onwardmahachi/derivmkts

R package for option pricing

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

• Compound 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 easy. The function greeks() accepts an option pricing function call as an argument, and returns a vectorized set of greeks for any pricing function that uses standard input names (i.e., the asset price is s, the volatility is v, etc.).

As an example, the following calculation will produce the full complement of greeks for a call, for each of three strike prices. You can access the delta values, for example, using x['Delta', ].

x1 <- greeks(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

My favorite example is this small bit of code, which computes and plots all call and put Greeks for 500 options. This produces 16 plots in all:

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

This is a great illustration of how powerful R can be.

Binomial calculations

binomopt

By default the binomopt function returns the price of an American call. In adddition:

• putopt=TRUE returns the price of an American put.

• returngreeks=TRUE returns a subset of the Greeks along with the binomial parameters.

• returntrees=TRUE 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 binomopt function can return:

x <- binomopt(41, 40, .3, .08, 1, 0, 3, putopt=TRUE, american=TRUE,
              returntrees=TRUE)
x
$price
   price 
3.292948 

$greeks
       delta        gamma        theta 
-0.331656818  0.037840906 -0.005106465 

$params
         s          k          v          r         tt          d 
41.0000000 40.0000000  0.3000000  0.0800000  1.0000000  0.0000000 
     nstep          p         up         dn          h 
 3.0000000  0.4568067  1.2212461  0.8636926  0.3333333 

$oppricetree
         [,1]      [,2]     [,3]      [,4]
[1,] 3.292948 0.7409412 0.000000  0.000000
[2,] 0.000000 5.6029294 1.400911  0.000000
[3,] 0.000000 0.0000000 9.415442  2.648727
[4,] 0.000000 0.0000000 0.000000 13.584345

$stree
     [,1]     [,2]     [,3]     [,4]
[1,]   41 50.07109 61.14913 74.67813
[2,]    0 35.41139 43.24603 52.81404
[3,]    0  0.00000 30.58456 37.35127
[4,]    0  0.00000  0.00000 26.41565

$probtree
     [,1]      [,2]      [,3]       [,4]
[1,]    1 0.4568067 0.2086723 0.09532291
[2,]    0 0.5431933 0.4962687 0.34004825
[3,]    0 0.0000000 0.2950590 0.40435476
[4,]    0 0.0000000 0.0000000 0.16027409

$exertree
      [,1]  [,2]  [,3]  [,4]
[1,] FALSE FALSE FALSE FALSE
[2,] FALSE FALSE FALSE FALSE
[3,] FALSE FALSE  TRUE  TRUE
[4,] FALSE FALSE FALSE  TRUE

$deltatree
           [,1]        [,2]       [,3]
[1,] -0.3316568 -0.07824964  0.0000000
[2,]  0.0000000 -0.63298582 -0.1712971
[3,]  0.0000000  0.00000000 -1.0000000

$bondtree
         [,1]      [,2]      [,3]
[1,] 16.89088  4.658986  0.000000
[2,]  0.00000 28.017840  8.808828
[3,]  0.00000  0.000000 38.947430

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.

binomplot(41, 40, .3, .08, 1, 0, 3, putopt=TRUE, american=TRUE)

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():

quincunx(n=11, numballs=250, delay=0, probright=0.5)

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!