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:
– 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 minimum or maximum 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 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:
– 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 volatility of the underlying asset. Note that a rate of 15% should be entered as .
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:
and , where , the machine precision.
-
10:
– double
Input
-
On entry: the annual continuous yield rate. Note that a rate of 8% should be entered as .
Constraint:
and , where , the machine precision.
-
11:
– 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 minimum or maximum observed price or at expiry for and .
-
12:
– 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.
-
13:
– 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.
-
14:
– 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 .
-
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 time, i.e., , for and , where .
-
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 the annual risk-free interest rate, i.e., , for and .
-
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 cost of carry rate, i.e., , for and , where .
-
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,
. 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 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 .