nag_jumpdiff_merton_greeks (s30jbc) (PDF version)
s Chapter Contents
s Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_jumpdiff_merton_greeks (s30jbc)

+ Contents

    1  Purpose
    7  Accuracy

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,
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=EY-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 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; CjS,X,T,r,σj is the Black–Scholes–Merton option pricing formula for a European call (see nag_bsm_price (s30aac)).
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 .

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:     orderNag_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:     optionNag_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 or Nag_Put.
3:     mIntegerInput
On entry: the number of strike prices to be used.
Constraint: m1.
4:     nIntegerInput
On entry: the number of times to expiry to be used.
Constraint: n1.
5:     x[m]const doubleInput
On entry: x[i-1] must contain Xi, the ith strike price, for i=1,2,,m.
Constraint: x[i-1]z ​ and ​ x[i-1] 1 / z , where z = nag_real_safe_small_number , the safe range parameter, for i=1,2,,m.
6:     sdoubleInput
On entry: S, the price of the underlying asset.
Constraint: sz ​ and ​s1.0/z, where z=nag_real_safe_small_number, the safe range parameter.
7:     t[n]const doubleInput
On entry: t[i-1] must contain Ti, the ith time, in years, to expiry, for i=1,2,,n.
Constraint: t[i-1]z, where z = nag_real_safe_small_number , the safe range parameter, for i=1,2,,n.
8:     sigmadoubleInput
On entry: σ, the annual total volatility, including jumps.
Constraint: sigma>0.0.
9:     rdoubleInput
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.
10:   lambdadoubleInput
On entry: λ, the number of expected jumps per year.
Constraint: lambda>0.0.
11:   jvoldoubleInput
On entry: the proportion of the total volatility associated with jumps.
Constraint: 0.0jvol<1.0.
12:   p[m×n]doubleOutput
Note: the i,jth element of the matrix P is stored in
  • p[j-1×m+i-1] when order=Nag_ColMajor;
  • p[i-1×n+j-1] when order=Nag_RowMajor.
On exit: the m×n array p contains the computed option prices.
13:   delta[m×n]doubleOutput
Note: the i,jth element of the matrix is stored in
  • delta[j-1×m+i-1] when order=Nag_ColMajor;
  • delta[i-1×n+j-1] when order=Nag_RowMajor.
On exit: the m×n array delta contains the sensitivity, PS, of the option price to change in the price of the underlying asset.
14:   gamma[m×n]doubleOutput
Note: the i,jth element of the matrix is stored in
  • gamma[j-1×m+i-1] when order=Nag_ColMajor;
  • gamma[i-1×n+j-1] when order=Nag_RowMajor.
On exit: the m×n array gamma contains the sensitivity, 2PS2, of delta to change in the price of the underlying asset.
15:   vega[m×n]doubleOutput
Note: the i,jth element of the matrix is stored in
  • vega[j-1×m+i-1] when order=Nag_ColMajor;
  • vega[i-1×n+j-1] when order=Nag_RowMajor.
On exit: the m×n array vega contains the sensitivity, Pσ, of the option price to change in the volatility of the underlying asset.
16:   theta[m×n]doubleOutput
Note: the i,jth element of the matrix is stored in
  • theta[j-1×m+i-1] when order=Nag_ColMajor;
  • theta[i-1×n+j-1] when order=Nag_RowMajor.
On exit: the m×n array theta contains the sensitivity, -PT, of the option price to change in the time to expiry of the option.
17:   rho[m×n]doubleOutput
Note: the i,jth element of the matrix is stored in
  • rho[j-1×m+i-1] when order=Nag_ColMajor;
  • rho[i-1×n+j-1] when order=Nag_RowMajor.
On exit: the m×n array rho contains the sensitivity, Pr, of the option price to change in the annual risk-free interest rate.
18:   vanna[m×n]doubleOutput
Note: the i,jth element of the matrix is stored in
  • vanna[j-1×m+i-1] when order=Nag_ColMajor;
  • vanna[i-1×n+j-1] when order=Nag_RowMajor.
On exit: the m×n array vanna contains the sensitivity, 2PSσ, 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×n]doubleOutput
Note: the i,jth element of the matrix is stored in
  • charm[j-1×m+i-1] when order=Nag_ColMajor;
  • charm[i-1×n+j-1] when order=Nag_RowMajor.
On exit: the m×n array charm contains the sensitivity, -2PS T, of delta to change in the time to expiry of the option.
20:   speed[m×n]doubleOutput
Note: the i,jth element of the matrix is stored in
  • speed[j-1×m+i-1] when order=Nag_ColMajor;
  • speed[i-1×n+j-1] when order=Nag_RowMajor.
On exit: the m×n array speed contains the sensitivity, 3PS3, of gamma to change in the price of the underlying asset.
21:   colour[m×n]doubleOutput
Note: the i,jth element of the matrix is stored in
  • colour[j-1×m+i-1] when order=Nag_ColMajor;
  • colour[i-1×n+j-1] when order=Nag_RowMajor.
On exit: the m×n array colour contains the sensitivity, -3PS2 T, of gamma to change in the time to expiry of the option.
22:   zomma[m×n]doubleOutput
Note: the i,jth element of the matrix is stored in
  • zomma[j-1×m+i-1] when order=Nag_ColMajor;
  • zomma[i-1×n+j-1] when order=Nag_RowMajor.
On exit: the m×n array zomma contains the sensitivity, 3PS2σ, of gamma to change in the volatility of the underlying asset.
23:   vomma[m×n]doubleOutput
Note: the i,jth element of the matrix is stored in
  • vomma[j-1×m+i-1] when order=Nag_ColMajor;
  • vomma[i-1×n+j-1] when order=Nag_RowMajor.
On exit: the m×n array vomma contains the sensitivity, 2Pσ2, of vega to change in the volatility of the underlying asset.
24:   failNagError *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: m1.
On entry, n=value.
Constraint: n1.
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: jvol0.0 and jvol<1.0.
On entry, lambda=value.
Constraint: lambda>0.0.
On entry, r=value.
Constraint: r0.0.
On entry, s=value.
Constraint: svalue and svalue.
On entry, sigma=value.
Constraint: sigma>0.0.
NE_REAL_ARRAY
On entry, t[value]=value.
Constraint: t[i]value.
On entry, x[value]=value.
Constraint: x[i]value and x[i]value.

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 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.

9.1  Program Text

Program Text (s30jbce.c)

9.2  Program Data

Program Data (s30jbce.d)

9.3  Program Results

Program Results (s30jbce.r)


nag_jumpdiff_merton_greeks (s30jbc) (PDF version)
s Chapter Contents
s Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012