The function may be called by the names: s30fac, nag_specfun_opt_barrier_std_price or nag_barrier_std_price.
3Description
s30fac computes the price of a standard barrier option, where the exercise, for a given strike price, , depends on the underlying asset price, , reaching or crossing a specified barrier level, . Barrier options of type In only become active (are knocked in) if the underlying asset price attains the pre-determined barrier level during the lifetime of the contract. Those of type Out start active and are knocked out if the underlying asset price attains the barrier level during the lifetime of the contract. A cash rebate, , may be paid if the option is inactive at expiration. The option may also be described as Up (the underlying price starts below the barrier level) or Down (the underlying price starts above the barrier level). This gives the following options which can be specified as put or call contracts.
Down-and-In: the option starts inactive with the underlying asset price above the barrier level. It is knocked in if the underlying price moves down to hit the barrier level before expiration.
Down-and-Out: the option starts active with the underlying asset price above the barrier level. It is knocked out if the underlying price moves down to hit the barrier level before expiration.
Up-and-In: the option starts inactive with the underlying asset price below the barrier level. It is knocked in if the underlying price moves up to hit the barrier level before expiration.
Up-and-Out: the option starts active with the underlying asset price below the barrier level. It is knocked out if the underlying price moves up to hit the barrier level before expiration.
The payoff is for a call or for a put, if the option is active at expiration, otherwise it may pay a pre-specified cash rebate, . Following Haug (2007), the prices of the various standard barrier options can be written as shown below. The volatility, , risk-free interest rate, , and annualised dividend yield, , are constants. The integer parameters, and , take the values , depending on the type of barrier.
with
and where denotes the cumulative Normal distribution function,
Down-and-In ():
When , with ,
and with ,
When , with
and with ,
Down-and-Out ():
When , with ,
and with ,
When , with ,
and with ,
Up-and-In ():
When , with , ,
and with ,
When , with , ,
and with ,
Up-and-Out ():
When , with , ,
and with ,
When , with , ,
and with ,
The option price is computed for each strike price in a set , , and for each expiry time in a set , .
4References
Haug E G (2007) The Complete Guide to Option Pricing Formulas (2nd Edition) McGraw-Hill
5Arguments
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 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint:
or .
2: – Nag_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: – Nag_BarrierInput
On entry: indicates the barrier type as In or Out and its relation to the price of the underlying asset as Up or Down.
Down-and-In.
Down-and-Out.
Up-and-In.
Up-and-Out.
Constraint:
, , or .
4: – IntegerInput
On entry: the number of strike prices to be used.
Constraint:
.
5: – IntegerInput
On entry: the number of times to expiry to be used.
Constraint:
.
6: – const doubleInput
On entry: must contain
, the th strike price, for .
Constraint:
, where , the safe range parameter, for .
7: – doubleInput
On entry: , the price of the underlying asset.
Constraint:
, where , the safe range parameter.
8: – doubleInput
On entry: the barrier price.
Constraint:
, where , the safe range parameter.
9: – doubleInput
On entry: the value of a possible cash rebate to be paid if the option has not been knocked in (or out) before expiration.
Constraint:
.
10: – const doubleInput
On entry: must contain
, the th time, in years, to expiry, for .
Constraint:
, where , the safe range parameter, for .
11: – doubleInput
On entry: , the volatility of the underlying asset. Note that a rate of 15% should be entered as .
Constraint:
.
12: – doubleInput
On entry: , the annual risk-free interest rate, continuously compounded. Note that a rate of 5% should be entered as .
Constraint:
.
13: – doubleInput
On entry: , the annual continuous yield rate. Note that a rate of 8% should be entered as .
Constraint:
.
14: – doubleOutput
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 .
15: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument had an illegal value.
NE_ENUM_REAL_2
On entry, s and h are inconsistent with type: and .
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 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_REAL
On entry, .
Constraint: and .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: and .
On entry, .
Constraint: .
NE_REAL_ARRAY
On entry, .
Constraint: .
On entry, .
Constraint: and .
7Accuracy
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 s15abcands15adc). An accuracy close to machine precision can generally be expected.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
s30fac 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.
9Further Comments
None.
10Example
This example computes the price of a Down-and-In put with a time to expiry of months, a stock price of and a strike price of . The barrier value is and there is a cash rebate of , payable on expiry if the option has not been knocked in. The risk-free interest rate is per year, there is an annual dividend return of and the volatility is per year.