NAG CL Interface
s30jac (opt_jumpdiff_merton_price)
1
Purpose
s30jac computes the European option price using the Merton jump-diffusion model.
2
Specification
void |
s30jac (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[],
NagError *fail) |
|
The function may be called by the names: s30jac, nag_specfun_opt_jumpdiff_merton_price or nag_jumpdiff_merton_price.
3
Description
s30jac 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,
Here is the instantaneous expected return on the asset price, ; is the instantaneous variance of the return when the Poisson event does not occur; is a standard Brownian motion; is the independent Poisson process and where is the random variable change in the stock price if the Poisson event occurs and is the expectation operator over the random variable .
This leads to the following price for a European option (see
Haug (2007))
where
is the time to expiry;
is the strike price;
is the annual risk-free interest rate;
is the Black–Scholes–Merton option pricing formula for a European call (see
s30aac).
where
is the total volatility including jumps;
is the expected number of jumps given as an average per year;
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 .
The option price is computed for each strike price in a set , , and for each expiry time in a set , .
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:
– Nag_OrderType
Input
-
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
. See
Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint:
or .
-
2:
– Nag_CallPut
Input
-
On entry: determines whether the option is a call or a put.
- A call; the holder has a right to buy.
- A put; the holder has a right to sell.
Constraint:
or .
-
3:
– Integer
Input
-
On entry: the number of strike prices to be used.
Constraint:
.
-
4:
– Integer
Input
-
On entry: the number of times to expiry to be used.
Constraint:
.
-
5:
– const double
Input
-
On entry: must contain
, the th strike price, for .
Constraint:
, where , the safe range parameter, for .
-
6:
– double
Input
-
On entry: , the price of the underlying asset.
Constraint:
, where , the safe range parameter.
-
7:
– const double
Input
-
On entry: must contain
, the th time, in years, to expiry, for .
Constraint:
, where , the safe range parameter, for .
-
8:
– double
Input
-
On entry: , the annual total volatility, including jumps.
Constraint:
.
-
9:
– double
Input
-
On entry: , the annual risk-free interest rate, continuously compounded. Note that a rate of 5% should be entered as .
Constraint:
.
-
10:
– double
Input
-
On entry: , the number of expected jumps per year.
Constraint:
.
-
11:
– double
Input
-
On entry: the proportion of the total volatility associated with jumps.
Constraint:
.
-
12:
– double
Output
-
Note: where
appears in this document, it refers to the array element
- when ;
- when .
On exit: contains , the option price evaluated for the strike price at expiry for and .
-
13:
– NagError *
Input/Output
-
The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_INT
-
On entry, .
Constraint: .
On entry, .
Constraint: .
- 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.
See
Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library CL Interface for further information.
- NE_REAL
-
On entry, .
Constraint: and .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: and .
On entry, .
Constraint: .
- NE_REAL_ARRAY
-
On entry, .
Constraint: .
On entry, .
Constraint: and .
7
Accuracy
The accuracy of the output is dependent on the accuracy of the cumulative Normal distribution function,
, occurring in
. 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
s15abc and
s15adc). An accuracy close to
machine precision can generally be expected.
8
Parallelism and Performance
s30jac is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
s30jac makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the
X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the
Users' Note for your implementation for any additional implementation-specific information.
None.
10
Example
This example computes the price of a European call with jumps. The time to expiry is months, the stock price is and the strike price is . The number of jumps per year is and the percentage of the total volatility due to jumps is . The risk-free interest rate is per year and the total volatility is per year.
10.1
Program Text
10.2
Program Data
10.3
Program Results