void	s30qcc (Nag_OrderType order, Nag_CallPut option, Integer m, Integer n, const double x[], double s, const double t[], double sigma, double r, double q, double p[], NagError *fail)

The function may be called by the names: s30qcc, nag_specfun_opt_amer_bs_price or nag_amer_bs_price.

3 Description

s30qcc computes the price of an American option using the closed form approximation of Bjerksund and Stensland (2002). The time to maturity,

T

, is divided into two periods, each with a flat early exercise boundary, by choosing a time

t \in [0, T]

, such that

t = \frac{1}{2} (\sqrt{5} - 1) T

. The two boundary values are defined as

\tilde{x} = \tilde{X} (t)

\tilde{X} = \tilde{X} (T)

with

\tilde{X} (τ) = B_{0} + (B_{\infty} - B_{0}) (1 - \exp \{h (τ)\}),

where

h (τ) = - (b τ + 2 σ \sqrt{τ}) (\frac{X^{2}}{(B_{\infty} - B_{0}) B_{0}}),

B_{\infty} \equiv \frac{β}{β - 1} X, B_{0} \equiv \max \{X, (\frac{r}{r - b}) X\},

β = (\frac{1}{2} - \frac{b}{σ^{2}}) + \sqrt{{(\frac{b}{σ^{2}} - \frac{1}{2})}^{2} + 2 \frac{r}{σ^{2}}} .

with

b = r - q

, the cost of carry, where

r

is the risk-free interest rate and

q

is the annual dividend rate. Here

X

is the strike price and

σ

is the annual volatility.

The price of an American call option is approximated as

\begin{matrix} P_{call} & = & α (\tilde{X}) S^{β} - α (\tilde{X}) ϕ (S, t | β, \tilde{X}, \tilde{X}) + \\ ϕ (S, t | 1, \tilde{X}, \tilde{X}) - ϕ (S, t | 1, \tilde{x}, \tilde{X}) - \\ X ϕ (S, t | 0, \tilde{X}, \tilde{X}) + X ϕ (S, t | 0, \tilde{x}, \tilde{X}) + \\ α (\tilde{x}) ϕ (S, t | β, \tilde{x}, \tilde{X}) - α (\tilde{x}) Ψ (S, T | β, \tilde{x}, \tilde{X}, \tilde{x}, t) + \\ Ψ (S, T | 1, \tilde{x}, \tilde{X}, \tilde{x}, t) - Ψ (S, T | 1, X, \tilde{X}, \tilde{x}, t) - \\ X Ψ (S, T | 0, \tilde{x}, \tilde{X}, \tilde{x}, t) + X Ψ (S, T | 0, X, \tilde{X}, \tilde{x}, t), \end{matrix}

where

α

ϕ

and

Ψ

are as defined in Bjerksund and Stensland (2002).

The price of a put option is obtained by the put-call transformation,

P_{put} (X, S, T, σ, r, q) = P_{call} (S, X, T, σ, q, r) .

The option price

P_{i j} = P (X = X_{i}, T = T_{j})

is computed for each strike price in a set

X_{i}

i = 1, 2, \dots, m

, and for each expiry time in a set

T_{j}

j = 1, 2, \dots, n

4 References

Bjerksund P and Stensland G (2002) Closed form valuation of American options Discussion Paper 2002/09 NHH Bergen Norway

Genz A (2004) Numerical computation of rectangular bivariate and trivariate Normal and

t

probabilities Statistics and Computing 14 151–160

5 Arguments

1: $order$ – 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

order = Nag_RowMajor

. 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:

order = Nag_RowMajor

Nag_ColMajor

2: $option$ – Nag_CallPut Input

On entry: determines whether the option is a call or a put.

$option = Nag_Call$: A call; the holder has a right to buy.
$option = Nag_Put$: A put; the holder has a right to sell.

Constraint:

option = Nag_Call

Nag_Put

3: $m$ – Integer Input

On entry: the number of strike prices to be used.

Constraint:

m \geq 1

4: $n$ – Integer Input

On entry: the number of times to expiry to be used.

Constraint:

n \geq 1

5: $x [m]$ – const double Input

On entry:

x [i - 1]

must contain

X_{i}

, the

i

th strike price, for

i = 1, 2, \dots, m

Constraint:

x [i - 1] \geq z ​ and ​ x [i - 1] \leq 1 / z

, where

z = nag_real_safe_small_number

, the safe range parameter, for

i = 1, 2, \dots, m

6: $s$ – double Input

On entry:

S

, the price of the underlying asset.

Constraint:

s \geq z ​ and ​ s \leq \frac{1}{z}

, where

z = nag_real_safe_small_number

, the safe range parameter and

s^{β} < \frac{1}{z}

where

β

is as defined in Section 3.

7: $t [n]$ – const double Input

On entry:

t [i - 1]

must contain

T_{i}

, the

i

th time, in years, to expiry, for

i = 1, 2, \dots, n

Constraint:

t [i - 1] \geq z

, where

z = nag_real_safe_small_number

, the safe range parameter, for

i = 1, 2, \dots, n

8: $sigma$ – double Input

On entry:

σ

, the volatility of the underlying asset. Note that a rate of 15% should be entered as

0.15

Constraint:

sigma > 0.0

9: $r$ – double Input

On entry:

r

, the annual risk-free interest rate, continuously compounded. Note that a rate of 5% should be entered as

0.05

Constraint:

r \geq 0.0

10: $q$ – double Input

On entry:

q

, the annual continuous yield rate. Note that a rate of 8% should be entered as

0.08

Constraint:

q \geq 0.0

11: $p [m \times n]$ – double Output

Note: where

P (i, j)

appears in this document, it refers to the array element

$p [(j - 1) \times m + i - 1]$ when $order = Nag_ColMajor$ ;
$p [(i - 1) \times n + j - 1]$ when $order = Nag_RowMajor$ .

On exit:

P (i, j)

contains

P_{i j}

, the option price evaluated for the strike price

x_{i}

at expiry

t_{j}

for

i = 1, 2, \dots, m

and

j = 1, 2, \dots, n

12: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error 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 $〈value〉$ had an illegal value.
NE_INT: On entry, $m = 〈value〉$ .
Constraint: $m \geq 1$ .

On entry, $n = 〈value〉$ .
Constraint: $n \geq 1$ .
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, $q = 〈value〉$ .
Constraint: $q \geq 0.0$ .

On entry, $r = 〈value〉$ .
Constraint: $r \geq 0.0$ .

On entry, $s = 〈value〉$ .
Constraint: $s \geq 〈value〉$ and $s \leq 〈value〉$ .

On entry, $s = 〈value〉$ and $β = 〈value〉$ .
Constraint: $s^{β} < 〈value〉$ .

On entry, $sigma = 〈value〉$ .
Constraint: $sigma > 0.0$ .
NE_REAL_ARRAY: On entry, $t [〈value〉] = 〈value〉$ .
Constraint: $t [i] \geq 〈value〉$ .

On entry, $x [〈value〉] = 〈value〉$ .
Constraint: $x [i] \geq 〈value〉$ and $x [i] \leq 〈value〉$ .

7 Accuracy

The accuracy of the output will be bounded by the accuracy of the cumulative bivariate Normal distribution function. The algorithm of Genz (2004) is used, as described in the document for g01hac, giving a maximum absolute error of less than

5 \times 10^{- 16}

. The univariate cumulative Normal distribution function also forms part of the evaluation (see s15abc and s15adc).

8 Parallelism and Performance

s30qcc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

s30qcc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

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.

9 Further Comments

None.

10 Example

This example computes the price of an American call with a time to expiry of

3

months, a stock price of

110

and a strike price of

100

. The risk-free interest rate is

8 %

per year, there is an annual dividend return of

12 %

and the volatility is

20 %

per year.

Interfaces: FL CL AD

NAG CL Interface Introduction

S (Specfun) Chapter Contents

S (Specfun) Chapter Introduction

s30qc: FL CL AD

NAG CL Interfaces30qcc (opt_​amer_​bs_​price)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG CL Interface
s30qcc (opt_amer_bs_price)