# NAG Library Function Document

## 1Purpose

nag_lookback_fls_price (s30bac) computes the price of a floating-strike lookback option.

## 2Specification

 #include #include
 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)

## 3Description

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, ${S}_{\mathrm{min}}$, 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 ${T}_{j}$, $j=1,2,\dots ,n$.

## 4References

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

## 5Arguments

1:    $\mathbf{order}$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 ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2:    $\mathbf{option}$Nag_CallPutInput
On entry: determines whether the option is a call or a put.
${\mathbf{option}}=\mathrm{Nag_Call}$
A call; the holder has a right to buy.
${\mathbf{option}}=\mathrm{Nag_Put}$
A put; the holder has a right to sell.
Constraint: ${\mathbf{option}}=\mathrm{Nag_Call}$ or $\mathrm{Nag_Put}$.
3:    $\mathbf{m}$IntegerInput
On entry: the number of minimum or maximum prices to be used.
Constraint: ${\mathbf{m}}\ge 1$.
4:    $\mathbf{n}$IntegerInput
On entry: the number of times to expiry to be used.
Constraint: ${\mathbf{n}}\ge 1$.
5:    $\mathbf{sm}\left[{\mathbf{m}}\right]$const doubleInput
On entry: ${\mathbf{sm}}\left[i-1\right]$ must contain ${S}_{\mathrm{min}}\left(\mathit{i}\right)$, the $\mathit{i}$th minimum observed price of the underlying asset when ${\mathbf{option}}=\mathrm{Nag_Call}$, or ${S}_{\mathrm{max}}\left(\mathit{i}\right)$, the maximum observed price when ${\mathbf{option}}=\mathrm{Nag_Put}$, for $\mathit{i}=1,2,\dots ,{\mathbf{m}}$.
Constraints:
• ${\mathbf{sm}}\left[\mathit{i}-1\right]\ge z\text{​ and ​}{\mathbf{sm}}\left[\mathit{i}-1\right]\le 1/z$, where $z={\mathbf{nag_real_safe_small_number}}$, the safe range parameter, for $\mathit{i}=1,2,\dots ,{\mathbf{m}}$;
• if ${\mathbf{option}}=\mathrm{Nag_Call}$, ${\mathbf{sm}}\left[\mathit{i}-1\right]\le S$, for $\mathit{i}=1,2,\dots ,{\mathbf{m}}$;
• if ${\mathbf{option}}=\mathrm{Nag_Put}$, ${\mathbf{sm}}\left[\mathit{i}-1\right]\ge S$, for $\mathit{i}=1,2,\dots ,{\mathbf{m}}$.
6:    $\mathbf{s}$doubleInput
On entry: $S$, the price of the underlying asset.
Constraint: ${\mathbf{s}}\ge z\text{​ and ​}{\mathbf{s}}\le 1.0/z$, where $z={\mathbf{nag_real_safe_small_number}}$, the safe range parameter.
7:    $\mathbf{t}\left[{\mathbf{n}}\right]$const doubleInput
On entry: ${\mathbf{t}}\left[i-1\right]$ must contain ${T}_{\mathit{i}}$, the $\mathit{i}$th time, in years, to expiry, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
Constraint: ${\mathbf{t}}\left[\mathit{i}-1\right]\ge z$, where $z={\mathbf{nag_real_safe_small_number}}$, the safe range parameter, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
8:    $\mathbf{sigma}$doubleInput
On entry: $\sigma$, the volatility of the underlying asset. Note that a rate of 15% should be entered as 0.15.
Constraint: ${\mathbf{sigma}}>0.0$.
9:    $\mathbf{r}$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: ${\mathbf{r}}\ge 0.0$.
10:  $\mathbf{q}$doubleInput
On entry: $q$, the annual continuous yield rate. Note that a rate of 8% should be entered as 0.08.
Constraint: ${\mathbf{q}}\ge 0.0$.
11:  $\mathbf{p}\left[{\mathbf{m}}×{\mathbf{n}}\right]$doubleOutput
Note: where ${\mathbf{P}}\left(i,j\right)$ appears in this document, it refers to the array element
• ${\mathbf{p}}\left[\left(j-1\right)×{\mathbf{m}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{p}}\left[\left(i-1\right)×{\mathbf{n}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: ${\mathbf{P}}\left(i,j\right)$ 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 $j=1,2,\dots ,{\mathbf{n}}$.
12:  $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}\ge 1$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 1$.
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.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_REAL
On entry, ${\mathbf{q}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{q}}\ge 0.0$.
On entry, ${\mathbf{r}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{r}}\ge 0.0$.
On entry, ${\mathbf{s}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{s}}\ge 〈\mathit{\text{value}}〉$ and ${\mathbf{s}}\le 〈\mathit{\text{value}}〉$.
On entry, ${\mathbf{sigma}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{sigma}}>0.0$.
NE_REAL_ARRAY
On entry, ${\mathbf{sm}}\left[〈\mathit{\text{value}}〉\right]=〈\mathit{\text{value}}〉$.
Constraint: $〈\mathit{\text{value}}〉\le {\mathbf{sm}}\left[i\right]\le 〈\mathit{\text{value}}〉$ for all $i$.
On entry, ${\mathbf{t}}\left[〈\mathit{\text{value}}〉\right]=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{t}}\left[i\right]\ge 〈\mathit{\text{value}}〉$ for all $i$.
On entry with a call option, ${\mathbf{sm}}\left[〈\mathit{\text{value}}〉\right]=〈\mathit{\text{value}}〉$.
Constraint: for call options, ${\mathbf{sm}}\left[i\right]\le 〈\mathit{\text{value}}〉$ for all $i$.
On entry with a put option, ${\mathbf{sm}}\left[〈\mathit{\text{value}}〉\right]=〈\mathit{\text{value}}〉$.
Constraint: for put options, ${\mathbf{sm}}\left[i\right]\ge 〈\mathit{\text{value}}〉$ for all $i$.

## 7Accuracy

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

## 8Parallelism and Performance

nag_lookback_fls_price (s30bac) 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

None.

## 10Example

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 $6%$.

### 10.1Program Text

Program Text (s30bace.c)

### 10.2Program Data

Program Data (s30bace.d)

### 10.3Program Results

Program Results (s30bace.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017