiple ways of solving the same problem.