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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_specfun_opt_lookback_fls_price (s30ba)

## Purpose

nag_specfun_opt_lookback_fls_price (s30ba) computes the price of a floating-strike lookback option.

## Syntax

[p, ifail] = s30ba(calput, sm, s, t, sigma, r, q, 'm', m, 'n', n)
[p, ifail] = nag_specfun_opt_lookback_fls_price(calput, sm, s, t, sigma, r, q, 'm', m, 'n', n)

## Description

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, ${S}_{\mathrm{min}}$, observed during the lifetime of the contract. A put option gives the holder the right to sell the underlying asset at the maximum price, ${S}_{\mathrm{max}}$, observed during the lifetime of the contract. Thus, at expiry, the payoff for a call option is $S-{S}_{\mathrm{min}}$, and for a put, ${S}_{\mathrm{max}}-S$.
For a given minimum value the price of a floating-strike lookback call with underlying asset price, $S$, and time to expiry, $T$, is
where $b=r-q\ne 0$. The volatility, $\sigma$, risk-free interest rate, $r$, and annualised dividend yield, $q$, are constants. When $r=q$, the option price is given by
 $Pcall = S e-qT Φ a1 - Smin e-rT Φ a2 + S e-rT σ⁢T ϕ a1 + a1 Φ a1 -1 .$
The corresponding put price is (for $b\ne 0$),
When $r=q$,
 $Pput = Smax e-rT Φ -a2 - S e-qT Φ -a1 + S e-rT σ⁢T ϕa1 + a1 Φa1 .$
In the above, $\Phi$ denotes the cumulative Normal distribution function,
 $Φx = ∫ -∞ x ϕy dy$
where $\varphi$ denotes the standard Normal probability density function
 $ϕy = 12π exp -y2/2$
and
 $a1 = ln S / Sm + b + σ2 / 2 T σ⁢T a2=a1-σ⁢T$
where ${S}_{m}$ is taken to be the minimum price attained by the underlying asset, ${S}_{\mathrm{min}}$, for a call and the maximum price, ${S}_{\mathrm{max}}$, for a put.
The option price ${P}_{ij}=P\left(X={X}_{i},T={T}_{j}\right)$ is computed for each minimum or maximum observed price in a set ${S}_{\mathrm{min}}\left(\mathit{i}\right)$ or ${S}_{\mathrm{max}}\left(\mathit{i}\right)$, $i=1,2,\dots ,m$, 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:     $\mathrm{calput}$ – string (length ≥ 1)
Determines whether the option is a call or a put.
${\mathbf{calput}}=\text{'C'}$
A call; the holder has a right to buy.
${\mathbf{calput}}=\text{'P'}$
A put; the holder has a right to sell.
Constraint: ${\mathbf{calput}}=\text{'C'}$ or $\text{'P'}$.
2:     $\mathrm{sm}\left({\mathbf{m}}\right)$ – double array
${\mathbf{sm}}\left(i\right)$ must contain ${S}_{\mathrm{min}}\left(\mathit{i}\right)$, the $\mathit{i}$th minimum observed price of the underlying asset when ${\mathbf{calput}}=\text{'C'}$, or ${S}_{\mathrm{max}}\left(\mathit{i}\right)$, the maximum observed price when ${\mathbf{calput}}=\text{'P'}$, for $\mathit{i}=1,2,\dots ,{\mathbf{m}}$.
Constraints:
• ${\mathbf{sm}}\left(\mathit{i}\right)\ge z\text{​ and ​}{\mathbf{sm}}\left(\mathit{i}\right)\le 1/z$, where $z={\mathbf{x02am}}\left(\right)$, the safe range parameter, for $\mathit{i}=1,2,\dots ,{\mathbf{m}}$;
• if ${\mathbf{calput}}=\text{'C'}$, ${\mathbf{sm}}\left(\mathit{i}\right)\le S$, for $\mathit{i}=1,2,\dots ,{\mathbf{m}}$;
• if ${\mathbf{calput}}=\text{'P'}$, ${\mathbf{sm}}\left(\mathit{i}\right)\ge S$, for $\mathit{i}=1,2,\dots ,{\mathbf{m}}$.
3:     $\mathrm{s}$ – double scalar
$S$, the price of the underlying asset.
Constraint: ${\mathbf{s}}\ge z\text{​ and ​}{\mathbf{s}}\le 1.0/z$, where $z={\mathbf{x02am}}\left(\right)$, the safe range parameter.
4:     $\mathrm{t}\left({\mathbf{n}}\right)$ – double array
${\mathbf{t}}\left(i\right)$ must contain ${T}_{\mathit{i}}$, the $\mathit{i}$th time, in years, to expiry, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
Constraint: ${\mathbf{t}}\left(\mathit{i}\right)\ge z$, where $z={\mathbf{x02am}}\left(\right)$, the safe range parameter, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
5:     $\mathrm{sigma}$ – double scalar
$\sigma$, the volatility of the underlying asset. Note that a rate of 15% should be entered as 0.15.
Constraint: ${\mathbf{sigma}}>0.0$.
6:     $\mathrm{r}$ – double scalar
$r$, the annual risk-free interest rate, continuously compounded. Note that a rate of 5% should be entered as 0.05.
Constraint: ${\mathbf{r}}\ge 0.0$.
7:     $\mathrm{q}$ – double scalar
$q$, the annual continuous yield rate. Note that a rate of 8% should be entered as 0.08.
Constraint: ${\mathbf{q}}\ge 0.0$.

### Optional Input Parameters

1:     $\mathrm{m}$int64int32nag_int scalar
Default: the dimension of the array sm.
The number of minimum or maximum prices to be used.
Constraint: ${\mathbf{m}}\ge 1$.
2:     $\mathrm{n}$int64int32nag_int scalar
Default: the dimension of the array t.
The number of times to expiry to be used.
Constraint: ${\mathbf{n}}\ge 1$.

### Output Parameters

1:     $\mathrm{p}\left(\mathit{ldp},{\mathbf{n}}\right)$ – double array
$\mathit{ldp}={\mathbf{m}}$.
${\mathbf{p}}\left(i,j\right)$ contains ${P}_{ij}$, the option price evaluated for the minimum or maximum observed price ${S}_{\mathrm{min}}\left(\mathit{i}\right)$ or ${S}_{\mathrm{max}}\left(\mathit{i}\right)$ at expiry ${{\mathbf{t}}}_{j}$ for $i=1,2,\dots ,{\mathbf{m}}$ and $j=1,2,\dots ,{\mathbf{n}}$.
2:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Errors or warnings detected by the function:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{calput}}=_$ was an illegal value.
${\mathbf{ifail}}=2$
Constraint: ${\mathbf{m}}\ge 1$.
${\mathbf{ifail}}=3$
Constraint: ${\mathbf{n}}\ge 1$.
${\mathbf{ifail}}=4$
Constraint: $_\le {\mathbf{sm}}\left(i\right)\le _$ for all $i$.
${\mathbf{ifail}}=5$
Constraint: ${\mathbf{s}}\ge _$ and ${\mathbf{s}}\le _$.
${\mathbf{ifail}}=6$
Constraint: ${\mathbf{t}}\left(i\right)\ge _$ for all $i$.
${\mathbf{ifail}}=7$
Constraint: ${\mathbf{sigma}}>0.0$.
${\mathbf{ifail}}=8$
Constraint: ${\mathbf{r}}\ge 0.0$.
${\mathbf{ifail}}=9$
Constraint: ${\mathbf{q}}\ge 0.0$.
${\mathbf{ifail}}=11$
Constraint: $\mathit{ldp}\ge {\mathbf{m}}$.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

The accuracy of the output is dependent on the accuracy of the cumulative Normal distribution function, $\Phi$. 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.

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 $6%$.
```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
```