NAG Library Function Document

nag_heston_greeks (s30nbc)

nag_heston_greeks (Nag_OrderType order, Nag_CallPut option, Integer m, Integer n, const double x[], double s, const double t[], double sigmav, double kappa, double corr, double var0, double eta, double grisk, double r, double q, double p[], double delta[], double gamma[], double vega[], double theta[], double rho[], double vanna[], double charm[], double speed[], double zomma[], double vomma[], NagError *fail)

3 Description

nag_heston_greeks (s30nbc) computes the price and sensitivities 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.

P_{put} = P_{call} + X e^{- r T} - S e^{- q T} .

Writing the expression for the price of a call option as

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

then the sensitivities or Greeks can be obtained in the following manner,

Delta

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

Vega

\frac{\partial P}{\partial v} = - X e^{- r T} \frac{1}{π} Re [\int_{0 - i / 2}^{0 + i / 2} f_{2} I (k, r, j, S, T, v) d k],  where ​ f_{2} = g [\frac{1 - e^{- ξ t}}{1 - h e^{- ξ t}}],

Rho

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

4 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 6 347–343 Review of Financial Studies

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

5 Arguments

1: order – Nag_OrderTypeInput

On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by

order = Nag_RowMajor

. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

or Nag_ColMajor.

2: 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

3: m – IntegerInput

On entry: the number of strike prices to be used.

Constraint:

m \geq 1

4: n – IntegerInput

On entry: the number of times to expiry to be used.

Constraint:

n \geq 1

5: x[m] – const doubleInput

On entry:

x [i - 1]

must contain

X_{i}

, 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

6: s – doubleInput

On entry:

S

, the price of the underlying asset.

Constraint:

s \geq z ​ and ​ s \leq 1.0 / z

, where

z = nag_real_safe_small_number

, the safe range parameter.

7: t[n] – const doubleInput

On entry:

t [i - 1]

must contain

T_{i}

, the

i

th time, in years, to expiry, for

i = 1, 2, \dots, n

Constraint:

t [i - 1] \geq z

, where

z = nag_real_safe_small_number

, the safe range parameter, for

i = 1, 2, \dots, n

8: sigmav – doubleInput

On entry: 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

9: kappa – doubleInput

On entry:

κ

, the long term mean reversion rate of the volatility.

Constraint:

kappa > 0.0

10: corr – doubleInput

On entry: the correlation between the two standard Brownian motions for the asset price and the volatility.

Constraint:

- 1.0 \leq corr \leq 1.0

11: var0 – doubleInput

On entry: the initial value of the variance,

v_{t}

, of the asset price.

Constraint:

var0 \geq 0.0

12: eta – doubleInput

On entry:

η

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

Constraint:

eta > 0.0

13: grisk – doubleInput

On entry: the risk aversion parameter,

γ

, of the representative agent.

Constraint:

0.0 \leq grisk \leq 1.0

and

grisk \times (1 - grisk) \times sigmav \times sigmav \leq kappa \times kappa

14: r – doubleInput

On entry:

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

15: q – doubleInput

On entry:

q

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

Constraint:

q \geq 0.0

16: p[ $m \times n$ ] – doubleOutput

Note: the

(i, j)

th element of the matrix

P

is stored in

$p [(j - 1) \times m + i - 1]$ when $order = Nag_ColMajor$ ;
$p [(i - 1) \times n + j - 1]$ when $order = Nag_RowMajor$ .

On exit: the

m \times n

array p contains the computed option prices.

17: delta[ $m \times n$ ] – doubleOutput

Note: the

(i, j)

th element of the matrix is stored in

$delta [(j - 1) \times m + i - 1]$ when $order = Nag_ColMajor$ ;
$delta [(i - 1) \times n + j - 1]$ when $order = Nag_RowMajor$ .

On exit: the

m \times n

array delta contains the sensitivity,

\frac{\partial P}{\partial S}

, of the option price to change in the price of the underlying asset.

18: gamma[ $m \times n$ ] – doubleOutput

Note: the

(i, j)

th element of the matrix is stored in

$gamma [(j - 1) \times m + i - 1]$ when $order = Nag_ColMajor$ ;
$gamma [(i - 1) \times n + j - 1]$ when $order = Nag_RowMajor$ .

On exit: the

m \times n

array gamma contains the sensitivity,

\frac{\partial^{2} P}{\partial S^{2}}

, of delta to change in the price of the underlying asset.

19: vega[ $m \times n$ ] – doubleOutput

Note: the

(i, j)

th element of the matrix is stored in

$vega [(j - 1) \times m + i - 1]$ when $order = Nag_ColMajor$ ;
$vega [(i - 1) \times n + j - 1]$ when $order = Nag_RowMajor$ .

On exit: the

m \times n

array vega contains the sensitivity,

\frac{\partial P}{\partial σ}

, of the option price to change in the volatility of the underlying asset.

20: theta[ $m \times n$ ] – doubleOutput

Note: the

(i, j)

th element of the matrix is stored in

$theta [(j - 1) \times m + i - 1]$ when $order = Nag_ColMajor$ ;
$theta [(i - 1) \times n + j - 1]$ when $order = Nag_RowMajor$ .

On exit: the

m \times n

array theta contains the sensitivity,

- \frac{\partial P}{\partial T}

, of the option price to change in the time to expiry of the option.

21: rho[ $m \times n$ ] – doubleOutput

Note: the

(i, j)

th element of the matrix is stored in

$rho [(j - 1) \times m + i - 1]$ when $order = Nag_ColMajor$ ;
$rho [(i - 1) \times n + j - 1]$ when $order = Nag_RowMajor$ .

On exit: the

m \times n

array rho contains the sensitivity,

\frac{\partial P}{\partial r}

, of the option price to change in the annual risk-free interest rate.

22: vanna[ $m \times n$ ] – doubleOutput

Note: the

(i, j)

th element of the matrix is stored in

$vanna [(j - 1) \times m + i - 1]$ when $order = Nag_ColMajor$ ;
$vanna [(i - 1) \times n + j - 1]$ when $order = Nag_RowMajor$ .

On exit: the

m \times n

array vanna contains the sensitivity,

\frac{\partial^{2} P}{\partial S \partial σ}

, of vega to change in the price of the underlying asset or, equivalently, the sensitivity of delta to change in the volatility of the asset price.

23: charm[ $m \times n$ ] – doubleOutput

Note: the

(i, j)

th element of the matrix is stored in

$charm [(j - 1) \times m + i - 1]$ when $order = Nag_ColMajor$ ;
$charm [(i - 1) \times n + j - 1]$ when $order = Nag_RowMajor$ .

On exit: the

m \times n

array charm contains the sensitivity,

- \frac{\partial^{2} P}{\partial S \partial T}

, of delta to change in the time to expiry of the option.

24: speed[ $m \times n$ ] – doubleOutput

Note: the

(i, j)

th element of the matrix is stored in

$speed [(j - 1) \times m + i - 1]$ when $order = Nag_ColMajor$ ;
$speed [(i - 1) \times n + j - 1]$ when $order = Nag_RowMajor$ .

On exit: the

m \times n

array speed contains the sensitivity,

\frac{\partial^{3} P}{\partial S^{3}}

, of gamma to change in the price of the underlying asset.

25: zomma[ $m \times n$ ] – doubleOutput

Note: the

(i, j)

th element of the matrix is stored in

$zomma [(j - 1) \times m + i - 1]$ when $order = Nag_ColMajor$ ;
$zomma [(i - 1) \times n + j - 1]$ when $order = Nag_RowMajor$ .

On exit: the

m \times n

array zomma contains the sensitivity,

\frac{\partial^{3} P}{\partial S^{2} \partial σ}

, of gamma to change in the volatility of the underlying asset.

26: vomma[ $m \times n$ ] – doubleOutput

Note: the

(i, j)

th element of the matrix is stored in

$vomma [(j - 1) \times m + i - 1]$ when $order = Nag_ColMajor$ ;
$vomma [(i - 1) \times n + j - 1]$ when $order = Nag_RowMajor$ .

On exit: the

m \times n

array vomma contains the sensitivity,

\frac{\partial^{2} P}{\partial σ^{2}}

, of vega to change in the volatility of the underlying asset.

27: fail – NagError *Input/Output

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

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_CONVERGENCE: Quadrature has not converged to the required accuracy. However, the result should be a reasonable approximation.; Quadrature has not converged to the required accuracy. The values returned cannot be relied upon.
NE_INT: On entry, $m = 〈value〉$ .
Constraint: $m \geq 1$ .; On entry, $n = 〈value〉$ .
Constraint: $n \geq 1$ .
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, $corr = 〈value〉$ .
Constraint: $|corr| \leq 1.0$ .; On entry, $eta = 〈value〉$ .
Constraint: $eta > 0.0$ .; On entry, $grisk = 〈value〉$ , $sigmav = 〈value〉$ and $kappa = 〈value〉$ .
Constraint: $0.0 \leq grisk \leq 1.0$ and $grisk \times (1.0 - grisk) \times {sigmav}^{2} \leq {kappa}^{2}$ .; On entry, $kappa = 〈value〉$ .
Constraint: $kappa > 0.0$ .; On entry, $q = 〈value〉$ .
Constraint: $q \geq 0.0$ .; On entry, $r = 〈value〉$ .
Constraint: $r \geq 0.0$ .; On entry, $s = 〈value〉$ .
Constraint: $s \geq 〈value〉$ and $s \leq 〈value〉$ .; On entry, $sigmav = 〈value〉$ .
Constraint: $sigmav > 0.0$ .; On entry, $var0 = 〈value〉$ .
Constraint: $var0 \geq 0.0$ .
NE_REAL_ARRAY: On entry, $t [〈value〉] = 〈value〉$ .
Constraint: $t [i - 1] \geq 〈value〉$ .; On entry, $x [〈value〉] = 〈value〉$ .
Constraint: $x [i - 1] \geq 〈value〉$ and $x [i - 1] \leq 〈value〉$ .

7 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.

8 Further Comments

None.

9 Example

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

1

year, the stock price is

100

and the strike price is

100

. The risk-free interest rate is

2.5 %

per year, the volatility of the variance,

σ_{v}

, is

57.51 %

per year, the mean reversion parameter,

κ

, is

1.5768

, the long term mean of the variance,

η

, is

0.0398

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

ρ

, is

-0.5711

. The risk aversion parameter,

γ

, is

1.0

and the initial value of the variance, var0, is

0.0175

NAG Library Function Documentnag_heston_greeks (s30nbc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Function Document

nag_heston_greeks (s30nbc)