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Chapter Contents
Chapter Introduction
NAG Toolbox

# 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,
 $dS S = α-λk dt + σ^ dWt + dqt .$
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))
 $Pcall = ∑ j=0 ∞ e-λT λTj j! Cj S, X, T, r, σj′ ,$
where $T$ is the time to expiry; $X$ is the strike price; $r$ is the annual risk-free interest rate; ${C}_{j}\left(S,X,T,r,{\sigma }_{j}^{\prime }\right)$ 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 $\sigma$ 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 ${T}_{j}$, $j=1,2,\dots ,n$.

## 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:     $\mathrm{calput}$ – string (length ≥ 1)
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:     $\mathrm{x}\left({\mathbf{m}}\right)$ – double array
${\mathbf{x}}\left(i\right)$ 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 $z={\mathbf{x02am}}\left(\right)$, the safe range parameter, for $\mathit{i}=1,2,\dots ,{\mathbf{m}}$.
3:     $\mathrm{s}$ – double scalar
$S$, the price of the underlying asset.
Constraint: ${\mathbf{s}}\ge z\text{​ and ​}{\mathbf{s}}\le 1.0/z$, where $z={\mathbf{x02am}}\left(\right)$, the safe range parameter.
4:     $\mathrm{t}\left({\mathbf{n}}\right)$ – double array
${\mathbf{t}}\left(i\right)$ 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 $z={\mathbf{x02am}}\left(\right)$, the safe range parameter, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
5:     $\mathrm{sigma}$ – double scalar
$\sigma$, the annual total volatility, including jumps.
Constraint: ${\mathbf{sigma}}>0.0$.
6:     $\mathrm{r}$ – double scalar
$r$, the annual risk-free interest rate, continuously compounded. Note that a rate of 5% should be entered as 0.05.
Constraint: ${\mathbf{r}}\ge 0.0$.
7:     $\mathrm{lambda}$ – double scalar
$\lambda$, the number of expected jumps per year.
Constraint: ${\mathbf{lambda}}>0.0$.
8:     $\mathrm{jvol}$ – double scalar
The proportion of the total volatility associated with jumps.
Constraint: $0.0\le {\mathbf{jvol}}<1.0$.

### Optional Input Parameters

1:     $\mathrm{m}$int64int32nag_int scalar
Default: the dimension of the array x.
The number of strike prices to be used.
Constraint: ${\mathbf{m}}\ge 1$.
2:     $\mathrm{n}$int64int32nag_int scalar
Default: the dimension of the array t.
The number of times to expiry to be used.
Constraint: ${\mathbf{n}}\ge 1$.

### Output Parameters

1:     $\mathrm{p}\left(\mathit{ldp},{\mathbf{n}}\right)$ – double array
$\mathit{ldp}={\mathbf{m}}$.
${\mathbf{p}}\left(i,j\right)$ 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 $j=1,2,\dots ,{\mathbf{n}}$.
2:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Errors or warnings detected by the function:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{calput}}=_$ was an illegal value.
${\mathbf{ifail}}=2$
Constraint: ${\mathbf{m}}\ge 1$.
${\mathbf{ifail}}=3$
Constraint: ${\mathbf{n}}\ge 1$.
${\mathbf{ifail}}=4$
Constraint: ${\mathbf{x}}\left(i\right)\ge _$ and ${\mathbf{x}}\left(i\right)\le _$.
${\mathbf{ifail}}=5$
Constraint: ${\mathbf{s}}\ge _$ and ${\mathbf{s}}\le _$.
${\mathbf{ifail}}=6$
Constraint: ${\mathbf{t}}\left(i\right)\ge _$.
${\mathbf{ifail}}=7$
Constraint: ${\mathbf{sigma}}>0.0$.
${\mathbf{ifail}}=8$
Constraint: ${\mathbf{r}}\ge 0.0$.
${\mathbf{ifail}}=9$
Constraint: ${\mathbf{lambda}}>0.0$.
${\mathbf{ifail}}=10$
Constraint: ${\mathbf{jvol}}\ge 0.0$ and ${\mathbf{jvol}}<1.0$.
${\mathbf{ifail}}=12$
Constraint: $\mathit{ldp}\ge {\mathbf{m}}$.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

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 nag_specfun_cdf_normal (s15ab) and nag_specfun_erfc_real (s15ad)). An accuracy close to machine precision can generally be expected.

None.

## Example

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));
end
end

```
```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
```