​​​Pat Hagan
Numerix
hagan@numerix.com
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 Sk(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 Sk(t) be the price of
asset k in its own currency
1.1a Sk(t) = number of units of currency k for each
share of asset k
and define the basket
1.1b X(t) = Sk
wkSk(t) - sum over k of
wkSk(t)
where wk 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(tex) — K]+ paid on the
settlement date tset (call)
[K — X(tex)]+ paid on the
settlement date tset (put)
b) Asian options on the basket. These options have a pre-set
series of observation dates t1, t2, …,
tm. Define the Xavg to be the average value of
the basket on these dates
Xavg = [X(t1) + X(t2) +
… + X(tm)]/m
Asian options are like the above options, except they pay off on
the average:
[Xavg — K]+ paid on the
settlement date tset (call)
[K — Xavg]+ paid on the
settlement date tset (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 tset if
X(tex) > K (digital call)
1 is paid on the settlement date tset if
X(tex) < K (digital put)
d) European (arbitrary payoffs) on the basket. These options pay
G(X(tex) ) paid on the settlement date tset
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 <tex,
then these options pa
y
[X(tex) — K]+ paid on the
settlement date tset (call)
[K — X(tex)]+ paid on the
settlement date tset (put)
Otherwise they pay nothing
ii) Up & out. If the (upper) barrier is not breached,
X(t) < B for all t < t <tex,
then these options pay
[X(tex) — K]+ paid on the
settlement date tset (call)
[K — X(tex)]+ paid on the
settlement date tset (put)
Otherwise they pay nothing
iii) Double barrier. If neither barrier is breached,
B1 < X(t) <
B2 for all t < t
<tex,
then these options pay
[X(tex) — K]+ paid on the
settlement date tset (call)
[K — X(tex)]+ paid on the
settlement date tset (put)
Otherwise they pay nothing.
​