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

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

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://www.nag.com/numeric/nl/nagdoc_28.5/flhtml/s/s30jaf.html

Parameters
calputstr, length 1

Determines whether the option is a call or a put.

A call; the holder has a right to buy.

A put; the holder has a right to sell.

xfloat, array-like, shape

must contain , the th strike price, for .

sfloat

, the price of the underlying asset.

tfloat, array-like, shape

must contain , the th time, in years, to expiry, for .

sigmafloat

, the annual total volatility, including jumps.

rfloat

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

lamdafloat

, the number of expected jumps per year.

jvolfloat

The proportion of the total volatility associated with jumps.

Returns
pfloat, ndarray, shape

contains , the option price evaluated for the strike price at expiry for and .

Raises
NagValueError
(errno )

On entry, .

Constraint: .

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On entry, .

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On entry, .

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On entry, .

Constraint: and .

(errno )

On entry, .

Constraint: and .

(errno )

On entry, .

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(errno )

On entry, .

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(errno )

On entry, .

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(errno )

On entry, .

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(errno )

On entry, .

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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,

Here is the instantaneous expected return on the asset price, ; is the instantaneous variance of the return when the Poisson event does not occur; is a standard Brownian motion; is the independent Poisson process and where is the random variable change in the stock price if the Poisson event occurs and is the expectation operator over the random variable .

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

where is the time to expiry; is the strike price; is the annual risk-free interest rate; is the Black–Scholes–Merton option pricing formula for a European call (see opt_bsm_price()).

where is the total volatility including jumps; is the expected number of jumps given as an average per year; 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 .

The option price is computed for each strike price in a set , , and for each expiry time in a set , .

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