c++lanquage for financial engineering

8/18/2019 C++lanquage for financial engineering

1/113

Financial Numerical Recipes in C++

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Bernt Arne degaard

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/ Contents

• $%

• &

• '()*(cash flow)

'+ ,-

./ #01(IRR)

23 4563 78

• 9:;8 ?@; 2(2@2)

o AB

o 4CD E> F LM NOPGHI- JK 2@

§


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§ xy

§ xyg z\ qr3 4CD


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o N/3 ,Q •u

o N/


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o &0; ef6(stochastic volatility)

• JK ?r ?@

o &

o JK ?r LMNcO

o ELP(short sales)3 Cl BQ

• 9R AB3 abcd

o 9R AB3 0r

o STU 9R AB3 gV

§ ND tR

§ #043 ND tR

§ NP(c3 ND tR

o 9R AB3 &u

§ Nelson Siegel W#D (1987)

§ Bliss R~ (1989)

§ 3 % X%8

o 9RAB RD(Term structure models)

§ Vasicek R~

§ Cox Ingersoll Ross R~

§ Y813 0r

§ ^_̀ X!"

§ CIR R~k 3l &u

§ Y81

• bu #0 R~Z(BQ/ [AP\ ]^WX)

o Black Scholes RDk 3l bond pricing.

§^_ ̀ X!"




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o Rendleman Bartter R~

§ _X ̀ ab\ c2&

o Vasicek R~

• ud ?L3 ij-

o .efX

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o p, ij3 k#

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o ( CnnX!opq cl Cnn2•

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o #include 3 AB

o xoX3 Ay

o z{ |Dk zD&3 abcd A'(implementation)

o .efX

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• }b ~•




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/ Introduction

+! "3q Y # q$ %~&\ .Vg $D' #-q 0r2$\ EFk G2 () %~\ &\z.*CX D+,

0r-\[q q.3 /WX 78-\ 0) LT ]B-z. g ~&k&\, (1k jV2\ 3A3 abcdk G2

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Aq\C++X!o9\ NOBz. Aq\ Y # z\ l AC |Dk 3D-[ E, abcd\ FV;WX(generic )

9G-5b gHB[}, 3A3 abcdk G2&\ _X ̀ C++3 STL (The Standard Template Library )q gVY

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b g 0r3 C++p, ij\


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double cash_flow_pv_discrete( vector& cflow_times,

vector& cflow_amounts,

double r){double PV=0.0;

for (unsigned int t=0; t


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double x1=0. 0;

double x2 = 0.2;

// create an initial bracket, with a root somewhere between bot, top

double f1 = cash_flow_pv(cflow_times, cflow_amounts, x1);

double f2 = cash_flow_pv(cflow_times, cflow_amounts, x2);

int i;

for (i=0;i


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// author Bernt A Oedegaard

#include #include

inline int sgn(double& r){ if (r>=0) {return 1;} else {return - 1;}; };

bool cash_flow_unique_irr(vector& cflow_times, vector& cflow_amounts) {

// check whether the cash flow has a unique irr.

int sign_changes=0; // first check Descartes rule

for (unsigned t=1;t,[X ?;Y # zz. gr,R RD3 |l) g0\ vA-7Y, *k& jVY # o?l #a\ jV-[ E\z\ lgz. z{ lU, vw,R RD) 4 o?l #c; [F\ BAlz.

vw,R RDk zD V?IuF\ gV-2 AY # >\ ST #-2@9m(numerical method ) y, gr,R RDWX 2q P@Y

# zz. vw,R RD\ G]-\ RDWX\ Black- Scholes(1973)RDg zb, gr,R RD\ G]-\ RD) Cox, Ross, and

Rubinstein(1979) RDg zz. *C;WX gr,R RDWX 0rU


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• AB

• 4CDE> F F


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o " (Gamma)

o #(Theta)

o $ (Vega)

o % (Rho)

o Black Scholes UV? abcdA'

• .+ef6(Implied volatility)

o ghijk 3l abcd A'

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#include // mathematical library

#include "normdist.h" // this defines the normal distribution

double option_price_call_black_scholes( double S, // spot price

double X, // Strike (exercise) price,

double r, // interest rate

double sigma,

double time)

{

double time_sqrt = sqrt(time);

double d1 = (log(S/X) +r*time)/(sigma*time_sqrt) + 0.5*sigma*time_sqrt;

double d2 = d1- (sigma*time_sqrt);

double c = S * N(d1) - X * exp(- r*time) * N(d2);

return c;

};

LM=NO

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// author: Bernt A Oedegaard

// The delta of the Black - Scholes formula

#include

#include "normdist.h"

double option_price_delta_call_black_scholes(double S, // spot price



double sigma, // volatility

double time){ // time to maturity



double delta = N(d1);

return delta;

};

z{ []\ PQR jV2[ E[}, !0Z) &X S0, zz.

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X'( , Vega)

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double time, // time to maturity

double& Delta, // partial wrt S

double& Gamma, // second prt wrt Sdouble& Theta, // partial wrt time

double& Vega, // partial wrt sigma

double& Rho){ // partial wrt r




Delta = N(d1);

Gamma = n(d1)/(S*sigma*time_sqrt);

Theta =- (S*sigma*n(d1)) / (2*time_sqrt) - r*X*exp( - r*time) *N(d2);

Vega = S * time_sqrt * n(d1);

Rho = X * time * exp(- r * time) * N(d2);

};

)&à2

(implied volatility)

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', pp 102~103, unpublished manuscript )




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)&`a2(IV)/ YZ[ \]^_ (binomial search) / b=c(bisection method)

// file black_scholes_imp_vol_bisect.cc// author: Bernt A Oedegaard

// calculate implied volatility of Black Scholes formula

#include "fin_algoritms.h"

#include

double option_price_implied_volatility_call_black_scholes_bisections(

double S, double X, double r, double time, double option_price)

{ // check for arbitrage violations:// if price at almost zero volatility greater than price, return 0

double sigma_low=0.0001;

double price = option_price_call_black_scholes(S, X, r, sigma_low, time);

if (price>option_price) return 0.0;

// simple binomial search for the implied volatility.

// relies on the value of the option increasing in volatility

const double ACCURACY = 1.0e- 5; // make this smaller for higher accuracy

const int MAX_ITERATIONS = 100;

const double HIGH_VALUE = 1e10;

const double ERROR = - 1e40;

// want to bracket sigma. first find a maximum sigma by finding a sigma

// with a estimated price higher than the actual price.

double sigma_high=0.3;

price = option_price_call_black_scholes(S, X, r, sigma_high, time);

while (price HIGH_VALUE) return ERROR; // panic, something wrong.

};

for (int i=0;i


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else { sigma_high = sigma; }

};

return ERROR;};

Newton- Raphsonc

ghijk 3l ^;Im G.k, *C g_3 Img 0r=3 wP, `% jV W# zz.65 e#3 IuF3 2

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)&`a2(IV)/ YZ[ \]^_ / de- fg hc(Newton steps)

// file black_scholes_imp_vol_newt.cc// author: Bernt A Oedegaard

// calculate implied volatility of Black Scholes formula using newton steps



#include

double option_price_implied_volatility_call_black_scholes_newton(

double S, double X, double r, double time, double option_price){

// check for arbitrage violations:

// if price at almost zero volatility greater than price, return 0

double sigma_low = 1e- 5;

double price = option_price_call_black_scholes(S, X, r, sigma_low, time);

if (price >option_price) return 0.0;

const int MAX_ITERATIONS = 100;

const double ACCURACY = 1.0e- 4;

double t_sqrt = sqrt(time);

double sigma = (option_price/S)/(0.398*t_sqrt); // find initial value

for (int i=0;i


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// file: black_scholes_price_payout_call.cc


// Calculation of the Black Scholes option price formula,

// special case where the underlying is paying out a yield of q.




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#include // mathematical library

#include "normdist.h" // this defines the normal distribution

double option_price_european_call_payout( double S, // spot price



double q, // yield on underlying


double time) { // time to maturity

double sigma_sqr = pow(sigma, 2);


double d1 = (log(S/X) + (r- q + 0.5*sigma_sqr) *time)/(sigma*time_sqrt);


double call_price = S * exp(- q*time)* N(d1) - X * exp(- r*time) * N(d2);

return call_price;

};

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vector& dividend_times,

vector& dividend_amounts )

// reduce the current stock price by the amount of dividends.{

for (int i=0;i


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• :xy [s5

• :


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g # -

Uz.

Roll- Geske- Whaley call89 |v \]^_ }"(RGW)

// file anal_price_am_call_div.cc


// calculate dividend adjusted formula for american call option.

#include

#include "normdist.h" // define the normal distribution functions

#include "fin_algoritms.h" // the regular black sholes formula

double option_price_american_call_dividend(double S,

double X,

double r,

double sigma,

double tau,

double D1,

double tau1)

{




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if (D1 0. 0)

&& (S_high1e10) { // early exercise never optimal, find BS value

return option_price_call_black_scholes(S- D1*exp(- r*tau1), X, r, sigma, tau);

};

S_bar = 0.5 * S_high; // now find S_bar that solves c=S_bar- D+X

c = option_price_call_black_scholes(S_bar, X, r, sigma, tau- tau1);

test = c- S_bar- D1+X;

while ( (fabs(test) >ACCURACY)

&& ((S_high- S_low) >ACCURACY) ) {

if (test


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/ (sigma*tau1_sqrt);

double b2 = b1 - sigma * tau1_sqrt;

double C = (S- D1*exp(- r*tau1)) * N(b1)+ (S- D1*exp(- r*tau1)) * N(a1, - b1, rho)

- (X*exp(- r*tau)) *N(a2, - b2, rho)

- (X- D1) *exp(- r*tau1) *N(b2);

return C;

};

Chapter4. b~I@l /Q 89 ': ;


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4.1

gh ?imWX


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3 W#X+ E


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4CD


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for (int step=steps- 1; step>=0; - - step) {

for (int i=0; i


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for (int i=1; i


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vectorprices (no_steps+1);

vectorcall_values (no_steps+1);

double R = exp(r*(t/no_steps));double Rinv = 1.0/R;

double u = exp(sigma*sqrt(t/no_steps));

double d = 1.0/u;

double uu= u*u;

double pUp = (R- d)/(u- d);

double pDown = 1.0 - pUp;

prices[0] = S*pow(d, no_steps);

int i;

for (i=1; i


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double& gamma, // second prt wrt S

double& theta, // partial wrt time

double& vega, // partial wrt sigmadouble& rho) // partial wrt r

{

vectorprices(no_steps+1);

vectorcall_values(no_steps+1);

double delta_t =(time/no_steps);

double R = exp(r*delta_t);

double Rinv = 1.0/R;

double u = exp(sigma*sqrt(delta_t));

double d = 1.0/u;

double uu= u*u;



prices[0] = S*pow(d, no_steps);

for (int i=1; i


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double f00 = call_values[0];

delta = (f11- f10)/(S*u- S*d);double h = 0.5 * S * ( uu - d*d);

gamma = ( (f22- f21)/(S*(uu- 1)) - (f21- f20)/(S*(1- d*d)) ) / h;

theta = (f21- f00) / (2*delta_t);

double diff = 0.02;

double tmp_sigma = sigma+diff;

double tmp_prices

= option_price_call_american_binomial(S, X, r, tmp_sigma, time, no_steps);

vega = (tmp_prices- f00) /diff;

diff = 0.05;

double tmp_r = r+diff;

tmp_prices = option_price_call_american_binomial(S, X, tmp_r, sigma, time, no_steps);

rho = (tmp_prices- f00) /diff;

};

Subsections

• () xy

• gr xy

o ghRDk 3l gr xy */3 ,Q •u3 ^_̀ X!"

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// start a new tree at each ex- dividend date.

// do this recursively, at each ex dividend date, at each step, call the

// binomial formula starting at that point to calculate the value of the live// option, and compare that to the value of exercising now.

{

int no_dividends = dividend_times.size();

if (no_dividends == 0) // just take the regular binomial

return option_price_call_american_binomial(S, X, r, sigma, t, steps);

int steps_before_dividend = (int)(dividend_times[0]/t*steps);

double R = exp(r*(t/steps));

double Rinv = 1.0/R;

double u = exp(sigma*sqrt(t/steps));

double uu= u*u;

double d = 1.0/u;



double dividend_amount = dividend_amounts[0];

vectortmp_dividend_times(no_dividends- 1); // temporaries with

vectortmp_dividend_amounts(no_dividends- 1); // one less dividend

for (int i=0;i


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// less the dividend.

call_values[i] = max(value_alive, (prices[i]- X)); // compare to exercising now

};for (int step=steps_before_dividend- 1; step>=0; - - step) {

for (int i=0; i


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4CD [}, (e3 $;WX D+, 0rg g6D[\[q YZ.9

c2& •4lz.

+,7 0Q*=c (implicit finite differences)

g-3 UX\ Hull (1993, P.354) 3 g\ i7X -b zz.g abcd 672 -9 c2 NDG#(linear algebra )q

jV-Lz.

#include

#include "newmat.h" // definitions for newmat matrix library

#include // standard STL vector template

#include

double option_price_put_european_finite_diff_implicit(

double S, double X, double r, double sigma, double time,

int no_S_steps, int no_t_steps)

{

double sigma_sqr = sigma*sigma;

// need no_S_steps to be even:

int M; if ((no_S_steps%2)==1) { M=no_S_steps+1; } else { M=no_S_steps; };

double delta_S = 2.0*S/M;

vectorS_values(M+1);

for (int m=0;m


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};

return F.element(M/2);

};

-.7 0Q*=c(explicit finite differences)

;j,

#A2[ E\z\ ~Clg zz.g-3 UX\ Hull (1993, P.356) \ i7X -b zz.

// findiff_exp_eur_put.cc


#include

#include

#include

double option_price_put_european_finite_diff_explicit(double S,

double X,

double r,

double sigma,

double time,

int no_S_steps,

int no_t_steps) {


unsigned int M; // need M = no_S_steps to be even:

if ((no_S_steps%2)==1) { M=no_S_steps+1; } else { M=no_S_steps; };



for (unsigned m=0;m


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b[j] = r1*(1.0- sigma_sqr*j*j*delta_t);

c[j] = r2*0.5*j*(r+sigma_sqr*j);

};vectorf_next(M+1);

for (unsigned m=0;m=0;- - t) {

f[0]=X;

for (unsigned m=1;m


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// need no_S_steps to be even:

int M; if ((no_S_steps%2)==1) { M=no_S_steps+1; } else { M=no_S_steps; };

double delta_S = 2.0*S/M;double S_values[M+1];

for (int m=0;m


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double r,

double sigma,

double time,int no_S_steps,

int no_t_steps) {


int M; // need M = no_S_steps to be even:

if ((no_S_steps%2)==1) { M=no_S_steps+1; } else { M=no_S_steps; };



for (int m=0;m


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NZR, t ,1ε ] t ,2ε m L1LAJhifB ?>2 hi+; j'.@. Rx( U{{W6 ?m BX- g2 gfJ( NO* JN+;

Kw.q, 3kl .JFB.1p Kh 8m nu+,i WXW ' ?@. Z\BX. e; `a( XYZVi ^W\.O( 2 -i, 0"

1 6^, NO( European asian spread option)( 6FpL- .Kw. 0MY# )MP ' ?@. .-* jZ- R6y BX( XoQ^_(

k.- ST3n6 (+Ym NO.@. .-- {2 sMz VBA Codem F The Complete Guide To Option Pricing FormulasG , Espen Gaarder

Haug, 1998, p142 V pnY$ U@. (ABC)




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g0) @0AB, 9R MHq 1l qr3 MAk 3D-\ ST3 gr2 Im(discretization )gz.g0k S2&\

g; Fk G2&\, SXk 3D;g[ E9 #~k- D: ]R3

SGSk 3D-[ E\- 9/qr3 f: SGS}\ ,-.g= Y AB6g zz.

Monte Carle methodl /Q 0?@ 89/ 7k

// file: simulated_call_euro.cc

// author: Bernt Arne Oedegaard

#include // standard mathematical functions

#include "random.h" // definition of random number generator

#include // define the max() function

double option_price_call_european_simulated( double S,

double X,

double r,

double sigma,

double time,

int no_sims)

{

double sigma_sqr = sigma * sigma;

double R = (r - 0.5 * sigma_sqr) *time;

double SD = sigma * sqrt(time);

double sum_payoffs = 0.0;

for (int n=1; n


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g\,

\


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q z, abm\ gV2 0\ 0WX


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double nn = 2.0*b/sigma_sqr;

double m = 2.0*r/sigma_sqr;

double K = 1.0- exp(- r*time);double q2 = (- (nn- 1) +sqrt(pow((nn- 1), 2.0)+(4*m/K)))*0.5;

// seed value from paper

double q2_inf = 0.5 * ( (- nn- 1.0) + sqrt(pow((nn- 1), 2.0)+4.0*m));

double S_star_inf = X / (1.0 - 1.0/q2_inf);

double h2 = - (b*time+2.0*sigma*time_sqrt)*(X/(S_star_inf- X));

double S_seed = X + (S_star_inf- X)*(1.0- exp(h2));

int no_iterations=0; // iterate on S to find S_star, using Newton steps

double Si=S_seed;

double g=1;

double gprime=1.0;

while ((fabs(g) >ACCURACY)

&& (fabs(gprime) >ACCURACY) // to avoid exploding Newton's

&& ( no_iterations++0.0)) {

double c = option_price_european_call_payout(Si, X, r, b, sigma, time);

double d1 = (log(Si/X)+(b+0.5*sigma_sqr) *time)/(sigma*time_sqrt);

g=(1.0- 1.0/q2) *Si- X- c+(1.0/q2) *Si*exp((b- r) *time) *N(d1);

gprime=( 1.0- 1.0/q2)*(1.0- exp((b- r) *time) *N(d1))

+(1.0/q2) *exp((b- r) *time) *n(d1)*(1.0/(sigma*time_sqrt));

Si=Si- (g/gprime);

};

double S_star = 0;

if (fabs(g) >ACCURACY) { S_star = S_seed; } // did not converge

else { S_star = Si; };

double C=0;

double c = option_price_european_call_payout(S, X, r, b, sigma, time);

if (S>=S_star) {

C=S- X;

}

else {

double d1 = (log(S_star/X)+(b+0.5*sigma_sqr) *time)/(sigma*time_sqrt);

double A2 = (1.0- exp((b- r) *time) *N(d1))* (S_star/q2);

C=c+A2*pow((S/S_star), q2);

};

return max(C, c); // know value will never be less than BS value




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}

q!-,\ Barone Adesi 2% ijq }B. p, ij\ p {~ Barone- Adesi and Whaley (1987) k zz.g ijk G2&\ Hull (1993) 3 >= 14A k&P ?Y # zz.

7.2 Geske- Johnsoncl :;t Mw@89 C ':/ 89

Geske and J ohnson (1984) k&\


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Chapter 8. ?@ \]^_(Futures algoritms)

: |k&\ N/ 0S3 ,-J,k jV2\ abcdk G2 {32tbq lz.

Subsections

• N/0S3 ,Q+u(Pricing of futures contract)

• N/


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• ghR~k 3l &u

8.2?@89

(Options on futures)

N/k Gl


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#include // mathematics library

#include "normdist.h" // normal distribution

double futures_option_price_call_european_black(

double F, // futures price

double X, // exercise price



double time) // time to maturity

{



double d1 = (log (F/X) + 0.5 * sigma_sqr * time) / (sigma * time_sqrt);

double d2 = d1 - sigma * time_sqrt;

return exp(- r*time)*(F * N(d1) - X * N(d2));

};

8.2.2b~I@l /Q #


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futures_prices[0] = F*pow(d, no_steps);

int i;

for (i=1; i


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BC 89

0?@

89

4CD


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double X, // exercise,

double r, // r_domestic,

double r_f, // r_foreign,double sigma, // volatility,

double time) // time to maturity

{



double d1 = (log(S/X) + (r- r_f+ (0.5*sigma_sqr)) * time)/(sigma*time_sqrt);

double d2 = d1 - sigma * time_sqrt;

return S * exp(- r_f*time) * N(d1) - X * exp(- r*time) * N(d2);

};

D] jE

4CD 12


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double u = exp(sigma*sqrt(t_delta));

double d = 1.0/u;

double uu= u*u;double pUp = (exp((r- r_f) *t_delta) - d)/(u- d); // adjust for foreign int.rate


exchange_rates[0] = S*pow(d, no_steps);

int i;

for (i=1; i


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2D, vwD3 ST

Uz.z} gq1 ) 5u2, g BQ\ 9R AB, WJR(flat)Xgzb lz.

vwD3 ST3 3P ,Q3 0rIm\ ,[IF\ ]92:z.

// file bonds_price_both.cc

// author: Bernt A Oedegaard.

#include

#include

double bonds_price(const vector& coupon_times,

const vector& coupon_amounts,

const vector& principal_times,

const vector& principal_amounts,

const double& r)

// calculate bond price when term structure is flat,

// given both coupon and principals

{

double p = 0;

for (unsigned i=0;i


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const vector& cashflows,

const double& r) {

// calculate bond price when term structure is flat,double p=0;

for (unsigned i=0;i p( .`q gV-b z[}, lU) '()*(cash flow ) -k gV-b z[

Ez.ST> p(3 %g\ !z[ ]B-[ Eb, ST3 2h p(8 '()*q nM-@ &z.g %g, ]B-, 2\

STP zD, !(\ LWl ST, ! ]3 -Ygz.! ST\ D: 0g gqg\, D: 0g p:gI\[q ?4J Y

AB, zz.!mY 29k&\ 67l STq nM-b zW^X '()*(cash flow )}\ tb zW@

q$(.).X!") UV9 #W[b zz.g-k gr(v%) oc0r3 STq tq.

// file bonds_price.cc


#include

#include

double bonds_price_discrete(const vector& cashflow_times,


const double& r) {

// calculate bond price when term structure is flat,

double p=0;

for (unsigned i=0;i


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z} ) J,q *,Y # M k zD&3 [s\ Y8-\ W#gz.8q ZD 9R AB, JRl ST\ 67l

Y8W#, Uz.

9R ABq LWl o7term_structurexoXq ,u2P z{z.abcd\ YZ.\ Yg[3 $;WX 2 Y81

b\ 5jq +5$\ 9F\ &@Y # zW@ &Wd(gz.

// file bonds_price_termstru.cc

// author: Bernt Arne Odegaard

#include

#include "term_structure_class.h"

double bonds_price(const vector& cashflow_times,


const term_structure_class& d) {

double p = 0;

for (unsigned i=0;i


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10.3H6b9

10.3.1 IJ H6b9

29&,

• \ ,lk zD& '()*(cash flow)g\,

• \ Y81g\,$1 q [s-\ Y833 ,lk zD&3 '+,Q

TN, j)k&3 9RAB\ YZ(flat)-zb :z.

// file bonds_duration.cc

// author: Bernt Arne Odegaard

#include

#include

double bonds_duration(const vector& cashflow_times,


const double& r)

// calculate the duration of a bond, simple case where the term

// structure is flat, interest rate r.

{

double S=0;

double D1=0;

for (unsigned i=0;i;WX\




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// file duration.cc


#include

#include "term_structure_class.h"

double bonds_duration(const vector& cashflow_times,

const vector& cashflow_amounts,

const term_structure_class& d ) {

double S=0;

double D1=0;

for (unsigned i=0;i


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const double& bond_price) {

double y = bonds_yield_to_maturity(cashflow_times, cashflows, bond_price);return bonds_duration(cashflow_times, cashflows, y); // use YTM in duration

};

10.3.3 +


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// file bonds_convexity.cc

// author: Bernt A Oedegaard.// calculate convexity of a bond.

#include

#include

double bonds_convexity(const vector& cashflow_times,

const vector& cashflow_amounts,

const double& y )

{

double C=0;

for (unsigned i=0;i


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// file exotics_lookback_call.cc


#include


double exotics_lookback_european_call(double S,

double Smin,

double r,

double q,

double sigma,

double time){

if (r==q) return 0;

double sigma_sqr=sigma*sigma;


double a1 = (log(S/Smin) + (r- q+sigma_sqr/2.0) *time)/(sigma*time_sqrt);

double a2 = a1- sigma*time_sqrt;

double a3 = (log(S/Smin) + (- r+q+sigma_sqr/2.0) *time)/(sigma*time_sqrt);

double Y1 = 2.0 * (r- q- sigma_sqr/2.0) *log(S/Smin) /sigma_sqr;

return S * exp(- q*time) *N(a1)

- S * exp(- q*time)*(sigma_sqr/(2.0*(r- q))) *N(- a1)

- Smin * exp(- r*time)*(N(a2)- (sigma_sqr/(2*(r- q))) *exp(Y1) *N(- a3));

};

2m,[ g; ;8 Im\

YZ[z.

D] jE

Hull (1993) 3 16|k gml


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Subsections

• qr ,Q3 ,-.g=

• @0AB3 u3

• ,Q •u

• &u-3 ANe#3 CD

jk ':/ .56b9

`|a\ D+, ,Qg ,-.g= 2\[q \btq. 2 A STq nMlz. 1 A\ ~3 ,lk zD&3 f:

,Q}g ABl STgz. g0) g 9/qrk} 3D-\ pg f: ,Q}g ]Bl

ut, 2\ STgz.

// file simulate_underlying_terminal.cc

#include

#include "random.h"

double simulate_terminal_price(double S, // current value of underlying


double sigma, // volatitily

double time) // time to final date

{

double R = (r - 0.5 * pow(sigma, 2) ) *time;

double SD = sigma * sqrt(time);

return S * exp(R + SD * random_normal());

};

tz 5>;WX\ $n=l3 M9Rk Uc& zdl ,lk zD&3 ,Q\ ,-.g= Y AB, zz. abcd3

A')


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void simulate_price_sequence(double S, // current value of underlying

double r, // interest ratedouble sigma, // volatitily

double time, // time to final date

int no_steps, // number of steps

vector& prices ) {

if (prices.size()


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double payoff_european_put(double& price, double& X){

return max(0.0, X- price);};

zI3 8X& ,Q3 u3 fG b\ fUk 3D-\ l e#3 fG N1\

r@lz. g0) fik C++ |Dk &,U /?g\, STL(Standard Template Library)3 abcdgz.

bz{ 8: JK


72/113


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double derivative_price_european_simulated ( double S, double r, double sigma, double time,

double payoff(vector& prices),int no_steps, int no_sims) {

double sum_payoffs=0;

vector prices(no_steps);

for (int n=0; n


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double payoff(vector& prices),

int no_steps,

int no_sims){

double X = S;

double c_bs = option_price_call_black_scholes(S, X, r, sigma, time);

vector prices(no_steps, 0.0);

double sum_payoffs=0;

double sum_payoffs_bs=0;

for (int n=0; n


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g Uz. 29&,

\ Black Scholes EFgz. l3 4

g F\ A'-9 c2&\, D: ,l3 4l2q BAY AB, zz.!e[} fgcu W#\ LT :o, h[9

#~k, g l) ~C, 2[ Eb, iQk& i.@ &z. 0r=3 D5,\ Q-9 c2&\


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double sigma,

double time_to_maturity,

double lambda,double kappa,

double delta) {

const int MAXN=50;

double tau=time_to_maturity;


double delta_sqr = delta*delta;

double lambdaprime = lambda * (1+kappa);

double gamma = log(1+kappa);

double c = exp(- lambdaprime*tau)*

option_price_call_black_scholes(S, X, sigma, r- lambda*kappa, tau);

double log_n = 0;

for (int n=1;n


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JK ?r NOg%b -\ 9m) (1 ?sk&P\ k/ ̀ gV2b zR 0WX, vilm(Markowitz), C"l

0gz.9:;8 ,u), cX%\ ?rWX Tu 2\, NO99WX/`3 +u) $D' 9G #0k& ?r\ fU2 -7Y b\ $D' ?r pg YZF #, zz.

// file mv_calc.cc





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// basic calculations for mean variance analysis

#include "newmat.h" #include

double mv_calculate_mean(const Matrix& e, const Matrix& w){

Matrix tmp = e.t() *w;

return tmp.element(0,0);

};

double mv_calculate_variance(const Matrix& V, const Matrix& w){

Matrix tmp = w.t() *V*w;

return tmp.element(0,0);

};

double mv_calculate_st_dev(const Matrix& V, const Matrix& w){

double var = mv_calculate_variance(V, w);

return sqrt(var);

};

VW =k XYZ^[

ELP3 CS BQg >\ STk\, D: $D' 9G#043 fU?r LMNcO 2@2(analytic solution)g\,

}, r@W# zz. $D'.9G #04 ) g-3 F\ 3T\ Sgz.

BSBQ >\ MV(JK?r) foD3 r@Fk S2&\ Huang and Litzenberger (1988)3 3|\ }B-%.

// file mv_calc_port_unconstrained.cc


#include "newmat.h"

ReturnMatrix mv_calculate_portfolio_given_mean_unconstrained(const Matrix& e,

const Matrix& V,

double r){

int no_assets=e.Nrows();

Matrix ones = Matrix(no_assets, 1); for (int i=0;i


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Matrix Vinv = V.i(); // inverse of V

Matrix A = (ones.t() *Vinv*e); double a = A.element(0,0);

Matrix B = e.t() *Vinv*e; double b = B.element(0,0);Matrix C = ones.t() *Vinv*ones; double c = C.element(0,0);

Matrix D = B*C - A*A; double d = D.element(0,0);

Matrix Vinv1=Vinv*ones;

Matrix Vinve=Vinv*e;

Matrix g = (Vinv1*b - Vinve*a)*(1.0/d);

Matrix h = (Vinve*c - Vinv1*a)*(1.0/d);

Matrix w = g + h*r;

w.Release();

return w;

};

|\](short sales)/ ^Ap_

*C3 7ok&\ ELP, O|-, ;2[[ E\z.! STk\ LMNcO3 ,]-k 0 A) ! g=g%b -\

CS BQ\ "g[ EW@ r Uz.

ELP3 Cl\ "L\ STk\, g-3 f;2 ~Cq J AB, zz.

29&,

g 0r (2 % X!") X!"3 MRq 9G-\ 0) g gM3 $;\ pD& q8z.!m^Xk g-3 ij\

cfsqp X sc\ 2 % #I9(quadratic solver)q gV2 0r\ -b zz. '+ OR /&k& g0) M;WX gV2b

zz. (: ! ij\ MD; 60, 5>;8 f;2 67WX f6Uz. Tc, Y# z\ 0) $; W#q u3-2

Cl BQ3 N1\ C 3 &l67WX&


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// calculate a mean variance portfolio under short sales constraints

#include "mv_calc.h"

#include "cfsqp_op.h" // optimize using CFSQP#include "mymatrix.h" //

int MV_SHORT_no_assets; // global values because they need to be

double MV_SHORT_r; // available in criterion function

Matrix* MV_SHORT_e;

Matrix* MV_SHORT_V;

double MV_SHORT_objective(double x[]){

Matrix X(MV_SHORT_no_assets, 1);

for (int i=0;i


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// expected return equals r

MV_SHORT_no_assets = e.Nrows();

MV_SHORT_V = &V;MV_SHORT_e = &e;

MV_SHORT_r = r;

Matrix w = null_matrix(MV_SHORT_no_assets, 1);

// check whether it is feasible to find optimum. Only if r is between

// min and max in e;

double r_min = e.element(0,0);

double r_max = e.element(0,0);

int i_min = 0;

int i_max = 0;

int i;

for (i=1;ir_max) | | (r


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double min_var = CFSQP_optimize_with_constraints_and_bounds(

x,

MV_SHORT_no_assets,MV_SHORT_objective,

0,0,0,2, // 2

linear equality constraints

MV_SHORT_constraints,

x_min,

x_max);

for (i=0; i gq1R3 S0)q &u-\ 2m abcd\ \bt9X lz.

Subsections

• 9RAB3 0r

• '+ STU 9RAB3 gV

o NDtR(qrst )

§ #043 NDtR

§ NP(c3 NDtR(qrst )

14 gq1 9RAB(term structure )\ NDtR, McCulloch(1975, 3 % XY%8), Nelson- Siegel RDi 3P,|k& SGU q[q g

V-2 9RABq . YZ.\ W#DKq I\ 10;Im> 69 gq13 74RD(Stocastic proc ess )\ •ulz !k n{ (c

9RABq &u-\ g!; Im ((cRD)g zz. (*q$)




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84/113


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if (time >= t_max) return obs_forwards[no_obs- 1]; // later than latest obs

int t=1; // find which two observations we are betweenwhile ( (tobs_times[t])) { ++t; };

double lambda = (obs_times[t]- time)/(obs_times[t]- obs_times[t- 1]);

// by ordering assumption, time is between t- 1, t

double r = obs_forwards[t- 1] * lambda + obs_forwards[t] * (1.0- lambda);

return r;

};

Subsections

• Nelson Siegel W#D(1987)

• Bliss R~(1989)

• 3 % X%8

5` }p/ #,[ 8\ #04\, McCulloch R~) Y81\

R~2-b zz.

Nelson Siegel g+@ (1987)

// file nels_sie.cc

// author: Bernt A Oedegaard// purpose: Calculate the term structure proposed by Nelson and Siegel

// Parsimonious Modeling of Yield Curves, J ournal of Business, (1987)

#include

double term_structure_yield_nelson_siegel(double t,

double beta0, double beta1, double beta2,

double lambda ) {

if (t==0.0) return beta0;




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double tl = t/lambda;

double r = beta0 + (beta1+beta2) * ((1- exp(- tl)) /tl) + beta2 * exp(- tl);

return r;};

Bliss IS(1989)

Nelson and Siegel (1987) R~3 7|WX&Bliss (1989), C"2Iz.

g0k\ 5 A3 $%V` 3 &ug AB-z.

// file termstru_yield_bliss.cc

// author: Bernt A Oedgaard

#include

double term_structure_yield_bliss(double t, double gamma0, double gamma1, double gamma2,

double lambda1, double lambda2) {

double r;

double t1 = t/lambda1;

double t2 = t/lambda2;

r = gamma0

+ gamma1 * ( (1- exp(- t1)) / t1 )

+ gamma2 * ( ( 1- exp(- t2)) / t2 )

+ gamma2 * ( - exp(- t2));

return r;

};

Cubic spline( 3* h•&4)

3 % XY%8 N\ gVl 9R AB &u3 Im) McCulloch (1971) k 32 C"2Iz.zk !(\ b5l R~

McCulloch (1975) P C"Bz.b g0>\ CPX Litzenberger and Rolfo (1984) P 3 % XY%8 N\ gV-b

zz.




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29k q jQ k Gl [,W#gz. ij3 l(Knot)g%b lz.N\ &u-9 c2&\ A3

#, AB-z.

X%8 ij3 l(knot)g a5'z@, g0) 67l ND P7, Uz.

// file: cu_spline.cc


#include

#include

double term_structure_discount_factor_cubic_spline(double t,

double b1,

double c1,

double d1,

const vector& f,

const vector& knots)

{

// calculate the discount factor for the spline functional form.

double d = 1.0

+ b1*t

+ c1*(pow(t, 2))

+ d1*(pow(t, 3));

for (int i=0;i= knots[i]) { d += f[i] * (pow((t- knots[i]), 3)); }

else { break; };

};

return d;

};

Subsections

• Vasicek R~




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• Cox Ingersoll Ross R~

o Y813 0r

o ^_ ̀ X!"

• CIR R~k 3l &u

o Y81

5` }p IS15

gM @( section )k&\ 9R AB3 W#q ( R#;8 ImWX &u-\ ?iIm\ t8z.zIWX 9R AB3

SCR~(economic model )\ \bt9X lz.

Vasicek IS

Vasicek (1977)k 32 Am2Iz.

// file vasicek.cc


#include "math.h"

double term_structure_discount_factor_vasicek(double time,

double r,

double a, double b, double sigma){

double A, B;


double aa = a*a;

15 4Z.Bu( |2zL(stoch astic process )- gh .1a Yt g- wm .Bu Z_3nV ?LYm

Kw(.2+ Kw)-m ^1.1z xkTr


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if (a==0.0){

B = time;

A = exp(sigma_sqr*pow(time, 3))/6.0;}

else {

B = (1.0 - exp(- a*time)) /a;

A = exp( ((B- time)*(aa*b- 0.5*sigma_sqr)) /aa - ((sigma_sqr*B*B)/(4*a)));

};

double d = A*exp(- B*r);

return d;

}

Cox Ingersoll Ross IS16

g R~) Cox et al. (1985) k 9:2b zD, 5> KD R~(equilibrium mo del )\ 9/X A2b zD aG;8

?sk&\ . gVUz.

i4-/ 7k

69 gq1)

T ,lk zD&3 Y81)

29&,

16 Vasicek.1zm b9 CIR.1( `am dwJ. r σ i L(9m k.6 ?@. [• zL( square root process)* 4Z.Bu( '1

. !p'" `O~; Z!Y# 9Gi Vasicek.1( 4T,i y+U .Bu. (̀- ). P' ?m E[V #W' ?@.(ABC)




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bl z!,[ R#(parameter)3 3V\ g-h pz.

1. - 69 gq1

2. - JKP7 $%V ̀

3. - ,| cX% $%V ̀

4. - |9 JK

5. - ?r

YZ[ \]^_

// file termstru_discfact_cir.cc


#include // mathematics library

double term_structure_discount_factor_cir(double t,

double r,

double kappa,

double lambda,

double theta,

double sigma)

// this is the original CIR formulation of their term structure.

{

double sigma_sqr=pow(sigma, 2);

double gamma = sqrt(pow((kappa+lambda), 2)+2.0*sigma_sqr);

double denum = (gamma+kappa+lambda)*(exp(gamma*t)- 1) +2*gamma;

double p=2*kappa*theta/sigma_sqr;

double enum1= 2*gamma*exp(0.5*(kappa+lambda+gamma) *t);

double A = pow((enum1/denum), p);

double B = (2*(exp(gamma*t)- 1)) /denum;

double dfact=A*exp(- B*r);

return dfact;

};




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Brown and Dybvig l /. #


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Chapter 16. ]


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// file bondopt_call_coupon.cc


#include



double bond_option_price_call_coupon_bond_black_scholes(

double B, double X, double r, double sigma, double time,

vectorcoupon_times, vectorcoupon_amounts){

for (unsigned int i=0;i


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int no_steps) // call on a zero coupon

bond.

{double delta_t = bond_maturity/no_steps;

double u=exp(S*sqrt(delta_t));

double d=1/u;

double p_up = (exp(M*delta_t) - d)/(u- d);

double p_down = 1.0- p_up;

vectorr(no_steps+1);

r[0]=interest*pow(d, no_steps);

double uu=u*u;

int i;

for (i=1;i


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Vasicek IS/ FG ': Q pg ,ulz.

29&, , , \ =#X lz.gV Y813 0rIm) ab zz, 4mQ| E3 0r) g-h pg Uz.

\ ,lk zD&3 Y81 3 g](zero - coupon )33 ,QWX lz. }9 , }95k zD&3

Y81g 8 4mQ| E3 ,lk zD&3 ,Q) g-h pz.(J amshidan (1989) # Hull (1993) }B.)

29&,

gz.

b 3 STk\




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Uz.

// file bondopt_call_vasicek.cc


#include



double bond_option_price_call_zero_vasicek(double X, // exercise price

double r, // current interest rate

double option_time_to_maturity,

double bond_time_to_maturity,

double a, // parameters

double b,

double sigma)

{

double T_t = option_time_to_maturity;

double s_t = bond_time_to_maturity;

double T_s = s_t- T_t;

double v_t_T;

double sigma_P;

if (a==0.0) {

v_t_T = sigma * sqrt ( T_t ) ;

sigma_P = sigma*T_s*sqrt(T_t);

}

else {

v_t_T = sqrt (sigma*sigma*(1- exp(- 2*a*T_t))/(2*a));

double B_T_s = (1- exp(- a*T_s)) /a;

sigma_P = v_t_T*B_T_s;

};

double h = (1.0/sigma_P) * log (

term_structure_discount_factor_vasicek(s_t, r, a, b, sigma)/

(term_structure_discount_factor_vasicek(T_t, r, a, b, sigma) *X) )

+ sigma_P/2.0;

double c =

term_structure_discount_factor_vasicek(s_t, r, a, b, sigma) *N(h)




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- X*term_structure_discount_factor_vasicek(T_t, r, a, b, sigma) *N(h- sigma_P);

return c;

};


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// normdist.cc

// numerical approximations to univariate N(z) and bivariate N(a, b, rho)// normal distributions

#include // math functions.

#ifndef PI

#define PI 3.141592653589793238462643

#endif

double n(double z) { // normal distribution function

return (1.0/sqrt(2.0*PI)) *exp(- 0.5*z*z);

};

double n(double r, double mu, double sigma) { // normal distribution function

double nv = 1.0/(sqrt(2.0*PI) *sigma);

double z=(r- mu) /sigma;

nv *= exp(- 0.5*z*z);

return nv;

};

// cumulative univariate normal distribution.

// This is a numerical approximation to the normal distribution.

// See Abramowitz and Stegun: Handbook of Mathemathical functions

// for description. The arguments to the functions are assumed

// normalized to a (0,1 ) distribution.

double N(double z) {

double b1 = 0.31938153;

double b2 = - 0.356563782;

double b3 = 1.781477937;

double b4 = - 1.821255978;

double b5 = 1.330274429;

double p = 0.2316419;

double c2 = 0.3989423;

if (z > 6.0) { return 1.0; }; // this guards against overflow




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if (z


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if ( ( a=0.0 ) && (rho>=0.0) ) {

return N(a) - N(a, - b, - rho);

}else if ( (a>=0.0) && (b=0.0) ) {

return N(b) - N(- a, b, - rho);

}

else if ( (a>=0.0) && (b>=0.0) && (rho= 0.0 ) {

double denum = sqrt(a*a - 2*rho*a*b + b*b);

double rho1 = ((rho * a - b) * sgn(a)) /denum;

double rho2 = ((rho * b - a) * sgn(b)) /denum;

double delta=(1.0- sgn(a) *sgn(b))/4.0;

return N(a, 0.0, rho1) + N(b, 0.0, rho2) - delta;

};

return - 99.9; // should never get here

};

D]op / References

65 e$ udij\ Abramowiz and Stegun (1964)q }B. Hull, 1993, ('+ 5" 2000)g3 ge$ ud?Lk

Sl Ku abcd A'\ }BY 0.

Cqql aQ rYs t. uv / A note on C++ and the source code

Subsections

• p, ij3 k#

• %glmc

• ( CnnYm! %vq cl Cnn2•

• #a 9r

o [#W#




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o s?t ?

• xu3 f; u3

• for XvgMwM3 &F

• #include 3 AB

• xoX3 Ay

• z{ |Dk zD&3 *|

• .efX

•

wh

bd

/ Source availability

p, ij\ Aq3 X Yg[)*q$GAH;WX\ http://finance.bi.no/~bernt/gcc_prog/*X/ ̀ZIP DFk&

$U $5X& k#Y # zz.g0Z)GNU C++ 3 f. q%)*$GAq, g pbq &1999/9/9,lk&3

gs9.*!cb ff 78g i' =Kgz.p, ijk&\ C++3 STL(Standard Template Library)q gV-b z9

#~k, z{ ^$5mk&\ Rw $5g ^$5 ,F-[ E\[P Roz.}S y.g jV3 ^$5m, ABl #4

$53 /Z\ Sb2 tz@, STL #4 $5g z{ $54(g pg 9G-\ 0WX X!"\ M #,

zz. :~&X gV-b z\ NDG#3 %glmc\ newmat X scb zW\, g0) eH- %g.Xgz.f.3

q%) $Bl FTP jgMk& newmat09.zip b\ newmat09.tar.gz X I


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=9k Gu2 ;n A h u b q u3-@(q),

*;u) : Matrix Ainv = A.i(); h

;) : Matrix C = A * b.t(); h

v' IuFWX&3 3 2: Matrix x = A.i() *b; X& 9GY #, zz.

NDG#3 ST\ ;n vr3 ]9, ,F-@ LT 4V-, Uz.T$l %glmcq NO-@, 9 ]>q m/lz.

' Cqq•O z{| 4Q Cqq.Q

C b\ Cnnq Ro\ STk\ . Rq 0gz, 3A3 X!"WX jV-b z\ CnnK43 5k G2 •4lz.

Subsections

• [#W#

• s?> t ?

+1 5} / Mathematical operators

R+g+ / Exponentiation

jB vr 9r ( )\ !GX jVY # z[}, [#W#\ |D3 5/k&\ >b, #a S03 %glmc

]k& u32\ W#X& A'2b zz.

• pow(x, n) \ \ 0rlz.

• exp(r) \ \ 0rlz.

•

~=B Uw / Increment and decrement.

e#q 10 s,(b\ tU),[b o\ #, () |Dk&\

int i=0

i = i+1

i = i- 1

X 9G-[}, C++k&\ ?, g-h pg 9GY #, zz.

int i=0




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i++

i- -

g0) ,t;g%b\ D-9 Dw[}, &=;8 rg ̀ ABq mn2 #\ Uc-, W #P zD, '?J *V;8 0) Ezb DY # zz.8q Z@date %b -\ xoXq u32, ! WzI1 X3 S\ 2b oW@ g-h pg

9G-\ 0}WX i'z.

date d(1,1,1995)

d++

+>\ 2jan95 , Uz.

n•/ a7 j, jV-9 Mk u32D z[ EW@ r Uz.! #~k C |DY Cnnk&\ W#Y L%Xq

u3l #4 $5(header file)\ gVlz. 8q Z@ :~&, () EFk& gV-\ ud ?L3 N; ud ;P W#

N(z)k&\ normdist.h X N|-b zz.


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#ifndef _NORMAL_DIST_H_

#define _NORMAL_DIST_H_

double n(double z); // normal distribution function

double n(double r, double mu, double sigmasqr); // normal distribution function

double N(double z); // cumulative probability of normal

double N(double a, double b, double rho); // cum prob of bivariate normal

#endif

g #4 $5k&\ 2 A3 ud ?L3 74 ;P W#(1 e#3 STh 2 e#3 ST)q u3-b zz. }S N()k

1 A3 S} Q\ 0)


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tj3 D> / Acknowledgements.

g ~&q +=k& EAl go, 5J () %5\ &8z.X!"3 q!q [;-\ 0P zIz.! ]k&P KJ 4Vl ,jY [;\ 2 9 ?Z3 g*\ 49-2 tj3 ?\ Mlz.

EARAUJ [email protected]

Michael L Locher

Lars Gregori

Steve Bellantoni

d

( N4 ()*+,)

Gamma

Gamma

Theta

Theta

Rho

Rho

Barone- Adesi and Whaley

Barone- Ades- Whaleymk no\


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CIR

estimated

CIR R~k 3l &u class

xoX3 Ay

Cox Ingersoll Ross, see CIR

CIR R~k 3l &u

currency

'{ 12


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explicit

4,; 2m3 ^_ ̀ X!"(4mQ|


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Merton

Jump Diffusion

J ump diffusion R~ modified duration

#u 5.g=

np

'+ ,-

option

currency

'{ 12


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,-.g=k 3l ; ?im

theta

Thetaunique irr

23 4563 78

vega

Vega

volatility

implied

.+ef6(implied volatility)




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D]op / Bibliography

Milton Abramowiz and Irene A Stegun.

Handbo ok of Mathematical Functions .

National Bureau of Standards, 1964.

Giovanni Barone- Adesi and Robert E Whaley.

Efficient analytic approximation of American option values.

Journal of Finance , 42(2):- 20, J une 1987.

Per Berck and Knut Syds!ter.

Matematisk Formelsamling fo r konomer .

Universitetsforlaget, 2 edition, 1995.

Fisher Black.

The pricing of commodity contracts.

Journal of Financial Econom ics , 3:- 79, 1976.

Fisher Black and M S Scholes.

The pricing of options and corporate liabilities.

Journal of Political Economy , 7:- 54, 1973.

Robert R Bliss.

Fitting term structures to bond prices.

Working paper, University of Chicago, J anuary 1989.

Phelim P Boyle.

Options: A Monte Carlo approach.


Richard A Brealey and Stewart C Myers.

Principles of Co rporate Financ e .

McGraw- Hill, fourth edition, 1996.

Michael Brennan and Eduardo Schwartz.

Finite difference methods and jump processes arising in the pricing of contingent claims: A synthesis.

Journal of Financial and Quantitative Analysis , 13:- 74, 1978.

Stephen J Brown and Philip H Dybvig.

The empirical implications of the Cox, Ingersoll, Ross theory of the term structure of interest rates.

Journal of Finance , 41:- 32, 1986.

J Cox, S Ross, and M Rubinstein.

Option pricing: A simplified approach.


J ohn Cox and Mark Rubinstein.

Options m arkets .

Prentice- Hall, 1985.




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J ohn C Cox, J onathan E Ingersoll, and Stephen A Ross.

A theory of the term structure of interest rates.

Econometrica , 53:- 408, 1985.M B Garman and S W Kohlhagen.

Foreign currency option values.

Journal of International Money and Finance , 2:- 37, 1983.

Robert Geske.

The valuation of compound options.

Journal of Financial Econom ics , 7:- 81, March 1979.

Robert Geske and H E J ohnson.

The american put valued analytically.

Journal of Finance , XXXIX(5), December 1984.

M Barry Goldman, Howard B Sosin, and Mary Ann Gatto.

Path- dependent options: Buy at the low, sell at the high.

Journal of Finance , 34, December 1979.

J O Grabbe.

The pricing of call and put options on foreign exchange.

Journal of International Money and Finance , 2:- 53, 1983.

Chi- fu Huang and Robert H. Litzenberger.

Foundations for financial econo mics .

North- Holland, 1988.

J ohn Hull.

Option s, Futures and other Derivative Securities .

Prentice- Hall, second edition, 1993.

J ohn Hull.

Option s, Futures and ot her Derivatives .

Prentice- Hall, third edition, 1997.

F J amshidan.

An exact bond option pricing formula.

Journal of Finance , 44:- 9, March 1989.

Stanley B Lippman.

C+ + primer .

Addison- Wesley, 2 edition, 1992.

Robert H. Litzenberger and J aques Rolfo.

An international study of tax effects on government bonds.

Journal of Finance , 39:- 22, 1984.

J Houston McCulloch.

Measuring the term structure of interest rates.

Journal of Business , 44:- 31, 1971.




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J Houston McCulloch.

The tax adjusted yield curve.

Journal of Finance , 30:- 829, 1975.Robert C Merton.

The theory of rational option pricing.

Bell Journal , 4:- 183, 1973.

David R Musser and Atul Saini.

STL Tutorial and Reference G uide: C+ + programming with the standard template library .

Addison- Wesley, 1996.

Charles R Nelson and Andrew F Siegel.

Parsimonious modelling of yield curves.

Journal of Business , 60(4):- 89, 1987.

Carl J Norstrom.

A sufficient conditions for a unique nonnegative internal rate of return.

Journal of Financial and Quantitative Analysis , 7(3):- 39, 1972.

William Press, Saul A Teukolsky, William T Vetterling, and Brian P Flannery.

Numerical Recipes in C .

Cambridge University Press, second edition, 1992.

Richard J Rendleman and Brit J Bartter.

Two- state option pricing.

Journal of Finance , 34(5):- 1110, December 1979.

Richard J Rendleman and Brit J Bartter.

The pricing of options on debt securities.

Journal of Financial and Quantitative Analysis , 15(1):- 24, March 1980.

Richard Roll.

An analytical formula for unprotected American call options on stocks with known dividends.


Stephen A Ross, Randolph Westerfield, and J effrey F J affe.

Corporate Financ e .

Irwin, fourth edition, 1996.

Mark Rubinstein.

Exotic options.

University of California, Berkeley, working paper, 1993.

William F Sharpe and Gordon J Alexander.

Investments .

Prentice- Hall, 4 edition, 1990.

Bjarne Stroustrup.

The C+ + Programm ing language .

Addison- Wesley, 2 edition, 1991.

O Vasicek.




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An equilibrium characterization of the term structure.


Robert E Whaley.On the valuation of American call options on stocks with known dividends.


Paul Wilmott, J eff Dewynne, and Sam Howison.

Option Pricing, Mathematical models and c om putation .

Oxford Financial Press, 1994.

ISBN 0 9522082 02.

c++lanquage for financial engineering

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