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Chapter Contents

Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_specfun_opt_heston_term (s30nc)

▸▿ 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_term (s30nc) computes the European option price given by Heston's stochastic volatility model with term structure.

Syntax

[p, ifail] = s30nc(calput, x, fwd, disc, ts, t, alpha, lambda, corr, sigmat, var0, 'm', m, 'numts', numts)

[p, ifail] = nag_specfun_opt_heston_term(calput, x, fwd, disc, ts, t, alpha, lambda, corr, sigmat, var0, 'm', m, 'numts', numts)

Description

nag_specfun_opt_heston_term (s30nc) computes the price of a European option for Heston's stochastic volatility model with time-dependent parameters which are piecewise constant. Starting from the stochastic volatility model given by the Stochastic Differential Equation (SDE) system defined by Heston (1993), a scaling of the variance process is introduced, together with a normalization, setting the long run variance,

η

, equal to

1

. This leads to

\frac{d S_{t}}{S_{t}} = μ_{t} d t + σ_{t} \sqrt{ν_{t}} d W_{t}^{(1)},

(1)

d ν_{t} = λ_{t} (1 - ν_{t}) d t + α_{t} \sqrt{ν_{t}} d W_{t}^{(2)},

(2)

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

(3)

where

μ_{t} = r_{t} - q_{t}

is the drift term representing the contribution of interest rates,

r_{t}

, and dividends,

q_{t}

, while

σ_{t}

is the scaling parameter,

ν_{t}

is the scaled variance,

λ_{t}

is the mean reversion rate and

α_{t}

is the volatility of the scaled volatility,

\sqrt{ν_{t}}

. Then,

W_{t}^{(i)}

, for

i = 1, 2

, are two standard Brownian motions with correlation parameter

ρ_{t}

. Without loss of generality, the drift term,

μ_{t}

, is eliminated by modelling the forward price,

F_{t}

, directly, instead of the spot price,

S_{t}

, with

F_{t} = S_{0} \exp (\int_{0}^{t} μ_{s} d s) .

(4)

If required, the spot can be expressed as,

S_{0} = D F_{t}

, where

D

is the discount factor.

The option price is computed by dividing the time to expiry,

T

, into

n_{s}

subintervals

[t_{0}, t_{1}], \dots, [t_{i - 1}, t_{i}], \dots, [t_{n_{s} - 1}, T]

and applying the method of characteristic functions to each subinterval, with appropriate initial conditions. Thus, a pair of ordinary differential equations (one of which is a Riccati equation) is solved on each subinterval as outlined in Elices (2008) and Mikhailov and Nögel (2003). Reversing time by taking

τ = T - t

, the characteristic function solution for the first time subinterval, starting at

τ = 0

, is given by Heston (1993), while the solution on each following subinterval uses the solution of the preceding subinterval as initial condition to compute the value of the characteristic function.

In the case of a ‘flat’ term structure, i.e., the parameters are constant over the time of the option,

T

, the form of the SDE system given by Heston (1993)

\frac{d S_{t}}{S_{t}} = μ_{t} d t + \sqrt{V_{t}} d W_{t}^{(1)},

(5)

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

(6)

can be recovered by setting

V_{0} = η = σ_{t}^{2}

κ = λ_{t}

σ_{v} = σ_{t} α_{t}

Conversely, given the Heston form of the SDE pair, to get the term structure form set

σ = \sqrt{η}

α_{t} = σ_{v} / \sqrt{η}

λ_{t} = κ

References

Bain A (2011) Private communication

Elices A (2008) Models with time-dependent parameters using transform methods: application to Heston’s model arXiv:0708.2020v2

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

Mikhailov S and Nögel U (2003) Heston’s Stochastic Volatility Model Implementation, Calibration and Some Extensions Wilmott Magazine July/August 74–79

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)

contains 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: $fwd$ – double scalar

The forward price of the asset.

Constraint:

fwd \geq z

and

fwd \leq 1 / z

, where

z = x02am ()

, the safe range parameter.

4: $disc$ – double scalar

The discount factor, where the current price of the underlying asset,

S_{0}

, is given as

S_{0} = disc \times fwd

Constraint:

disc \geq z

and

disc \leq 1 / z

, where

z = x02am ()

, the safe range parameter.

5: $ts (numts)$ – double array

ts (i)

must contain the length of the time intervals for which the corresponding element of alpha, lambda, corr and sigmat apply. These should be ordered as they occur in time i.e.,

Δ t_{i} = t_{i} - t_{i - 1}

Constraint:

ts (i) \geq z ​ and ​ ts (i) \leq 1 / z

, where

z = x02am ()

, the safe range parameter, for

i = 1, 2, \dots, numts

6: $t$ – double scalar

t contains the time to expiry. If

T > \sum Δ t_{i}

then the parameters associated with the last time interval are extended to the expiry time. If

T < \sum Δ t_{i}

then the parameters specified are used up until the expiry time. The rest are ignored.

Constraint:

t \geq z

, where

z = x02am ()

, the safe range parameter.

7: $alpha (numts)$ – double array

alpha (i)

must contain the value of

α_{t}

, the value of the volatility of the scaled volatility,

\sqrt{ν}

, over time subinterval

Δ t_{i}

Constraint:

alpha (i) \geq z ​ and ​ alpha (i) \leq 1 / z

, where

z = x02am ()

, the safe range parameter, for

i = 1, 2, \dots, numts

8: $lambda (numts)$ – double array

lambda (i)

must contain the value,

λ_{t}

, of the mean reversion parameter over the time subinterval

Δ t_{i}

Constraint:

lambda (i) \geq z ​ and ​ lambda (i) \leq 1 / z

, where

z = x02am ()

, the safe range parameter, for

i = 1, 2, \dots, numts

9: $corr (numts)$ – double array

corr (i)

must contain the value,

ρ_{t}

, of the correlation parameter over the time subinterval

Δ t_{i}

Constraint:

- 1.0 \leq corr (i) \leq 1.0

, for

i = 1, 2, \dots, numts

10: $sigmat (numts)$ – double array

sigmat (i)

must contain the value,

σ_{t}

, of the variance scale factor over the time subinterval

Δ t_{i}

Constraint:

sigmat (i) \geq z ​ and ​ sigmat (i) \leq 1 / z

, where

z = x02am ()

, the safe range parameter, for

i = 1, 2, \dots, numts

11: $var0$ – double scalar

ν_{0}

, the initial scaled variance.

Constraint:

var0 \geq 0.0

Optional Input Parameters

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

Constraint: $m \geq 1$ .
2: $numts$ – int64int32nag_int scalar: Default: the dimension of the arrays corr, lambda, sigmat, ts, alpha. (An error is raised if these dimensions are not equal.)
$n_{s}$ , the number of subintervals into which the time to expiry, $T$ , is divided.

Constraint: $numts \geq 1$ .

Output Parameters

1: $p (m)$ – double array: $p (i)$ contains the computed option price at the expiry time, $T$ , corresponding to strike $x (i)$ for the specified term structure, for $i = 1, 2, \dots, m$ .
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: $numts \geq 1$ .

$ifail = 4$: Constraint: $_\leq x (i) \leq_$ .

$ifail = 5$: Constraint: $_\leq fwd \leq_$ .

$ifail = 6$: Constraint: $_\leq disc \leq_$ .

$ifail = 7$: Constraint: $_\leq ts (i) \leq_$ .

$ifail = 8$: t is too small.

$ifail = 9$: Constraint: $_\leq alpha (i) \leq_$ .

$ifail = 10$: Constraint: $_\leq lambda (i) \leq_$ .

$ifail = 11$: Constraint: $|corr (i)| \leq 1.0$ .

$ifail = 12$: Constraint: $_\leq sigmat (i) \leq_$ .

$ifail = 13$: Constraint: $var0 > 0.0$ .

$ifail = 14$: Quadrature has not converged to the specified accuracy. However, the result should be a reasonable approximation.

$ifail = 15$: 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 solution is obtained by integrating the pair of ordinary differential equations over each subinterval in time. The accuracy is controlled by a relative tolerance over each time subinterval, which is set to

10^{- 8}

. Over a number of subintervals in time the error may accumulate and so the overall error in the computation may be greater than this. A threshold of

10^{- 10}

is used and solutions smaller than this are not accurately evaluated.

Further Comments

None.

Example

This example computes the price of a European call using Heston's stochastic volatility model with a term structure of interest rates.

Open in the MATLAB editor: s30nc_example

function s30nc_example


fprintf('s30nc example results\n\n');

% Price of a European call using Heston model with a term structure.

calput = 'C';
fwd    = 100;
disc   = 1;
var0   = 1;
fprintf('European Call :\n');
fprintf(' %-22s = %9.4f\n','Forward',fwd);
fprintf(' %-22s = %9.4f\n','Discount factor',disc);
fprintf(' %-22s = %9.4f\n','Variance',var0);

m      = 1;
x      = [100];
n      = 2;
ts     = [ 0.35 0.65];
t      = 1;
alpha  = [ 2.25 1.5];
corr   = [-0.05 0.1];
lambda = [ 2.0  1.5];
sigmat = [ 0.04 0.13];
fprintf('\n%6s%13s%11s%8s%12s\n','ts','alpha','lambda','corr','sigmat');
for i = 1:n
  fprintf('%10.4f%10.4f%10.4f%10.4f%10.4f\n',...
          ts(i),alpha(i),lambda(i),corr(i),sigmat(i));
end

[p, ifail] = s30nc(...
                   calput, x, fwd, disc, ts, t, ...
                   alpha, lambda, corr, sigmat, var0);

fprintf('\n%10s%10s%10s\n','strike','expiry','price');
for i = 1:m
  fprintf('%10.4f%10.4f%10.4f\n',x(i),t,p(i));
end

s30nc example results

European Call :
 Forward                =  100.0000
 Discount factor        =    1.0000
 Variance               =    1.0000

    ts        alpha     lambda    corr      sigmat
    0.3500    2.2500    2.0000   -0.0500    0.0400
    0.6500    1.5000    1.5000    0.1000    0.1300

    strike    expiry     price
  100.0000    1.0000    4.0074

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Chapter Contents

Chapter Introduction

NAG Toolbox