NAG Library Routine Document

S30QCF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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

S30QCF computes the Bjerksund and Stensland (2002) approximation to the price of an American option.

2 Specification

SUBROUTINE S30QCF (

CALPUT, M, N, X, S, T, SIGMA, R, Q, P, LDP, IFAIL)

INTEGER	M, N, LDP, IFAIL
REAL (KIND=nag_wp)	X(M), S, T(N), SIGMA, R, Q, P(LDP,N)
CHARACTER(1)	CALPUT

3 Description

S30QCF 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 Parameters

1: CALPUT – CHARACTER(1)Input

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

$CALPUT ='C'$: A call. The holder has a right to buy.
$CALPUT ='P'$: A put. The holder has a right to sell.

Constraint:

CALPUT ='C'

'P'

2: M – INTEGERInput

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

Constraint:

M \geq 1

3: N – INTEGERInput

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

Constraint:

N \geq 1

4: X(M) – REAL (KIND=nag_wp) arrayInput

On entry:

X (i)

must contain

X_{i}

, the

i

th strike price, for

i = 1, 2, \dots, M

Constraint:

X (i) \geq z ​ and ​ X (i) \leq 1 / z

, where

z = X02AMF ()

, the safe range parameter, for

i = 1, 2, \dots, M

5: S – REAL (KIND=nag_wp)Input

On entry:

S

, the price of the underlying asset.

Constraint:

S \geq z ​ and ​ S \leq \frac{1}{z}

, where

z = X02AMF ()

, the safe range parameter and

S^{β} < \frac{1}{z}

where

β

is as defined in Section 3.

6: T(N) – REAL (KIND=nag_wp) arrayInput

On entry:

T (i)

must contain

T_{i}

, the

i

th time, in years, to expiry, for

i = 1, 2, \dots, N

Constraint:

T (i) \geq z

, where

z = X02AMF ()

, the safe range parameter, for

i = 1, 2, \dots, N

7: SIGMA – REAL (KIND=nag_wp)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

8: R – REAL (KIND=nag_wp)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

9: Q – REAL (KIND=nag_wp)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

10: P(LDP,N) – REAL (KIND=nag_wp) arrayOutput

On exit: the leading

M \times N

part of the array P contains the computed option prices.

11: LDP – INTEGERInput

On entry: the first dimension of the array P as declared in the (sub)program from which S30QCF is called.

Constraint:

LDP \geq M

12: IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$: On entry, $CALPUT \neq'C'$ or $'P'$ .

$IFAIL = 2$: On entry, $M \leq 0$ .

$IFAIL = 3$: On entry, $N \leq 0$ .

$IFAIL = 4$: On entry, $X (i) < z$ or $X (i) > 1 / z$ , where $z = X02AMF ()$ , the safe range parameter.

$IFAIL = 5$: On entry, $S < z$ or $S > 1.0 / z$ , where $z = X02AMF ()$ , the safe range parameter.

$IFAIL = 6$: On entry, $T (i) < z$ , where $z = X02AMF ()$ , the safe range parameter.

$IFAIL = 7$: On entry, $SIGMA \leq 0.0$ .

$IFAIL = 8$: On entry, $R < 0.0$ .

$IFAIL = 9$: On entry, $Q < 0.0$ .

$IFAIL = 11$: On entry, $LDP < M$ .

$IFAIL = 14$: On entry, $β \geq \frac{1}{z}$ , where $z = X02AMF ()$ , the safe range parameter (see Section 3).

$IFAIL = 15$: Internal memory allocation failed.

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 G01HAF, 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 S15ABF and S15ADF).

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 Routine DocumentS30QCF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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 Routine Document

S30QCF