NAG Library Routine Document

S30BBF

INTEGER	M, N, LDP, IFAIL
REAL (KIND=nag_wp)	SM(M), S, T(N), SIGMA, R, Q, P(LDP,N), DELTA(LDP,N), GAMMA(LDP,N), VEGA(LDP,N), THETA(LDP,N), RHO(LDP,N), CRHO(LDP,N), VANNA(LDP,N), CHARM(LDP,N), SPEED(LDP,N), COLOUR(LDP,N), ZOMMA(LDP,N), VOMMA(LDP,N)
CHARACTER(1)	CALPUT

3 Description

S30BBF computes the price of a floating-strike lookback call or put option, together with the Greeks or sensitivities, which are the partial derivatives of the option price with respect to certain of the other input parameters. A call option of this type confers the right to buy the underlying asset at the lowest price,

S_{\min}

, observed during the lifetime of the contract. A put option gives the holder the right to sell the underlying asset at the maximum price,

S_{\max}

, observed during the lifetime of the contract. Thus, at expiry, the payoff for a call option is

S - S_{\min}

, and for a put,

S_{\max} - S

For a given minimum value the price of a floating-strike lookback call with underlying asset price,

S

, and time to expiry,

T

, is

P_{call} = S e^{- q T} Φ (a_{1}) - S_{\min} e^{- r T} Φ (a_{2}) + S e^{- r T} \frac{σ^{2}}{2 b} [{(\frac{S}{S_{\min}})}^{- 2 b / σ^{2}} Φ ({- a}_{1} + \frac{2 b}{σ} \sqrt{T}) {- e}^{b T} Φ ({- a}_{1})],

where

b = r - q \neq 0

. The volatility,

σ

, risk-free interest rate,

r

, and annualised dividend yield,

q

, are constants.

The corresponding put price is

P_{put} = S_{\max} e^{- r T} Φ ({- a}_{2}) - S e^{- q T} Φ ({- a}_{1}) + S e^{- r T} \frac{σ^{2}}{2 b} [- {(\frac{S}{S_{\max}})}^{- 2 b / σ^{2}} Φ (a_{1} - \frac{2 b}{σ} \sqrt{T}) + e^{b T} Φ (a_{1})] .

In the above,

Φ

denotes the cumulative Normal distribution function,

Φ (x) = \int_{- \infty}^{x} ϕ (y) d y

where

ϕ

denotes the standard Normal probability density function

ϕ (y) = \frac{1}{\sqrt{2 π}} \exp (- y^{2} / 2)

and

\begin{array}{l} a_{1} = \frac{\ln (S / S_{m}) + (b + σ^{2} / 2) T}{σ \sqrt{T}} \\ a_{2} = a_{1} - σ \sqrt{T} \end{array}

where

S_{m}

is taken to be the minimum price attained by the underlying asset,

S_{\min}

, for a call and the maximum price,

S_{\max}

, for a put.

The option price

P_{i j} = P (X = X_{i}, T = T_{j})

is computed for each minimum or maximum observed price in a set

S_{\min} (i)

S_{\max} (i)

i = 1, 2, \dots, m

, and for each expiry time in a set

T_{j}

j = 1, 2, \dots, n

4 References

Goldman B M, Sosin H B and Gatto M A (1979) Path dependent options: buy at the low, sell at the high Journal of Finance 34 1111–1127

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 minimum or maximum 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: $SM (M)$ – REAL (KIND=nag_wp) arrayInput

On entry:

SM (i)

must contain

S_{\min} (i)

, the

i

th minimum observed price of the underlying asset when

CALPUT ='C'

, or

S_{\max} (i)

, the maximum observed price when

CALPUT ='P'

, for

i = 1, 2, \dots, M

Constraints:

$SM (i) \geq z and SM (i) \leq 1 / z$ , where $z = X02AMF ()$ , the safe range parameter, for $i = 1, 2, \dots, M$ ;
if $CALPUT ='C'$ , $SM (i) \leq S$ , for $i = 1, 2, \dots, M$ ;
if $CALPUT ='P'$ , $SM (i) \geq S$ , 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: $SIGMA$ – REAL (KIND=nag_wp)Input

On entry:

σ

, the volatility of the underlying asset. Note that a rate of 15% should be entered as 0.15.

Constraint:

SIGMA > 0.0

8: $R$ – REAL (KIND=nag_wp)Input

On entry: the annual risk-free interest rate,

r

, continuously compounded. Note that a rate of 5% should be entered as 0.05.

Constraint:

R \geq 0.0

and

abs (R - Q) > 10 \times eps \times \max (abs (R), 1)

, where

eps = X02AJF ()

, the machine precision.

9: $Q$ – REAL (KIND=nag_wp)Input

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

Constraint:

Q \geq 0.0

and

abs (R - Q) > 10 \times eps \times \max (abs (R), 1)

, where

eps = X02AJF ()

, the machine precision.

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

On exit:

P (i, j)

contains

P_{i j}

, the option price evaluated for the minimum or maximum observed price

S_{\min} (i)

S_{\max} (i)

at expiry

T_{j}

for

i = 1, 2, \dots, M

and

j = 1, 2, \dots, N

11: $LDP$ – INTEGERInput

On entry: the first dimension of the arrays P, DELTA, GAMMA, VEGA, THETA, RHO, CRHO, VANNA, CHARM, SPEED, COLOUR, ZOMMA and VOMMA as declared in the (sub)program from which S30BBF is called.

Constraint:

LDP \geq M

12: $DELTA (LDP, N)$ – 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.

13: $GAMMA (LDP, N)$ – 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.

14: $VEGA (LDP, N)$ – 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

15: $THETA (LDP, N)$ – 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

16: $RHO (LDP, N)$ – 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

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

On exit:

CRHO (i, j)

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

P_{i j}

to change in the annual cost of carry rate, i.e.,

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

, for

i = 1, 2, \dots, M

and

j = 1, 2, \dots, N

, where

b = r - q

18: $VANNA (LDP, N)$ – 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

19: $CHARM (LDP, N)$ – 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

20: $SPEED (LDP, N)$ – 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

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

On exit:

COLOUR (i, j)

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

Γ_{i j}

to change in the time, i.e.,

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

, for

i = 1, 2, \dots, M

and

j = 1, 2, \dots, N

22: $ZOMMA (LDP, N)$ – 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

23: $VOMMA (LDP, N)$ – 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

24: $IFAIL$ – 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, $SM (〈value〉) = 〈value〉$ .
Constraint: $〈value〉 \leq SM (i) \leq 〈value〉$ for all $i$ .

On entry with a call option, $SM (〈value〉) = 〈value〉$ .
Constraint: for call options, $SM (i) \leq 〈value〉$ for all $i$ .

On entry with a put option, $SM (〈value〉) = 〈value〉$ .
Constraint: for put options, $SM (i) \geq 〈value〉$ for all $i$ .

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

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

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

$IFAIL = 8$: On entry, $R = 〈value〉$ .
Constraint: $R \geq 0.0$ .

$IFAIL = 9$: On entry, $Q = 〈value〉$ .
Constraint: $Q \geq 0.0$ .

$IFAIL = 11$: On entry, $LDP = 〈value〉$ and $M = 〈value〉$ .
Constraint: $LDP \geq M$ .

$IFAIL = 12$: On entry, $R = 〈value〉$ and $Q = 〈value〉$ .
Constraint: $|R - Q| > 10 \times eps \times \max (|R|, 1)$ , where $eps$ is the machine precision.

$IFAIL = - 99$: 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 dependent on the accuracy of the cumulative Normal distribution function,

Φ

. This is evaluated using a rational Chebyshev expansion, chosen so that the maximum relative error in the expansion is of the order of the machine precision (see S15ABF and S15ADF). An accuracy close to machine precision can generally be expected.

8 Parallelism and Performance

S30BBF 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 of a floating-strike lookback put with a time to expiry of

6

months and a stock price of

87

. The maximum price observed so far is

100

. The risk-free interest rate is

6 %

per year and the volatility is

30 %

per year with an annual dividend return of

4 %

NAG Library Routine DocumentS30BBF

▸▿ 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

S30BBF