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# 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 + σ̂ dWt + dqt . $dS S = (α-λk) dt + σ^ dWt + dqt .$
Here α$\alpha$ is the instantaneous expected return on the asset price, S$S$; σ̂2${\stackrel{^}{\sigma }}^{2}$ is the instantaneous variance of the return when the Poisson event does not occur; dWt${dW}_{t}$ is a standard Brownian motion; qt${q}_{t}$ is the independent Poisson process and k = E[Y1]$k=E\left[Y-1\right]$ where Y1$Y-1$ is the random variable change in the stock price if the Poisson event occurs and E$E$ is the expectation operator over the random variable Y$Y$.
This leads to the following price for a European option (see Haug (2007))
 ∞ Pcall = ∑ ( e − λT (λT)j )/(j ! )Cj(S,X,T,r,σj′), j = 0
$Pcall = ∑ j=0 ∞ e-λT (λT)j j! Cj ( S, X, T, r, σj′ ) ,$
where T$T$ is the time to expiry; X$X$ is the strike price; r$r$ is the annual risk-free interest rate; Cj(S,X,T,r,σj)${C}_{j}\left(S,X,T,r,{\sigma }_{j}^{\prime }\right)$ is the Black–Scholes–Merton option pricing formula for a European call (see nag_specfun_opt_bsm_price (s30aa)).
 σj′ = sqrt( z2 + δ2 (j/T) ) , z2 = σ2 − λ δ2 , δ2 = (γ σ2)/(λ) ,
$σj′ = z2 + δ2 ( j T ) , z2 = σ2 - λ δ2 , δ2 = γ σ2 λ ,$
where σ$\sigma$ is the total volatility including jumps; λ$\lambda$ is the expected number of jumps given as an average per year; γ$\gamma$ 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 Cj (S,X,T,r,σj)${C}_{j}\left(S,X,T,r,{\sigma }_{j}^{\prime }\right)$.

## 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:     calput – string (length ≥ 1)
Determines whether the option is a call or a put.
calput = 'C'${\mathbf{calput}}=\text{'C'}$
A call. The holder has a right to buy.
calput = 'P'${\mathbf{calput}}=\text{'P'}$
A put. The holder has a right to sell.
Constraint: calput = 'C'${\mathbf{calput}}=\text{'C'}$ or 'P'$\text{'P'}$.
2:     x(m) – double array
m, the dimension of the array, must satisfy the constraint m1${\mathbf{m}}\ge 1$.
x(i)${\mathbf{x}}\left(i\right)$ must contain Xi${X}_{\mathit{i}}$, the i$\mathit{i}$th strike price, for i = 1,2,,m$\mathit{i}=1,2,\dots ,{\mathbf{m}}$.
Constraint: x(i)z ​ and ​ x(i) 1 / z ${\mathbf{x}}\left(\mathit{i}\right)\ge z\text{​ and ​}{\mathbf{x}}\left(\mathit{i}\right)\le 1/z$, where z = x02am () $z=\mathbf{x02am}\left(\right)$, the safe range parameter, for i = 1,2,,m$\mathit{i}=1,2,\dots ,{\mathbf{m}}$.
3:     s – double scalar
S$S$, the price of the underlying asset.
Constraint: sz ​ and ​s1.0 / z${\mathbf{s}}\ge z\text{​ and ​}{\mathbf{s}}\le 1.0/z$, where z = x02am()$z=\mathbf{x02am}\left(\right)$, the safe range parameter.
4:     t(n) – double array
n, the dimension of the array, must satisfy the constraint n1${\mathbf{n}}\ge 1$.
t(i)${\mathbf{t}}\left(i\right)$ must contain Ti${T}_{\mathit{i}}$, the i$\mathit{i}$th time, in years, to expiry, for i = 1,2,,n$\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
Constraint: t(i)z${\mathbf{t}}\left(\mathit{i}\right)\ge z$, where z = x02am () $z=\mathbf{x02am}\left(\right)$, the safe range parameter, for i = 1,2,,n$\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
5:     sigma – double scalar
σ$\sigma$, the annual total volatility, including jumps.
Constraint: sigma > 0.0${\mathbf{sigma}}>0.0$.
6:     r – double scalar
r$r$, the annual risk-free interest rate, continuously compounded. Note that a rate of 5% should be entered as 0.05.
Constraint: r0.0${\mathbf{r}}\ge 0.0$.
7:     lambda – double scalar
λ$\lambda$, the number of expected jumps per year.
Constraint: lambda > 0.0${\mathbf{lambda}}>0.0$.
8:     jvol – double scalar
The proportion of the total volatility associated with jumps.
Constraint: 0.0jvol < 1.0$0.0\le {\mathbf{jvol}}<1.0$.

### Optional Input Parameters

1:     m – int64int32nag_int scalar
Default: The dimension of the array x.
The number of strike prices to be used.
Constraint: m1${\mathbf{m}}\ge 1$.
2:     n – int64int32nag_int scalar
Default: The dimension of the array t.
The number of times to expiry to be used.
Constraint: n1${\mathbf{n}}\ge 1$.

ldp

### Output Parameters

1:     p(ldp,n) – double array
ldpm$\mathit{ldp}\ge {\mathbf{m}}$.
The leading m × n${\mathbf{m}}×{\mathbf{n}}$ part of the array p contains the computed option prices.
2:     delta(ldp,n) – double array
ldpm$\mathit{ldp}\ge {\mathbf{m}}$.
The leading m × n${\mathbf{m}}×{\mathbf{n}}$ part of the array delta contains the sensitivity, (P)/(S)$\frac{\partial P}{\partial S}$, of the option price to change in the price of the underlying asset.
3:     gamma(ldp,n) – double array
ldpm$\mathit{ldp}\ge {\mathbf{m}}$.
The leading m × n${\mathbf{m}}×{\mathbf{n}}$ part of the array gamma contains the sensitivity, (2P)/(S2)$\frac{{\partial }^{2}P}{\partial {S}^{2}}$, of delta to change in the price of the underlying asset.
4:     vega(ldp,n) – double array
ldpm$\mathit{ldp}\ge {\mathbf{m}}$.
The leading m × n${\mathbf{m}}×{\mathbf{n}}$ part of the array vega contains the sensitivity, (P)/(σ)$\frac{\partial P}{\partial \sigma }$, of the option price to change in the volatility of the underlying asset.
5:     theta(ldp,n) – double array
ldpm$\mathit{ldp}\ge {\mathbf{m}}$.
The leading m × n${\mathbf{m}}×{\mathbf{n}}$ part of the array theta contains the sensitivity, (P)/(T)$-\frac{\partial P}{\partial T}$, of the option price to change in the time to expiry of the option.
6:     rho(ldp,n) – double array
ldpm$\mathit{ldp}\ge {\mathbf{m}}$.
The leading m × n${\mathbf{m}}×{\mathbf{n}}$ part of the array rho contains the sensitivity, (P)/(r)$\frac{\partial P}{\partial r}$, of the option price to change in the annual risk-free interest rate.
7:     vanna(ldp,n) – double array
ldpm$\mathit{ldp}\ge {\mathbf{m}}$.
The leading m × n${\mathbf{m}}×{\mathbf{n}}$ part of the array vanna contains the sensitivity, (2P)/(Sσ)$\frac{{\partial }^{2}P}{\partial S\partial \sigma }$, of vega to change in the price of the underlying asset or, equivalently, the sensitivity of delta to change in the volatility of the asset price.
8:     charm(ldp,n) – double array
ldpm$\mathit{ldp}\ge {\mathbf{m}}$.
The leading m × n${\mathbf{m}}×{\mathbf{n}}$ part of the array charm contains the sensitivity, (2P)/(S T)$-\frac{{\partial }^{2}P}{\partial S\partial T}$, of delta to change in the time to expiry of the option.
9:     speed(ldp,n) – double array
ldpm$\mathit{ldp}\ge {\mathbf{m}}$.
The leading m × n${\mathbf{m}}×{\mathbf{n}}$ part of the array speed contains the sensitivity, (3P)/(S3)$\frac{{\partial }^{3}P}{\partial {S}^{3}}$, of gamma to change in the price of the underlying asset.
10:   colour(ldp,n) – double array
ldpm$\mathit{ldp}\ge {\mathbf{m}}$.
The leading m × n${\mathbf{m}}×{\mathbf{n}}$ part of the array colour contains the sensitivity, (3P)/(S2 T)$-\frac{{\partial }^{3}P}{\partial {S}^{2}\partial T}$, of gamma to change in the time to expiry of the option.
11:   zomma(ldp,n) – double array
ldpm$\mathit{ldp}\ge {\mathbf{m}}$.
The leading m × n${\mathbf{m}}×{\mathbf{n}}$ part of the array zomma contains the sensitivity, (3P)/(S2σ)$\frac{{\partial }^{3}P}{\partial {S}^{2}\partial \sigma }$, of gamma to change in the volatility of the underlying asset.
12:   vomma(ldp,n) – double array
ldpm$\mathit{ldp}\ge {\mathbf{m}}$.
The leading m × n${\mathbf{m}}×{\mathbf{n}}$ part of the array vomma contains the sensitivity, (2P)/(σ2)$\frac{{\partial }^{2}P}{\partial {\sigma }^{2}}$, of vega to change in the volatility of the underlying asset.
13:   ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

## Error Indicators and Warnings

Errors or warnings detected by the function:
ifail = 1${\mathbf{ifail}}=1$
On entry, calput = 'C'${\mathbf{calput}}=\text{'C'}$ or 'P'$\text{'P'}$.
ifail = 2${\mathbf{ifail}}=2$
On entry, m0${\mathbf{m}}\le 0$.
ifail = 3${\mathbf{ifail}}=3$
On entry, n0${\mathbf{n}}\le 0$.
ifail = 4${\mathbf{ifail}}=4$
On entry, x(i) < z${\mathbf{x}}\left(\mathit{i}\right) or x(i) > 1 / z${\mathbf{x}}\left(\mathit{i}\right)>1/z$, where z = x02am()$z=\mathbf{x02am}\left(\right)$, the safe range parameter.
ifail = 5${\mathbf{ifail}}=5$
On entry, s < z${\mathbf{s}} or s > 1.0 / z${\mathbf{s}}>1.0/z$, where z = x02am()$z=\mathbf{x02am}\left(\right)$, the safe range parameter.
ifail = 6${\mathbf{ifail}}=6$
On entry, t(i) < z${\mathbf{t}}\left(\mathit{i}\right), where z = x02am()$z=\mathbf{x02am}\left(\right)$, the safe range parameter.
ifail = 7${\mathbf{ifail}}=7$
On entry, sigma0.0${\mathbf{sigma}}\le 0.0$.
ifail = 8${\mathbf{ifail}}=8$
On entry, r < 0.0${\mathbf{r}}<0.0$.
ifail = 9${\mathbf{ifail}}=9$
On entry, lambda0.0${\mathbf{lambda}}\le 0.0$.
ifail = 10${\mathbf{ifail}}=10$
On entry, jvol < 0.0${\mathbf{jvol}}<0.0$ or jvol1.0${\mathbf{jvol}}\ge 1.0$.
ifail = 12${\mathbf{ifail}}=12$
On entry, ldp < m$\mathit{ldp}<{\mathbf{m}}$.

## Accuracy

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

```function nag_specfun_opt_jumpdiff_merton_greeks_example
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] = ...
nag_specfun_opt_jumpdiff_merton_greeks(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(' Strike    Price    Delta    Gamma     Vega    Theta      Rho\n');
for i=1:2
fprintf('%8.4f %8.4f %8.4f %8.4f %8.4f %8.4f %8.4f\n', x(i), ...
p(i,1), delta(i,1), gamma(i,1), vega(i,1), theta(i,1), rho(i,1));
end

fprintf('\n Strike    Price    Vanna    Charm    Speed   Colour    Zomma    Vomma\n');
for i=1:2
fprintf('%8.4f %8.4f %8.4f %8.4f %8.4f %8.4f %8.4f %8.4f\n', x(i), ...
p(i,1), vanna(i,1), charm(i,1), speed(i,1), colour(i,1), ...
zomma(i,1), vomma(i,1));
end
```
```

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

Strike    Price    Vanna    Charm    Speed   Colour    Zomma    Vomma
80.0000  23.6090  -0.6334   0.1080  -0.0006  -0.0035   0.0315  70.6824
90.0000  15.4193  -0.7726   0.0770  -0.0009   0.0109  -0.0186  49.7161

```
```function s30jb_example
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(' Strike    Price    Delta    Gamma     Vega    Theta      Rho\n');
for i=1:2
fprintf('%8.4f %8.4f %8.4f %8.4f %8.4f %8.4f %8.4f\n', x(i), ...
p(i,1), delta(i,1), gamma(i,1), vega(i,1), theta(i,1), rho(i,1));
end

fprintf('\n Strike    Price    Vanna    Charm    Speed   Colour    Zomma    Vomma\n');
for i=1:2
fprintf('%8.4f %8.4f %8.4f %8.4f %8.4f %8.4f %8.4f %8.4f\n', x(i), ...
p(i,1), vanna(i,1), charm(i,1), speed(i,1), colour(i,1), ...
zomma(i,1), vomma(i,1));
end
```
```

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

Strike    Price    Vanna    Charm    Speed   Colour    Zomma    Vomma
80.0000  23.6090  -0.6334   0.1080  -0.0006  -0.0035   0.0315  70.6824
90.0000  15.4193  -0.7726   0.0770  -0.0009   0.0109  -0.0186  49.7161

```

Chapter Contents
Chapter Introduction
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