NAG Library Function Document

nag_lookback_fls_price (s30bac)

void	nag_lookback_fls_price (Nag_OrderType order, Nag_CallPut option, Integer m, Integer n, const double sm[], double s, const double t[], double sigma, double r, double q, double p[], NagError *fail)

3 Description

nag_lookback_fls_price (s30bac) computes the price of a floating-strike lookback call or put option. 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. When

r = q

, the option price is given by

P_{call} = S e^{- q T} Φ (a_{1}) - S_{\min} e^{- r T} Φ (a_{2}) + S e^{- r T} σ \sqrt{T} [ϕ (a_{1}) + a_{1} (Φ (a_{1}) - 1)] .

The corresponding put price is (for

b \neq 0

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})] .

When

r = q

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

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 minimum or maximum 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: sm[m] – const doubleInput

On entry:

sm [i - 1]

must contain

S_{\min} (i)

, the

i

th minimum observed price of the underlying asset when

option = Nag_Call

, or

S_{\max} (i)

, the maximum observed price when

option = Nag_Put

, for

i = 1, 2, \dots, m

Constraints:

$sm [i - 1] \geq z and sm [i - 1] \leq 1 / z$ , where $z = nag_real_safe_small_number$ , the safe range parameter, for $i = 1, 2, \dots, m$ ;
if $option = Nag_Call$ , $sm [i - 1] \leq S$ , for $i = 1, 2, \dots, m$ ;
if $option = Nag_Put$ , $sm [i - 1] \geq S$ , 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: sigma – doubleInput

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

9: 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

10: 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

11: p[ $m \times n$ ] – doubleOutput

Note: where

P (i, j)

appears in this document, it refers to the array element

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

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

12: 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_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, $q = ⟨value⟩$ .
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, $sigma = ⟨value⟩$ .
Constraint: $sigma > 0.0$ .
NE_REAL_ARRAY: On entry, $sm [⟨value⟩] = ⟨value⟩$ .
Constraint: $⟨value⟩ \leq sm [i] \leq ⟨value⟩$ for all $i$ .
On entry, $t [⟨value⟩] = ⟨value⟩$ .
Constraint: $t [i] \geq ⟨value⟩$ 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 $i$ .

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 nag_cumul_normal (s15abc) and nag_erfc (s15adc)). An accuracy close to machine precision can generally be expected.

8 Parallelism and Performance

nag_lookback_fls_price (s30bac) 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 floating-strike lookback call with a time to expiry of

6

months and a stock price of

120

. The minimum price observed so far is

100

. The risk-free interest rate is

10 %

per year and the volatility is

30 %

per year with an annual dividend return of

6 %

NAG Library Function Documentnag_lookback_fls_price (s30bac)

+− 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_lookback_fls_price (s30bac)