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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_pde_1d_blackscholes_means (d03ne)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_pde_1d_blackscholes_means (d03ne) computes average values of a continuous function of time over the remaining life of an option. It is used together with nag_pde_1d_blackscholes_closed (d03nd) to value options with time-dependent arguments.


[phiav, ifail] = d03ne(t0, tmat, td, phid, 'ntd', ntd)
[phiav, ifail] = nag_pde_1d_blackscholes_means(t0, tmat, td, phid, 'ntd', ntd)


nag_pde_1d_blackscholes_means (d03ne) computes the quantities
ϕt0,   ϕ^=1T-t0 t0Tϕζdζ,   ϕ-= 1T-t0 t0Tϕ2ζdζ 1/2  
from a given set of values phid of a continuous time-dependent function ϕt at a set of discrete points td, where t0 is the current time and T is the maturity time. Thus ϕ^ and ϕ- are first and second order averages of ϕ over the remaining life of an option.
The function may be used in conjunction with nag_pde_1d_blackscholes_closed (d03nd) in order to value an option in the case where the risk-free interest rate r, the continuous dividend q, or the stock volatility σ is time-dependent and is described by values at a set of discrete times (see Use with ). This is illustrated in Example.




Compulsory Input Parameters

1:     t0 – double scalar
The current time t0.
Constraint: td1t0tdntd.
2:     tmat – double scalar
The maturity time T.
Constraint: td1tmattdntd.
3:     tdntd – double array
The discrete times at which ϕ is specified.
Constraint: td1<td2<<tdntd.
4:     phidntd – double array
phidi must contain the value of ϕ at time tdi, for i=1,2,,ntd.

Optional Input Parameters

1:     ntd int64int32nag_int scalar
Default: the dimension of the arrays td, phid. (An error is raised if these dimensions are not equal.)
The number of discrete times at which ϕ is given.
Constraint: ntd2.

Output Parameters

1:     phiav3 – double array
phiav1 contains the value of ϕ interpolated to t0, phiav2 contains the first-order average ϕ^ and phiav3 contains the second-order average ϕ-, where:
ϕ^=1T-t0 t0Tϕζdζ ,   ϕ-= 1T-t0 t0Tϕ2ζdζ 1/2 .  
2:     ifail int64int32nag_int scalar
ifail=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:
On entry,t0 lies outside the range [td1,tdntd],
ortmat lies outside the range [td1,tdntd],
ortd badly ordered,
Unexpected failure in internal call to nag_interp_1d_spline (e01ba) or nag_fit_1dspline_eval (e02bb).
An unexpected error has been triggered by this routine. Please contact NAG.
Your licence key may have expired or may not have been installed correctly.
Dynamic memory allocation failed.


If ϕC4t0,T then the error in the approximation of ϕt0 and ϕ^ is OH4, where H=maxiTi+1-Ti, for i=1,2,,ntd-1. The approximation is exact for polynomials of degree up to 3.
The third quantity ϕ- is OH2, and exact for linear functions.

Further Comments


The time taken is proportional to ntd.

Use with nag_pde_1d_blackscholes_closed (d03nd)

Suppose you wish to evaluate the analytic solution of the Black–Scholes equation in the case when the risk-free interest rate r is a known function of time, and is represented as a set of values at discrete times. A call to nag_pde_1d_blackscholes_means (d03ne) providing these values in phid produces an output array phiav suitable for use as the argument r in a subsequent call to nag_pde_1d_blackscholes_closed (d03nd).
Time-dependent values of the continuous dividend Q and the volatility σ may be handled in the same way.

Algorithmic Details

The ntd data points are fitted with a cubic B-spline using the function nag_interp_1d_spline (e01ba). Evaluation is then performed using nag_fit_1dspline_eval (e02bb), and the definite integrals are computed using direct integration of the cubic splines in each interval. The special case of T=to is handled by interpolating ϕ at that point.


This example demonstrates the use of the function in conjunction with nag_pde_1d_blackscholes_closed (d03nd) to solve the Black–Scholes equation for valuation of a 5-month American call option on a non-dividend-paying stock with an exercise price of $50. The risk-free interest rate varies linearly with time and the stock volatility has a quadratic variation. Since these functions are integrated exactly by nag_pde_1d_blackscholes_means (d03ne) the solution of the Black–Scholes equation by nag_pde_1d_blackscholes_closed (d03nd) is also exact.
The option is valued at a range of times and stock prices.
function d03ne_example

fprintf('d03ne example results\n\n');

tmat = 0.4166667;
td   = [0:0.1:0.5];
rd   = [0.1:0.01:0.15];
sigd = [0.3 0.46 0.54 0.54 0.36 0.3];

% American 5-month call option, exercise price 50

kopt = int64(2);
x = 50;
ns    = 21;  nt    = 4;
s_beg = 0;   t_beg = 0;
s_end = 100; t_end = 0.125;
tmat  = 0.4166667;
tdpar = [true; false; true];
q     = [0];

% Discretize s and t
ds = (s_end-s_beg)/(ns-1);
dt = (t_end-t_beg)/(nt-1);
s = [s_beg:ds:s_end];
t = [t_beg:dt:t_end];

f = zeros(ns,nt);
theta = f; delta = f; gamma = f; lambda = f; rho = f;

% Loop over times and prices
for j = 1:nt

  % Find average values of r and sigma
  [ra, ifail] = d03ne( ...
                          t(j), tmat, td, rd);
  [siga, ifail] = d03ne( ...
                          t(j), tmat, td, sigd);

  for i = 1:ns
    [f(i,j),theta(i,j),delta(i,j),gamma(i,j),lambda(i,j),rho(i,j),ifail] = ...
    d03nd( ...
           kopt, x, s(i), t(j), tmat, tdpar, ra, q, siga);

% Tabulate option values only
print_greek(ns,nt,tmat,s,t,'Option Values',f);
% print_greek(ns,nt,tmat,s,t,'Theta',theta);
% print_greek(ns,nt,tmat,s,t,'Delta',delta);
% print_greek(ns,nt,tmat,s,t,'Gamma',gamma);
% print_greek(ns,nt,tmat,s,t,'Lambda',lambda);
% print_greek(ns,nt,tmat,s,t,'Rho',rho);

% plot initial and final option values and greeks 
fig1 = figure;
title('Option values and greeks at 5 months to maturity');
xlabel('stock price');
ylabel('values and derivatives');
fig2 = figure;
title('Option values and greeks at 3.5 months to maturity');
xlabel('stock price');
ylabel('values and derivatives');

function print_greek(ns,nt,tmat,s,t,grname,greek)

  fprintf('  Stock Price  |   Time to Maturity (months)\n');
  fprintf('%16s %12.4e%12.4e%12.4e%12.4e\n', '|', 12*(tmat-t));
  fprintf('%15s+%48s\n', '---------------', ...
  for i = 1:ns
    fprintf('%12.4e%4s %12.4e%12.4e%12.4e%12.4e\n', s(i), '|', greek(i,:));
d03ne example results

Option Values

  Stock Price  |   Time to Maturity (months)
               |   5.0000e+00  4.5000e+00  4.0000e+00  3.5000e+00
  0.0000e+00   |   0.0000e+00  0.0000e+00  0.0000e+00  0.0000e+00
  5.0000e+00   |   8.3570e-14  1.6081e-14  1.1216e-15  1.9522e-17
  1.0000e+01   |   2.5775e-07  1.0991e-07  2.8081e-08  3.5715e-09
  1.5000e+01   |   1.7581e-04  1.0511e-04  4.6513e-05  1.3698e-05
  2.0000e+01   |   6.9193e-03  4.9696e-03  2.9591e-03  1.3701e-03
  2.5000e+01   |   7.0752e-02  5.6767e-02  4.0397e-02  2.4520e-02
  3.0000e+01   |   3.4255e-01  2.9499e-01  2.3506e-01  1.6927e-01
  3.5000e+01   |   1.0512e+00  9.4849e-01  8.1382e-01  6.5475e-01
  4.0000e+01   |   2.3997e+00  2.2341e+00  2.0134e+00  1.7424e+00
  4.5000e+01   |   4.4829e+00  4.2630e+00  3.9702e+00  3.6055e+00
  5.0000e+01   |   7.2786e+00  7.0226e+00  6.6859e+00  6.2677e+00
  5.5000e+01   |   1.0687e+01  1.0414e+01  1.0063e+01  9.6324e+00
  6.0000e+01   |   1.4580e+01  1.4305e+01  1.3959e+01  1.3546e+01
  6.5000e+01   |   1.8832e+01  1.8563e+01  1.8236e+01  1.7855e+01
  7.0000e+01   |   2.3337e+01  2.3079e+01  2.2774e+01  2.2429e+01
  7.5000e+01   |   2.8016e+01  2.7768e+01  2.7485e+01  2.7173e+01
  8.0000e+01   |   3.2809e+01  3.2573e+01  3.2308e+01  3.2022e+01
  8.5000e+01   |   3.7678e+01  3.7450e+01  3.7201e+01  3.6935e+01
  9.0000e+01   |   4.2595e+01  4.2374e+01  4.2136e+01  4.1885e+01
  9.5000e+01   |   4.7543e+01  4.7327e+01  4.7097e+01  4.6856e+01
  1.0000e+02   |   5.2510e+01  5.2298e+01  5.2074e+01  5.1840e+01

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Chapter Contents
Chapter Introduction
NAG Toolbox

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