# NAG Library Routine Document

## 1Purpose

s30baf computes the price of a floating-strike lookback option.

## 2Specification

Fortran Interface
 Subroutine s30baf ( m, n, sm, s, t, r, q, p, ldp,
 Integer, Intent (In) :: m, n, ldp Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: sm(m), s, t(n), sigma, r, q Real (Kind=nag_wp), Intent (Inout) :: p(ldp,n) Character (1), Intent (In) :: calput
#include nagmk26.h
 void s30baf_ (const char *calput, const Integer *m, const Integer *n, const double sm[], const double *s, const double t[], const double *sigma, const double *r, const double *q, double p[], const Integer *ldp, Integer *ifail, const Charlen length_calput)

## 3Description

s30baf 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:     $\mathbf{calput}$ – Character(1)Input
On entry: 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:     $\mathbf{m}$ – IntegerInput
On entry: the number of minimum or maximum prices to be used.
Constraint: ${\mathbf{m}}\ge 1$.
3:     $\mathbf{n}$ – IntegerInput
On entry: the number of times to expiry to be used.
Constraint: ${\mathbf{n}}\ge 1$.
4:     $\mathbf{sm}\left({\mathbf{m}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${\mathbf{sm}}\left(i\right)$ 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 $z={\mathbf{x02amf}}\left(\right)$, 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 $\mathit{i}=1,2,\dots ,{\mathbf{m}}$.
5:     $\mathbf{s}$ – Real (Kind=nag_wp)Input
On entry: $S$, the price of the underlying asset.
Constraint: ${\mathbf{s}}\ge z\text{​ and ​}{\mathbf{s}}\le 1.0/z$, where $z={\mathbf{x02amf}}\left(\right)$, the safe range parameter.
6:     $\mathbf{t}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${\mathbf{t}}\left(i\right)$ 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 $z={\mathbf{x02amf}}\left(\right)$, the safe range parameter, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
7:     $\mathbf{sigma}$ – Real (Kind=nag_wp)Input
On entry: $\sigma$, the volatility of the underlying asset. Note that a rate of 15% should be entered as 0.15.
Constraint: ${\mathbf{sigma}}>0.0$.
8:     $\mathbf{r}$ – Real (Kind=nag_wp)Input
On entry: $r$, the annual risk-free interest rate, continuously compounded. Note that a rate of 5% should be entered as 0.05.
Constraint: ${\mathbf{r}}\ge 0.0$.
9:     $\mathbf{q}$ – Real (Kind=nag_wp)Input
On entry: $q$, the annual continuous yield rate. Note that a rate of 8% should be entered as 0.08.
Constraint: ${\mathbf{q}}\ge 0.0$.
10:   $\mathbf{p}\left({\mathbf{ldp}},{\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
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 $j=1,2,\dots ,{\mathbf{n}}$.
11:   $\mathbf{ldp}$ – IntegerInput
On entry: the first dimension of the array p as declared in the (sub)program from which s30baf is called.
Constraint: ${\mathbf{ldp}}\ge {\mathbf{m}}$.
12:   $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{calput}}=〈\mathit{\text{value}}〉$ was an illegal value.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}\ge 1$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 1$.
${\mathbf{ifail}}=4$
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 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 $i$.
${\mathbf{ifail}}=5$
On entry, ${\mathbf{s}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{s}}\ge 〈\mathit{\text{value}}〉$ and ${\mathbf{s}}\le 〈\mathit{\text{value}}〉$.
${\mathbf{ifail}}=6$
On entry, ${\mathbf{t}}\left(〈\mathit{\text{value}}〉\right)=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{t}}\left(i\right)\ge 〈\mathit{\text{value}}〉$ for all $i$.
${\mathbf{ifail}}=7$
On entry, ${\mathbf{sigma}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{sigma}}>0.0$.
${\mathbf{ifail}}=8$
On entry, ${\mathbf{r}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{r}}\ge 0.0$.
${\mathbf{ifail}}=9$
On entry, ${\mathbf{q}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{q}}\ge 0.0$.
${\mathbf{ifail}}=11$
On entry, ${\mathbf{ldp}}=〈\mathit{\text{value}}〉$ and ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ldp}}\ge {\mathbf{m}}$.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 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 s15abf and s15adf). An accuracy close to machine precision can generally be expected.

## 8Parallelism and Performance

s30baf 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 routine. 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 (s30bafe.f90)

### 10.2Program Data

Program Data (s30bafe.d)

### 10.3Program Results

Program Results (s30bafe.r)

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