# Using RcppArmadillo to price European Put Options

February 27, 2018
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### 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
incredibly powerful linear algebra library, Armadillo.

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

where

and

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

[1] 5.52021

[1] 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:

[1] 5.52021

[1] 5.52021 4.58142 3.68485 2.85517 2.11883 1.49793


Finally, let’s look at
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
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.

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