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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_specfun_opt_jumpdiff_merton_greeks (s30jb)

## Purpose

nag_specfun_opt_jumpdiff_merton_greeks (s30jb) computes the European option price together with its sensitivities (Greeks) using the Merton jump-diffusion model.

## Syntax

[p, delta, gamma, vega, theta, rho, vanna, charm, speed, colour, zomma, vomma, ifail] = s30jb(calput, x, s, t, sigma, r, lambda, jvol, 'm', m, 'n', n)
[p, delta, gamma, vega, theta, rho, vanna, charm, speed, colour, zomma, vomma, ifail] = nag_specfun_opt_jumpdiff_merton_greeks(calput, x, s, t, sigma, r, lambda, jvol, 'm', m, 'n', n)

## Description

nag_specfun_opt_jumpdiff_merton_greeks (s30jb) uses Merton's jump-diffusion model (Merton (1976)) to compute the price of a European option, together with the Greeks or sensitivities, which are the partial derivatives of the option price with respect to certain of the other input parameters. Merton's model 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:     $\mathrm{delta}\left(\mathit{ldp},{\mathbf{n}}\right)$ – double array
$\mathit{ldp}={\mathbf{m}}$.
The leading ${\mathbf{m}}×{\mathbf{n}}$ part of the array delta contains the sensitivity, $\frac{\partial P}{\partial S}$, of the option price to change in the price of the underlying asset.
3:     $\mathrm{gamma}\left(\mathit{ldp},{\mathbf{n}}\right)$ – double array
$\mathit{ldp}={\mathbf{m}}$.
The leading ${\mathbf{m}}×{\mathbf{n}}$ part of the array gamma contains the sensitivity, $\frac{{\partial }^{2}P}{\partial {S}^{2}}$, of delta to change in the price of the underlying asset.
4:     $\mathrm{vega}\left(\mathit{ldp},{\mathbf{n}}\right)$ – double array
$\mathit{ldp}={\mathbf{m}}$.
${\mathbf{vega}}\left(i,j\right)$, contains the first-order Greek measuring the sensitivity of the option price ${P}_{ij}$ to change in the volatility of the underlying asset, i.e., $\frac{\partial {P}_{ij}}{\partial \sigma }$, for $i=1,2,\dots ,{\mathbf{m}}$ and $j=1,2,\dots ,{\mathbf{n}}$.
5:     $\mathrm{theta}\left(\mathit{ldp},{\mathbf{n}}\right)$ – double array
$\mathit{ldp}={\mathbf{m}}$.
${\mathbf{theta}}\left(i,j\right)$, contains the first-order Greek measuring the sensitivity of the option price ${P}_{ij}$ to change in time, i.e., $-\frac{\partial {P}_{ij}}{\partial T}$, for $i=1,2,\dots ,{\mathbf{m}}$ and $j=1,2,\dots ,{\mathbf{n}}$, where $b=r-q$.
6:     $\mathrm{rho}\left(\mathit{ldp},{\mathbf{n}}\right)$ – double array
$\mathit{ldp}={\mathbf{m}}$.
${\mathbf{rho}}\left(i,j\right)$, contains the first-order Greek measuring the sensitivity of the option price ${P}_{ij}$ to change in the annual risk-free interest rate, i.e., $-\frac{\partial {P}_{ij}}{\partial r}$, for $i=1,2,\dots ,{\mathbf{m}}$ and $j=1,2,\dots ,{\mathbf{n}}$.
7:     $\mathrm{vanna}\left(\mathit{ldp},{\mathbf{n}}\right)$ – double array
$\mathit{ldp}={\mathbf{m}}$.
${\mathbf{vanna}}\left(i,j\right)$, contains the second-order Greek measuring the sensitivity of the first-order Greek ${\Delta }_{ij}$ to change in the volatility of the asset price, i.e., $-\frac{\partial {\Delta }_{ij}}{\partial T}=-\frac{{\partial }^{2}{P}_{ij}}{\partial S\partial \sigma }$, for $i=1,2,\dots ,{\mathbf{m}}$ and $j=1,2,\dots ,{\mathbf{n}}$.
8:     $\mathrm{charm}\left(\mathit{ldp},{\mathbf{n}}\right)$ – double array
$\mathit{ldp}={\mathbf{m}}$.
${\mathbf{charm}}\left(i,j\right)$, contains the second-order Greek measuring the sensitivity of the first-order Greek ${\Delta }_{ij}$ to change in the time, i.e., $-\frac{\partial {\Delta }_{ij}}{\partial T}=-\frac{{\partial }^{2}{P}_{ij}}{\partial S\partial T}$, for $i=1,2,\dots ,{\mathbf{m}}$ and $j=1,2,\dots ,{\mathbf{n}}$.
9:     $\mathrm{speed}\left(\mathit{ldp},{\mathbf{n}}\right)$ – double array
$\mathit{ldp}={\mathbf{m}}$.
${\mathbf{speed}}\left(i,j\right)$, contains the third-order Greek measuring the sensitivity of the second-order Greek ${\Gamma }_{ij}$ to change in the price of the underlying asset, i.e., $-\frac{\partial {\Gamma }_{ij}}{\partial S}=-\frac{{\partial }^{3}{P}_{ij}}{\partial {S}^{3}}$, for $i=1,2,\dots ,{\mathbf{m}}$ and $j=1,2,\dots ,{\mathbf{n}}$.
10:   $\mathrm{colour}\left(\mathit{ldp},{\mathbf{n}}\right)$ – double array
$\mathit{ldp}={\mathbf{m}}$.
${\mathbf{colour}}\left(i,j\right)$, contains the third-order Greek measuring the sensitivity of the second-order Greek ${\Gamma }_{ij}$ to change in the time, i.e., $-\frac{\partial {\Gamma }_{ij}}{\partial T}=-\frac{{\partial }^{3}{P}_{ij}}{\partial S\partial T}$, for $i=1,2,\dots ,{\mathbf{m}}$ and $j=1,2,\dots ,{\mathbf{n}}$.
11:   $\mathrm{zomma}\left(\mathit{ldp},{\mathbf{n}}\right)$ – double array
$\mathit{ldp}={\mathbf{m}}$.
${\mathbf{zomma}}\left(i,j\right)$, contains the third-order Greek measuring the sensitivity of the second-order Greek ${\Gamma }_{ij}$ to change in the volatility of the underlying asset, i.e., $-\frac{\partial {\Gamma }_{ij}}{\partial \sigma }=-\frac{{\partial }^{3}{P}_{ij}}{\partial {S}^{2}\partial \sigma }$, for $i=1,2,\dots ,{\mathbf{m}}$ and $j=1,2,\dots ,{\mathbf{n}}$.
12:   $\mathrm{vomma}\left(\mathit{ldp},{\mathbf{n}}\right)$ – double array
$\mathit{ldp}={\mathbf{m}}$.
${\mathbf{vomma}}\left(i,j\right)$, contains the second-order Greek measuring the sensitivity of the first-order Greek ${\Delta }_{ij}$ to change in the volatility of the underlying asset, i.e., $-\frac{\partial {\Delta }_{ij}}{\partial \sigma }=-\frac{{\partial }^{2}{P}_{ij}}{\partial {\sigma }^{2}}$, for $i=1,2,\dots ,{\mathbf{m}}$ and $j=1,2,\dots ,{\mathbf{n}}$.
13:   $\mathrm{ifail}$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 two European calls with jumps. The time to expiry is $6$ months, the stock price is $100$ and strike prices are $80$ and $90$ respectively. The number of jumps per year is $5$ and the percentage of the total volatility due to jumps is $25%$. The risk-free interest rate is $8%$ per year while the total volatility is $25%$ per year.
```function s30jb_example

fprintf('s30jb example results\n\n');

put    = 'C';
lambda = 5;
s      = 100.0;
sigma  = 0.25;
r      = 0.08;
jvol   = 0.25;
x      = [80.0, 90.0];
t      = [0.5];

[p, delta, gamma, vega,   theta, rho, ...
vanna, charm, speed, colour, zomma, vomma, ifail] = ...
s30jb(...
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(' Time to Expiry : %8.4f\n', t(1));

fprintf('%8s%9s%9s%9s%9s%9s%9s\n','Strike','Price','Delta','Gamma',...
'Vega','Theta','Rho');
for i=1:2
fprintf('%8.4f%9.4f%9.4f%9.4f%9.4f%9.4f%9.4f\n', x(i), p(i,1), ...
delta(i,1), gamma(i,1), vega(i,1), theta(i,1), rho(i,1));
end

fprintf('\n%26s%9s%9s%9s%9s%9s\n','Vanna','Charm','Speed','Colour',...
'Zomma','Vomma');
for i=1:2
fprintf('%17s%9.4f%9.4f%9.4f%9.4f%9.4f%9.4f\n', ' ', vanna(i,1), ...
charm(i,1), speed(i,1), colour(i,1), zomma(i,1), vomma(i,1));
end

```
```s30jb example results

Merton Jump-Diffusion Model
European Call :
Spot       =    100.0000
Volatility =      0.2500
Rate       =      0.0800
Jumps      =      5.0000
Jump Vol   =      0.2500

Time to Expiry :   0.5000
Strike    Price    Delta    Gamma     Vega    Theta      Rho
80.0000  23.6090   0.9431   0.0064   8.1206  -7.6718  35.3480
90.0000  15.4193   0.8203   0.0149  18.5256  -9.9695  33.3037

Vanna    Charm    Speed   Colour    Zomma    Vomma
-0.6334   0.1080  -0.0006  -0.0035   0.0315  70.6824
-0.7726   0.0770  -0.0009   0.0109  -0.0186  49.7161
```