For a given minimum value the price of a floating-strike lookback call with underlying asset price,
, and time to expiry,
, is
where
. The volatility,
, risk-free interest rate,
, and annualised dividend yield,
, are constants.
The corresponding put price is
In the above,
denotes the cumulative Normal distribution function,
where
denotes the standard Normal probability density function
and
where
is taken to be the minimum price attained by the underlying asset,
, for a call and the maximum price,
, for a put.
Goldman B M, Sosin H B and Gatto M A (1979) Path dependent options: buy at the low, sell at the high Journal of Finance 34 1111–1127
-
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 minimum or maximum 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 minimum observed price of the underlying asset when , or , the maximum observed price when , for .
Constraints:
- , where , the safe range parameter, for ;
- if ,
, for ;
- if ,
, 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 volatility of the underlying asset. Note that a rate of 15% should be entered as .
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:
and , where , the machine precision.
-
9:
– Real (Kind=nag_wp)
Input
-
On entry: the annual continuous yield rate. Note that a rate of 8% should be entered as .
Constraint:
and , where , the machine precision.
-
10:
– Real (Kind=nag_wp) array
Output
-
On exit: contains , the option price evaluated for the minimum or maximum observed price or at expiry for and .
-
11:
– Integer
Input
-
On entry: the first dimension of the arrays
p,
delta,
gamma,
vega,
theta,
rho,
crho,
vanna,
charm,
speed,
colour,
zomma and
vomma as declared in the (sub)program from which
s30bbf is called.
Constraint:
.
-
12:
– 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.
-
13:
– 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.
-
14:
– 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 .
-
15:
– 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 .
-
16:
– 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 .
-
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 cost of carry rate, i.e., , for and , where .
-
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,
. 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 a floating-strike lookback put with a time to expiry of months and a stock price of . The maximum price observed so far is . The risk-free interest rate is per year and the volatility is per year with an annual dividend return of .