NAG Library Routine Document

S30NBF

INTEGER	M, N, LDP, IFAIL
REAL (KIND=nag_wp)	X(M), S, T(N), SIGMAV, KAPPA, CORR, VAR0, ETA, GRISK, R, Q, P(LDP,N), DELTA(LDP,N), GAMMA(LDP,N), VEGA(LDP,N), THETA(LDP,N), RHO(LDP,N), VANNA(LDP,N), CHARM(LDP,N), SPEED(LDP,N), ZOMMA(LDP,N), VOMMA(LDP,N)
CHARACTER(1)	CALPUT

3 Description

S30NBF 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] .

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

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 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

5 Parameters

1: $CALPUT$ – CHARACTER(1)Input

On entry: 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: $M$ – INTEGERInput

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

Constraint:

M \geq 1

3: $N$ – INTEGERInput

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

Constraint:

N \geq 1

4: $X (M)$ – REAL (KIND=nag_wp) arrayInput

On entry:

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 = X02AMF ()

, the safe range parameter, for

i = 1, 2, \dots, M

5: $S$ – REAL (KIND=nag_wp)Input

On entry:

S

, the price of the underlying asset.

Constraint:

S \geq z ​ and ​ S \leq 1.0 / z

, where

z = X02AMF ()

, the safe range parameter.

6: $T (N)$ – REAL (KIND=nag_wp) arrayInput

On entry:

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 = X02AMF ()

, the safe range parameter, for

i = 1, 2, \dots, N

7: $SIGMAV$ – REAL (KIND=nag_wp)Input

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

8: $KAPPA$ – REAL (KIND=nag_wp)Input

On entry:

κ

, the long term mean reversion rate of the volatility.

Constraint:

KAPPA > 0.0

9: $CORR$ – REAL (KIND=nag_wp)Input

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

10: $VAR0$ – REAL (KIND=nag_wp)Input

On entry: the initial value of the variance,

v_{t}

, of the asset price.

Constraint:

VAR0 \geq 0.0

11: $ETA$ – REAL (KIND=nag_wp)Input

On entry:

η

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

Constraint:

ETA > 0.0

12: $GRISK$ – REAL (KIND=nag_wp)Input

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

13: $R$ – REAL (KIND=nag_wp)Input

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

14: $Q$ – REAL (KIND=nag_wp)Input

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

15: $P (LDP, N)$ – REAL (KIND=nag_wp) arrayOutput

On exit:

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

16: $LDP$ – INTEGERInput

On entry: the first dimension of the arrays P, DELTA, GAMMA, VEGA, THETA, RHO, VANNA, CHARM, SPEED, ZOMMA and VOMMA as declared in the (sub)program from which S30NBF is called.

Constraint:

LDP \geq M

17: $DELTA (LDP, N)$ – REAL (KIND=nag_wp) arrayOutput

On exit: the leading

M \times N

part of the array DELTA contains the sensitivity,

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

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

18: $GAMMA (LDP, N)$ – REAL (KIND=nag_wp) arrayOutput

On exit: the leading

M \times N

part of the array GAMMA contains the sensitivity,

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

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

19: $VEGA (LDP, N)$ – REAL (KIND=nag_wp) arrayOutput

On exit:

VEGA (i, j)

, contains the first-order Greek measuring the sensitivity of the option price

P_{i j}

to change in the volatility of the underlying asset, i.e.,

\frac{\partial P_{i j}}{\partial σ}

, for

i = 1, 2, \dots, M

and

j = 1, 2, \dots, N

20: $THETA (LDP, N)$ – REAL (KIND=nag_wp) arrayOutput

On exit:

THETA (i, j)

, contains the first-order Greek measuring the sensitivity of the option price

P_{i j}

to change in time, i.e.,

- \frac{\partial P_{i j}}{\partial T}

, for

i = 1, 2, \dots, M

and

j = 1, 2, \dots, N

, where

b = r - q

21: $RHO (LDP, N)$ – REAL (KIND=nag_wp) arrayOutput

On exit:

RHO (i, j)

, contains the first-order Greek measuring the sensitivity of the option price

P_{i j}

to change in the annual risk-free interest rate, i.e.,

- \frac{\partial P_{i j}}{\partial r}

, for

i = 1, 2, \dots, M

and

j = 1, 2, \dots, N

22: $VANNA (LDP, N)$ – REAL (KIND=nag_wp) arrayOutput

On exit:

VANNA (i, j)

, contains the second-order Greek measuring the sensitivity of the first-order Greek

Δ_{i j}

to change in the volatility of the asset price, i.e.,

- \frac{\partial Δ_{i j}}{\partial T} = - \frac{\partial^{2} P_{i j}}{\partial S \partial σ}

, for

i = 1, 2, \dots, M

and

j = 1, 2, \dots, N

23: $CHARM (LDP, N)$ – REAL (KIND=nag_wp) arrayOutput

On exit:

CHARM (i, j)

, contains the second-order Greek measuring the sensitivity of the first-order Greek

Δ_{i j}

to change in the time, i.e.,

- \frac{\partial Δ_{i j}}{\partial T} = - \frac{\partial^{2} P_{i j}}{\partial S \partial T}

, for

i = 1, 2, \dots, M

and

j = 1, 2, \dots, N

24: $SPEED (LDP, N)$ – REAL (KIND=nag_wp) arrayOutput

On exit:

SPEED (i, j)

, contains the third-order Greek measuring the sensitivity of the second-order Greek

Γ_{i j}

to change in the price of the underlying asset, i.e.,

- \frac{\partial Γ_{i j}}{\partial S} = - \frac{\partial^{3} P_{i j}}{\partial S^{3}}

, for

i = 1, 2, \dots, M

and

j = 1, 2, \dots, N

25: $ZOMMA (LDP, N)$ – REAL (KIND=nag_wp) arrayOutput

On exit:

ZOMMA (i, j)

, contains the third-order Greek measuring the sensitivity of the second-order Greek

Γ_{i j}

to change in the volatility of the underlying asset, i.e.,

- \frac{\partial Γ_{i j}}{\partial σ} = - \frac{\partial^{3} P_{i j}}{\partial S^{2} \partial σ}

, for

i = 1, 2, \dots, M

and

j = 1, 2, \dots, N

26: $VOMMA (LDP, N)$ – REAL (KIND=nag_wp) arrayOutput

On exit:

VOMMA (i, j)

, contains the second-order Greek measuring the sensitivity of the first-order Greek

Δ_{i j}

to change in the volatility of the underlying asset, i.e.,

- \frac{\partial Δ_{i j}}{\partial σ} = - \frac{\partial^{2} P_{i j}}{\partial σ^{2}}

, for

i = 1, 2, \dots, M

and

j = 1, 2, \dots, N

27: $IFAIL$ – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$: On entry, $CALPUT = 〈value〉$ was an illegal value.

$IFAIL = 2$: On entry, $M = 〈value〉$ .
Constraint: $M \geq 1$ .

$IFAIL = 3$: On entry, $N = 〈value〉$ .
Constraint: $N \geq 1$ .

$IFAIL = 4$: On entry, $X (〈value〉) = 〈value〉$ .
Constraint: $X (i) \geq 〈value〉$ and $X (i) \leq 〈value〉$ .

$IFAIL = 5$: On entry, $S = 〈value〉$ .
Constraint: $S \geq 〈value〉$ and $S \leq 〈value〉$ .

$IFAIL = 6$: On entry, $T (〈value〉) = 〈value〉$ .
Constraint: $T (i) \geq 〈value〉$ .

$IFAIL = 7$: On entry, $SIGMAV = 〈value〉$ .
Constraint: $SIGMAV > 0.0$ .

$IFAIL = 8$: On entry, $KAPPA = 〈value〉$ .
Constraint: $KAPPA > 0.0$ .

$IFAIL = 9$: On entry, $CORR = 〈value〉$ .
Constraint: $|CORR| \leq 1.0$ .

$IFAIL = 10$: On entry, $VAR0 = 〈value〉$ .
Constraint: $VAR0 \geq 0.0$ .

$IFAIL = 11$: On entry, $ETA = 〈value〉$ .
Constraint: $ETA > 0.0$ .

$IFAIL = 12$: On entry, $GRISK = 〈value〉$ , $SIGMAV = 〈value〉$ and $KAPPA = 〈value〉$ .
Constraint: $0.0 \leq GRISK \leq 1.0$ and $GRISK \times (1.0 - GRISK) \times {SIGMAV}^{2} \leq {KAPPA}^{2}$ .

$IFAIL = 13$: On entry, $R = 〈value〉$ .
Constraint: $R \geq 0.0$ .

$IFAIL = 14$: On entry, $Q = 〈value〉$ .
Constraint: $Q \geq 0.0$ .

$IFAIL = 16$: On entry, $LDP = 〈value〉$ and $M = 〈value〉$ .
Constraint: $LDP \geq M$ .

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

$IFAIL = 18$: Solution cannot be computed accurately. Check values of input parameters.

$IFAIL = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.8 in the Essential Introduction for further information.

$IFAIL = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.7 in the Essential Introduction for further information.

$IFAIL = - 999$: Dynamic memory allocation failed.
See Section 3.6 in the Essential Introduction for further information.

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 Parallelism and Performance

S30NBF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

None.

10 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 Routine DocumentS30NBF

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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 Routine Document

S30NBF