naginterfaces.library.specfun.opt_​lookback_​fls_​greeks

naginterfaces.library.specfun.opt_lookback_fls_greeks(calput, sm, s, t, sigma, r, q)

opt_lookback_fls_greeks computes the price of a floating-strike lookback option together with its sensitivities (Greeks).

For full information please refer to the NAG Library document for s30bb

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/s/s30bbf.html

Parameters

calputstr, length 1

Determines whether the option is a call or a put.

$calput=\text{‘C'}$

A call; the holder has a right to buy.

$calput=\text{‘P'}$

A put; the holder has a right to sell.

smfloat, array-like, shape $\left(m\right)$

$sm[i-1]$ must contain $S_{min}\left(\text{i}\right)$, the $\text{i}$th minimum observed price of the underlying asset when $calput=\text{‘C'}$, or $S_{max}\left(\text{i}\right)$, the maximum observed price when $calput=\text{‘P'}$, for $\text{i}=1,2,\dots ,m$.

sfloat

$S$, the price of the underlying asset.

tfloat, array-like, shape $\left(n\right)$

$t[i-1]$ must contain $T_{\text{i}}$, the $\text{i}$th time, in years, to expiry, for $\text{i}=1,2,\dots ,n$.

sigmafloat

$\sigma$, the volatility of the underlying asset. Note that a rate of 15% should be entered as $0.15$.

rfloat

The annual risk-free interest rate, $r$, continuously compounded. Note that a rate of 5% should be entered as $0.05$.

qfloat

The annual continuous yield rate. Note that a rate of 8% should be entered as $0.08$.

Returns

pfloat, ndarray, shape $(m,n)$

$p[i-1,j-1]$ contains $P_{ij}$, the option price evaluated for the minimum or maximum observed price $S_{min}\left(\text{i}\right)$ or $S_{max}\left(\text{i}\right)$ at expiry $t_j$ for $i=1,2,\dots ,m$ and $j=1,2,\dots ,n$.

deltafloat, ndarray, shape $(m,n)$

The leading $m\times n$ 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.

gammafloat, ndarray, shape $(m,n)$

The leading $m\times n$ 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.

vegafloat, ndarray, shape $(m,n)$

$vega[i-1,j-1]$, 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 $i=1,2,\dots ,m$ and $j=1,2,\dots ,n$.

thetafloat, ndarray, shape $(m,n)$

$theta[i-1,j-1]$, 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 $i=1,2,\dots ,m$ and $j=1,2,\dots ,n$, where $b=r-q$.

rhofloat, ndarray, shape $(m,n)$

$rho[i-1,j-1]$, 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 $i=1,2,\dots ,m$ and $j=1,2,\dots ,n$.

crhofloat, ndarray, shape $(m,n)$

$crho[i-1,j-1]$, 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 $i=1,2,\dots ,m$ and $j=1,2,\dots ,n$, where $b=r-q$.

vannafloat, ndarray, shape $(m,n)$

$vanna[i-1,j-1]$, 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 $i=1,2,\dots ,m$ and $j=1,2,\dots ,n$.

charmfloat, ndarray, shape $(m,n)$

$charm[i-1,j-1]$, 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 $i=1,2,\dots ,m$ and $j=1,2,\dots ,n$.

speedfloat, ndarray, shape $(m,n)$

$speed[i-1,j-1]$, 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 $i=1,2,\dots ,m$ and $j=1,2,\dots ,n$.

colourfloat, ndarray, shape $(m,n)$

$colour[i-1,j-1]$, 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 $i=1,2,\dots ,m$ and $j=1,2,\dots ,n$.

zommafloat, ndarray, shape $(m,n)$

$zomma[i-1,j-1]$, 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 $i=1,2,\dots ,m$ and $j=1,2,\dots ,n$.

vommafloat, ndarray, shape $(m,n)$

$vomma[i-1,j-1]$, 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 $i=1,2,\dots ,m$ and $j=1,2,\dots ,n$.

Raises

NagValueError

(errno $1$)

On entry, $calput=⟨value⟩$.

Constraint: $calput=\text{‘C'}\text{or}\text{‘P'}$.

(errno $2$)

On entry, $m=⟨value⟩$.

Constraint: $m\ge 1$.

(errno $3$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 1$.

(errno $4$)

On entry with a put option, $sm[⟨value⟩]=⟨value⟩$.

Constraint: for put options, $sm\left[i\right]\ge ⟨value⟩$ for all $i$.

(errno $4$)

On entry with a call option, $sm[⟨value⟩]=⟨value⟩$.

Constraint: for call options, $sm\left[i\right]\le ⟨value⟩$ for all $i$.

(errno $4$)

On entry, $sm[⟨value⟩]=⟨value⟩$.

Constraint: $⟨value⟩\le sm\left[i\right]\le ⟨value⟩$ for all $i$.

(errno $5$)

On entry, $s=⟨value⟩$.

Constraint: $s\ge ⟨value⟩$ and $s\le ⟨value⟩$.

(errno $6$)

On entry, $t[⟨value⟩]=⟨value⟩$.

Constraint: $t\left[i\right]\ge ⟨value⟩$ for all $i$.

(errno $7$)

On entry, $sigma=⟨value⟩$.

Constraint: $sigma>0.0$.

(errno $8$)

On entry, $r=⟨value⟩$.

Constraint: $r\ge 0.0$.

(errno $9$)

On entry, $q=⟨value⟩$.

Constraint: $q\ge 0.0$.

(errno $12$)

On entry, $r=⟨value⟩$ and $q=⟨value⟩$.

Constraint: $|r-q|>10\times \text{eps}\times max(\left|r\right|,1)$, where $\text{eps}$ is the machine precision.

Notes

opt_lookback_fls_greeks computes the price of a floating-strike lookback call or put option, together with the Greeks or sensitivities, which are the partial derivatives of the option price with respect to certain of the other input parameters. A call option of this type confers the right to buy the underlying asset at the lowest price, $S_{min}$, observed during the lifetime of the contract. A put option gives the holder the right to sell the underlying asset at the maximum price, $S_{max}$, observed during the lifetime of the contract. Thus, at expiry, the payoff for a call option is $S-S_{min}$, and for a put, $S_{max}-S$.

For a given minimum value the price of a floating-strike lookback call with underlying asset price, $S$, and time to expiry, $T$, is

$$P_{call}=S{e}^{-qT}\Phi \left(a_1\right)-S_{min}{e}^{-rT}\Phi \left(a_2\right)+S{e}^{-rT}\frac{{\sigma }^2}{2b}\left[{\left(\frac{S}{S_{min}}\right)}^{-2b/{\sigma }^2}\Phi ({-a}_1+\frac{2b}{\sigma }\sqrt{T}){-e}^{bT}\Phi \left({-a}_1\right)\right]\text{,}$$

where $b=r-q\ne 0$. The volatility, $\sigma$, risk-free interest rate, $r$, and annualised dividend yield, $q$, are constants.

The corresponding put price is

$$P_{put}=S_{max}{e}^{-rT}\Phi \left({-a}_2\right)-S{e}^{-qT}\Phi \left({-a}_1\right)+S{e}^{-rT}\frac{{\sigma }^2}{2b}[-{\left(\frac{S}{S_{max}}\right)}^{-2b/{\sigma }^2}\Phi (a_1-\frac{2b}{\sigma }\sqrt{T})+e^{bT}\Phi \left(a_1\right)]\text{.}$$

In the above, $\Phi$ denotes the cumulative Normal distribution function,

$$\Phi \left(x\right)={\int }_{-\infty }^x\varphi \left(y\right)dy$$

where $\varphi$ denotes the standard Normal probability density function

$$\varphi \left(y\right)=\frac{1}{\sqrt{2\pi }}exp(-y^2/2)$$

and

$$\begin{array}{c}\begin{array}{c}{a}_1=\frac{ln\left(S/S_m\right)+(b+{\sigma }^2/2)T}{\sigma \sqrt{T}}\\ \\ a_2=a_1-\sigma \sqrt{T}\end{array}\end{array}$$

where $S_m$ is taken to be the minimum price attained by the underlying asset, $S_{min}$, for a call and the maximum price, $S_{max}$, for a put.

The option price $P_{ij}=P(X=X_i,T=T_j)$ is computed for each minimum or maximum observed price in a set $S_{min}\left(\text{i}\right)$ or $S_{max}\left(\text{i}\right)$, $i=1,2,\dots ,m$, 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