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Chapter Contents

Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_specfun_opt_asian_geom_price (s30sa)

▸▿ Contents

1 Purpose

2 Syntax

3 Description

4 References

▸▿ 5 Parameters

5.1 Compulsory Input Parameters

5.2 Optional Input Parameters

5.3 Output Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

Purpose

nag_specfun_opt_asian_geom_price (s30sa) computes the Asian geometric continuous average-rate option price.

Syntax

[p, ifail] = s30sa(calput, x, s, t, sigma, r, b, 'm', m, 'n', n)

[p, ifail] = nag_specfun_opt_asian_geom_price(calput, x, s, t, sigma, r, b, 'm', m, 'n', n)

Description

nag_specfun_opt_asian_geom_price (s30sa) 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

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

Parameters

Compulsory Input Parameters

1: $calput$ – string (length ≥ 1)

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: $x (m)$ – double array

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 = x02am ()

, the safe range parameter, for

i = 1, 2, \dots, m

3: $s$ – double scalar

S

, the price of the underlying asset.

Constraint:

s \geq z ​ and ​ s \leq 1.0 / z

, where

z = x02am ()

, the safe range parameter.

4: $t (n)$ – double array

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 = x02am ()

, the safe range parameter, for

i = 1, 2, \dots, n

5: $sigma$ – double scalar

σ

, the volatility of the underlying asset. Note that a rate of 15% should be entered as 0.15.

Constraint:

sigma > 0.0

6: $r$ – double scalar

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

7: $b$ – double scalar

b

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

0.08

Optional Input Parameters

1: $m$ – int64int32nag_int scalar: Default: the dimension of the array x.
The number of strike prices to be used.

Constraint: $m \geq 1$ .
2: $n$ – int64int32nag_int scalar: Default: the dimension of the array t.
The number of times to expiry to be used.

Constraint: $n \geq 1$ .

Output Parameters

1: $p (ldp, n)$ – double array: $ldp = m$ .
$p (i, j)$ 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 $j = 1, 2, \dots, n$ .
2: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$: On entry, $calput =_$ was an illegal value.

$ifail = 2$: Constraint: $m \geq 1$ .

$ifail = 3$: Constraint: $n \geq 1$ .

$ifail = 4$: Constraint: $x (i) \geq_$ and $x (i) \leq_$ .

$ifail = 5$: Constraint: $s \geq_$ and $s \leq_$ .

$ifail = 6$: Constraint: $t (i) \geq_$ .

$ifail = 7$: Constraint: $sigma > 0.0$ .

$ifail = 8$: Constraint: $r \geq 0.0$ .

$ifail = 11$: Constraint: $ldp \geq m$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

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_specfun_cdf_normal (s15ab) and nag_specfun_erfc_real (s15ad)). An accuracy close to machine precision can generally be expected.

Further Comments

None.

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.

Open in the MATLAB editor: s30sa_example

function s30sa_example


fprintf('s30sa example results\n\n');

put = 'P';
s = 80.0;
sigma = 0.2;
r = 0.05;
b = 0.08;
x = [85.0];
t = [0.25];


[p, ifail] = s30sa( ...
                    put, x, s, t, sigma, r, b);


fprintf('\nAsian Option: Geometric Continuous Average-Rate\nAsian Put :\n');
fprintf('  Spot          =   %9.4f\n', s);
fprintf('  Volatility    =   %9.4f\n', sigma);
fprintf('  Rate          =   %9.4f\n', r);
fprintf('  Cost of carry =   %9.4f\n\n', b);

fprintf('   Strike    Expiry   Option Price\n');

for i=1:1
  for j=1:1
    fprintf('%9.4f %9.4f %9.4f\n', x(i), t(j), p(i,j));
  end
end

s30sa example results


Asian Option: Geometric Continuous Average-Rate
Asian Put :
  Spot          =     80.0000
  Volatility    =      0.2000
  Rate          =      0.0500
  Cost of carry =      0.0800

   Strike    Expiry   Option Price
  85.0000    0.2500    4.6922

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Chapter Contents

Chapter Introduction

NAG Toolbox