NAG Library Function Document

nag_asian_geom_price (s30sac)

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

3 Description

nag_asian_geom_price (s30sac) computes the price of an Asian geometric continuous average-rate option for constant volatility,

σ

, risk-free rate,

r

, and cost of carry,

b

(see Kemna and Vorst (1990)). For a given strike price,

X

, the price of a call option with underlying price,

S

, and time to expiry,

T

, is

P_{call} = S e^{(\bar{b} - r) T} Φ ({\bar{d}}_{1}) - X e^{- r T} Φ ({\bar{d}}_{2}),

and the corresponding put option price is

P_{put} = X e^{- r T} Φ (- {\bar{d}}_{2}) - S e^{(\bar{b} - r) T} Φ (- {\bar{d}}_{1}),

where

{\bar{d}}_{1} = \frac{\ln (S / X) + (\bar{b} + {\bar{σ}}^{2} / 2) T}{\bar{σ} \sqrt{T}}

and

{\bar{d}}_{2} = {\bar{d}}_{1} - \bar{σ} \sqrt{T},

with

\bar{σ} = \frac{σ}{\sqrt{3}}, \bar{b} = \frac{1}{2} (r - \frac{σ^{2}}{6}) .

Φ

is the cumulative Normal distribution function,

Φ (x) = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{x} \exp (- y^{2} / 2) d y .

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

Kemna A and Vorst A (1990) A pricing method for options based on average asset values Journal of Banking and Finance 14 113–129

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

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 1.0 / z

, where

z = nag_real_safe_small_number

, the safe range parameter.

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: b – doubleInput

On entry:

b

, the annual cost of carry rate. Note that a rate of 8% should be entered as

0.08

11: p[ $m \times n$ ] – doubleOutput

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 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, $r = ⟨value⟩$ .
Constraint: $r \geq 0.0$ .
On entry, $s = ⟨value⟩$ .
Constraint: $s \geq ⟨value⟩$ and $s \leq ⟨value⟩$ .
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 is dependent on the accuracy of the cumulative Normal distribution function,

Φ

. 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 nag_cumul_normal (s15abc) and nag_erfc (s15adc)). An accuracy close to machine precision can generally be expected.

8 Parallelism and Performance

nag_asian_geom_price (s30sac) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

Please 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 Asian geometric continuous average-rate put with a time to expiry of

3

months, a stock price of

80

and a strike price of

85

. The risk-free interest rate is

5 %

per year, the cost of carry is

8 %

and the volatility is

20 %

per year.

NAG Library Function Documentnag_asian_geom_price (s30sac)

+− 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 Library Function Document

nag_asian_geom_price (s30sac)