NAG Library Function Document

nag_pde_bs_1d_analytic (d03ndc)

void	nag_pde_bs_1d_analytic (Nag_OptionType kopt, double x, double s, double t, double tmat, const Nag_Boolean tdpar[], const double r[], const double q[], const double sigma[], double f, double theta, double delta, double gamma, double lambda, double rho, NagError *fail)

3 Description

nag_pde_bs_1d_analytic (d03ndc) computes an analytic solution to the Black–Scholes equation (see Hull (1989) and Wilmott et al. (1995))

\frac{\partial f}{\partial t} + (r - q) S \frac{\partial f}{\partial S} + \frac{σ^{2} S^{2}}{2} \frac{\partial^{2} f}{\partial S^{2}} = r f

(1)

S_{\min} < S < S_{\max}, t_{\min} < t < t_{\max},

(2)

for the value

f

of a European put or call option, or an American call option with zero dividend

q

. In equation (1)

t

is time,

S

is the stock price,

X

is the exercise price,

r

is the risk free interest rate,

q

is the continuous dividend, and

σ

is the stock volatility. The arguments

r

q

and

σ

may be either constant, or functions of time. In the latter case their average instantaneous values over the remaining life of the option should be provided to nag_pde_bs_1d_analytic (d03ndc). An auxiliary function nag_pde_bs_1d_means (d03nec) is available to compute such averages from values at a set of discrete times.

nag_pde_bs_1d_analytic (d03ndc) also returns values of the Greeks

Θ = \frac{\partial f}{\partial t}, Δ = \frac{\partial f}{\partial x}, Γ = \frac{\partial^{2} f}{\partial x^{2}}, Λ = \frac{\partial f}{\partial σ}, ρ = \frac{\partial f}{\partial r} .

nag_bsm_greeks (s30abc) also computes the European option price given by the Black–Scholes–Merton formula together with a more comprehensive set of sensitivities (Greeks).

Further details of the analytic solution returned are given in Section 8.1.

4 References

Hull J (1989) Options, Futures and Other Derivative Securities Prentice–Hall

Wilmott P, Howison S and Dewynne J (1995) The Mathematics of Financial Derivatives Cambridge University Press

5 Arguments

1: kopt – Nag_OptionTypeInput

On entry: specifies the kind of option to be valued:

$kopt = Nag_EuropeanCall$: A European call option.
$kopt = Nag_AmericanCall$: An American call option.
$kopt = Nag_EuropeanPut$: A European put option.

Constraints:

$kopt = Nag_EuropeanCall$ , $Nag_AmericanCall$ or $Nag_EuropeanPut$ ;
if $q \neq 0$ , $kopt \neq Nag_AmericanCall$ .

2: x – doubleInput

On entry: the exercise price

X

Constraint:

x \geq 0.0

3: s – doubleInput

On entry: the stock price at which the option value and the Greeks should be evaluated.

Constraint:

s \geq 0.0

4: t – doubleInput

On entry: the time at which the option value and the Greeks should be evaluated.

Constraint:

t \geq 0.0

5: tmat – doubleInput

On entry: the maturity time of the option.

Constraint:

tmat \geq t

6: tdpar[ $3$ ] – const Nag_BooleanInput

On entry: specifies whether or not various arguments are time-dependent. More precisely,

r

is time-dependent if

tdpar [0] = Nag_TRUE

and constant otherwise. Similarly,

tdpar [1]

specifies whether

q

is time-dependent and

tdpar [2]

specifies whether

σ

is time-dependent.

7: r[ $\dim$ ] – const doubleInput

Note: the dimension, dim, of the array r must be at least

$3$ when $tdpar [0] = Nag_TRUE$ ;
$1$ otherwise.

On entry: if

tdpar [0] = Nag_FALSE

then

r [0]

must contain the constant value of

r

. The remaining elements need not be set.

tdpar [0] = Nag_TRUE

then

r [0]

must contain the value of

r

at time t and

r [1]

must contain its average instantaneous value over the remaining life of the option:

\hat{r} = \int_{t}^{tmat} r (ζ) d ζ .

The auxiliary function nag_pde_bs_1d_means (d03nec) may be used to construct r from a set of values of

r

at discrete times.

8: q[ $\dim$ ] – const doubleInput

Note: the dimension, dim, of the array q must be at least

$3$ when $tdpar [1] = Nag_TRUE$ ;
$1$ otherwise.

On entry: if

tdpar [1] = Nag_FALSE

then

q [0]

must contain the constant value of

q

. The remaining elements need not be set.

tdpar [1] = Nag_TRUE

then

q [0]

must contain the constant value of

q

and

q [1]

must contain its average instantaneous value over the remaining life of the option:

\hat{q} = \int_{t}^{tmat} q (ζ) d ζ .

The auxiliary function nag_pde_bs_1d_means (d03nec) may be used to construct q from a set of values of

q

at discrete times.

9: sigma[ $\dim$ ] – const doubleInput

Note: the dimension, dim, of the array sigma must be at least

$3$ when $tdpar [2] = Nag_TRUE$ ;
$1$ otherwise.

On entry: if

tdpar [2] = Nag_FALSE

then

sigma [0]

must contain the constant value of

σ

. The remaining elements need not be set.

tdpar [2] = Nag_TRUE

then

sigma [0]

must contain the value of

σ

at time t,

sigma [1]

the average instantaneous value

\hat{σ}

, and

sigma [2]

the second-order average

\bar{σ}

, where:

\hat{σ} = \int_{t}^{tmat} σ (ζ) d ζ,

\bar{σ} = {(\int_{t}^{tmat} σ^{2} (ζ) d ζ)}^{1 / 2} .

The auxiliary function nag_pde_bs_1d_means (d03nec) may be used to compute sigma from a set of values at discrete times.

Constraints:

if $tdpar [2] = Nag_FALSE$ , $sigma [0] > 0.0$ ;
if $tdpar [2] = Nag_TRUE$ , $sigma [i - 1] > 0.0$ , for $i = 1, 2, 3$ .

10: f – double *Output

On exit: the value

f

of the option at the stock price s and time t.

11: theta – double *Output

12: delta – double *Output

13: gamma – double *Output

14: lambda – double *Output

15: rho – double *Output

On exit: the values of various Greeks at the stock price s and time t, as follows:

\begin{array}{l} theta = Θ = \frac{\partial f}{\partial t}, & delta = Δ = \frac{\partial f}{\partial s}, & gamma = Γ = \frac{\partial^{2} f}{\partial s^{2}}, \\ lambda = Λ = \frac{\partial f}{\partial σ}, & rho = ρ = \frac{\partial f}{\partial r} . \end{array}

16: 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_INCOMPAT_PARAM: On entry, $q [0]$ is not equal to $0.0$ with American call option. $q [0] = 〈value〉$ .; On entry, $sigma [i - 1] = 〈value〉$ and $i = 〈value〉$ .
Constraint: $sigma [i - 1] > 0.0$ .
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, $s = 〈value〉$ .
Constraint: $s \geq 0.0$ .; On entry, $t = 〈value〉$ .
Constraint: $t \geq 0.0$ .; On entry, $x = 〈value〉$ .
Constraint: $x \geq 0.0$ .
NE_REAL_2: On entry, $tmat = 〈value〉$ and $t = 〈value〉$ .
Constraint: $tmat \geq t$ .

7 Accuracy

Given accurate values of r, q and sigma no further approximations are made in the evaluation of the Black–Scholes analytic formulae, and the results should therefore be within machine accuracy. The values of r, q and sigma returned from nag_pde_bs_1d_means (d03nec) are exact for polynomials of degree up to

3

8 Further Comments

8.1 Algorithmic Details

The Black–Scholes analytic formulae are used to compute the solution. For a European call option these are as follows:

f = S e^{- \hat{q} (T - t)} N (d_{1}) - X e^{- \hat{r} (T - t)} N (d_{2})

where

\begin{array}{lcl} d_{1} & = & \frac{\log (S / X) + (\hat{r} - \hat{q} + {\bar{σ}}^{2} / 2) (T - t)}{\bar{σ} \sqrt{T - t}}, \\ d_{2} & = & \frac{\log (S / X) + (\hat{r} - \hat{q} - {\bar{σ}}^{2} / 2) (T - t)}{\bar{σ} \sqrt{T - t}} = d_{1} - \bar{σ} \sqrt{T - t}, \end{array}

N (x)

is the cumulative Normal distribution function and

N^{'} (x)

is its derivative

\begin{array}{lcl} N (x) & = & \frac{1}{\sqrt{2 π}} \int_{- \infty}^{x} e^{- ζ^{2} / 2} d ζ, \\ N^{'} (x) & = & \frac{1}{\sqrt{2 π}} e^{- x^{2} / 2} . \end{array}

The functions

\hat{q}

\hat{r}

\hat{σ}

and

\bar{σ}

are average values of

q

r

and

σ

over the time to maturity:

\begin{array}{lcl} \hat{q} & = & \frac{1}{T - t} \int_{t}^{T} q (ζ) d ζ, \\ \hat{r} & = & \frac{1}{T - t} \int_{t}^{T} r (ζ) d ζ, \\ \hat{σ} & = & \frac{1}{T - t} \int_{t}^{T} σ (ζ) d ζ, \\ \bar{σ} & = & {(\frac{1}{T - t} \int_{t}^{T} σ^{2} (ζ) d ζ)}^{1 / 2} . \end{array}

The Greeks are then calculated as follows:

\begin{array}{lcl} Δ & = & \frac{\partial f}{\partial S} = e^{- \hat{q} (T - t)} N (d_{1}) + \frac{S e^{- \hat{q} (T - t)} N^{'} (d_{1}) - X e^{- \hat{r} (T - t)} N^{'} (d_{2})}{\bar{σ} S \sqrt{T - t}}, \\ Γ & = & \frac{\partial^{2} f}{\partial S^{2}} = \frac{S e^{- \hat{q} (T - t)} N^{'} (d_{1}) + X e^{- \hat{r} (T - t)} N^{'} (d_{2})}{\bar{σ} S^{2} \sqrt{T - t}} + \frac{S e^{- \hat{q} (T - t)} N^{'} (d_{1}) - X e^{- \hat{r} (T - t)} N^{'} (d_{2})}{{\bar{σ}}^{2} S^{2} (T - t)}, \\ Θ & = & \frac{\partial f}{\partial t} = r f + (q - r) S Δ - \frac{σ^{2} S^{2}}{2} Γ, \\ Λ & = & \frac{\partial f}{\partial σ} = (\frac{X d_{1} e^{- \hat{r} (T - t)} N^{'} (d_{2}) - S d_{2} e^{- \hat{q} (T - t)} N^{'} (d_{1})}{{\bar{σ}}^{2}}) \hat{σ}, \\ ρ & = & \frac{\partial f}{\partial r} = X (T - t) e^{- \hat{r} (T - t)} N (d_{2}) + \frac{(S e^{- \hat{q} (T - t)} N^{'} (d_{1}) - X e^{- \hat{r} (T - t)} N^{'} (d_{2})) \sqrt{T - t}}{\bar{σ}} . \end{array}

Note: that

Θ

is obtained from substitution of other Greeks in the Black–Scholes partial differential equation, rather than differentiation of

f

. The values of

q

r

and

σ

appearing in its definition are the instantaneous values, not the averages. Note also that both the first-order average

\hat{σ}

and the second-order average

\bar{σ}

appear in the expression for

Λ

. This results from the fact that

Λ

is the derivative of

f

with respect to

σ

, not

\hat{σ}

For a European put option the equivalent equations are:

\begin{array}{lcl} f & = & X e^{- \hat{r} (T - t)} N (- d_{2}) - S e^{- \hat{q} (T - t)} N (- d_{1}), \\ Δ & = & \frac{\partial f}{\partial S} = - e^{- \hat{q} (T - t)} N (- d_{1}) + \frac{S e^{- \hat{q} (T - t)} N^{'} (- d_{1}) - X e^{- \hat{r} (T - t)} N^{'} (- d_{2})}{\bar{σ} S \sqrt{T - t}}, \\ Γ & = & \frac{\partial^{2} f}{\partial S^{2}} = \frac{X e^{- \hat{r} (T - t)} N^{'} (- d_{2}) + S e^{- \hat{q} (T - t)} N^{'} (- d_{1})}{\bar{σ} S^{2} \sqrt{T - t}} + \frac{X e^{- \hat{r} (T - t)} N^{''} (- d_{2}) - S e^{- \hat{q} (T - t)} N^{''} (- d_{1})}{{\bar{σ}}^{2} S^{2} (T - t)}, \\ Θ & = & \frac{\partial f}{\partial t} = r f + (q - r) S Δ - \frac{σ^{2} S^{2}}{2} Γ, \\ Λ & = & \frac{\partial f}{\partial σ} = (\frac{X d_{1} e^{- \hat{r} (T - t)} N^{'} (- d_{2}) - S d_{2} e^{- \hat{q} (T - t)} N^{'} (- d_{1})}{{\bar{σ}}^{2}}) \hat{σ}, \\ ρ & = & \frac{\partial f}{\partial r} = - X (T - t) e^{- \hat{r} (T - t)} N (- d_{2}) + \frac{(S e^{- \hat{q} (T - t)} N^{'} (- d_{1}) - X e^{- \hat{r} (T - t)} N^{'} (- d_{2})) \sqrt{T - t}}{\hat{σ}} . \end{array}

The analytic solution for an American call option with

q = 0

is identical to that for a European call, since early exercise is never optimal in this case. For all other cases no analytic solution is known.

9 Example

This example solves the Black–Scholes equation for valuation of a

5

-month American call option on a non-dividend-paying stock with an exercise price of $50. The risk-free interest rate is 10% per annum, and the stock volatility is 40% per annum.

The option is valued at a range of times and stock prices.

NAG Library Function Documentnag_pde_bs_1d_analytic (d03ndc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

8.1 Algorithmic Details

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Function Document

nag_pde_bs_1d_analytic (d03ndc)