Double-Sided Parisian Options

Winter School Mathematical Finance - January 22th 2007 – Jasper Anderluh

joint work with J.A.M. van der Weide

Overview

The Parisian Option Contract : Introduction and notation

Contract Pay-off, Parisian Stopping time,

Contract types, Applications.

Fourier Transform

Transform of damped probability,

Transform of the Parisian stopping time.

Numerical Examples

Price behavior of double-sided Parisian in call,

Greek behavior, various contract types.

Introduction and Notation (2)

In general, the pay-off of a path-dependent option depends on the whole stock price path.

A Parisian option knocks in or out as soon as the stock price S makes an excursion below or above some barrier L for time D.

K

L

0.08

Let be a filtered probability space with a Brownian motion. We use the Black-Scholes economy, given by the following dynamics :

Using classical results we can compute the value at time t of a claim , the payoff at expiry time T by :

The pay-off should represent the Parisian option pay-off, so we introduce the Parisian stopping time.

Introduction and Notation (3)

Introduce the following notation for the last time before t we hit level L,

For the single-sided Parisian stopping time,

And for the double-sided Parisian stopping time,

Introduction and Notation (4)

Consider the following example D is 10 days,

Introduction and Notation (5)

For the double-sided Parisian in call the pay-off is given by,

For the single-sided Parisian options, the following contract types can be constructed,

Introduction and Notation (6)

Laplace Transform

(1997) Chesney, Jeanblanc and Yor,

“Brownian Excursions and Parisian Barrier Options”

PDE approach

(1999) Haber, Schonbucher and Wilmott,

“Pricing Parisian Options”

Even in the Black-Scholes world,

obtaining accurate prices is not trivial.

Introduction and Notation (7)

Introduction and Notation (8)

At present time not exchange traded, so are there applications?

Building block of convertible bonds with soft-call constraint [Kwok].

Appear in investment problems when considered from the point of view of real options [Gauthier].

Used in modeling credit risk [Moraux].

Application in life-insurance [Chen and Suchanecki].

Fourier Transform (1)

In the Black-Scholes world is given by a GBM. Like Carr and Madan we find by using Girsanov / Change of Numeraire,

It is convenient to use the following short-hand notation,

So, the quantity of interest is the following,

Now we compute the following Fourier Transform,

Substitute and use we get

Note, that in order to proceed, we restate everything in terms of the underlying standard Brownian motion W,

Fourier Transform (2)

Now we have the following lemma,

where,

For the Fourier transforms and the following holds,

Note the independence between and . Where does it come from? Can we compute the left-hand side expectations?

Fourier Transform (3)

The Brownian meander at time t>0 is defined by,

We are only interested in its final value (u=1), denoted by,

By CJY this final value is for every t>0 independent of the pair and , where N has the following density,

Fourier Transform (4)

For the double-sided case we introduce,

And we have the following lemma,

where,

For any bounded measurable function f we have,

Fourier Transform (5)

Now we have a martingale argument,

And by the previous lemma we can compute,

Fourier Transform (6)

Finally resulting in the following theorem,

where,

For the restricted Laplace transforms of the following holds,

By taking limits, we can show that the probability that a standard Brownian motion makes a positive excursion of length D2 before making a negative excursion of length D1 is given by,

also follows from excursion theory

Fourier Transform (7)

Combining these results gives,

Now everything is known and the formulas for Fourier transforms follow by elaborate computation.

Fourier Transform (8)

Numerical Examples (1)

Double-sided Parisian in call prices.

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ce

Numerical Examples (2)

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Double-sided Parisian in call prices.

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Double-sided Parisian in call prices.

Numerical Examples (3)

75 75.5 76 76.5 77 77.5 78 78.5 79 79.5 80-0.15

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lta

d=0 d=1 d=2 d=3 d=4d=5

d=6

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d=8

d=9

plain-vanilla

Double-sided Parisian in call delta, NOTE : negative gamma.

Numerical Examples (4)

Numerical Examples (5)

Negative gamma? Then there should be negative theta.

Now you can construct the following contract types:

. The double-sided Parisian in call. Most expensive.

with or .

The single-sided Parisian up-and-in call. Price inbetween.

.The single-sided Parisian down-before-up-in call. Cheapest.

Numerical Examples (6)