NAG FL Interface
s30jbf (opt_​jumpdiff_​merton_​greeks)

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1 Purpose

s30jbf computes the European option price together with its sensitivities (Greeks) using the Merton jump-diffusion model.

2 Specification

Fortran Interface
Subroutine s30jbf ( calput, m, n, x, s, t, sigma, r, lambda, jvol, p, ldp, delta, gamma, vega, theta, rho, vanna, charm, speed, colour, zomma, vomma, ifail)
Integer, Intent (In) :: m, n, ldp
Integer, Intent (Inout) :: ifail
Real (Kind=nag_wp), Intent (In) :: x(m), s, t(n), sigma, r, lambda, jvol
Real (Kind=nag_wp), Intent (Inout) :: p(ldp,n), delta(ldp,n), gamma(ldp,n), vega(ldp,n), theta(ldp,n), rho(ldp,n), vanna(ldp,n), charm(ldp,n), speed(ldp,n), colour(ldp,n), zomma(ldp,n), vomma(ldp,n)
Character (1), Intent (In) :: calput
C Header Interface
#include <nag.h>
void  s30jbf_ (const char *calput, const Integer *m, const Integer *n, const double x[], const double *s, const double t[], const double *sigma, const double *r, const double *lambda, const double *jvol, double p[], const Integer *ldp, double delta[], double gamma[], double vega[], double theta[], double rho[], double vanna[], double charm[], double speed[], double colour[], double zomma[], double vomma[], Integer *ifail, const Charlen length_calput)
The routine may be called by the names s30jbf or nagf_specfun_opt_jumpdiff_merton_greeks.

3 Description

s30jbf 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 α is the instantaneous expected return on the asset price, S; σ^2 is the instantaneous variance of the return when the Poisson event does not occur; dWt is a standard Brownian motion; qt 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))
Pcall = j=0 e-λT (λT)j 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; Cj(S,X,T,r,σj) is the Black–Scholes–Merton option pricing formula for a European call (see s30aaf).
σj = z2 + δ2 ( j T ) , z2 = σ2 - λ δ2 , δ2 = γ σ2 λ ,  
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 Cj (S,X,T,r,σj).
The option price Pij=P(X=Xi,T=Tj) is computed for each strike price in a set Xi, i=1,2,,m, and for each expiry time in a set Tj, j=1,2,,n.

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: 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' or 'P'.
2: m Integer Input
On entry: the number of strike prices to be used.
Constraint: m1.
3: n Integer Input
On entry: the number of times to expiry to be used.
Constraint: n1.
4: x(m) Real (Kind=nag_wp) array Input
On entry: x(i) must contain Xi, the ith strike price, for i=1,2,,m.
Constraint: x(i)z ​ and ​ x(i) 1 / z , where z = x02amf () , the safe range parameter, for i=1,2,,m.
5: s Real (Kind=nag_wp) Input
On entry: S, the price of the underlying asset.
Constraint: sz ​ and ​s1.0/z, where z=x02amf(), the safe range parameter.
6: t(n) Real (Kind=nag_wp) array Input
On entry: t(i) must contain Ti, the ith time, in years, to expiry, for i=1,2,,n.
Constraint: t(i)z, where z = x02amf () , the safe range parameter, for i=1,2,,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: r0.0.
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: 0.0jvol<1.0.
11: p(ldp,n) Real (Kind=nag_wp) array Output
On exit: p(i,j) contains Pij, the option price evaluated for the strike price xi at expiry tj for i=1,2,,m and j=1,2,,n.
12: ldp Integer Input
On entry: the first dimension of the arrays p, delta, gamma, vega, theta, rho, vanna, charm, speed, colour, zomma and vomma as declared in the (sub)program from which s30jbf is called.
Constraint: ldpm.
13: delta(ldp,n) Real (Kind=nag_wp) array Output
On exit: the leading m×n part of the array delta contains the sensitivity, PS, of the option price to change in the price of the underlying asset.
14: gamma(ldp,n) Real (Kind=nag_wp) array Output
On exit: the leading m×n part of the array gamma contains the sensitivity, 2PS2, of delta to change in the price of the underlying asset.
15: vega(ldp,n) Real (Kind=nag_wp) array Output
On exit: vega(i,j), contains the first-order Greek measuring the sensitivity of the option price Pij to change in the volatility of the underlying asset, i.e., Pij σ , for i=1,2,,m and j=1,2,,n.
16: theta(ldp,n) Real (Kind=nag_wp) array Output
On exit: theta(i,j), contains the first-order Greek measuring the sensitivity of the option price Pij to change in time, i.e., - Pij T , for i=1,2,,m and j=1,2,,n, where b=r-q.
17: rho(ldp,n) Real (Kind=nag_wp) array Output
On exit: rho(i,j), contains the first-order Greek measuring the sensitivity of the option price Pij to change in the annual risk-free interest rate, i.e., - Pij r , for i=1,2,,m and j=1,2,,n.
18: vanna(ldp,n) Real (Kind=nag_wp) array Output
On exit: vanna(i,j), contains the second-order Greek measuring the sensitivity of the first-order Greek Δij to change in the volatility of the asset price, i.e., - Δij T = - 2 Pij Sσ , for i=1,2,,m and j=1,2,,n.
19: charm(ldp,n) Real (Kind=nag_wp) array Output
On exit: charm(i,j), contains the second-order Greek measuring the sensitivity of the first-order Greek Δij to change in the time, i.e., - Δij T = - 2 Pij ST , for i=1,2,,m and j=1,2,,n.
20: speed(ldp,n) Real (Kind=nag_wp) array Output
On exit: speed(i,j), contains the third-order Greek measuring the sensitivity of the second-order Greek Γij to change in the price of the underlying asset, i.e., - Γij S = - 3 Pij S3 , for i=1,2,,m and j=1,2,,n.
21: colour(ldp,n) Real (Kind=nag_wp) array Output
On exit: colour(i,j), contains the third-order Greek measuring the sensitivity of the second-order Greek Γij to change in the time, i.e., - Γij T = - 3 Pij ST , for i=1,2,,m and j=1,2,,n.
22: zomma(ldp,n) Real (Kind=nag_wp) array Output
On exit: zomma(i,j), contains the third-order Greek measuring the sensitivity of the second-order Greek Γij to change in the volatility of the underlying asset, i.e., - Γij σ = - 3 Pij S2σ , for i=1,2,,m and j=1,2,,n.
23: vomma(ldp,n) Real (Kind=nag_wp) array Output
On exit: vomma(i,j), contains the second-order Greek measuring the sensitivity of the first-order Greek Δij to change in the volatility of the underlying asset, i.e., - Δij σ = - 2 Pij σ2 , for i=1,2,,m and j=1,2,,n.
24: ifail Integer Input/Output
On entry: ifail must be set to 0, −1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of −1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value −1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value 0 is recommended. 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 or −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=value was an illegal value.
ifail=2
On entry, m=value.
Constraint: m1.
ifail=3
On entry, n=value.
Constraint: n1.
ifail=4
On entry, x(value)=value.
Constraint: x(i)value and x(i)value.
ifail=5
On entry, s=value.
Constraint: svalue and svalue.
ifail=6
On entry, t(value)=value.
Constraint: t(i)value.
ifail=7
On entry, sigma=value.
Constraint: sigma>0.0.
ifail=8
On entry, r=value.
Constraint: r0.0.
ifail=9
On entry, lambda=value.
Constraint: lambda>0.0.
ifail=10
On entry, jvol=value.
Constraint: jvol0.0 and jvol < 1.0.
ifail=12
On entry, ldp=value and m=value.
Constraint: ldpm.
ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
ifail=-399
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
ifail=-999
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

The accuracy of the output is dependent on the accuracy of the cumulative Normal distribution function, Φ, occurring in Cj. 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 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
s30jbf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

None.

10 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.

10.1 Program Text

Program Text (s30jbfe.f90)

10.2 Program Data

Program Data (s30jbfe.d)

10.3 Program Results

Program Results (s30jbfe.r)