NAG Library Function Document

nag_jumpdiff_merton_greeks (s30jbc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

nag_jumpdiff_merton_greeks (s30jbc) computes the European option price together with its sensitivities (Greeks) using the Merton jump-diffusion model.

2 Specification

#include <nag.h>

#include <nags.h>

void

nag_jumpdiff_merton_greeks (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[], double delta[], double gamma[], double vega[], double theta[], double rho[], double vanna[], double charm[], double speed[], double colour[], double zomma[], double vomma[], NagError *fail)

3 Description

nag_jumpdiff_merton_greeks (s30jbc) 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,

\frac{d S}{S} = (α - λ k) d t + \hat{σ} {d W}_{t} + {d q}_{t} .

Here

α

is the instantaneous expected return on the asset price,

S

;

{\hat{σ}}^{2}

is the instantaneous variance of the return when the Poisson event does not occur;

{d W}_{t}

is a standard Brownian motion;

q_{t}

is the independent Poisson process and

k = E [Y - 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))

P_{call} = \sum_{j = 0}^{\infty} \frac{e^{- λ T} {(λ T)}^{j}}{j!} C_{j} (S, X, T, r, σ_{j}^{'}),

where

T

is the time to expiry;

X

is the strike price;

r

is the annual risk-free interest rate;

C_{j} (S, X, T, r, σ_{j}^{'})

is the Black–Scholes–Merton option pricing formula for a European call (see nag_bsm_price (s30aac)).

\begin{array}{l} σ_{j}^{'} = \sqrt{z^{2} + δ^{2} (\frac{j}{T})}, \\ z^{2} = σ^{2} - λ δ^{2}, \\ δ^{2} = \frac{γ σ^{2}}{λ}, \end{array}

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

C_{j} (S, X, T, r, σ_{j}^{'})

4 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

5 Arguments

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 3.2.1.3 in the Essential Introduction 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.

$option = Nag_Call$: A call. The holder has a right to buy.
$option = Nag_Put$: A put. The holder has a right to sell.

Constraint:

option = Nag_Call

Nag_Put

3: m – IntegerInput

On entry: the number of strike prices to be used.

Constraint:

m \geq 1

4: n – IntegerInput

On entry: the number of times to expiry to be used.

Constraint:

n \geq 1

5: x[m] – const doubleInput

On entry:

x [i - 1]

must contain

X_{i}

, the

i

th strike price, for

i = 1, 2, \dots, m

Constraint:

x [i - 1] \geq z ​ and ​ x [i - 1] \leq 1 / z

, where

z = nag_real_safe_small_number

, the safe range parameter, for

i = 1, 2, \dots, m

6: s – doubleInput

On entry:

S

, the price of the underlying asset.

Constraint:

s \geq z ​ and ​ s \leq 1.0 / z

, where

z = nag_real_safe_small_number

, the safe range parameter.

7: t[n] – const doubleInput

On entry:

t [i - 1]

must contain

T_{i}

, the

i

th time, in years, to expiry, for

i = 1, 2, \dots, n

Constraint:

t [i - 1] \geq z

, where

z = nag_real_safe_small_number

, the safe range parameter, for

i = 1, 2, \dots, 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:

r \geq 0.0

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:

0.0 \leq jvol < 1.0

12: p[ $m \times n$ ] – doubleOutput

Note: the

(i, j)

th element of the matrix

P

is stored in

$p [(j - 1) \times m + i - 1]$ when $order = Nag_ColMajor$ ;
$p [(i - 1) \times n + j - 1]$ when $order = Nag_RowMajor$ .

On exit: the

m \times n

array p contains the computed option prices.

13: delta[ $m \times n$ ] – doubleOutput

Note: the

(i, j)

th element of the matrix is stored in

$delta [(j - 1) \times m + i - 1]$ when $order = Nag_ColMajor$ ;
$delta [(i - 1) \times n + j - 1]$ when $order = Nag_RowMajor$ .

On exit: the

m \times n

array delta contains the sensitivity,

\frac{\partial P}{\partial S}

, of the option price to change in the price of the underlying asset.

14: gamma[ $m \times n$ ] – doubleOutput

Note: the

(i, j)

th element of the matrix is stored in

$gamma [(j - 1) \times m + i - 1]$ when $order = Nag_ColMajor$ ;
$gamma [(i - 1) \times n + j - 1]$ when $order = Nag_RowMajor$ .

On exit: the

m \times n

array gamma contains the sensitivity,

\frac{\partial^{2} P}{\partial S^{2}}

, of delta to change in the price of the underlying asset.

15: vega[ $m \times n$ ] – doubleOutput

Note: the

(i, j)

th element of the matrix is stored in

$vega [(j - 1) \times m + i - 1]$ when $order = Nag_ColMajor$ ;
$vega [(i - 1) \times n + j - 1]$ when $order = Nag_RowMajor$ .

On exit: the

m \times n

array vega contains the sensitivity,

\frac{\partial P}{\partial σ}

, of the option price to change in the volatility of the underlying asset.

16: theta[ $m \times n$ ] – doubleOutput

Note: the

(i, j)

th element of the matrix is stored in

$theta [(j - 1) \times m + i - 1]$ when $order = Nag_ColMajor$ ;
$theta [(i - 1) \times n + j - 1]$ when $order = Nag_RowMajor$ .

On exit: the

m \times n

array theta contains the sensitivity,

- \frac{\partial P}{\partial T}

, of the option price to change in the time to expiry of the option.

17: rho[ $m \times n$ ] – doubleOutput

Note: the

(i, j)

th element of the matrix is stored in

$rho [(j - 1) \times m + i - 1]$ when $order = Nag_ColMajor$ ;
$rho [(i - 1) \times n + j - 1]$ when $order = Nag_RowMajor$ .

On exit: the

m \times n

array rho contains the sensitivity,

\frac{\partial P}{\partial r}

, of the option price to change in the annual risk-free interest rate.

18: vanna[ $m \times n$ ] – doubleOutput

Note: the

(i, j)

th element of the matrix is stored in

$vanna [(j - 1) \times m + i - 1]$ when $order = Nag_ColMajor$ ;
$vanna [(i - 1) \times n + j - 1]$ when $order = Nag_RowMajor$ .

On exit: the

m \times n

array vanna contains the sensitivity,

\frac{\partial^{2} P}{\partial S \partial σ}

, 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.

19: charm[ $m \times n$ ] – doubleOutput

Note: the

(i, j)

th element of the matrix is stored in

$charm [(j - 1) \times m + i - 1]$ when $order = Nag_ColMajor$ ;
$charm [(i - 1) \times n + j - 1]$ when $order = Nag_RowMajor$ .

On exit: the

m \times n

array charm contains the sensitivity,

- \frac{\partial^{2} P}{\partial S \partial T}

, of delta to change in the time to expiry of the option.

20: speed[ $m \times n$ ] – doubleOutput

Note: the

(i, j)

th element of the matrix is stored in

$speed [(j - 1) \times m + i - 1]$ when $order = Nag_ColMajor$ ;
$speed [(i - 1) \times n + j - 1]$ when $order = Nag_RowMajor$ .

On exit: the

m \times n

array speed contains the sensitivity,

\frac{\partial^{3} P}{\partial S^{3}}

, of gamma to change in the price of the underlying asset.

21: colour[ $m \times n$ ] – doubleOutput

Note: the

(i, j)

th element of the matrix is stored in

$colour [(j - 1) \times m + i - 1]$ when $order = Nag_ColMajor$ ;
$colour [(i - 1) \times n + j - 1]$ when $order = Nag_RowMajor$ .

On exit: the

m \times n

array colour contains the sensitivity,

- \frac{\partial^{3} P}{\partial S^{2} \partial T}

, of gamma to change in the time to expiry of the option.

22: zomma[ $m \times n$ ] – doubleOutput

Note: the

(i, j)

th element of the matrix is stored in

$zomma [(j - 1) \times m + i - 1]$ when $order = Nag_ColMajor$ ;
$zomma [(i - 1) \times n + j - 1]$ when $order = Nag_RowMajor$ .

On exit: the

m \times n

array zomma contains the sensitivity,

\frac{\partial^{3} P}{\partial S^{2} \partial σ}

, of gamma to change in the volatility of the underlying asset.

23: vomma[ $m \times n$ ] – doubleOutput

Note: the

(i, j)

th element of the matrix is stored in

$vomma [(j - 1) \times m + i - 1]$ when $order = Nag_ColMajor$ ;
$vomma [(i - 1) \times n + j - 1]$ when $order = Nag_RowMajor$ .

On exit: the

m \times n

array vomma contains the sensitivity,

\frac{\partial^{2} P}{\partial σ^{2}}

, of vega to change in the volatility of the underlying asset.

24: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $m = 〈value〉$ .
Constraint: $m \geq 1$ .; On entry, $n = 〈value〉$ .
Constraint: $n \geq 1$ .
NE_INTERNAL_ERROR: 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.
NE_REAL: On entry, $jvol = 〈value〉$ .
Constraint: $jvol \geq 0.0$ and $jvol < 1.0$ .; On entry, $lambda = 〈value〉$ .
Constraint: $lambda > 0.0$ .; On entry, $r = 〈value〉$ .
Constraint: $r \geq 0.0$ .; On entry, $s = 〈value〉$ .
Constraint: $s \geq 〈value〉$ and $s \leq 〈value〉$ .; On entry, $sigma = 〈value〉$ .
Constraint: $sigma > 0.0$ .
NE_REAL_ARRAY: On entry, $t [〈value〉] = 〈value〉$ .
Constraint: $t [i] \geq 〈value〉$ .; On entry, $x [〈value〉] = 〈value〉$ .
Constraint: $x [i] \geq 〈value〉$ and $x [i] \leq 〈value〉$ .

7 Accuracy

The accuracy of the output is dependent on the accuracy of the cumulative Normal distribution function,

Φ

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

8 Further Comments

None.

9 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.

NAG Library Function Documentnag_jumpdiff_merton_greeks (s30jbc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Function Document

nag_jumpdiff_merton_greeks (s30jbc)