Homework 4
Stochastic Volatility Option Pricing

Computational Finance

Copyright © Philip H. Dybvig 1996

Scenario You are working as a consultant in an accounting firm. One of your charges is the valuation of employee stock options. The question has been raised of the effect of stochastic volatility on valuation of the options granted to officers of the company.

Action You will be given a program that uses simulations to evaluate options when the volatility of the stock is random. You will use the program to evaluate some call options, and you will perform a statistical analysis of the resultant values to form an opinion of the accuracy of the simulation.

Concept Simulation is a powerful tool for computing option values. When the binomial model can be used, it is much quicker, but simulation is easier to program and more flexible and is the only practical technique for many hard problems in practice.

Instructions

1. Put the program files (simu2test.cc, simu2.cc, and simu2.h ) in a directory on your machine where you want to work. Rename them as necessary. Compile the programs.
2. Write to a file the results of 100 valuations of 1000 simulations each. Use your favorite statistical package or spreadsheet to compute the mean (best estimate) and standard error (= standard deviation/sqrt(100.0)) of the values from the 100 runs. (If you prefer compute the mean and standard error in C++ instead.) You may use either the default parameters (for interest rate, volatility, etc.) or choose your own.
3. (thought question) What is the precision given the 100,000 (=100*1000) draws? Given that the typical error is proportional to the reciprocal of the square root of the number of simulations, how many simulations should be required for pricing to the nearest penny?

1. Modify the program to allow the random part of the stock price change to be correlated with the random part of the change in volatility.

1. Estimate a GARCH or ARCH model for volatility, using actual data from a common stock or common stock index. (This is probably meaningful to you only if you have taken Advanced Econometrics.) Modify the C++ programs to use the model and parameters you estimate.

//
// simu2.h  Stochastic Volatility Option Pricing Model
//

class svprice {
  public:
    svprice(double ttm=0.25,int nper=12,double r=.05,double initstd=.15,
      double meanstd=.2,double k=6.0,double sigstd=.5);
    double eurcall(double stock,double strike,long int nsimu=1000);
  private:
    int npers;
    double tinc, r1per, stockP, sigma0, sigma, meansigma, sigmasigma, kappa,
      c0, c1, c2, c3, c4, c5;
    double stocktotret();};

//
// simu2test.cc Stochastic Volatility Option Pricing Test File
//
#include <iostream.h>
#include <math.h>
#include "simu2.h"

main() {
  long int i,nsimus;
  svprice sim2;
  for(nsimus=10;nsimus<=100000l;nsimus *= 10)
    cout << nsimus << " " <<
      floor(sim2.eurcall((double) 100.0,(double) 100.0,nsimus)*100.0+0.5)/100.0
      << endl;
  cout << endl;
  for(i=0;i<20;i++)
    cout << "100000 " << 
      floor(sim2.eurcall((double) 100.0,(double) 100.0,(long int) 100000l)*100.0+0.5)/100.0
      << endl;
  return(0);}

//
// simu2.cc  Stochastic Volatility Option Pricing Model
//
#include <math.h>
#include <stdlib.h>
#include <iostream.h>
#include "simu2.h"
#define unifrand() ((double) rand()/((double) RAND_MAX))
#define MAX(a,b) (((a) > (b)) ? (a) : (b))

svprice::svprice(double ttm,int nper,double r,double initstd, double
    meanstd, double k, double sigstd) {
  npers = nper;
  tinc = ttm/(double) nper;
  r1per = 1.0 + r*tinc;
  sigma0 = initstd;
  meansigma = meanstd;
  sigmasigma = sigstd;
  kappa = k;
  c0 = kappa * tinc * meansigma;
  c1 = 1.0 - kappa * tinc;
  c2 = 1.0 - sigmasigma * sqrt(tinc)*0.5*sqrt((double) 12);
  c3 = sigmasigma * sqrt(tinc) * sqrt((double) 12);
  c4 = sqrt(tinc)*sqrt((double) 12);}

double svprice::eurcall(double stock,double strike,long int nsimu) {
  long int i,n;
  double x;
  x=0.0;
  for(n=0;n<nsimu;n++) {
    stockP = stock;
    sigma = sigma0;
    for(i=0;i<npers;i++) {
      stockP *= stocktotret();
      }
    x += MAX(stockP-strike,0);}
  return(x/((double) nsimu * pow(r1per,(double) npers)));}

double svprice::stocktotret() {
//
//  The following straightforward computations are algebraically the same as
//  the ones used below but are much slower because many more calculations are
//  performed in each pass through the loop.
//
//  sigma = (kappa*tinc * meansigma + (1.0 - kappa * tinc) * sigma) *
//    (1 + sigmasigma * sqrt(tinc) * (unifrand()-0.5) * sqrt((double) 12));
//  return(r1per + sigma * sqrt(tinc) * (unifrand()-0.5) * sqrt((double) 12));}
//
  sigma = (c0 + c1 * sigma) * (c2 + c3 * unifrand());
  return(r1per + sigma * c4 * (unifrand()-0.5));}