naginterfaces.library.specfun.opt_​jumpdiff_​merton_​price

naginterfaces.library.specfun.opt_jumpdiff_merton_price(calput, x, s, t, sigma, r, lamda, jvol)

opt_jumpdiff_merton_price computes the European option price using the Merton jump-diffusion model.

For full information please refer to the NAG Library document for s30ja

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/s/s30jaf.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 annual total volatility, including jumps.

rfloat

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

lamdafloat

$\lambda$, the number of expected jumps per year.

jvolfloat

The proportion of the total volatility associated with jumps.

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, $lamda=⟨value⟩$.

Constraint: $lamda>0.0$.

(errno $10$)

On entry, $jvol=⟨value⟩$.

Constraint: $jvol\ge 0.0$ and $jvol<1.0$.

Notes

opt_jumpdiff_merton_price uses Merton’s jump-diffusion model (Merton (1976)) to compute the price of a European option. This assumes that the asset price is described by a Brownian motion with drift, as in the Black–Scholes–Merton case, together with a compound Poisson process to model the jumps. The corresponding stochastic differential equation is,

$$\frac{dS}{S}=(\alpha -\lambda k)dt+\hat{\sigma }{dW}_t+{dq}_t\text{.}$$

Here $\alpha$ is the instantaneous expected return on the asset price, $S$; ${\hat{\sigma }}^2$ is the instantaneous variance of the return when the Poisson event does not occur; ${dW}_t$ is a standard Brownian motion; $q_t$ is the independent Poisson process and $k=E[Y-1]$ where $Y-1$ is the random variable change in the stock price if the Poisson event occurs and $E$ is the expectation operator over the random variable $Y$.

This leads to the following price for a European option (see Haug (2007))

$$P_{call}=\sum _{j=0}^{\infty }\frac{e^{-\lambda T}{\left(\lambda T\right)}^j}{j!}{C}_j(S,X,T,r,{\sigma }_j^{\prime })\text{,}$$

where $T$ is the time to expiry; $X$ is the strike price; $r$ is the annual risk-free interest rate; $C_j(S,X,T,r,{\sigma }_j^{\prime })$ is the Black–Scholes–Merton option pricing formula for a European call (see opt_bsm_price()).

$$\begin{array}{c}\begin{array}{c}{\sigma }_j^{\prime }=\sqrt{z^2+{\delta }^2\left(\frac{j}{T}\right)}\text{,}\\ z^2={\sigma }^2-\lambda {\delta }^2\text{,}\\ {\delta }^2=\frac{\gamma {\sigma }^2}{\lambda }\text{,}\end{array}\end{array}$$

where $\sigma$ is the total volatility including jumps; $\lambda$ is the expected number of jumps given as an average per year; $\gamma$ is the proportion of the total volatility due to jumps.

The value of a put is obtained by substituting the Black–Scholes–Merton put price for $C_j(S,X,T,r,{\sigma }_j^{\prime })$.

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

Haug, E G, 2007, The Complete Guide to Option Pricing Formulas, (2nd Edition), McGraw-Hill

Merton, R C, 1976, Option pricing when underlying stock returns are discontinuous, Journal of Financial Economics (3), 125–144