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Wilmott
magazine
Christian Wallner
Ostpreußenstraße 6A
21391 Reppenstedt
GERMANY
christian.wallner@gmx.net
Uwe Wystup
HfB—Business School of Finance and Management
Sonnemannstrasse 9–11
60314 Frankfurt am Main
GERMANY
http://www.mathfinance.de
Efficient Computation of Option Price
Sensitivities for Options of American Style
an example. For contract parameters maturity in years
T
, strike
K
and
put/call indicator
φ
, which is
+
1 for a call and
−
1 for a put, the payoff of
the option is
[
φ(
S
T
−
K
)
]
+
=
max[0
,φ(
S
T
−
K
)
]
.
(2)
We denote by
V
(
t
,
x
)
the value of an American style put or call at time
t
if
the spot
S
t
takes the value
x
. It is well known (see e.g. Karatzas and Shreve
1998) that in this model the value at time zero is given by
V
(
0
,
S
0
)
=
sup
τ
∈
T
IE
[
e
−
r
d
τ
[
S
τ
−
K
)
]
+
]
,
(3)
1.
Introduction
We examine which is a suitable method to compute Greeks for American
style call and put options in the BlackScholes model. We choose an
exchange rate for the underlying following a geometric Brownian motion,
dS
t
=
S
t
[
(
r
d
−
r
f
)
dt
+
σ
dW
t
]
,
(1)
under the riskneutral measure. As usual
r
d
denotes the domestic interest
rate,
r
f
the foreign interest rate,
σ
the volatility. The analysis we do is also
applicable to equity options, but we take the foreign exchange market as
Abstract:
No frontoffice software can survive without providing derivatives of option prices with respect to underlying market or model parameters, the so called Greeks. If
a closed form solution for an option exists, Greeks can be computed analytically and they are numerically stable. However, for American style options, there is no closedform
solution. The price is computed by binomial trees, finite difference methods or an analytic approximation. Taking derivatives of these prices leads to instable numerics or mis
leading results, specially for Greeks of higher order. We compare the computation of the Greeks in various pricing methods and conclude with the recommendation
to use
LeisenReimer trees
.
Keywords:
American Options, Greeks, LeisenReimer trees.
JEL classification: C63, F31
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where
T
is the set of all stopping times taking values in [0
,
T
]. A closed
form solution for this optimization problem has not yet been found.
1.1
Option Price Sensitivities
Option price sensitivities, the socalled
Greeks
of option values are deriva
tives with respect to market variables or model parameters. The most
commonly used Greeks are listed in Table 1. Numerous relationships and
properties of the Greeks for European style options are presented in Reiss
and Wystup (2001). Other relevant publications include the work by Carr
(2001), Broadie and Glasserman (1996) in the case of Monte Carlo simula
tions, Pelsser and Vorst (1994) in the case of binomial trees, the work by
Eric Benhamou (2003) and (2004), who uses Malliavin calculus, and the
contribution by Rogers and Stapleton (1998) using binomial trees with a
random number of steps. Joubert and Rogers (1997) use a lookup table for
a fast, accurate and inelegant valuation of American options. Formulae
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 Fall '09
 Weinberger,Edward

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