COMPUTATIONAL FINANCE

Philip H. Dybvig
Washington University
Saint Louis, Missouri

Copyright © Philip H. Dybvig 1996

Some Simple Interest Derivatives

• Riskless Bonds
• Bond Options
  • Puts, Calls, Straddles, etc.
  • American, European, Down-and-Out, etc.
• Bond Futures and Futures Options
• Caps Floors, Collars
• Riskless Inverse Floaters

Some Complex Interest Derivatives

• Mortgages and CMOs
• Structured Loans
• Risky Corporate Bonds
• Callable and/or Convertible Bonds
• Foreign Exchange Futures Options
• Hybrid Securities, e.g. a binary option paying off if at maturity 3-mo LIBOR > 12% and the dollar is stronger against the yen than it was at the start of the contract

Why Not Simply Use Black-Scholes?

• The interest rate is not constant.
• The volatility is not constant.
• ``Today's price'' is not an asset price.
• We may want to value claims that are not simple combinations of puts and calls.

Binomial Pricing of Interest Derivatives

   |- r + delta

r -|

   |- r - delta

The interest rate is not an asset! Therefore, we can't use the formula from the previous lecture to compute the risk-neutral probabilities or state prices. There are several approaches:

• Make an assumption about the price process for some asset (e.g. a perpetuity).
• Make an assumption about the nature of supply and demand in the economy and compute equilibrium prices.
• Make an assumption about the interest rate process in the risk-neutral probabilities, e.g.
  • Random walk
  • Modest mean reversion (my preference)

A Random Walk or Modest Mean Reversion

E[Delta r]=k(r*-r)Delta t,

then

pi*_u delta+(1-pi*_u)(-delta)=k(r*-r) Delta t

or

pi*_u=(1/2)+(k(r^*-r)Delta t)/2 delta.

A random walk (good for short maturities and non-critical applications) corresponds to k=0.

Another issue: use uneven spacing to make interest rate volatility a function of the interest rate. Fudge factors can fit today's yield curve. Stochastic volatility and additional factors are harder.

Two Observations

About Intermediate Cash Flows When a claim includes intermediate cash flows (as for a coupon bond or a cap), the claim is simply added in at the appropriate time. For example, if V indicates the ex-cashflow value and C the cashflow, we have, at some node at time t, V(t,r)=(pi*_u(V(t,r+delta)+C(t,r+delta))+pi*_d(V(t,r-delta)+ C(t,r-delta)))/(1+r)

In-class Exercise: Bond Prices

In-Class Exercise: Bond Option Evaluation

The Header File cap.h

// cap.h
//
// Fixed-income binomial option pricing include file
//
#ifndef MAXTERNODES
#define MAXTERNODES 200
#endif
class f_i_bin {
public:
  f_i_bin(double ttm=.25,int npers=4,double sigma=.02,
    double rrbar=.07,double k=.125);
  double bprice(double r0);
  double cap(double level,double r0);
private:
  int nper;
  double tinc,up,down,sig,rbar,kappa,prfact;
  double r[MAXTERNODES],val[MAXTERNODES];};

The Test File captest.cc

// captest.cc
//
// Fixed income binomial option pricing model: test
//
#include <iostream.h>
#include "cap.h"
main() {
  f_i_bin c2;
  double caprate,rzero;
  cout << "\nType the cap rate, a space, the initial"
    << " rate, and then ENTER." << "\n"
    << "Make the cap rate negative to terminate." << "\n\n";
  cout << "State rates in percent per annum (without the "
    << "percent sign).\n\n";
  while(1) {
    cout << "cap-rate initial-rate: ";
    cin >> caprate >> rzero;
    if(!cin) {
      cout << "\nError: expected cap-rate initial-rate\n"
        << "Terminating\n" << endl;
      return(1);}
    if(caprate<0.0) {
      cout << endl;
      return(0);}
    cout << "cap value = " << c2.cap(caprate/100.0,rzero/100.0)
      << "\n\n";}}

The Implementation File cap.cc: Part 1

// cap.cc
//
// Fixed-income binomial option pricing engine
//
#include <math.h>
#include <iostream.h>
#include "cap.h"
#define MAX(a,b) (((a) > (b)) ? (a) : (b))
#define MIN(a,b) (((a) < (b)) ? (a) : (b))

The Implementation File cap.cc: Part 2

f_i_bin::f_i_bin(double ttm,int npers,double sigma,double rrbar,double k) {
  nper=npers;
  tinc = ttm/(double) nper;
  sig = sigma;
  up = sigma*sqrt(tinc);
  rbar = rrbar;
  kappa = k;
  prfact = kappa*sqrt(tinc)/(2.0*sig);}

The Implementation File cap.cc: Part 3

double f_i_bin::bprice(double r0) {
  int i,j;
  double prup;
//initialize terminal payoffs
//i is the number of up moves
  for(i=0;i<=nper;i++) {
//  r[i] = r0 + up * (double)(2*i-nper);   not needed for this claim
    val[i] = 1.0;}
//compute prices back through the tree
//j is the number of periods from the end
//i is the number of up moves from the start
  for(j=1;j<=nper;j++) {for(i=0;i<=nper-j;i++) {
      r[i] = r0 + up * (double) (2*i-nper + j);
      prup = 0.5 + prfact*(rbar-r[i]);
      prup = MIN(1.0,MAX(0.0,prup));
      val[i] = (prup*val[i+1]+(1.0-prup)*val[i])*exp(-r[i]*tinc);}}
  return(val[0]);}

The Implementation File cap.cc: Part 4

double f_i_bin::cap(double level,double r0) {
  int i,j;
  double prup;
//initialize terminal payoffs
//i is the number of up moves
  for(i=0;i<=nper;i++) {
//  r[i] = r0 + up * (double)(2*i-nper);   not needed for this claim
    val[i] = 0.0;}
//compute prices back through the tree
//j is the number of periods from the end
//i is the number of up moves from the start
  for(j=1;j<=nper;j++) {for(i=0;i<=nper-j;i++) {
      r[i] = r0 + up * (double) (2*i-nper + j);
      prup = 0.5 + prfact*(rbar-r[i]);
      prup = MIN(1.0,MAX(0.0,prup));
      val[i] = (prup*val[i+1]+(1.0-prup)*val[i])*exp(-r[i]*tinc)
        + MAX(0.0,(r[i]-level)*tinc);}}
  return(val[0]);}