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Chapter Contents
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, Smin${S}_{\mathrm{min}}$, observed during the lifetime of the contract. A put option gives the holder the right to sell the underlying asset at the maximum price, Smax${S}_{\mathrm{max}}$, observed during the lifetime of the contract. Thus, at expiry, the payoff for a call option is SSmin$S-{S}_{\mathrm{min}}$, and for a put, SmaxS${S}_{\mathrm{max}}-S$.
For a given minimum value the price of a floating-strike lookback call with underlying asset price, S$S$, and time to expiry, T$T$, is
 Pcall = S e − qT Φ(a1) − Smin e − rT Φ(a2) + S e − rT   (σ2)/(2b) [(S/(Smin)) − 2b / σ2 Φ( − a1 + (2b)/σsqrt(T)) − ebTΦ( − a1)] ,
where b = rq0$b=r-q\ne 0$. The volatility, σ$\sigma$, risk-free interest rate, r$r$, and annualised dividend yield, q$q$, are constants. When r = q$r=q$, the option price is given by
 Pcall = S e − qT Φ (a1) − Smin e − rT Φ (a2) + S e − rT σ×sqrt(T) [φ(a1) + a1(Φ(a1) − 1)] . $Pcall = S e-qT Φ (a1) - Smin e-rT Φ (a2) + S e-rT σ⁢T [ ϕ (a1) + a1 ( Φ (a1) -1 ) ] .$
The corresponding put price is (for b0$b\ne 0$),
 Pput = Smax e − rT Φ ( − a2) − S e − qT Φ ( − a1) + S e − rT   (σ2)/(2b) [ − (S/(Smax)) − 2b / σ2 Φ(a1 − (2b)/σsqrt(T)) + ebTΦ(a1)] .
When r = q$r=q$,
 Pput = Smax e − rT Φ ( − a2) − S e − qT Φ ( − a1) + S e − rT σ×sqrt(T) [φ(a1) + a1Φ(a1)] . $Pput = Smax e-rT Φ (-a2) - S e-qT Φ (-a1) + S e-rT σ⁢T [ ϕ(a1) + a1 Φ(a1) ] .$
In the above, Φ$\Phi$ denotes the cumulative Normal distribution function,
 x Φ(x) = ∫ φ(y)dy − ∞
$Φ(x) = ∫ -∞ x ϕ(y) dy$
where φ$\varphi$ denotes the standard Normal probability density function
 φ(y) = 1/(sqrt(2π)) exp( − y2 / 2) $ϕ(y) = 12π exp( -y2/2 )$
and
 a1 = ( ln (S / Sm) + (b + σ2 / 2) T )/(σ×sqrt(T)) a2 = a1 − σ×sqrt(T)
$a1 = ln ( S / Sm ) + ( b + σ2 / 2 ) T σ⁢T a2=a1-σ⁢T$
where Sm${S}_{m}$ is taken to be the minimum price attained by the underlying asset, Smin${S}_{\mathrm{min}}$, for a call and the maximum price, Smax${S}_{\mathrm{max}}$, for a put.

## 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'${\mathbf{calput}}=\text{'C'}$
A call. The holder has a right to buy.
calput = 'P'${\mathbf{calput}}=\text{'P'}$
A put. The holder has a right to sell.
Constraint: calput = 'C'${\mathbf{calput}}=\text{'C'}$ or 'P'$\text{'P'}$.
2:     sm(m) – double array
m, the dimension of the array, must satisfy the constraint m1${\mathbf{m}}\ge 1$.
sm(i)${\mathbf{sm}}\left(i\right)$ must contain Smin (i) ${S}_{\mathrm{min}}\left(\mathit{i}\right)$, the i$\mathit{i}$th minimum observed price of the underlying asset when calput = 'C'${\mathbf{calput}}=\text{'C'}$, or Smax (i) ${S}_{\mathrm{max}}\left(\mathit{i}\right)$, the maximum observed price when calput = 'P'${\mathbf{calput}}=\text{'P'}$, for i = 1,2,,m$\mathit{i}=1,2,\dots ,{\mathbf{m}}$.
Constraints:
• sm(i)z ​ and ​ sm(i) 1 / z ${\mathbf{sm}}\left(\mathit{i}\right)\ge z\text{​ and ​}{\mathbf{sm}}\left(\mathit{i}\right)\le 1/z$, where z = x02am() $z=\mathbf{x02am}\left(\right)$, the safe range parameter, for i = 1,2,,m$\mathit{i}=1,2,\dots ,{\mathbf{m}}$;
• if calput = 'C'${\mathbf{calput}}=\text{'C'}$, sm(i)S${\mathbf{sm}}\left(\mathit{i}\right)\le S$, for i = 1,2,,m$\mathit{i}=1,2,\dots ,{\mathbf{m}}$;
• if calput = 'P'${\mathbf{calput}}=\text{'P'}$, sm(i)S${\mathbf{sm}}\left(\mathit{i}\right)\ge S$, for i = 1,2,,m$\mathit{i}=1,2,\dots ,{\mathbf{m}}$.
3:     s – double scalar
S$S$, the price of the underlying asset.
Constraint: sz ​ and ​s1.0 / z${\mathbf{s}}\ge z\text{​ and ​}{\mathbf{s}}\le 1.0/z$, where z = x02am()$z=\mathbf{x02am}\left(\right)$, the safe range parameter.
4:     t(n) – double array
n, the dimension of the array, must satisfy the constraint n1${\mathbf{n}}\ge 1$.
t(i)${\mathbf{t}}\left(i\right)$ must contain Ti${T}_{\mathit{i}}$, the i$\mathit{i}$th time, in years, to expiry, for i = 1,2,,n$\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
Constraint: t(i)z${\mathbf{t}}\left(\mathit{i}\right)\ge z$, where z = x02am () $z=\mathbf{x02am}\left(\right)$, the safe range parameter, for i = 1,2,,n$\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
5:     sigma – double scalar
σ$\sigma$, the volatility of the underlying asset. Note that a rate of 15% should be entered as 0.15.
Constraint: sigma > 0.0${\mathbf{sigma}}>0.0$.
6:     r – double scalar
r$r$, the annual risk-free interest rate, continuously compounded. Note that a rate of 5% should be entered as 0.05.
Constraint: r0.0${\mathbf{r}}\ge 0.0$.
7:     q – double scalar
q$q$, the annual continuous yield rate. Note that a rate of 8% should be entered as 0.08.
Constraint: q0.0${\mathbf{q}}\ge 0.0$.

### 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: m1${\mathbf{m}}\ge 1$.
2:     n – int64int32nag_int scalar
Default: The dimension of the array t.
The number of times to expiry to be used.
Constraint: n1${\mathbf{n}}\ge 1$.

ldp

### Output Parameters

1:     p(ldp,n) – double array
ldpm$\mathit{ldp}\ge {\mathbf{m}}$.
The leading m × n${\mathbf{m}}×{\mathbf{n}}$ part of the array p contains the computed option prices.
2:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

## Error Indicators and Warnings

Errors or warnings detected by the function:
ifail = 1${\mathbf{ifail}}=1$
On entry, calput = 'C'${\mathbf{calput}}=\text{'C'}$ or 'P'$\text{'P'}$.
ifail = 2${\mathbf{ifail}}=2$
On entry, m0${\mathbf{m}}\le 0$.
ifail = 3${\mathbf{ifail}}=3$
On entry, n0${\mathbf{n}}\le 0$.
ifail = 4${\mathbf{ifail}}=4$
 On entry, sm(i) < z${\mathbf{sm}}\left(\mathit{i}\right) or sm(i) > 1 / z${\mathbf{sm}}\left(\mathit{i}\right)>1/z$, where z = x02am()$z=\mathbf{x02am}\left(\right)$, the safe range parameter, or calput = 'C'${\mathbf{calput}}=\text{'C'}$ and sm(i) > S${\mathbf{sm}}\left(\mathit{i}\right)>S$, or calput = 'P'${\mathbf{calput}}=\text{'P'}$ and sm(i) < S${\mathbf{sm}}\left(\mathit{i}\right).
ifail = 5${\mathbf{ifail}}=5$
On entry, s < z${\mathbf{s}} or s > 1.0 / z${\mathbf{s}}>1.0/z$, where z = x02am()$z=\mathbf{x02am}\left(\right)$, the safe range parameter.
ifail = 6${\mathbf{ifail}}=6$
On entry, t(i) < z${\mathbf{t}}\left(\mathit{i}\right), where z = x02am()$z=\mathbf{x02am}\left(\right)$, the safe range parameter.
ifail = 7${\mathbf{ifail}}=7$
On entry, sigma0.0${\mathbf{sigma}}\le 0.0$.
ifail = 8${\mathbf{ifail}}=8$
On entry, r < 0.0${\mathbf{r}}<0.0$.
ifail = 9${\mathbf{ifail}}=9$
On entry, q < 0.0${\mathbf{q}}<0.0$.
ifail = 11${\mathbf{ifail}}=11$
On entry, ldp < m$\mathit{ldp}<{\mathbf{m}}$.

## Accuracy

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

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

[p, ifail] = nag_specfun_opt_lookback_fls_price(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
```
```

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

```
```function s30ba_example
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
```
```

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

```