# NAG FL Interfaces30jaf (opt_​jumpdiff_​merton_​price)

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## 1Purpose

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

## 2Specification

Fortran Interface
 Subroutine s30jaf ( m, n, x, s, t, r, jvol, p, ldp,
 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
#include <nag.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)
The routine may be called by the names s30jaf or nagf_specfun_opt_jumpdiff_merton_price.

## 3Description

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 $\alpha$ is the instantaneous expected return on the asset price, $S$; ${\stackrel{^}{\sigma }}^{2}$ is the instantaneous variance of the return when the Poisson event does not occur; ${dW}_{t}$ is a standard Brownian motion; ${q}_{t}$ is the independent Poisson process and $k=E\left[Y-1\right]$ 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))
 $Pcall = ∑ j=0 ∞ e-λT (λT)j j! Cj (S,X,T,r,σj′) ,$
where $T$ is the time to expiry; $X$ is the strike price; $r$ is the annual risk-free interest rate; ${C}_{j}\left(S,X,T,r,{\sigma }_{j}^{\prime }\right)$ 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 $\sigma$ is the total volatility including jumps; $\lambda$ is the expected number of jumps given as an average per year; $\gamma$ 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 ${C}_{j}\left(S,X,T,r,{\sigma }_{j}^{\prime }\right)$.
The option price ${P}_{ij}=P\left(X={X}_{i},T={T}_{j}\right)$ is computed for each strike price in a set ${X}_{i}$, $i=1,2,\dots ,m$, and for each expiry time in a set ${T}_{j}$, $j=1,2,\dots ,n$.

## 4References

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

## 5Arguments

1: $\mathbf{calput}$Character(1) Input
On entry: determines whether the option is a call or a put.
${\mathbf{calput}}=\text{'C'}$
A call; the holder has a right to buy.
${\mathbf{calput}}=\text{'P'}$
A put; the holder has a right to sell.
Constraint: ${\mathbf{calput}}=\text{'C'}$ or $\text{'P'}$.
2: $\mathbf{m}$Integer Input
On entry: the number of strike prices to be used.
Constraint: ${\mathbf{m}}\ge 1$.
3: $\mathbf{n}$Integer Input
On entry: the number of times to expiry to be used.
Constraint: ${\mathbf{n}}\ge 1$.
4: $\mathbf{x}\left({\mathbf{m}}\right)$Real (Kind=nag_wp) array Input
On entry: ${\mathbf{x}}\left(i\right)$ must contain ${X}_{\mathit{i}}$, the $\mathit{i}$th strike price, for $\mathit{i}=1,2,\dots ,{\mathbf{m}}$.
Constraint: ${\mathbf{x}}\left(\mathit{i}\right)\ge z\text{​ and ​}{\mathbf{x}}\left(\mathit{i}\right)\le 1/z$, where $z={\mathbf{x02amf}}\left(\right)$, the safe range parameter, for $\mathit{i}=1,2,\dots ,{\mathbf{m}}$.
5: $\mathbf{s}$Real (Kind=nag_wp) Input
On entry: $S$, the price of the underlying asset.
Constraint: ${\mathbf{s}}\ge z\text{​ and ​}{\mathbf{s}}\le 1.0/z$, where $z={\mathbf{x02amf}}\left(\right)$, the safe range parameter.
6: $\mathbf{t}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Input
On entry: ${\mathbf{t}}\left(i\right)$ must contain ${T}_{\mathit{i}}$, the $\mathit{i}$th time, in years, to expiry, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
Constraint: ${\mathbf{t}}\left(\mathit{i}\right)\ge z$, where $z={\mathbf{x02amf}}\left(\right)$, the safe range parameter, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
7: $\mathbf{sigma}$Real (Kind=nag_wp) Input
On entry: $\sigma$, the annual total volatility, including jumps.
Constraint: ${\mathbf{sigma}}>0.0$.
8: $\mathbf{r}$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: ${\mathbf{r}}\ge 0.0$.
9: $\mathbf{lambda}$Real (Kind=nag_wp) Input
On entry: $\lambda$, the number of expected jumps per year.
Constraint: ${\mathbf{lambda}}>0.0$.
10: $\mathbf{jvol}$Real (Kind=nag_wp) Input
On entry: the proportion of the total volatility associated with jumps.
Constraint: $0.0\le {\mathbf{jvol}}<1.0$.
11: $\mathbf{p}\left({\mathbf{ldp}},{\mathbf{n}}\right)$Real (Kind=nag_wp) array Output
On exit: ${\mathbf{p}}\left(i,j\right)$ contains ${P}_{ij}$, the option price evaluated for the strike price ${{\mathbf{x}}}_{i}$ at expiry ${{\mathbf{t}}}_{j}$ for $i=1,2,\dots ,{\mathbf{m}}$ and $j=1,2,\dots ,{\mathbf{n}}$.
12: $\mathbf{ldp}$Integer Input
On entry: the first dimension of the array p as declared in the (sub)program from which s30jaf is called.
Constraint: ${\mathbf{ldp}}\ge {\mathbf{m}}$.
13: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{calput}}=⟨\mathit{\text{value}}⟩$ was an illegal value.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{m}}\ge 1$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 1$.
${\mathbf{ifail}}=4$
On entry, ${\mathbf{x}}\left(⟨\mathit{\text{value}}⟩\right)=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{x}}\left(i\right)\ge ⟨\mathit{\text{value}}⟩$ and ${\mathbf{x}}\left(i\right)\le ⟨\mathit{\text{value}}⟩$.
${\mathbf{ifail}}=5$
On entry, ${\mathbf{s}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{s}}\ge ⟨\mathit{\text{value}}⟩$ and ${\mathbf{s}}\le ⟨\mathit{\text{value}}⟩$.
${\mathbf{ifail}}=6$
On entry, ${\mathbf{t}}\left(⟨\mathit{\text{value}}⟩\right)=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{t}}\left(i\right)\ge ⟨\mathit{\text{value}}⟩$.
${\mathbf{ifail}}=7$
On entry, ${\mathbf{sigma}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{sigma}}>0.0$.
${\mathbf{ifail}}=8$
On entry, ${\mathbf{r}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{r}}\ge 0.0$.
${\mathbf{ifail}}=9$
On entry, ${\mathbf{lambda}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{lambda}}>0.0$.
${\mathbf{ifail}}=10$
On entry, ${\mathbf{jvol}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{jvol}}\ge 0.0$ and ${\mathbf{jvol}}<1.0$.
${\mathbf{ifail}}=12$
On entry, ${\mathbf{ldp}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ldp}}\ge {\mathbf{m}}$.
${\mathbf{ifail}}=-99$
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

## 7Accuracy

The accuracy of the output is dependent on the accuracy of the cumulative Normal distribution function, $\Phi$, occurring in ${C}_{j}$. 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.

## 8Parallelism 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.

None.

## 10Example

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.

### 10.1Program Text

Program Text (s30jafe.f90)

### 10.2Program Data

Program Data (s30jafe.d)

### 10.3Program Results

Program Results (s30jafe.r)