derivmkts: An R repository from DanielFMiller

Introduction

This is a collection of option pricing functions for a course in financial derivatives. The names of the functions mostly match those in my book Derivatives Markets, which explains the package name. This is the initial version; I expect to add and refine functions over time. Depending upon the level of the course, this package may or may not be helpful for students. I hope that at a minimum it will be helpful for instructors.

There are of course other option pricing packages in R, notably RQuantLib and fOptions. I don't claim to add significant pricing functionality to those. This package does, however, have a few aspects that might be unique, which I describe below.

The package includes functions for computing

Black-Scholes prices and greeks for European options
Option pricing and plotting using the binomial model
Barrier options
Pricing of options with jumps using the Merton model
Analytical and Monte Carlo pricing of Asian options

Things of note

Calculation of Greeks

I have tried to make calculation of Greeks especially easy. The function bsopt() is vectorized, and simultaneously computes prices and Greeks for European calls and puts. There are also two generic functions, greeks() and greeks2(), which allow vectorized calculation of greeks for any option pricing calculation based on the standard Black-Scholes or barrier option inputs.

As an example, the following two calculations will produce identical output, computing the full complement of greeks for a call, for three strike prices. You can access the delta values as, for example, x['Delta', ].

x1 <- greeks(bscall(s=40, k=c(35, 40, 45), v=0.3, r=0.08, tt=0.25, d=0))
x2 <- greeks2(bscall, s=40, k=c(35, 40, 45), v=0.3, r=0.08, tt=0.25, d=0)
x1

             bscall_35   bscall_40   bscall_45
Price       6.13481997  2.78473666  0.97436823
Delta       0.86401619  0.58251565  0.28200793
Gamma       0.03636698  0.06506303  0.05629794
Vega        0.04364029  0.07807559  0.06755753
Rho         0.07106457  0.05128972  0.02576487
Theta      -0.01340407 -0.01733098 -0.01336419
Psi        -0.08640162 -0.05825156 -0.02820079
Elasticity  5.63352271  8.36726367 11.57705764

The function

y <- bsopt(s=40, k=c(35, 40, 45), v=0.3, r=0.08, tt=0.25, d=0)

will do the same for both calls and puts simultanteously. The delta values for the call would be y[['Call']]['Delta', ]

My favorite example, which you should run, is this:

k <- 100; v <- 0.30; r <- 0.08; tt <- 2; d <- 0
S <- seq(.5, 250, by=.5)
x <- bsopt(S, k, v, r, tt, d)
par(mfrow=c(4, 4))  ## create a 4x4 plot
for (i in c('Call', 'Put')) {
    for (j in rownames(x[[i]])) {  ## loop over greeks
        plot(S, x[[i]][j, ], main=paste(i, j), ylab=j, type='l')
    }
}

This small bit of code computes and plots all call and put Greeks for 500 options. This is 16 plots in all. It's a great illustration of how powerful R can be.

Binomial calculations

binomopt

By default the binomopt function returns the price of a American call. In adddition, with:

putopt=TRUE it returns the price of an American put.
returngreeks=TRUE it returns a subset of the Greeks along with the binomial parameters.
returntrees=TRUE it returns as a list all of the above plus the full binomial tree ($stree), the probability of reaching each node ($probtree), whether or not the option is exercised at each node ($exertree), and the replicating portfolio at each node ($deltatree and $bondtree).

Here is an example illustrating everything that the function can return:

binomplot

This function plots the binomial tree, providing a visual depiction of the nodes, the probability of reaching each node, and whether exercise occurs at that node.

Galton board or quincunx

The Galton board is a pegboard that illustrates the central limit theorem. Balls drop from the top and randomly fall right or left, providing a physical simulation of a binomial distribution. (My physicist brother-in-law tells me that real-life Galton boards don't typically generate a normal distribution because, among other things, balls acquire momentum in the direction of their original travel. The distribution is thus likely to be fatter-tailed than normal.)

You can see the Galton board in action with quincunx().

Feedback

Please feel free to contact me with bug reports or suggestions. Best would be to file an issue on Github, but email is fine as well.

I hope you find this helpful!

DanielFMiller/derivmkts