Integer type: int32 int64 nag_int show int32 show int32 show int64 show int64 show nag_int show nag_int

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_specfun_opt_bsm_price (s30aa)

▸▿ 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_bsm_price (s30aa) computes the European option price given by the Black–Scholes–Merton formula.

Syntax

[p, ifail] = s30aa(calput, x, s, t, sigma, r, q, 'm', m, 'n', n)

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

Description

nag_specfun_opt_bsm_price (s30aa) computes the price of a European call (or put) option for constant volatility,

σ

, and risk-free interest rate,

r

, with a possible dividend yield,

q

, using the Black–Scholes–Merton formula (see Black and Scholes (1973) and Merton (1973)). For a given strike price,

X

, the price of a European call with underlying price,

S

, and time to expiry,

T

, is

P_{call} = S e^{- q T} Φ (d_{1}) - X e^{- r T} Φ (d_{2})

and the corresponding European put price is

P_{put} = X e^{- r T} Φ (- d_{2}) - S e^{- q T} Φ (- d_{1})

and where

Φ

denotes the cumulative Normal distribution function,

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

and

\begin{array}{l} d_{1} = \frac{\ln (S / X) + (r - q + σ^{2} / 2) T}{σ \sqrt{T}}, \\ d_{2} = d_{1} - σ \sqrt{T} . \end{array}

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

Black F and Scholes M (1973) The pricing of options and corporate liabilities Journal of Political Economy 81 637–654

Merton R C (1973) Theory of rational option pricing Bell Journal of Economics and Management Science 4 141–183

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: $q$ – double scalar

q

, the annual continuous yield rate. Note that a rate of 8% should be entered as 0.08.

Constraint:

q \geq 0.0

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: $r \geq 0.0$ .

$ifail = 9$: Constraint: $q \geq 0.0$ .

$ifail = 11$: 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 prices for six European call options using two expiry times and three strike prices as input. The times to expiry are taken as

0.7

and

0.8

years respectively. The stock price is

55

, with strike prices,

58

60

and

62

. The risk-free interest rate is

10 %

per year and the volatility is

30 %

per year.

Open in the MATLAB editor: s30aa_example

function s30aa_example


fprintf('s30aa example results\n\n');

put = 'c';
s = 55;
sigma = 0.3;
r = 0.1;
q = 0;
x = [58, 60, 62];
t = [0.7, 0.8];

[p, ifail] = s30aa( ...
                    put, x, s, t, sigma, r, q);


fprintf('\nBlack-Scholes-Merton formula\n European Call :\n');
fprintf('  Spot       =   %9.4f\n', s);
fprintf('  Volatility =   %9.4f\n', sigma);
fprintf('  Rate       =   %9.4f\n', r);
fprintf('  Dividend   =   %9.4f\n\n', q);

fprintf('   Strike    Expiry   Option Price\n');
for i=1:3
  for j=1:2
    fprintf('%9.4f %9.4f %9.4f\n', x(i), t(j), p(i,j));
  end
end

s30aa example results


Black-Scholes-Merton formula
 European Call :
  Spot       =     55.0000
  Volatility =      0.3000
  Rate       =      0.1000
  Dividend   =      0.0000

   Strike    Expiry   Option Price
  58.0000    0.7000    5.9198
  58.0000    0.8000    6.5506
  60.0000    0.7000    5.0809
  60.0000    0.8000    5.6992
  62.0000    0.7000    4.3389
  62.0000    0.8000    4.9379

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox