Computational Finance Lecture 3: Asset Allocation Simulation

COMPUTATIONAL FINANCE

Lecture 3: Asset Allocation Simulation
Brief Introduction to Simulation

Philip H. Dybvig
Washington University
Saint Louis, Missouri

Copyright © Philip H. Dybvig 1996

Some Leading Uses of Simulation in Finance

Analysis of portfolio strategies
Option pricing
Stochastic pro forma analysis

Some Simulation Strategies

Use pseudo-random numbers (Monte Carlo)
Use quasi-random numbers
Use historical data (back-testing)
Use pseudo-random numbers to draw from historical data (bootstrapping)

Representing Discrete Random Variables

A discrete random variable's distribution can be represented by listing the values and their probabilities:

   value    probability
     uS          p
     dS         1-p

This is implicit in our notation

 |- uS
-|
 |- dS

where the up move has probability p.

Representing a general random variable

For random variables that can take on a continuum of values, listing all outcomes is not feasible and we use instead the distribution function. The distribution function of a random variable y is defined by F(x)=prob(y<=x), i.e. the value of the distribution function at x is the probability that y is less than or equal to x.

Distribution Function: Example

Suppose dS<uS. Then the random variable

   value    probability
     uS          p
     dS         1-p

has the distribution function

        | 0    x < dS
F(x) = -| 1-p  dS <= x < uS
        | 1    uS <= x

Uniform Distribution

A uniform distribution is distributed uniformly on some interval (by default [0,1]). We have the shorthand y ~ U(a,b) to indicated that y is distributed uniformly on the interval [a,b]. For y ~ U(0,1), the distribution function is

 
        | 0    x < 0
F(x) = -| x    0 <= x < 1
        | 1    1 <= x

Therefore, for 0 <= a < b <= 1, The probability that y lies in the interval [a,b] is simply the length b-a of the interval.

In-class Exercise: Distribution Functions

Take y ~ U(0,1). Express the distribution of the random variable

 | u     z < p
-|
 | d     p <= z

both (1) as a list of values and probabilities and (2) as a distribution function. Assume that d < u.

Simulating a Uniform Distribution

The standard function rand returns a random int in the range from 0 to RAND_MAX. This random variable generator is available on all platforms but has a bit of a bad reputation (see, for example, Chapter 7 of the book Numerical Recipes in C, by Press et al, Cambridge, ISBN 0-521-35465-X); many compilers have better random number generators built-in. Using rand will be more than adequate for our purposes.

To generate a uniform variate, we take (double) rand()/(double) RAND_MAX. A variable that is U(0,1) has a mean of 1/2 and a variance of 1/12, so we can generate a uniform variate with mean m and standard deviation s as

m + s * (((double)rand()/(double)RAND_MAX)-0.5)*sqrt((double) 12).

This is implicit in our program this week.

Portfolio ``Accounting''

Make the following definitions.

C_t is the amount of cash after trading at t.
S^i_t is the position in stock i after trading at t.
r_t is the total return (1 + interest rate) on cash at t.
r^i_t is the total return (1 plus rate of return) on stock i at t.
d_i is the proportional cost for trading security i.

Then, absent taxes,

                    i   i    i            i   i    i
C  = C   r  + Sum (S   r  - S ) - Sum d |S   r  - S |.
 t    t-1 t      i  t-1 t    t       i i  t-1 t    t

With taxes, things are much messier. (See my paper with Hyeng-Keun Koo.)

The Header File simu1.h

//
// simu1.h Asset Allocation Header
//

struct termvals {
  double termstock;
  double termwealth;};

class aa {
  public:
    aa(double ttm=1.0,int nper=120,double r=.05,double mean=.15,double stddev=.2);
    int restart(double ttm,int nper,double r,double mean,double stddev);
    termvals fixprops(double inrisky,double initstock=0.0,double initcash=100.0);
  private:
    int npers;
    double tinc, r1per, mean1per, std1persqrt12, wealth, stockpos, cash, stockP, stockret;
    double stocktotret();};

The Test File simu1test.cc

//
// simu1test.cc Asset Allocation test file
//
#include <iostream.h>
#include "simu1.h"

main() {
  int i;
  aa sim1;
  termvals y;
  for(i=0;i<100;i++) {
    y = sim1.fixprops(.5);
    cout << y.termstock << " " << y.termwealth << endl;}
  return(0);}

The Implementation File simu1.cc: Part 1

//
// simu1.cc Asset Allocation Simulation
//
#include <math.h>
#include <stdlib.h>
#include <iostream.h>
#include "simu1.h"
#define unifrand() ((double) rand()/((double) RAND_MAX))

The Implementation File simu1.cc: Part 2

aa::aa(double ttm,int nper,double r,double mean,double stddev) {
  restart(ttm,nper,r,mean,stddev);}

int aa::restart(double ttm,int nper,double r,double mean,double stddev) {
  npers = nper;
  tinc = ttm/(double) nper;
  r1per = 1.0 + r*tinc;
  mean1per = 1.0 + mean*tinc;
  std1persqrt12 = sqrt((double) 12)*stddev*sqrt(tinc);
//  cout << " " << npers << " " << tinc << " " << r1per << " "
//    << mean1per << " " << std1persqrt12 << endl;
  return(0);}

The Implementation File simu1.cc: Part 3

termvals aa::fixprops(double inrisky,double initstock,double initcash) {
  int i;
  termvals x;
  stockpos = initstock;
  cash = initcash;
  stockP = 100.0;
//  cout << npers << endl;
  for(i=0;i<npers;i++) {
    wealth = stockpos + cash;
    stockpos = inrisky * wealth;
    cash = wealth - stockpos;
    stockret = stocktotret();
//cout << stockret << " " << tinc << " " << mean1per << " " << std1persqrt12 <<
//" " << unifrand() <<endl;
    stockP *= stockret;
    stockpos *= stockret;
    cash *= r1per;
//    cout << stockret << " " << stockP << " " << stockpos << " " << cash << endl;
    }
  x.termstock = stockP;
  x.termwealth = stockpos + cash;
  return(x);}

double aa::stocktotret() {
  return(mean1per + std1persqrt12 * (unifrand()-0.5));}