Brief Introduction to Simulation

Philip H. Dybvig

Washington University

Saint Louis, Missouri

- Analysis of portfolio strategies
- Option pricing
- Stochastic
*pro forma*analysis

- 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)

value probability uS p dS 1-pThis is implicit in our notation

|- uS -| |- dSwhere the up move has probability p.

value probability uS p dS 1-phas the distribution function

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

| 0 x < 0 F(x) = -| x 0 <= x < 1 | 1 1 <= xTherefore, for 0 <= a < b <= 1, The probability that y lies in the interval [a,b] is simply the length b-a of the interval.

| u z < p -| | d p <= zboth (1) as a list of values and probabilities and (2) as a distribution function. Assume that d < u.

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.

- 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.

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

// // 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();};

// // 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);}

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

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);}

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));}