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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_specfun_opt_jumpdiff_merton_price (s30ja)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


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


[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)


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 = α-λk dt + σ^ dWt + dqt .  
Here α is the instantaneous expected return on the asset price, S; σ^2 is the instantaneous variance of the return when the Poisson event does not occur; dWt is a standard Brownian motion; qt is the independent Poisson process and k=EY-1 where Y-1 is the random variable change in the stock price if the Poisson event occurs and E is the expectation operator over the random variable Y.
This leads to the following price for a European option (see Haug (2007))
Pcall = j=0 e-λT λTj j! Cj S, X, T, r, σj ,  
where T is the time to expiry; X is the strike price; r is the annual risk-free interest rate; CjS,X,T,r,σj is the Black–Scholes–Merton option pricing formula for a European call (see nag_specfun_opt_bsm_price (s30aa)).
σj = z2 + δ2 j T , z2 = σ2 - λ δ2 , δ2 = γ σ2 λ ,  
where σ is the total volatility including jumps; λ is the expected number of jumps given as an average per year; γ 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 .
The option price Pij=PX=Xi,T=Tj is computed for each strike price in a set Xi, i=1,2,,m, and for each expiry time in a set Tj, j=1,2,,n.


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


Compulsory Input Parameters

1:     calput – string (length ≥ 1)
Determines whether the option is a call or a put.
A call; the holder has a right to buy.
A put; the holder has a right to sell.
Constraint: calput='C' or 'P'.
2:     xm – double array
xi must contain Xi, the ith strike price, for i=1,2,,m.
Constraint: xiz ​ and ​ xi 1 / z , where z = x02am , the safe range parameter, for i=1,2,,m.
3:     s – double scalar
S, the price of the underlying asset.
Constraint: sz ​ and ​s1.0/z, where z=x02am, the safe range parameter.
4:     tn – double array
ti must contain Ti, the ith time, in years, to expiry, for i=1,2,,n.
Constraint: tiz, where z = x02am , the safe range parameter, for i=1,2,,n.
5:     sigma – double scalar
σ, the annual total volatility, including jumps.
Constraint: sigma>0.0.
6:     r – double scalar
r, the annual risk-free interest rate, continuously compounded. Note that a rate of 5% should be entered as 0.05.
Constraint: r0.0.
7:     lambda – double scalar
λ, the number of expected jumps per year.
Constraint: lambda>0.0.
8:     jvol – double scalar
The proportion of the total volatility associated with jumps.
Constraint: 0.0jvol<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: m1.
2:     n int64int32nag_int scalar
Default: the dimension of the array t.
The number of times to expiry to be used.
Constraint: n1.

Output Parameters

1:     pldpn – double array
pij contains Pij, the option price evaluated for the strike price xi at expiry tj for i=1,2,,m and j=1,2,,n.
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, calput=_ was an illegal value.
Constraint: m1.
Constraint: n1.
Constraint: xi_ and xi_.
Constraint: s_ and s_.
Constraint: ti_.
Constraint: sigma>0.0.
Constraint: r0.0.
Constraint: lambda>0.0.
Constraint: jvol0.0 and jvol < 1.0.
Constraint: ldpm.
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.


The accuracy of the output is dependent on the accuracy of the cumulative Normal distribution function, Φ, occurring in Cj. 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



This example computes the price of a European call with jumps. The time to expiry is 3 months, the stock price is 45 and the strike price is 55. The number of jumps per year is 3 and the percentage of the total volatility due to jumps is 40%. The risk-free interest rate is 10% per year and the total volatility is 25% per year.
function s30ja_example

fprintf('s30ja example results\n\n');

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));

s30ja example results

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

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Chapter Contents
Chapter Introduction
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