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A Water Wave Interaction Problem, Scott Rethorst, Vehicle Research Corportation

Scott Rethorst

Vehicle Research Corporation

1806 Foothill Street

S Pasadena CA 91030

A concentrated pressure distribution propagating steadily over a flat ocean generates a single mode wave shape of large amplitude locally, together with a trailing gravity wave (Lamb's Hydrodynamics, Art.242). Use of this theory in design has the local wave amplitude of the order of 20-50 ft.This theory assumes a flat, undisturbed ocean in the absence of the forcing pressure distribution. The problem posed relates to a real ocean surface containing waves from other disturbances. How do small amplitude ocean waves interact with the large amplitude wave? What happens as a 4ft. amplitude wave is made to climb a 20-50 ft. approaching wave? Does it lose energy and smooth out? Two-dimensional, irrotational, inviscid, but non-linear free surface hydrodynamics apply.

Options on Baskets, Pat Hagan, Numerix

Pat Hagan

Many equity options are written on the performance of a basket of
assets; most commonly the basket consists of 2-30 shares of stocks.
Let S_{k}_{}(t) be the value of asset k. To keep our European
customers happy, we cannot assume that the asset prices are all in
terms of the same currency. So, let S_{k}_{}(t) be the price of
asset k in its own currency

1.1a S_{k}_{}(t) = number of units of currency k for each
share of asset k

and define the basket

1.1b X(t) = S_{k}_{}
w_{k}S_{k}_{}(t) - sum over k of
w_{k}S_{k}_{}(t)

where w_{k}_{} are given weights. In decreasing order of
importance, the options we need to value are (these are defined
below):

a) standard European call/put options on the basket

b) Asian options on the basket

et

d) European (arbitrary payoffs) on the basket

e) barrier options on the basket.

There are two standard methods of pricing these options:

(i) Monte Carlo simulation — this is slow, inaccurate, and leads to "noisy" hedges

(ii) "moment matching" methods — this method can be stunningly bad.

We’d like to develop a very fast, highly accurate method to
determine the value of these options, and the correct hedges
(derivatives of the value). We’d like to see if an expansion
method (probably a perturbation method) can be used. A complete
"win" would result in something like effective media theory
which would replace the *n* assets in the basket by the
evolution of a single asset, the basket.

Options

a) standard European call/put options on the basket. These options are usually automatically exercised if they are in-the-money, and are usually cash-settled for

[X(t_{ex}_{}) — K]^{+} paid on the
settlement date t_{set}_{} (call)

[K — X(t_{ex}_{})]^{+} paid on the
settlement date t_{set}_{} (put)

b) Asian options on the basket. These options have a pre-set
series of observation dates t_{1}, t_{2}, …,
t_{m}. Define the X_{avg}_{} to be the average value of
the basket on these dates

X_{avg}_{} = [X(t_{1}) + X(t_{2}) +
… + X(t_{m})]/m

Asian options are like the above options, except they pay off on the average:

[X_{avg}_{} — K]^{+} paid on the
settlement date t_{set}_{} (call)

[K — X_{avg}_{}]^{+} paid on the
settlement date t_{set}_{} (put)

The most common type of Asian are far-from-the money puts, which are sold as protection on, for example, high tech portfolios

c) European digitals on the basket. These p

ay

1 is paid on the settlement date t_{set}_{} if
X(t_{ex}_{}) > K (digital call)

1 is paid on the settlement date t_{set}_{} if
X(t_{ex}_{}) < K (digital put)

d) European (arbitrary payoffs) on the basket. These options pay

G(X(t_{ex}_{}) ) paid on the settlement date t_{set}_{}

where G is defined in the contract. Examples are parabolic payoffs, range notes, …

e) Barrier options on the basket.

i.) __Down & out.__ If the (lower) barrier is not breached,

X(t) >B for all t < t <t_{ex}_{},

then these options pa

y

[X(t_{ex}_{}) — K]^{+} paid on the
settlement date t_{set}_{} (call)

[K — X(t_{ex}_{})]^{+} paid on the
settlement date t_{set}_{} (put)

Otherwise they pay nothing

ii) __Up & out.__ If the (upper) barrier is not breached,

X(t) < B for all t < t <t_{ex}_{},

then these options pay

[X(t_{ex}_{}) — K]^{+} paid on the
settlement date t_{set}_{} (call)

[K — X(t_{ex}_{})]^{+} paid on the
settlement date t_{set}_{} (put)

Otherwise they pay nothing

iii) __Double barrier.__ If neither barrier is breached,

B_{1} < X(t) <
B_{2} for all t < t
<t_{ex}_{},

then these options pay

[X(t_{ex}_{}) — K]^{+} paid on the
settlement date t_{set}_{} (call)

[K — X(t_{ex}_{})]^{+} paid on the
settlement date t_{set}_{} (put)

Otherwise they pay nothing.

Inverse Problems for Optical Fiber Devicemeasurement and Design, Gregory G. Luther, Corning Inc.

Gregory G. Luther

Corning Inc.

Optical fibers made of glass have now been developed that can transmit over 1Tb/s over thousands of kilometers. As telecommunications companies prepare to sell more and more transmission bandwidth, the demand for optical fibers has increased. Optical fibers are used in a variety of applications forming the key constituent not only in transmission, but in amplifiers and other telecommunications components. Techniques for characterizing optical fibers at various stages in the research, development and manufacturing cycle continue to be essential. The ability to quickly design fiber characteristics for fiber-based components is critical for delivering new products in the market place. We will investigate mathematical techniques in connection with the measurement of transverse index profiles of fibers at various stages in the manufacturing process. We will also consider the measurement and synthesis of longitudinal properties of fibers.

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