NAG Library Function Document

nag_jumpdiff_merton_price (s30jac)


    1  Purpose
    7  Accuracy


nag_jumpdiff_merton_price (s30jac) computes the European option price using the Merton jump-diffusion model.


#include <nag.h>
#include <nags.h>
void  nag_jumpdiff_merton_price (Nag_OrderType order, Nag_CallPut option, Integer m, Integer n, const double x[], double s, const double t[], double sigma, double r, double lambda, double jvol, double p[], NagError *fail)


nag_jumpdiff_merton_price (s30jac) 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_bsm_price (s30aac)).
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:     order Nag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by order=Nag_RowMajor. See Section in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     option Nag_CallPutInput
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: option=Nag_Call or Nag_Put.
3:     m IntegerInput
On entry: the number of strike prices to be used.
Constraint: m1.
4:     n IntegerInput
On entry: the number of times to expiry to be used.
Constraint: n1.
5:     x[m] const doubleInput
On entry: x[i-1] must contain Xi, the ith strike price, for i=1,2,,m.
Constraint: x[i-1]z ​ and ​ x[i-1] 1 / z , where z = nag_real_safe_small_number , the safe range parameter, for i=1,2,,m.
6:     s doubleInput
On entry: S, the price of the underlying asset.
Constraint: sz ​ and ​s1.0/z, where z=nag_real_safe_small_number, the safe range parameter.
7:     t[n] const doubleInput
On entry: t[i-1] must contain Ti, the ith time, in years, to expiry, for i=1,2,,n.
Constraint: t[i-1]z, where z = nag_real_safe_small_number , the safe range parameter, for i=1,2,,n.
8:     sigma doubleInput
On entry: σ, the annual total volatility, including jumps.
Constraint: sigma>0.0.
9:     r doubleInput
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.
10:   lambda doubleInput
On entry: λ, the number of expected jumps per year.
Constraint: lambda>0.0.
11:   jvol doubleInput
On entry: the proportion of the total volatility associated with jumps.
Constraint: 0.0jvol<1.0.
12:   p[m×n] doubleOutput
Note: where Pi,j appears in this document, it refers to the array element
  • p[j-1×m+i-1] when order=Nag_ColMajor;
  • p[i-1×n+j-1] when order=Nag_RowMajor.
On exit: Pi,j contains Pij, the option price evaluated for the strike price xi at expiry tj for i=1,2,,m and j=1,2,,n.
13:   fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

Error Indicators and Warnings

Dynamic memory allocation failed.
See Section in How to Use the NAG Library and its Documentation for further information.
On entry, argument value had an illegal value.
On entry, m=value.
Constraint: m1.
On entry, n=value.
Constraint: n1.
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 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 2.7.5 in How to Use the NAG Library and its Documentation for further information.
On entry, jvol=value.
Constraint: jvol0.0 and jvol < 1.0.
On entry, lambda=value.
Constraint: lambda>0.0.
On entry, r=value.
Constraint: r0.0.
On entry, s=value.
Constraint: svalue and svalue.
On entry, sigma=value.
Constraint: sigma>0.0.
On entry, t[value]=value.
Constraint: t[i]value.
On entry, x[value]=value.
Constraint: x[i]value and x[i]value.


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_cumul_normal (s15abc) and nag_erfc (s15adc)). An accuracy close to machine precision can generally be expected.

Parallelism and Performance

nag_jumpdiff_merton_price (s30jac) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_jumpdiff_merton_price (s30jac) 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 function. 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 (s30jace.c)

Program Data

Program Data (s30jace.d)

Program Results

Program Results (s30jace.r)

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