naginterfaces.library.specfun.opt_​amer_​bs_​price

naginterfaces.library.specfun.opt_amer_bs_price(calput, x, s, t, sigma, r, q)

opt_amer_bs_price computes the Bjerksund and Stensland (2002) approximation to the price of an American option.

For full information please refer to the NAG Library document for s30qc

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/s/s30qcf.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.

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

$x[i-1]$ must contain $X_{\text{i}}$, the $\text{i}$th strike price, 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

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

qfloat

$q$, 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 strike price $x_i$ at expiry $t_j$ 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, $x[⟨value⟩]=⟨value⟩$.

Constraint: $x\left[i\right]\ge ⟨value⟩$ and $x\left[i\right]\le ⟨value⟩$.

(errno $5$)

On entry, $s=⟨value⟩$.

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

(errno $6$)

On entry, $t[⟨value⟩]=⟨value⟩$.

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

(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 $14$)

On entry, $s=⟨value⟩$ and $\beta =⟨value⟩$.

Constraint: $s^{\beta }<⟨value⟩$.

Notes

opt_amer_bs_price computes the price of an American option using the closed form approximation of Bjerksund and Stensland (2002). The time to maturity, $T$, is divided into two periods, each with a flat early exercise boundary, by choosing a time $t\in [0,T]$, such that $t=\frac{1}{2}(\sqrt{5}-1)T$. The two boundary values are defined as $\tilde{x}=\tilde{X}\left(t\right)$, $\tilde{X}=\tilde{X}\left(T\right)$ with

$$\tilde{X}\left(\tau \right)=B_0+(B_{\infty }-B_0)(1-exp\left\{h\left(\tau \right)\right\})\text{,}$$

where

$$h\left(\tau \right)=-(b\tau +2\sigma \sqrt{\tau })\left(\frac{X^2}{(B_{\infty }-B_0)B_0}\right)\text{,}$$

$$B_{\infty }\equiv \frac{\beta }{\beta -1}X\text{,}{B}_0\equiv max\{X,\left(\frac{r}{r-b}\right)X\}\text{,}$$

$$\beta =(\frac{1}{2}-\frac{b}{{\sigma }^2})+\sqrt{{(\frac{b}{{\sigma }^2}-\frac{1}{2})}^2+2\frac{r}{{\sigma }^2}}\text{.}$$

with $b=r-q$, the cost of carry, where $r$ is the risk-free interest rate and $q$ is the annual dividend rate. Here $X$ is the strike price and $\sigma$ is the annual volatility.

The price of an American call option is approximated as

$$\begin{array}{c}\begin{array}{ccc}{P}_{call}& =& \alpha \left(\tilde{X}\right)S^{\beta }-\alpha \left(\tilde{X}\right)\varphi (S,t|\beta ,\tilde{X},\tilde{X})+\\ & & \varphi (S,t|1,\tilde{X},\tilde{X})-\varphi (S,t|1,\tilde{x},\tilde{X})-\\ & & X\varphi (S,t|0,\tilde{X},\tilde{X})+X\varphi (S,t|0,\tilde{x},\tilde{X})+\\ & & \alpha \left(\tilde{x}\right)\varphi (S,t|\beta ,\tilde{x},\tilde{X})-\alpha \left(\tilde{x}\right)\Psi (S,T|\beta ,\tilde{x},\tilde{X},\tilde{x},t)+\\ & & \Psi (S,T|1,\tilde{x},\tilde{X},\tilde{x},t)-\Psi (S,T|1,X,\tilde{X},\tilde{x},t)-\\ & & X\Psi (S,T|0,\tilde{x},\tilde{X},\tilde{x},t)+X\Psi (S,T|0,X,\tilde{X},\tilde{x},t)\text{,}\end{array}\end{array}$$

where $\alpha$, $\varphi$ and $\Psi$ are as defined in Bjerksund and Stensland (2002).

The price of a put option is obtained by the put-call transformation,

$$P_{put}(X,S,T,\sigma ,r,q)=P_{call}(S,X,T,\sigma ,q,r)\text{.}$$

The option price $P_{ij}=P(X=X_i,T=T_j)$ is computed for each strike price in a set $X_i$, $i=1,2,\dots ,m$, and for each expiry time in a set $T_j$, $j=1,2,\dots ,n$.

References

Bjerksund, P and Stensland, G, 2002, Closed form valuation of American options (Discussion Paper 2002/09), NHH Bergen Norway

Genz, A, 2004, Numerical computation of rectangular bivariate and trivariate Normal and $t$ probabilities, Statistics and Computing (14), 151–160