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,
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 .
Merton R C (1976) Option pricing when underlying stock returns are discontinuous Journal of Financial Economics 3 125–144
-
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:
– double
Output
-
Note: the
th element of the matrix is stored in
- when ;
- when .
On exit: the
array
delta contains the sensitivity,
, of the option price to change in the price of the underlying asset.
-
14:
– double
Output
-
Note: the
th element of the matrix is stored in
- when ;
- when .
On exit: the
array
gamma contains the sensitivity,
, of
delta to change in the price of the underlying asset.
-
15:
– double
Output
-
Note: where
appears in this document, it refers to the array element
- when ;
- when .
On exit: , contains the first-order Greek measuring the sensitivity of the option price to change in the volatility of the underlying asset, i.e., , for and .
-
16:
– double
Output
-
Note: where
appears in this document, it refers to the array element
- when ;
- when .
On exit: , contains the first-order Greek measuring the sensitivity of the option price to change in time, i.e., , for and , where .
-
17:
– double
Output
-
Note: where
appears in this document, it refers to the array element
- when ;
- when .
On exit: , contains the first-order Greek measuring the sensitivity of the option price to change in the annual risk-free interest rate, i.e., , for and .
-
18:
– double
Output
-
Note: where
appears in this document, it refers to the array element
- when ;
- when .
On exit: , contains the second-order Greek measuring the sensitivity of the first-order Greek to change in the volatility of the asset price, i.e., , for and .
-
19:
– double
Output
-
Note: where
appears in this document, it refers to the array element
- when ;
- when .
On exit: , contains the second-order Greek measuring the sensitivity of the first-order Greek to change in the time, i.e., , for and .
-
20:
– double
Output
-
Note: where
appears in this document, it refers to the array element
- when ;
- when .
On exit: , contains the third-order Greek measuring the sensitivity of the second-order Greek to change in the price of the underlying asset, i.e., , for and .
-
21:
– double
Output
-
Note: where
appears in this document, it refers to the array element
- when ;
- when .
On exit: , contains the third-order Greek measuring the sensitivity of the second-order Greek to change in the time, i.e., , for and .
-
22:
– double
Output
-
Note: where
appears in this document, it refers to the array element
- when ;
- when .
On exit: , contains the third-order Greek measuring the sensitivity of the second-order Greek to change in the volatility of the underlying asset, i.e., , for and .
-
23:
– double
Output
-
Note: where
appears in this document, it refers to the array element
- when ;
- when .
On exit: , contains the second-order Greek measuring the sensitivity of the first-order Greek to change in the volatility of the underlying asset, i.e., , for and .
-
24:
– NagError *
Input/Output
-
The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
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.
Background information to multithreading can be found in the
Multithreading documentation.
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.
This example computes the price of two European calls with jumps. The time to expiry is months, the stock price is and strike prices are and respectively. 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 while the total volatility is per year.