NAG Library Routine Document

s30jaf (opt_jumpdiff_merton_price)


    1  Purpose
    7  Accuracy


s30jaf computes the European option price using the Merton jump-diffusion model.


Fortran Interface
Subroutine s30jaf ( calput, m, n, x, s, t, sigma, r, lambda, jvol, p, ldp, ifail)
Integer, Intent (In):: m, n, ldp
Integer, Intent (Inout):: ifail
Real (Kind=nag_wp), Intent (In):: x(m), s, t(n), sigma, r, lambda, jvol
Real (Kind=nag_wp), Intent (Inout):: p(ldp,n)
Character (1), Intent (In):: calput
C Header Interface
#include nagmk26.h
void  s30jaf_ (const char *calput, const Integer *m, const Integer *n, const double x[], const double *s, const double t[], const double *sigma, const double *r, const double *lambda, const double *jvol, double p[], const Integer *ldp, Integer *ifail, const Charlen length_calput)


s30jaf 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 s30aaf).
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


1:     calput – Character(1)Input
On entry: 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:     m – IntegerInput
On entry: the number of strike prices to be used.
Constraint: m1.
3:     n – IntegerInput
On entry: the number of times to expiry to be used.
Constraint: n1.
4:     xm – Real (Kind=nag_wp) arrayInput
On entry: xi must contain Xi, the ith strike price, for i=1,2,,m.
Constraint: xiz ​ and ​ xi 1 / z , where z = x02amf , the safe range parameter, for i=1,2,,m.
5:     s – Real (Kind=nag_wp)Input
On entry: S, the price of the underlying asset.
Constraint: sz ​ and ​s1.0/z, where z=x02amf, the safe range parameter.
6:     tn – Real (Kind=nag_wp) arrayInput
On entry: ti must contain Ti, the ith time, in years, to expiry, for i=1,2,,n.
Constraint: tiz, where z = x02amf , the safe range parameter, for i=1,2,,n.
7:     sigma – Real (Kind=nag_wp)Input
On entry: σ, the annual total volatility, including jumps.
Constraint: sigma>0.0.
8:     r – Real (Kind=nag_wp)Input
On entry: r, the annual risk-free interest rate, continuously compounded. Note that a rate of 5% should be entered as 0.05.
Constraint: r0.0.
9:     lambda – Real (Kind=nag_wp)Input
On entry: λ, the number of expected jumps per year.
Constraint: lambda>0.0.
10:   jvol – Real (Kind=nag_wp)Input
On entry: the proportion of the total volatility associated with jumps.
Constraint: 0.0jvol<1.0.
11:   pldpn – Real (Kind=nag_wp) arrayOutput
On exit: 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.
12:   ldp – IntegerInput
On entry: the first dimension of the array p as declared in the (sub)program from which s30jaf is called.
Constraint: ldpm.
13:   ifail – IntegerInput/Output
On entry: ifail must be set to 0, -1​ or ​1. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value -1​ or ​1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, if you are not familiar with this argument, the recommended value is 0. When the value -1​ or ​1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

Error Indicators and Warnings

If on entry ifail=0 or -1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry, calput=value was an illegal value.
On entry, m=value.
Constraint: m1.
On entry, n=value.
Constraint: n1.
On entry, xvalue=value.
Constraint: xivalue and xivalue.
On entry, s=value.
Constraint: svalue and svalue.
On entry, tvalue=value.
Constraint: tivalue.
On entry, sigma=value.
Constraint: sigma>0.0.
On entry, r=value.
Constraint: r0.0.
On entry, lambda=value.
Constraint: lambda>0.0.
On entry, jvol=value.
Constraint: jvol0.0 and jvol < 1.0.
On entry, ldp=value and m=value.
Constraint: ldpm.
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.


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 s15abf and s15adf). An accuracy close to machine precision can generally be expected.

Parallelism and Performance

s30jaf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
s30jaf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

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.

Program Text

Program Text (s30jafe.f90)

Program Data

Program Data (s30jafe.d)

Program Results

Program Results (s30jafe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017