NAG Library Function Document
nag_lookback_fls_price (s30bac)
1
Purpose
nag_lookback_fls_price (s30bac) computes the price of a floating-strike lookback option.
2
Specification
#include <nag.h> |
#include <nags.h> |
void |
nag_lookback_fls_price (Nag_OrderType order,
Nag_CallPut option,
Integer m,
Integer n,
const double sm[],
double s,
const double t[],
double sigma,
double r,
double q,
double p[],
NagError *fail) |
|
3
Description
nag_lookback_fls_price (s30bac) computes the price of a floating-strike lookback call or put option. A call option of this type confers the right to buy the underlying asset at the lowest price, , observed during the lifetime of the contract. A put option gives the holder the right to sell the underlying asset at the maximum price, , observed during the lifetime of the contract. Thus, at expiry, the payoff for a call option is , and for a put, .
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. When
, the option price is given by
The corresponding put price is (for
),
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.
The option price is computed for each minimum or maximum observed price in a set or , , and for each expiry time in a set , .
4
References
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
5
Arguments
- 1:
– Nag_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
. See
Section 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint:
or .
- 2:
– Nag_CallPutInput
-
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:
– IntegerInput
-
On entry: the number of minimum or maximum prices to be used.
Constraint:
.
- 4:
– IntegerInput
-
On entry: the number of times to expiry to be used.
Constraint:
.
- 5:
– const doubleInput
-
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 .
- 6:
– doubleInput
-
On entry: , the price of the underlying asset.
Constraint:
, where , the safe range parameter.
- 7:
– const doubleInput
-
On entry: must contain
, the th time, in years, to expiry, for .
Constraint:
, where , the safe range parameter, for .
- 8:
– doubleInput
-
On entry: , the volatility of the underlying asset. Note that a rate of 15% should be entered as 0.15.
Constraint:
.
- 9:
– doubleInput
-
On entry: , the annual risk-free interest rate, continuously compounded. Note that a rate of 5% should be entered as 0.05.
Constraint:
.
- 10:
– doubleInput
-
On entry: , the annual continuous yield rate. Note that a rate of 8% should be entered as 0.08.
Constraint:
.
- 11:
– doubleOutput
-
Note: where
appears in this document, it refers to the array element
- when ;
- when .
On exit: contains , the option price evaluated for the minimum or maximum observed price or at expiry for and .
- 12:
– NagError *Input/Output
-
The NAG error argument (see
Section 3.7 in How to Use the NAG Library and its Documentation).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 2.3.1.2 in How to Use the NAG Library and its Documentation 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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
- NE_REAL
-
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: and .
On entry, .
Constraint: .
- NE_REAL_ARRAY
-
On entry, .
Constraint: for all .
On entry, .
Constraint: for all .
On entry with a call option, .
Constraint: for call options, for all .
On entry with a put option, .
Constraint: for put options, for all .
7
Accuracy
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
nag_cumul_normal (s15abc) and
nag_erfc (s15adc)). An accuracy close to
machine precision can generally be expected.
8
Parallelism and Performance
nag_lookback_fls_price (s30bac) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
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 floating-strike lookback call with a time to expiry of months and a stock price of . The minimum 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 .
10.1
Program Text
Program Text (s30bace.c)
10.2
Program Data
Program Data (s30bace.d)
10.3
Program Results
Program Results (s30bace.r)