NAG Library Routine Document

s30bbf (opt_lookback_fls_greeks)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

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 $S_{\min }\left(\mathit{i}\right)$, the $\mathit{i}$th minimum observed price of the underlying asset when $\mathbf{calput}=\text{'C'}$, or $S_{\max }\left(\mathit{i}\right)$, 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: the annual risk-free interest rate, $r$, continuously compounded. Note that a rate of 5% should be entered as $0.05$.

Constraint: $\mathbf{r}\ge 0.0$ and $\mathrm{abs}\left(\mathbf{r}-\mathbf{q}\right)>10\times \mathrm{eps}\times \max \left(\mathrm{abs}\left(\mathbf{r}\right),1\right)$, where $\mathrm{eps}=\mathbf{x02ajf}\left(\right)$, the machine precision.

9:     $\mathbf{q}$ – Real (Kind=nag_wp)Input

On entry: the annual continuous yield rate. Note that a rate of 8% should be entered as $0.08$.

Constraint: $\mathbf{q}\ge 0.0$ and $\mathrm{abs}\left(\mathbf{r}-\mathbf{q}\right)>10\times \mathrm{eps}\times \max \left(\mathrm{abs}\left(\mathbf{r}\right),1\right)$, where $\mathrm{eps}=\mathbf{x02ajf}\left(\right)$, the machine precision.

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 $S_{\min }\left(\mathit{i}\right)$ or $S_{\max }\left(\mathit{i}\right)$ 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 arrays p, delta, gamma, vega, theta, rho, crho, vanna, charm, speed, colour, zomma and vomma as declared in the (sub)program from which s30bbf is called.

Constraint: $\mathbf{ldp}\ge \mathbf{m}$.

12:   $\mathbf{delta}\left(\mathbf{ldp},\mathbf{n}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: the leading $\mathbf{m}\times \mathbf{n}$ part of the array delta contains the sensitivity, $\frac{\partial P}{\partial S}$, of the option price to change in the price of the underlying asset.

13:   $\mathbf{gamma}\left(\mathbf{ldp},\mathbf{n}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: the leading $\mathbf{m}\times \mathbf{n}$ part of the array gamma contains the sensitivity, $\frac{{\partial }^{2}P}{\partial S^2}$, of delta to change in the price of the underlying asset.

14:   $\mathbf{vega}\left(\mathbf{ldp},\mathbf{n}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: $\mathbf{vega}\left(i,j\right)$, contains the first-order Greek measuring the sensitivity of the option price $P_{ij}$ to change in the volatility of the underlying asset, i.e., $\frac{\partial P_{ij}}{\partial \sigma }$, for $i=1,2,\dots ,\mathbf{m}$ and $j=1,2,\dots ,\mathbf{n}$.

15:   $\mathbf{theta}\left(\mathbf{ldp},\mathbf{n}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: $\mathbf{theta}\left(i,j\right)$, contains the first-order Greek measuring the sensitivity of the option price $P_{ij}$ to change in time, i.e., $-\frac{\partial P_{ij}}{\partial T}$, for $i=1,2,\dots ,\mathbf{m}$ and $j=1,2,\dots ,\mathbf{n}$, where $b=r-q$.

16:   $\mathbf{rho}\left(\mathbf{ldp},\mathbf{n}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: $\mathbf{rho}\left(i,j\right)$, contains the first-order Greek measuring the sensitivity of the option price $P_{ij}$ to change in the annual risk-free interest rate, i.e., $-\frac{\partial P_{ij}}{\partial r}$, for $i=1,2,\dots ,\mathbf{m}$ and $j=1,2,\dots ,\mathbf{n}$.

17:   $\mathbf{crho}\left(\mathbf{ldp},\mathbf{n}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: $\mathbf{crho}\left(i,j\right)$, contains the first-order Greek measuring the sensitivity of the option price $P_{ij}$ to change in the annual cost of carry rate, i.e., $-\frac{\partial P_{ij}}{\partial b}$, for $i=1,2,\dots ,\mathbf{m}$ and $j=1,2,\dots ,\mathbf{n}$, where $b=r-q$.

18:   $\mathbf{vanna}\left(\mathbf{ldp},\mathbf{n}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: $\mathbf{vanna}\left(i,j\right)$, contains the second-order Greek measuring the sensitivity of the first-order Greek ${\Delta }_{ij}$ to change in the volatility of the asset price, i.e., $-\frac{\partial {\Delta }_{ij}}{\partial T}=-\frac{{\partial }^2{P}_{ij}}{\partial S\partial \sigma }$, for $i=1,2,\dots ,\mathbf{m}$ and $j=1,2,\dots ,\mathbf{n}$.

19:   $\mathbf{charm}\left(\mathbf{ldp},\mathbf{n}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: $\mathbf{charm}\left(i,j\right)$, contains the second-order Greek measuring the sensitivity of the first-order Greek ${\Delta }_{ij}$ to change in the time, i.e., $-\frac{\partial {\Delta }_{ij}}{\partial T}=-\frac{{\partial }^2{P}_{ij}}{\partial S\partial T}$, for $i=1,2,\dots ,\mathbf{m}$ and $j=1,2,\dots ,\mathbf{n}$.

20:   $\mathbf{speed}\left(\mathbf{ldp},\mathbf{n}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: $\mathbf{speed}\left(i,j\right)$, contains the third-order Greek measuring the sensitivity of the second-order Greek ${\Gamma }_{ij}$ to change in the price of the underlying asset, i.e., $-\frac{\partial {\Gamma }_{ij}}{\partial S}=-\frac{{\partial }^3{P}_{ij}}{\partial S^3}$, for $i=1,2,\dots ,\mathbf{m}$ and $j=1,2,\dots ,\mathbf{n}$.

21:   $\mathbf{colour}\left(\mathbf{ldp},\mathbf{n}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: $\mathbf{colour}\left(i,j\right)$, contains the third-order Greek measuring the sensitivity of the second-order Greek ${\Gamma }_{ij}$ to change in the time, i.e., $-\frac{\partial {\Gamma }_{ij}}{\partial T}=-\frac{{\partial }^3{P}_{ij}}{\partial S\partial T}$, for $i=1,2,\dots ,\mathbf{m}$ and $j=1,2,\dots ,\mathbf{n}$.

22:   $\mathbf{zomma}\left(\mathbf{ldp},\mathbf{n}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: $\mathbf{zomma}\left(i,j\right)$, contains the third-order Greek measuring the sensitivity of the second-order Greek ${\Gamma }_{ij}$ to change in the volatility of the underlying asset, i.e., $-\frac{\partial {\Gamma }_{ij}}{\partial \sigma }=-\frac{{\partial }^3{P}_{ij}}{\partial S^2\partial \sigma }$, for $i=1,2,\dots ,\mathbf{m}$ and $j=1,2,\dots ,\mathbf{n}$.

23:   $\mathbf{vomma}\left(\mathbf{ldp},\mathbf{n}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: $\mathbf{vomma}\left(i,j\right)$, contains the second-order Greek measuring the sensitivity of the first-order Greek ${\Delta }_{ij}$ to change in the volatility of the underlying asset, i.e., $-\frac{\partial {\Delta }_{ij}}{\partial \sigma }=-\frac{{\partial }^2{P}_{ij}}{\partial {\sigma }^2}$, for $i=1,2,\dots ,\mathbf{m}$ and $j=1,2,\dots ,\mathbf{n}$.

24:   $\mathbf{ifail}$ – 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).

6

Error Indicators and Warnings

$\mathbf{ifail}=1$

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: $\mathbf{ldp}\ge \mathbf{m}$.

$\mathbf{ifail}=12$

On entry, $\mathbf{r}=〈\mathit{\text{value}}〉$ and $\mathbf{q}=〈\mathit{\text{value}}〉$.
Constraint: $\left|\mathbf{r}-\mathbf{q}\right|>10\times \mathrm{eps}\times \max \left(\left|\mathbf{r}\right|,1\right)$, where $\mathrm{eps}$ is the machine precision.

$\mathbf{ifail}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

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.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (s30bbfe.f90)

10.2

Program Data

Program Data (s30bbfe.d)

10.3

Program Results

Program Results (s30bbfe.r)