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NAG Toolbox: nag_specfun_opt_jumpdiff_merton_price (s30ja)

Purpose

nag_specfun_opt_jumpdiff_merton_price (s30ja) computes the European option price using the Merton jump-diffusion model.

Syntax

[p, ifail] = s30ja(calput, x, s, t, sigma, r, lambda, jvol, 'm', m, 'n', n)
[p, ifail] = nag_specfun_opt_jumpdiff_merton_price(calput, x, s, t, sigma, r, lambda, jvol, 'm', m, 'n', n)

Description

nag_specfun_opt_jumpdiff_merton_price (s30ja) uses Merton's jump-diffusion model (Merton (1976)) to compute the price of a European option. This assumes that the asset price is described by a Brownian motion with drift, as in the Black–Scholes–Merton case, together with a compound Poisson process to model the jumps. The corresponding stochastic differential equation is,
(dS)/(S) = (αλk) dt + σ̂ dWt + dqt .
dS S = (α-λk) dt + σ^ dWt + dqt .
Here αα is the instantaneous expected return on the asset price, SS; σ̂2σ^2 is the instantaneous variance of the return when the Poisson event does not occur; dWtdWt is a standard Brownian motion; qtqt is the independent Poisson process and k = E[Y1]k=E[Y-1] where Y1Y-1 is the random variable change in the stock price if the Poisson event occurs and EE is the expectation operator over the random variable YY.
This leads to the following price for a European option (see Haug (2007))
Pcall = ( eλT (λT)j )/(j ! )Cj(S,X,T,r,σj),
j = 0
Pcall = j=0 e-λT (λT)j j! Cj ( S, X, T, r, σj ) ,
where TT is the time to expiry; XX is the strike price; rr is the annual risk-free interest rate; Cj(S,X,T,r,σj)Cj(S,X,T,r,σj) is the Black–Scholes–Merton option pricing formula for a European call (see nag_specfun_opt_bsm_price (s30aa)).
σj = sqrt( z2 + δ2 (j/T) ) ,
z2 = σ2 λ δ2 ,
δ2 = (γ σ2)/(λ) ,
σj = z2 + δ2 ( j T ) , z2 = σ2 - λ δ2 , δ2 = γ σ2 λ ,
where σσ is the total volatility including jumps; λλ is the expected number of jumps given as an average per year; γγ is the proportion of the total volatility due to jumps.
The value of a put is obtained by substituting the Black–Scholes–Merton put price for Cj (S,X,T,r,σj) Cj ( S, X, T, r, σj ).

References

Haug E G (2007) The Complete Guide to Option Pricing Formulas (2nd Edition) McGraw-Hill
Merton R C (1976) Option pricing when underlying stock returns are discontinuous Journal of Financial Economics 3 125–144

Parameters

Compulsory Input Parameters

1:     calput – string (length ≥ 1)
Determines whether the option is a call or a put.
calput = 'C'calput='C'
A call. The holder has a right to buy.
calput = 'P'calput='P'
A put. The holder has a right to sell.
Constraint: calput = 'C'calput='C' or 'P''P'.
2:     x(m) – double array
m, the dimension of the array, must satisfy the constraint m1m1.
x(i)xi must contain XiXi, the iith strike price, for i = 1,2,,mi=1,2,,m.
Constraint: x(i)z ​ and ​ x(i) 1 / z xiz ​ and ​ xi 1 / z , where z = x02am () z = x02am () , the safe range parameter, for i = 1,2,,mi=1,2,,m.
3:     s – double scalar
SS, the price of the underlying asset.
Constraint: sz ​ and ​s1.0 / zsz ​ and ​s1.0/z, where z = x02am()z=x02am(), the safe range parameter.
4:     t(n) – double array
n, the dimension of the array, must satisfy the constraint n1n1.
t(i)ti must contain TiTi, the iith time, in years, to expiry, for i = 1,2,,ni=1,2,,n.
Constraint: t(i)ztiz, where z = x02am () z = x02am () , the safe range parameter, for i = 1,2,,ni=1,2,,n.
5:     sigma – double scalar
σσ, the annual total volatility, including jumps.
Constraint: sigma > 0.0sigma>0.0.
6:     r – double scalar
rr, the annual risk-free interest rate, continuously compounded. Note that a rate of 5% should be entered as 0.05.
Constraint: r0.0r0.0.
7:     lambda – double scalar
λλ, the number of expected jumps per year.
Constraint: lambda > 0.0lambda>0.0.
8:     jvol – double scalar
The proportion of the total volatility associated with jumps.
Constraint: 0.0jvol < 1.00.0jvol<1.0.

Optional Input Parameters

1:     m – int64int32nag_int scalar
Default: The dimension of the array x.
The number of strike prices to be used.
Constraint: m1m1.
2:     n – int64int32nag_int scalar
Default: The dimension of the array t.
The number of times to expiry to be used.
Constraint: n1n1.

Input Parameters Omitted from the MATLAB Interface

ldp

Output Parameters

1:     p(ldp,n) – double array
ldpmldpm.
The leading m × nm×n part of the array p contains the computed option prices.
2:     ifail – int64int32nag_int scalar
ifail = 0ifail=0 unless the function detects an error (see [Error Indicators and Warnings]).

Error Indicators and Warnings

Errors or warnings detected by the function:
  ifail = 1ifail=1
On entry, calput = 'C'calput='C' or 'P''P'.
  ifail = 2ifail=2
On entry, m0m0.
  ifail = 3ifail=3
On entry, n0n0.
  ifail = 4ifail=4
On entry, x(i) < zxi<z or x(i) > 1 / zxi>1/z, where z = x02am()z=x02am(), the safe range parameter.
  ifail = 5ifail=5
On entry, s < zs<z or s > 1.0 / zs>1.0/z, where z = x02am()z=x02am(), the safe range parameter.
  ifail = 6ifail=6
On entry, t(i) < zti<z, where z = x02am()z=x02am(), the safe range parameter.
  ifail = 7ifail=7
On entry, sigma0.0sigma0.0.
  ifail = 8ifail=8
On entry, r < 0.0r<0.0.
  ifail = 9ifail=9
On entry, lambda0.0lambda0.0.
  ifail = 10ifail=10
On entry, jvol < 0.0jvol<0.0 or jvol1.0jvol1.0.
  ifail = 12ifail=12
On entry, ldp < mldp<m.

Accuracy

The accuracy of the output is dependent on the accuracy of the cumulative Normal distribution function, ΦΦ, occurring in CjCj. This is evaluated using a rational Chebyshev expansion, chosen so that the maximum relative error in the expansion is of the order of the machine precision (see nag_specfun_cdf_normal (s15ab) and nag_specfun_erfc_real (s15ad)). An accuracy close to machine precision can generally be expected.

Further Comments

None.

Example

function nag_specfun_opt_jumpdiff_merton_price_example
put = 'C';
lambda = 3;
s = 45;
sigma = 0.25;
r = 0.1;
jvol = 0.4;
x = [55.0];
t = [0.25];

[p, ifail] = nag_specfun_opt_jumpdiff_merton_price(put, x, s, t, sigma, r, lambda, jvol);


fprintf('\nMerton Jump-Diffusion Model\n European Call :\n');
fprintf('  Spot       =   %9.4f\n', s);
fprintf('  Volatility =   %9.4f\n', sigma);
fprintf('  Rate       =   %9.4f\n', r);
fprintf('  Jumps      =   %9.4f\n', lambda);
fprintf('  Jump Vol   =   %9.4f\n\n', jvol);

fprintf('   Strike    Expiry   Option Price\n');
for i=1:1
  for j=1:1
    fprintf('%9.4f %9.4f %9.4f\n', x(i), t(j), p(i,j));
  end
end
 

Merton Jump-Diffusion Model
 European Call :
  Spot       =     45.0000
  Volatility =      0.2500
  Rate       =      0.1000
  Jumps      =      3.0000
  Jump Vol   =      0.4000

   Strike    Expiry   Option Price
  55.0000    0.2500    0.2417

function s30ja_example
put = 'C';
lambda = 3;
s = 45;
sigma = 0.25;
r = 0.1;
jvol = 0.4;
x = [55.0];
t = [0.25];

[p, ifail] = s30ja(put, x, s, t, sigma, r, lambda, jvol);


fprintf('\nMerton Jump-Diffusion Model\n European Call :\n');
fprintf('  Spot       =   %9.4f\n', s);
fprintf('  Volatility =   %9.4f\n', sigma);
fprintf('  Rate       =   %9.4f\n', r);
fprintf('  Jumps      =   %9.4f\n', lambda);
fprintf('  Jump Vol   =   %9.4f\n\n', jvol);

fprintf('   Strike    Expiry   Option Price\n');
for i=1:1
  for j=1:1
    fprintf('%9.4f %9.4f %9.4f\n', x(i), t(j), p(i,j));
  end
end
 

Merton Jump-Diffusion Model
 European Call :
  Spot       =     45.0000
  Volatility =      0.2500
  Rate       =      0.1000
  Jumps      =      3.0000
  Jump Vol   =      0.4000

   Strike    Expiry   Option Price
  55.0000    0.2500    0.2417


PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

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