nag_specfun_opt_lookback_fls_price (s30ba) 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

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

Parameters

Compulsory Input Parameters

1: $calput$ – string (length ≥ 1)

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: $sm (m)$ – double array

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 $z = x02am ()$ , 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 $i = 1, 2, \dots, m$ .

3: $s$ – double scalar

S

, the price of the underlying asset.

Constraint:

s \geq z ​ and ​ s \leq 1.0 / z

, where

z = x02am ()

, the safe range parameter.

4: $t (n)$ – double array

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 = x02am ()

, the safe range parameter, for

i = 1, 2, \dots, n

5: $sigma$ – double scalar

σ

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

Constraint:

sigma > 0.0

6: $r$ – double scalar

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

7: $q$ – double scalar

q

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

Constraint:

q \geq 0.0

Optional Input Parameters

1: $m$ – int64int32nag_int scalar: Default: the dimension of the array sm.
The number of minimum or maximum prices to be used.

Constraint: $m \geq 1$ .
2: $n$ – int64int32nag_int scalar: Default: the dimension of the array t.
The number of times to expiry to be used.

Constraint: $n \geq 1$ .

Output Parameters

1: $p (ldp, n)$ – double array: $ldp = m$ .
$p (i, j)$ contains $P_{i j}$ , the option price evaluated for the minimum or maximum observed price $S_{\min} (i)$ or $S_{\max} (i)$ at expiry $t_{j}$ for $i = 1, 2, \dots, m$ and $j = 1, 2, \dots, n$ .
2: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$: On entry, $calput =_$ was an illegal value.

$ifail = 2$: Constraint: $m \geq 1$ .

$ifail = 3$: Constraint: $n \geq 1$ .

$ifail = 4$: Constraint: $_\leq sm (i) \leq_$ for all $i$ .

$ifail = 5$: Constraint: $s \geq_$ and $s \leq_$ .

$ifail = 6$: Constraint: $t (i) \geq_$ for all $i$ .

$ifail = 7$: Constraint: $sigma > 0.0$ .

$ifail = 8$: Constraint: $r \geq 0.0$ .

$ifail = 9$: Constraint: $q \geq 0.0$ .

$ifail = 11$: Constraint: $ldp \geq m$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

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_specfun_cdf_normal (s15ab) and nag_specfun_erfc_real (s15ad)). An accuracy close to machine precision can generally be expected.

Further Comments

None.

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 %

Open in the MATLAB editor: s30ba_example

function s30ba_example


fprintf('s30ba example results\n\n');

put = 'c';
s = 120;
sigma = 0.3;
r = 0.1;
q = 0.06;
sm = [100.0];
t = [0.5];

[p, ifail] = s30ba( ...
                    put, sm , s, t, sigma, r, q);

fprintf('\nFloating-strike Lookback\n European Call :\n');
fprintf('  Spot       =   %9.4f\n', s);
fprintf('  Volatility =   %9.4f\n', sigma);
fprintf('  Rate       =   %9.4f\n', r);
fprintf('  Dividend   =   %9.4f\n\n', q);

fprintf('   Strike    Expiry   Option Price\n');
for i=1:1
  for j=1:1
    fprintf('%9.4f %9.4f %9.4f\n', sm(i), t(j), p(i,j));
  end
end

s30ba example results


Floating-strike Lookback
 European Call :
  Spot       =    120.0000
  Volatility =      0.3000
  Rate       =      0.1000
  Dividend   =      0.0600

   Strike    Expiry   Option Price
 100.0000    0.5000   25.3534

PDF version (NAG web site, 64-bit version, 64-bit version)

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Chapter Introduction

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