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The current issue and full text archive of this journal is available at www.emeraldinsight.com/0002-1466.htm
AFR 69,3
A relaxed lattice option pricing model: implied skewness and kurtosis
Dasheng Ji
Fifth Third Bank, Cincinatti, Ohio, USA, and
268
B. Wade Brorsen
Department of Agricultural Economics, Oklahoma State University, Stillwater, Oklahoma, USA
Abstract
Purpose – The purpose of this paper is to develop an option pricing model applicable to US options. The lognormality assumption that

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AFR69,3
268
Agricultural Finance ReviewVol. 69 No. 3, 2009pp. 268-283
#
Emerald Group Publishing Limited
0002-1466DOI 10.1108/00021460911002662
A relaxed lattice option pricingmodel: implied skewness andkurtosis
Dasheng Ji
Fifth Third Bank, Cincinatti, Ohio, USA, and
B. Wade Brorsen
Department of Agricultural Economics, Oklahoma State University,Stillwater, Oklahoma, USA
Abstract
Purpose
– The purpose of this paper is to develop an option pricing model applicable to US options.The lognormality assumption that has typically been imposed with past binomial and trinomialoption pricing models is relaxed. The relaxed lattice model is then used to determine skewness andkurtosis of distributions of futures prices implied from option prices.
Design/methodology/approach
– The relaxed lattice is based on Gaussian quadrature. Themarkets studied include corn, soybeans, and wheat. Skewness and kurtosis are implied byminimizing the squared deviations of actual option premia from predicted premia.
Findings
– Positive skewness is the major source of nonnormality, but both skewness and kurtosisare important as the trinomial model that considers kurtosis has greater accuracy than the binomialmodel. The out-of-sample forecasting accuracy of the relaxed lattice models is better than the Black-Scholes model in most, but not all cases.
Research limitations/implications
– The model might benefit from using option prices frommore than one day. The implied skewness and kurtosis were quite variable and using more datamight reduce this variability.
Practical implications
– Empirical results mostly show positive implied skewness, which suggestsextreme price rises were more likely than extreme price decreases.
Originality/value
– The relaxed lattice is a new model and the results about implied highermoments are new for these commodities. There are competing models available that should be able toget similar accuracy, so one key advantage of the new approach is its simplicity and ease of use.
Keywords
United States of America, Option markets, Skewness, Kurtosis, Pricing, Modelling
Paper type
Research paper
Introduction
Among the many option pricing formulas developed, the Black-Scholes optionvaluation formula has been the most popular since its publication in 1973. Thelimitation of the Black-Scholes formula is mainly due to assuming a lognormalunderlying distribution, and that the formula is applicable only to European options.The inaccurate assumptions may explain the implied volatility smile (Tomek andPeterson, 2001) and the inaccuracy of forecasts based on implied volatility (Manfredoand Sanders, 2004). Numerous alternative distributions to the lognormal distributionhave been suggested (e.g. Boyle, 1986; Hull and White, 1989; Myers and Hanson, 1993),but our focus here is on nonparametric methods. Using the idea analogous to Taylorseries as anapproximation foranarbitraryanalyticalfunction,Jarrow andRudd(1982)developed an approximate option valuation formula, in which the biases of an option
The current issue and full text archive of this journal is available at
www.emeraldinsight.com/0002-1466.htm
The project was partially funded by Oklahoma State University’s Targeted Research InitiativeProgram and Hatch Project OKL02170. Considerable advice from Paul Preckel and helpfulcomments from Tim Krehbiel are gratefully acknowledged.
A relaxed latticeoption pricingmodel
269
formula are captured by a function of additional parameters measuring skewness andkurtosis (see Jurczenko
et al.
, 2002 for a review of related literature). The Jarrow-Ruddmodel has the potential to increase the accuracy of the Black-Scholes, but is limited toEuropean options since it is not a lattice model. In practice, various lattice models arewidely used to solve American option pricing problems. This article seeks to develop anoption pricing model that can capture the information in higher moments, but isapplicabletoAmericanoptions.Anewrelaxedlatticeapproachisusedtomeetthisgoal.Lattice modelscan becharacterizedinterms of the constraintsonthe lattice structureand the price information required for calculation. The binomial model of Cox, Ross andRubinstein (CRR) (1979) and the trinomial model of Boyle (1986) are two standard latticemodels. A standard lattice model requires constraints such as recombining, constantparameters, symmetry, and a lognormal underlying distribution.These standard modelsarejustdifferentwaysofsolvingthesameoptionpricingproblemsincetheyallassumealognormaldistribution.Theonlypriceinformationneededforcalculationisthevolatilityof the underlying asset return. Tian (1993) relaxed the constraint that the lattice must besymmetric, but holds the others. Since Tian’s (1993) models maintain the lognormalityassumption, the only potential advantage of his models over the standard models iscomputational accuracy. In the implied tree frameworks of Derman and Kani (1994),Rubinstein (1994, 1998), and Jackwerth (1997), the constraint of an underlying lognormaldistribution is relaxed. However, these implied trees require estimating a parameter foreachpointintheterminaldistribution.Theremustbe enough traded optionsavailabletocalculate the additional implied parameters or else additional restrictions are needed(Jackwerth and Rubinstein, 1996). Moreover, with the implied trees, the parameters inthelatticearenotconstants,andthismaybeanextracomplexityforpractitioners.The relaxed lattice method used here allows a non-lognormal underlying distributionand uses an asymmetric lattice structure, but keeps the advantages of recombining treesand constant parameters. In terms of the price information needed, this relaxed lattice isflexible and provides two possible applications. When the higher moments of theunderlyingassetareknown,anoptioncontractonthisassetcanbevalueddirectlyusingarelaxed lattice based only on the moments of the underlying asset’s returns. On the otherhand,ifoptionpricesareavailable,theseoptionpricescanbeusedtoimplytheparametersof the relaxed lattice model. This article explores the second application of the relaxedlattice. That is, the relaxed lattice model is used to imply the move sizes and transitionprobabilitiesoftheunderlyingdistributionofcorn,soybeans,andwheatfuturesoptions.Different from the implied trees, inwhich the move sizes and transition probabilitiesarecalculated by using an algorithm with the available option prices, the parametersinthe relaxed lattice are implied by a way similar to implying the volatility by ananalytical formula such as Sherrick
et al.
(1992) do with Black-Scholes. In practice, thevolatility is implied by an option pricing formula rather than being estimated by usinghistorical returns because ‘‘although the evidence is still unclear, it appears thatmethods based on calculating implicit volatility are better than methods based onhistorical returns’’ (Sharpe
et al.
, 1995, p. 695). The objective of the empirical work issimilar to Corrado and Su (1996), who used Jarrow and Rudd’s approximation formulato imply the volatility as well as skewness and kurtosis. The performance of such amethod can be examined in different ways. This study examines the out-of-sampleperformance of move sizes and transition probabilities implied directly by the relaxedbinomial and trinomial models. The out-of-sample performance of the relaxed latticemethod is compared with the performances of the parameters implied from otherrepresentativeanalyticalformulasandlattice models.
AFR69,3
270
The relaxed lattices
Generally,the predictedcall option price
c
andput option price
p
based onabinomialortrinomial model can be considered as the functions of several observed factors such asthe underlying asset price, interest rate, expiration date, and strike price, and someunknownparameters:
c
¼
c
ð
S
;
r
;
T
;
X
;
u
Þ ð
1
Þ
p
¼
p
ð
S
;
r
;
T
;
X
;
u
Þ ð
2
Þ
where
c
ðÁÞ
and
p
ðÁÞ
are the binomial or trinomial option formulas for calls and puts,respectively,
S
isthe observedunderlying asset price,
r
isthe observedrisk-freeinterestrate,
T
is the time of expiration of the option contract,
X
is the strike price, and
u
is avectorofunknownparameters.In a binomial model, the parameters
u
,
d
, and
q
, corresponding to upwardmovement, downward movement, and the upward probability in a binomial tree,respectively, constitute the
u
. For a standard binomial model and Tian’s binomialmodel, these parameters are constants. For an implied binomial tree or a generalizedbinomial tree, the parameters vary step by step. In the relaxed binomial modeldeveloped here, the parameters are constant over the steps. However, different from thestandard and Tian’s binomial model, the parameters are not uniquely determined bythe volatility, since no specific probability distribution is assumed. What we know isonly that there is a functional relationship between the option prices and theobservable factorsas defined in(1)and (2). Thisfunctional relationshipcan be obtainedbyabackwardprocedureillustrated below.An
n
-step binomial tree has
n
þ
1 nodes at the final, or the
n
th, step. Given currentunderlyingassetprice
S
andmovesizes
u
and
d
,theunderlyingprice atthe
i
thnodeofthe
n
thstepis
Su
i
d
n
À
i
.Byusing
c
i n
todenotethecallvalueand
p
i n
theputvalueatthe
i
thnodeofthe
n
thstep,theoptionvaluesatthe
n
þ
1
nodesofthefinalsteparecalculatedby
c
i n
¼
max
ð
0
;
Su
i
d
n
À
i
À
X
Þ ð
3
Þ
p
i n
¼
max
ð
0
;
X
À
Su
i
d
n
À
i
Þ ð
4
Þ
where
i
¼
0
;
1
;
. . .
;
n
.Then, for European options, the
n
call and
n
put values at the nodes of the
ð
n
À
1
Þ
thstep aregivenby
c
i n
À
1
¼ ð
qc
i
þ
1
n
þ ð
1
À
q
Þ
c
i n
Þ
e
À
r
Á
t
ð
5
Þ
p
i n
À
1
¼ ð
qp
i
þ
1
n
þ ð
1
À
q
Þ
p
i n
Þ
e
À
r
Á
t
ð
6
Þ
where
i
¼
0
;
1
;
. . .
;
n
À
1
.ForAmericanoptions,theoptionvalues atthe
n
À
1
thstep aregivenby
c
i n
À
1
¼
max
ð
Su
i
d
j
i
À
n
þ
1
j
À
X
;
ð
qc
i
þ
1
n
þ ð
1
À
q
Þ
c
i n
Þ
e
À
r
Á
t
Þ ð
7
Þ
A relaxed latticeoption pricingmodel
271
p
i n
À
1
¼
max
ð
X
À
Su
i
d
j
i
À
n
þ
1
j
;
ð
qp
i
þ
1
n
þ ð
1
À
q
Þ
p
i n
Þ
e
À
r
Á
t
Þ ð
8
Þ
where
i
¼
0
;
1
;
. . .
;
n
À
1
:
By continuing the backward computation until the current time, the call and putoptionvaluesas functions ofthefactorsmentionedabove areobtained.In atrinomial model, there aresixparameters,correspondingto the three jump sizesand three probabilities, to be determined. For the lattice to be recombining, the three jump parameters
u
,
m
, and
d
should satisfy the condition
ud
¼
m
2
. Also, the threeprobability parameters
q
u
,
q
m
, and
q
d
satisfy the condition
q
u
þ
q
m
þ
q
d
¼
1. Thus,there are four free parameters (only three once risk neutrality is imposed). For an
n
-period trinomial tree, there are
2
n
þ
1
nodes at the final period. The
2
n
þ
1
optionvalues atthe finalstep ofan
n
-period trinomialtree arecalculatedas
c
i n
¼
max
ð
Su
max
ð
i
À
n
;
0
Þ
m
ð
n
Àj
i
À
n
jÞ
d
max
ð
n
À
i
;
0
Þ
À
X
;
0
Þ ð
9
Þ
p
i n
¼
max
ð
X
À
Su
max
ð
i
À
n
;
0
Þ
m
ð
n
Àj
i
À
n
jÞ
d
max
ð
n
À
i
;
0
Þ
;
0
Þ ð
10
Þ
i
¼
0
;
1
. . .
;
2
n
ThentheEuropeanoptions atthe
n
À
1
thstep arevalued as
c
i n
À
1
¼ ð
q
u
c
i
þ
2
n
þ
q
m
c
i
þ
1
n
þ
q
d
c
i n
Þ
e
À
r
Á
t
ð
11
Þ
p
i n
À
1
¼ ð
q
u
p
i
þ
2
n
þ
q
m
p
i
þ
1
n
þ
q
d
p
i n
Þ
e
À
r
Á
t
ð
12
Þ
where
i
¼
0
;
1
;
. . .
;
n
À
1
.IftheoptionsareAmerican,the optionsatthe
ð
n
À
1
Þ
thstep arevalued as
c
i n
À
1
¼
max
Su
max
ð
i
À
n
þ
1
;
0
Þ
m
ð
n
À
1
Àj
i
À
n
þ
1
jÞ
d
max
ð
n
À
1
À
i
;
0
Þ
À
X
;
ð
q
u
c
i
þ
2
n
þ
q
m
c
i
þ
1
n
þ
q
d
c
i n
Þ
e
À
r
Á
t
Á
ð
13
Þ
p
i n
À
1
¼
max
X
À
Su
max
ð
i
À
n
þ
1
;
0
Þ
m
ð
n
À
1
Àj
i
À
n
þ
1
jÞ
d
max
ð
n
À
1
À
i
;
0
Þ
;
ð
q
u
p
i
þ
2
n
þ
q
m
p
i
þ
1
n
þ
q
d
p
i n
Þ
e
À
r
Á
t
Á
ð
14
Þ
i
¼
0
;
1
;
. . .
;
n
À
1
Similar to the procedure for the binomial tree models, the backward computationcontinues until the currenttime toobtain the call and put optionvalues as the functionsofseveralobserved or unknownarguments.Thus far there is no difference between the relaxed lattice approach and the previousstandard tree models. The difference between the relaxed lattices and the standard treemodels is the method to determine the move sizes and transition probabilities. Thestandard tree models add the additional restriction of lognormality on the latticeframework described above. Then the move sizes and transition probabilities aredetermined uniquely by the volatility of underlying prices. Furthermore, the standard

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