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,
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
s30aaf).
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:
– Character(1)
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 .
-
2:
– Integer
Input
-
On entry: the number of strike prices to be used.
Constraint:
.
-
3:
– Integer
Input
-
On entry: the number of times to expiry to be used.
Constraint:
.
-
4:
– Real (Kind=nag_wp) array
Input
-
On entry: must contain
, the th strike price, for .
Constraint:
, where , the safe range parameter, for .
-
5:
– Real (Kind=nag_wp)
Input
-
On entry: , the price of the underlying asset.
Constraint:
, where , the safe range parameter.
-
6:
– Real (Kind=nag_wp) array
Input
-
On entry: must contain
, the th time, in years, to expiry, for .
Constraint:
, where , the safe range parameter, for .
-
7:
– Real (Kind=nag_wp)
Input
-
On entry: , the annual total volatility, including jumps.
Constraint:
.
-
8:
– Real (Kind=nag_wp)
Input
-
On entry: , the annual risk-free interest rate, continuously compounded. Note that a rate of 5% should be entered as .
Constraint:
.
-
9:
– Real (Kind=nag_wp)
Input
-
On entry: , the number of expected jumps per year.
Constraint:
.
-
10:
– Real (Kind=nag_wp)
Input
-
On entry: the proportion of the total volatility associated with jumps.
Constraint:
.
-
11:
– Real (Kind=nag_wp) array
Output
-
On exit: contains , the option price evaluated for the strike price at expiry for and .
-
12:
– 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:
.
-
13:
– Real (Kind=nag_wp) array
Output
-
On exit: the leading
part of the array
delta contains the sensitivity,
, of the option price to change in the price of the underlying asset.
-
14:
– Real (Kind=nag_wp) array
Output
-
On exit: the leading
part of the array
gamma contains the sensitivity,
, of
delta to change in the price of the underlying asset.
-
15:
– Real (Kind=nag_wp) array
Output
-
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:
– Real (Kind=nag_wp) array
Output
-
On exit: , contains the first-order Greek measuring the sensitivity of the option price to change in time, i.e., , for and , where .
-
17:
– Real (Kind=nag_wp) array
Output
-
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:
– Real (Kind=nag_wp) array
Output
-
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:
– Real (Kind=nag_wp) array
Output
-
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:
– Real (Kind=nag_wp) array
Output
-
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:
– Real (Kind=nag_wp) array
Output
-
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:
– Real (Kind=nag_wp) array
Output
-
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:
– Real (Kind=nag_wp) array
Output
-
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:
– Integer
Input/Output
-
On entry:
ifail must be set to
,
or
to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of means that an error message is printed while a value of means that it is not.
If halting is not appropriate, the value
or
is recommended. If message printing is undesirable, then the value
is recommended. Otherwise, the value
is recommended.
When the value or is used it is essential to test the value of ifail on exit.
On exit:
unless the routine detects an error or a warning has been flagged (see
Section 6).
If on entry
or
, explanatory error messages are output on the current error message unit (as defined by
x04aaf).
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
s15abf and
s15adf). 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 routine. 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.