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Chapter Contents

Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_specfun_opt_heston_price (s30na)

▸▿ 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_heston_price (s30na) computes the European option price given by Heston's stochastic volatility model.

Syntax

[p, ifail] = s30na(calput, x, s, t, sigmav, kappa, corr, var0, eta, grisk, r, q, 'm', m, 'n', n)

[p, ifail] = nag_specfun_opt_heston_price(calput, x, s, t, sigmav, kappa, corr, var0, eta, grisk, r, q, 'm', m, 'n', n)

Description

nag_specfun_opt_heston_price (s30na) computes the price of a European option using Heston's stochastic volatility model. The return on the asset price,

S

, is

\frac{d S}{S} = (r - q) d t + \sqrt{v_{t}} d W_{t}^{(1)}

and the instantaneous variance,

v_{t}

, is defined by a mean-reverting square root stochastic process,

d v_{t} = κ (η - v_{t}) d t + σ_{v} \sqrt{v_{t}} d W_{t}^{(2)},

where

r

is the risk free annual interest rate;

q

is the annual dividend rate;

v_{t}

is the variance of the asset price;

σ_{v}

is the volatility of the volatility,

\sqrt{v_{t}}

;

κ

is the mean reversion rate;

η

is the long term variance.

d W_{t}^{(i)}

, for

i = 1, 2

, denotes two correlated standard Brownian motions with

ℂov [d W_{t}^{(1)}, d W_{t}^{(2)}] = ρ d t .

The option price is computed by evaluating the integral transform given by Lewis (2000) using the form of the characteristic function discussed by Albrecher et al. (2007), see also Kilin (2006).

P_{call} = S e^{- q T} - X e^{- r T} \frac{1}{π} Re [\int_{0 + i / 2}^{\infty + i / 2} e^{- i k \bar{X}} \frac{\hat{H} (k, v, T)}{k^{2} - i k} d k],

(1)

where

\bar{X} = \ln (S / X) + (r - q) T

and

\hat{H} (k, v, T) = \exp (\frac{2 κ η}{σ_{v}^{2}} [t g - \ln (\frac{1 - h e^{- ξ t}}{1 - h})] + v_{t} g [\frac{1 - e^{- ξ t}}{1 - h e^{- ξ t}}]),

g = \frac{1}{2} (b - ξ), h = \frac{b - ξ}{b + ξ}, t = σ_{v}^{2} T / 2,

ξ = {[b^{2} + 4 \frac{k^{2} - i k}{σ_{v}^{2}}]}^{\frac{1}{2}},

b = \frac{2}{σ_{v}^{2}} [(1 - γ + i k) ρ σ_{v} + \sqrt{κ^{2} - γ (1 - γ) σ_{v}^{2}}]

with

t = σ_{v}^{2} T / 2

. Here

γ

is the risk aversion parameter of the representative agent with

0 \leq γ \leq 1

and

γ (1 - γ) σ_{v}^{2} \leq κ^{2}

. The value

γ = 1

corresponds to

λ = 0

, where

λ

is the market price of risk in Heston (1993) (see Lewis (2000) and Rouah and Vainberg (2007)).

The price of a put option is obtained by put-call parity.

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

Albrecher H, Mayer P, Schoutens W and Tistaert J (2007) The little Heston trap Wilmott Magazine January 2007 83–92

Heston S (1993) A closed-form solution for options with stochastic volatility with applications to bond and currency options Review of Financial Studies 6 327–343

Kilin F (2006) Accelerating the calibration of stochastic volatility models MPRA Paper No. 2975 http://mpra.ub.uni-muenchen.de/2975/

Lewis A L (2000) Option valuation under stochastic volatility Finance Press, USA

Rouah F D and Vainberg G (2007) Option Pricing Models and Volatility using Excel-VBA John Wiley and Sons, Inc

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: $sigmav$ – double scalar

The volatility,

σ_{v}

, of the volatility process,

\sqrt{v_{t}}

. Note that a rate of 20% should be entered as

0.2

Constraint:

sigmav > 0.0

6: $kappa$ – double scalar

κ

, the long term mean reversion rate of the volatility.

Constraint:

kappa > 0.0

7: $corr$ – double scalar

The correlation between the two standard Brownian motions for the asset price and the volatility.

Constraint:

- 1.0 \leq corr \leq 1.0

8: $var0$ – double scalar

The initial value of the variance,

v_{t}

, of the asset price.

Constraint:

var0 \geq 0.0

9: $eta$ – double scalar

η

, the long term mean of the variance of the asset price.

Constraint:

eta > 0.0

10: $grisk$ – double scalar

The risk aversion parameter,

γ

, of the representative agent.

Constraint:

0.0 \leq grisk \leq 1.0

and

grisk \times (1.0 - grisk) \times sigmav \times sigmav \leq kappa \times kappa

11: $r$ – 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

12: $q$ – 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:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

$ifail = 1$: 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: $sigmav > 0.0$ .

$ifail = 8$: Constraint: $kappa > 0.0$ .

$ifail = 9$: Constraint: $|corr| \leq 1.0$ .

$ifail = 10$: Constraint: $var0 \geq 0.0$ .

$ifail = 11$: Constraint: $eta > 0.0$ .

$ifail = 12$: Constraint: $0.0 \leq grisk \leq 1.0$ and $grisk \times (1.0 - grisk) \times {sigmav}^{2} \leq {kappa}^{2}$ .

$ifail = 13$: Constraint: $r \geq 0.0$ .

$ifail = 14$: Constraint: $q \geq 0.0$ .

$ifail = 16$: Constraint: $ldp \geq m$ .

W $ifail = 17$: Quadrature has not converged to the specified accuracy. However, the result should be a reasonable approximation.

W $ifail = 18$: Solution cannot be computed accurately. Check values of input arguments.

$ifail = - 99$: 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 determined by the accuracy of the numerical quadrature used to evaluate the integral in (1). An adaptive method is used which evaluates the integral to within a tolerance of

\max (10^{- 8}, 10^{- 10} \times |I|)

, where

|I|

is the absolute value of the integral.

Further Comments

None.

Example

This example computes the price of a European call using Heston's stochastic volatility model. The time to expiry is

6

months, the stock price is

100

and the strike price is

100

. The risk-free interest rate is

5 %

per year, the volatility of the variance,

σ_{v}

, is

22.5 %

per year, the mean reversion parameter,

κ

, is

2.0

, the long term mean of the variance,

η

, is

0.01

and the correlation between the volatility process and the stock price process,

ρ

, is

0.0

. The risk aversion parameter,

γ

, is

1.0

and the initial value of the variance, var0, is

0.01

Open in the MATLAB editor: s30na_example

function s30na_example


fprintf('s30na example results\n\n');

put = 'C';
s = 100.0;
r = 0.05;
q = 0.0;
gamma=1.0;
kappa = 2.0;
eta = 0.01;
var0 = 0.01;
sigmav = 0.225;
corr = 0.0;
x = [100.0];
t = [0.5];


[p, ifail] = s30na( ...
                    put, x, s, t, sigmav, kappa, corr, var0, eta, gamma, r, q);


fprintf('\nHeston''s Stochastic Volatility Model\n European Call :\n');
fprintf(' Spot                   =  %9.4f\n', s);
fprintf(' Volatility of vol      =  %9.4f\n', sigmav);
fprintf(' Mean reversion         =  %9.4f\n', kappa);
fprintf(' Correlation            =  %9.4f\n', corr);
fprintf(' Variance               =  %9.4f\n', eta);
fprintf(' Mean of variance       =  %9.4f\n', var0);
fprintf(' Risk aversion          =  %9.4f\n', gamma);
fprintf(' Rate                   =  %9.4f\n', r);
fprintf(' Dividend               =  %9.4f\n\n', 1);

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

s30na example results


Heston's Stochastic Volatility Model
 European Call :
 Spot                   =   100.0000
 Volatility of vol      =     0.2250
 Mean reversion         =     2.0000
 Correlation            =     0.0000
 Variance               =     0.0100
 Mean of variance       =     0.0100
 Risk aversion          =     1.0000
 Rate                   =     0.0500
 Dividend               =     1.0000

   Strike    Expiry   Option Price
 100.0000    0.5000    4.0851

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Chapter Introduction

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