NAG Library Function Document

nag_heston_term (s30ncc)

void	nag_heston_term (Nag_CallPut option, Integer m, Integer numts, const double x[], double fwd, double disc, const double ts[], double t, const double alpha[], const double lambda[], const double corr[], const double sigmat[], double var0, double p[], NagError *fail)

3 Description

nag_heston_term (s30ncc) 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

4 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

5 Arguments

1: 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

2: m – IntegerInput

On entry:

m

, the number of strike prices to be used.

Constraint:

m \geq 1

3: numts – IntegerInput

On entry:

n_{s}

, the number of subintervals into which the time to expiry,

T

, is divided.

Constraint:

numts \geq 1

4: x[m] – const doubleInput

On entry:

x [i - 1]

contains 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

5: fwd – doubleInput

On entry: the forward price of the asset.

Constraint:

fwd \geq z

and

fwd \leq 1 / z

, where

z = nag_real_safe_small_number

, the safe range parameter.

6: disc – doubleInput

On entry: 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 = nag_real_safe_small_number

, the safe range parameter.

7: ts[numts] – const doubleInput

On entry:

ts [i - 1]

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 - 1] \geq z ​ and ​ ts [i - 1] \leq 1 / z

, where

z = nag_real_safe_small_number

, the safe range parameter, for

i = 1, 2, \dots, numts

8: t – doubleInput

On entry: 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 = nag_real_safe_small_number

, the safe range parameter.

9: alpha[numts] – const doubleInput

On entry:

alpha [i - 1]

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 - 1] \geq z ​ and ​ alpha [i - 1] \leq 1 / z

, where

z = nag_real_safe_small_number

, the safe range parameter, for

i = 1, 2, \dots, numts

10: lambda[numts] – const doubleInput

On entry:

lambda [i - 1]

must contain the value,

λ_{t}

, of the mean reversion parameter over the time subinterval

Δ t_{i}

Constraint:

lambda [i - 1] \geq z ​ and ​ lambda [i - 1] \leq 1 / z

, where

z = nag_real_safe_small_number

, the safe range parameter, for

i = 1, 2, \dots, numts

11: corr[numts] – const doubleInput

On entry:

corr [i - 1]

must contain the value,

ρ_{t}

, of the correlation parameter over the time subinterval

Δ t_{i}

Constraint:

- 1.0 \leq corr [i - 1] \leq 1.0

, for

i = 1, 2, \dots, numts

12: sigmat[numts] – const doubleInput

On entry:

sigmat [i - 1]

must contain the value,

σ_{t}

, of the variance scale factor over the time subinterval

Δ t_{i}

Constraint:

sigmat [i - 1] \geq z ​ and ​ sigmat [i - 1] \leq 1 / z

, where

z = nag_real_safe_small_number

, the safe range parameter, for

i = 1, 2, \dots, numts

13: var0 – doubleInput

On entry:

ν_{0}

, the initial scaled variance.

Constraint:

var0 \geq 0.0

14: p[m] – doubleOutput

On exit:

p [i - 1]

contains the computed option price at the expiry time,

T

, corresponding to strike

x [i - 1]

for the specified term structure, for

i = 1, 2, \dots, m

15: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ACCURACY: Solution cannot be computed accurately. Check values of input arguments.
NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_CONVERGENCE: Quadrature has not converged to the specified accuracy. However, the result should be a reasonable approximation.
NE_INT: On entry, $m = ⟨value⟩$ .
Constraint: $m \geq 1$ .
On entry, $numts = ⟨value⟩$ .
Constraint: $numts \geq 1$ .
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, $disc = ⟨value⟩$ .
Constraint: $⟨value⟩ \leq disc \leq ⟨value⟩$ .
On entry, $fwd = ⟨value⟩$ .
Constraint: $⟨value⟩ \leq fwd \leq ⟨value⟩$ .
On entry, $t = ⟨value⟩$ .
Constraint: $t \geq ⟨value⟩$ .
On entry, $var0 = ⟨value⟩$ .
Constraint: $var0 > 0.0$ .
NE_REAL_ARRAY: On entry, $alpha [⟨value⟩] = ⟨value⟩$ .
Constraint: $⟨value⟩ \leq alpha [i - 1] \leq ⟨value⟩$ .
On entry, $corr [⟨value⟩] = ⟨value⟩$ .
Constraint: $|corr [i - 1]| \leq 1.0$ .
On entry, $lambda [⟨value⟩] = ⟨value⟩$ .
Constraint: $⟨value⟩ \leq lambda [i - 1] \leq ⟨value⟩$ .
On entry, $sigmat [⟨value⟩] = ⟨value⟩$ .
Constraint: $⟨value⟩ \leq sigmat [i - 1] \leq ⟨value⟩$ .
On entry, $ts [⟨value⟩] = ⟨value⟩$ .
Constraint: $⟨value⟩ \leq ts [i - 1] \leq ⟨value⟩$ .
On entry, $x [⟨value⟩] = ⟨value⟩$ .
Constraint: $⟨value⟩ \leq x [i - 1] \leq ⟨value⟩$ .

7 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.

8 Parallelism and Performance

nag_heston_term (s30ncc) 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 a European call using Heston's stochastic volatility model with a term structure of interest rates.

NAG Library Function Documentnag_heston_term (s30ncc)

+− 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_heston_term (s30ncc)