NAG Library Function Document

nag_amer_bs_price (s30qcc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

nag_amer_bs_price (s30qcc) computes the Bjerksund and Stensland (2002) approximation to the price of an American option.

2 Specification

#include <nag.h>

#include <nags.h>

void	nag_amer_bs_price (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)

3 Description

nag_amer_bs_price (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) .

4 References

Bjerksund P and Stensland G (2002) Closed form valuation of American options Discussion Paper 2002/09 NHH Bergen Norway http://www.nhh.no/

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_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

order = Nag_RowMajor

. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

or Nag_ColMajor.

2: option – Nag_CallPutInput

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 – IntegerInput

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

Constraint:

m \geq 1

4: n – IntegerInput

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

Constraint:

n \geq 1

5: x[m] – const doubleInput

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 – doubleInput

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 doubleInput

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 – doubleInput

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 – doubleInput

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 – doubleInput

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[ $m \times n$ ] – doubleOutput

Note: the

(i, j)

th element of the matrix

P

is stored in

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

On exit: the

m \times n

array p contains the computed option prices.

12: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
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.
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 nag_bivariate_normal_dist (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 nag_cumul_normal (s15abc) and nag_erfc (s15adc)).

8 Further Comments

None.

9 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.

NAG Library Function Documentnag_amer_bs_price (s30qcc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Function Document

nag_amer_bs_price (s30qcc)