NAG Library Routine Document

S30JAF

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,

\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 S30AAF).

\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 Parameters

1: CALPUT – CHARACTER(1)Input

On entry: determines whether the option is a call or a put.

$CALPUT ='C'$: A call. The holder has a right to buy.
$CALPUT ='P'$: A put. The holder has a right to sell.

Constraint:

CALPUT ='C'

'P'

2: M – INTEGERInput

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

Constraint:

M \geq 1

3: N – INTEGERInput

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

Constraint:

N \geq 1

4: X(M) – REAL (KIND=nag_wp) arrayInput

On entry:

X (i)

must contain

X_{i}

, the

i

th strike price, for

i = 1, 2, \dots, M

Constraint:

X (i) \geq z ​ and ​ X (i) \leq 1 / z

, where

z = X02AMF ()

, the safe range parameter, for

i = 1, 2, \dots, M

5: S – REAL (KIND=nag_wp)Input

On entry:

S

, the price of the underlying asset.

Constraint:

S \geq z ​ and ​ S \leq 1.0 / z

, where

z = X02AMF ()

, the safe range parameter.

6: T(N) – REAL (KIND=nag_wp) arrayInput

On entry:

T (i)

must contain

T_{i}

, the

i

th time, in years, to expiry, for

i = 1, 2, \dots, N

Constraint:

T (i) \geq z

, where

z = X02AMF ()

, the safe range parameter, for

i = 1, 2, \dots, N

7: SIGMA – REAL (KIND=nag_wp)Input

On entry:

σ

, the annual total volatility, including jumps.

Constraint:

SIGMA > 0.0

8: 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:

R \geq 0.0

9: LAMBDA – REAL (KIND=nag_wp)Input

On entry:

λ

, the number of expected jumps per year.

Constraint:

LAMBDA > 0.0

10: JVOL – REAL (KIND=nag_wp)Input

On entry: the proportion of the total volatility associated with jumps.

Constraint:

0.0 \leq JVOL < 1.0

11: P(LDP,N) – REAL (KIND=nag_wp) arrayOutput

On exit: the leading

M \times N

part of the array P contains the computed option prices.

12: LDP – INTEGERInput

On entry: the first dimension of the array P as declared in the (sub)program from which S30JAF is called.

Constraint:

LDP \geq M

13: IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$: On entry, $CALPUT \neq'C'$ or $'P'$ .

$IFAIL = 2$: On entry, $M \leq 0$ .

$IFAIL = 3$: On entry, $N \leq 0$ .

$IFAIL = 4$: On entry, $X (i) < z$ or $X (i) > 1 / z$ , where $z = X02AMF ()$ , the safe range parameter.

$IFAIL = 5$: On entry, $S < z$ or $S > 1.0 / z$ , where $z = X02AMF ()$ , the safe range parameter.

$IFAIL = 6$: On entry, $T (i) < z$ , where $z = X02AMF ()$ , the safe range parameter.

$IFAIL = 7$: On entry, $SIGMA \leq 0.0$ .

$IFAIL = 8$: On entry, $R < 0.0$ .

$IFAIL = 9$: On entry, $LAMBDA \leq 0.0$ .

$IFAIL = 10$: On entry, $JVOL < 0.0$ or $JVOL \geq 1.0$ .

$IFAIL = 12$: On entry, $LDP < M$ .

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 S15ABF and S15ADF). An accuracy close to machine precision can generally be expected.

8 Further Comments

None.

9 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.

NAG Library Routine DocumentS30JAF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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 Routine Document

S30JAF