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DERIVING IMPLIED DISTRIBUTIONS FROM .  

COMMODITY OPTION PRICES: AN APPLICATION TO 1 '

SOYBEAN, CORN AND WHEAT USING
MIXTURE OF LOGNORMALS

Han B Paper for the Degree of M. S.
MICHIGAN STATE UNIVERSITY
RUI FAN
2001

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DERIVIN G IMPLIED DISTRIBUTIONS FROM
COMMODITY OPTION PRICES: AN APPLICATION TO
SOYBEAN, CORN AND WHEAT USING
MIXTURE OF LOGNORMALS

Rui Fan

A PLAN B PAPER

Submitted to
Agricultural Economics Department
In partial fulfillment of the requirement for the degree of

MASTER OF SICENCE

Department of Agricultural Economics

2001

ACKNOLEDGEMENTS

I would like to acknowledge all the help from Dr. Steven Hanson and Dr. Robert
Myers, from the initial ideas for the foundation for this paper, to the proof reading of the
final draft. Without them, this would not have been possible.

Also I would like to thank Dr. James Hilker, Dr. Roy Black for providing

guidance to my study.

TABLE OF CONTENTS

Section 1 Introduction ........................................................................ 1
Section 2 Literature Review ................................................................. 6
Section 3 Option Pricing and Implied Distribution Forecast ..................... 10
3.1. Black’s Commodity Option Pricing Model ....................................... 10
3.2. A Mixture of Loghormals Approach .............................................. 14

3.3 Using Market Traded Option Prices To Generate Implied Distribution
Forecast ................................................................................ 17
Section 4 Application to Corn, Soybean and Wheat Futures Prices ............... 21
4.1 Data source ............................................................................. 21
4.2 Estimating the Implied Distribution .................................................. 23
Section 5 Soybean Estimation Results .................................................... 26
Section 6 Corn Estimation Results ....................................................... 32

Section 7 Further Comparisons of Enlarged Graphs of Black and MLN Models

....................................................................................... 45

Section 8 Wheat Estimation Results ...................................................... 48

Section 9 Conclusion ......................................................................... 53

9.1 Comparison of the Two Model ...................................................... 53

9.2 Conclusion ............................................................................. 56

Appendix I Derivation of Formulas of MLN Method ................................... 58
Appendix II PDFs, CDFs And Option Pricing Ermrs of Storage Contracts of

Soybean, Corn and Wheat ..................................................... 68

Reference ........................................................................................... 80

ii

LIST OF TABLES

Table 4.1: Data Source .............................................................................. 22
Table 4.2: Estimated parameters forpall contracts/days by MLN methods .................. 24
Table 7.1: Contribution to variance by different range of futures price —by Black and
MLN models ............................................................................ 47
Table 9.1: moments of the implied distribution and mean square deviation of objective

function for all contracts/days ........................................................ 54

iii

Figure 5.1:
Figure 5.2:
Figure 5.3:
Figure 6.1:
Figure 6.2:
Figure 6.3:
Figure 6.4:
Figure 6.5:
Figure 6.6:
Figure 6.7:
Figure 6.8:
Figure 6.9:
Figure 7.1:
Figure 8.1:
Figure 8.2:

Figure 8.3:

LIST OF FIGURES

Estimated PDFs — November (harvest) contract of soybean ..................... 28
Estimated CDFs — November (harvest) contract of soybean .................... 29
Option pricing errors — November (harvest) contract of soybean ............... 30
Estimated PDFs — December (harvest) contract of corn. . . . . . . . . . . . . . ....33
Estimated CDFs — December (harvest) contract of corn .......................... 34
Option pricing errors — December (harvest) contract of corn .................... 35
Estimated PDFs — December (harvest) contract of corn — 2000 ................. 38
Estimated CDFs — December (harvest) contract of com - 2000 ................. 39
Estimated PDFs — December (harvest) contract of corn — 1998 ................. 41
Estimated CDFs — December (harvest) contract of corn - 1998 ................. 42
Estimated PDFs — December (harvest) contract of corn — 1999 ................. 43
Estimated CDFs — December (harvest) contract of corn - 1999 ................. 44
Enlarged DF of Black and MLN models — December contract of com ........ 46

Estimated PDFs — July (harvest) contract of wheat. . 49
Estimated CDFs — July (harvest) contract of wheat ................................ 50
Option pricing errors — December (harvest) contract of wheat ................... 51

iv

Section 1. Introduction

Commodity prices are highly volatile, particularly compared to prices of
manufactured consumer goods (Newbery and Stiglitz 1981). This high volatility is a
result of both seasonal changes and daily supply and demand shocks. When important
information about supply and demand conditions arrives, significant price changes may
occur. These price movements generate price uncertainty, which can pose major
problems to policy makers and industry participants, particularly in countries where
export earnings and GDP depend heavily on sales of primary commodities.

In order to reduce the likelihood of unfavorable outcomes, agricultural producers
use various strategies to manage price risks. One strategy is to use forward pricing
instruments such as forward contracts, futures and options to hedge the risk in selling or
buying the physical commodity. Futures and options markets provide important hedging
instruments for many of the major commodities in the United States including com,
soybean, wheat, hogs, cattle, etc. These contracts offer speculative instruments to
investors and additional pricing strategies to agricultural producers and agribusinesses.

A good estimate of the distribution of commodity prices is important for several
reasons. First, efficient pricing of derivatives assets such as commodity options and the
effective use of these derivatives requires a good estimate of the underlying commodity
price distributions. Second, policy makers need to be able to observe market reactions to
their policies or to exogenous shocks. Third, knowledge of the perceived commodity
price distribution could also be of value to the private sector. In managing risk in physical
, commodity trading, investors could make decisions based on the market perception of

commodity price risk. Corporations could make hedging decisions based on the

uncertainty perceived by the market. Therefore, a lot of effort has been put into
forecasting the distribution of commodity prices. However, forecasting commodity price
distributions is extremely difficult due to two important features displayed by most
commodity prices. First, they often displayinon-stationarity. Second, there are periods
when price jumps abruptly to very high levels (or falls to very low levels) relative to their
long run average. These two features of commodity price series make forecasting the
distribution of commodity price extremely difficult.

Traditional price forecasting approaches use historical data. One approach is to
develop market structure models of commodity prices according to economic theory and
supply demand conditions. Much of this research relies on a standard set of economic
methods. Forecasts can be generated from either a single equation or from the
unrestricted or restricted reduced-form models. Streeter and Tomek (1992) found that the
variance of futures price changes depends on a variety of factors, including time-to-
maturity, seasonality, economic conditions and market structure. Structural models have
.the potential to provide useful information in explaining commodity price behavior.
However, one difficulty in generating forecasts from structural models is the timing of
when information is known. Variables whose value will not be known at the time that
actual forecasts are made cannot be incorporated into the model. If there is such a
variable in the model, then its value must be forecasted independently and, hence, treated
asiexogenous in the model.

Traditional approaches also include time series models. These models have been
successfully applied to capture the time-varying volatility of commodity prices, such as

GARCH (Myers and Hanson, 1993), time-independent mixture-of-normals distribution

(Kim and Kon, 1994) and exponentially weighted moving average (EWMA) models
(Venkateswanan and Neenakshi, 1993). However, all these methods only involve
historical price data and don’t include all available market information. In commodity
price time series even when stochastic volatility is explicitly modeled, as in a GARCH
framework, parameters are often updated quite slowly even when major regime changes
have occurred.

A well-known procedure that is different from the traditional historical data
approach incorporates available market information into forecasts of commodity price
distributions. This method derives the distribution imbedded in the price of options
written on commodity futures prices traded on an exchange. An option is a contract
which permits, but does not require, the holder to buy (sell) the underlying asset at a
predetermined strike price on a given expiration date.1 Because the option price itself is
observable, and'because the option price is a function of the future price distribution,
then, given a theoretical model, the market prices of options can be used to derive the
market’s expectation of the future price distribution. This method is referred as a
“market-based” forecast because it is based entirely on the expectation of participants in
the option market. Black-Scholes (1973) option pricing formula has been widely used to
back out the distribution estimate implied by the model, given market determined option
premia. In commodity markets, Black’s extension of the Black-Scholes model to price
options written on futures contracts has been widely used to derive distributions of the

underlying commodity prices.

 

l A call option gives the holder the right to buy the underlying asset by a certain date for a certain price. A
put option gives the holder the right to sell the underlying asset by a certain date for a certain price.

Compared with using historical data, Black’s formula has several advantages.
First, this method incorporates market information. The implied distribution reflects the
expectation of market participants about the future price distribution using all information
available on' the day the option is traded. Second, this approach has the advantages of
simplicity and tractability. In Black’s model, all information required to implement the
option pricing formulas is directly observable except for the volatility of the underlying
asset price.

Even though Black’s model has been widely accepted, this approach is
inconsistent in the following sense. In order to compute the current implied volatility
from the Black model, the volatility is assumed deterministic over time. But resulting
implied estimates, together with the historical time series data, suggest that volatilities are
in fact time varying and stochastic. Thus the purpose of this research is to develop and
apply an alternative method that will impose fewer restrictions than Black’s model.

The option pricing model derived in this research assumes that futures price
changes follow a mixture of lognormal distributions (MLN). This method imposes a
flexible distributional assumption which is capable of capturing the fat-tailed, peaked,
and skewed characteristics of the underlying asset price distribution. This general class of
densities is shown by Ritchey (1986) to be descriptive of the majority of common stock
returns. Furthermore, the density is mathematically tractable as well as economically

plausible. In the context of risk neutral valuationz, this model provides an exact solution.

 

2 Note that the probability derived from option prices will be the risk-neutral probabilities, because
contingent claims are priced accordingly to the risk-neutral distribution rather than the actual distribution.
These two distributions will be equal if commodity price risk is not priced. For the remainder of this paper,
the term probability, CDF, PDF, and the expected value will refer to these measures under the risk-neutral
distribution.

This paper develops a method to estimate the distribution of futures price
movements from traded European options using a flexible mixture of lognormals
distributional assumption. More specifically, the objectives are: 1) develop an alternative
method for pricing commodity options when excess skewness and fat-tailed distributions
in the underlying futures price can be modeled as a mixture of two lognormal processes.
2) apply this method to elicit implied distributions from corn, soybean, and wheat futures
option prices and test how well this forecasting model predicts distribution. 3) evaluate
the performance and forecasting ability of this alternative method.

The rest of this paper is organized as follows: Section 2 is literature review.
Section 3 describes a mixture of lognormals method for estimating an asset’s PDF from
European option prices. Section 4 discusses the application of the method to the corn,
soybean, and wheat futures market, and compares the estimated distribution with those

derived from Black’s model. Section 5 concludes.

Section 2: Literature Review

In their path-breaking paper, Black and Scholes succeeded in solving a
differential equation to obtain exact formulas for pricing of European call and put
options. Based on this option pricing formula for non—dividend paying stock, Black
derived the option pricing formula for a commodity futures option3. In recent years, the
Black-Scholes option valuation formula has achieved wide acceptance by both
theoreticians and market practitioners. The distribution underlying the formula is that the
asset price follows geometric Brownian motion with constant volatility. Of the five
variables", which are necessary to specify the model and determine the equilibrium
market premium, all are directly observable except volatility (the standard derivation of
return from the underlying assets), which can be estimated using historical data or other
approaches. 1

Therefore, simultaneous observation of option prices and the price of the
underlying asset can be used to estimate volatility that is expected over the remaining life
of the option. A large amount of empirical research has been done to use market
determined option premia to back out the distribution estimate implied by the option
pricing model. Option prices have been used to describe the distribution of stock returns

(Ait-Sahalia and Lo 1996, Jackwerth and Rubinstein 1996), commodity prices (Turvey

 

3 Futures price F of commodity can be related to its spot prices by an expression of the form F =Sea’T‘”,

where T is maturity date, I is current date and. In the case of financial asset, or is the risk-free rate of interest
less the yield on the asset. In the case of commodity, or is the risk-free rate of interest plus the storage costs
per dollar per unit time less the convenience yield. It is shown in Black (1976) that a futures price can be
treated in the same way as a security paying a continuous dividend yield at risk free interest rate r. Thus,
the expected growth rate in a futures price in a risk-neutral world would be zero. The expected gain to the
holder of a futures contract in a risk-neutral world must be zero.

4 The five variables in Black-Scholes model are strike price, current future price, time to maturity, risk-free
interest rate and volatility of the underlying asset.

1990, Hauser and Neff 1985), oil prices (Melick and Thomas 1997), exchange rate
(Campa, Chang, and Reinder 1997; Malz 1996), and interest rates (Abken 1995,
McCauley and Mclick 1996). These studies find that the variances implied from market
option premium and the Black-Scholes model (or extensions of the Black-Scholes model
in different markets) are often better predictors of variances of underlying assets than
those obtained from historical data.

Other studies on the information content of implied volatilities from different
underlying assets, including commodity prices, yield disappointing results. Canina and
Figlewski (1992) find that “implied volatility has virtually no correlation with future
volatility, and it does not incorporate the information contained in recent observed
volatility.” Day and Lewis (1992) find the coefficient of the implied volatility is barely
significant in a volatility forecast equation. In addition, some studies reported that market
prices of options appear to deviate systematically from theoretical prices. These
aberrations are documented in empirical studies by Black (1975), MacBeth and Merville
. (1980), Rubinstein (1985), and Whaley (1982). A main explanation for these observed
discrepancies is that the distribution of the underlying asset’s return is not lognormal, as
assumed by Black-Scholes model. In the case of commodity futures, the available
evidence doesn’t support identical normal distributions for the distribution of commodity
futures price changes. I

Some researchers have tried to overcome this inconsistency by specifying a
pricing model that deals rigorously with the stochastic nature of volatility as in Hull and
White (1987) or Wiggins (1987). StOchastic volatility models require the investor to

forecast the entire joint probability distribution for asset returns including changes in

volatility and'also the market price of volatility risk. These requirements make these
models significantly more difficult to implement than Black—Scholes and other constant-
volatility models. Therefore, some researchers have specified deterministic volatility
functions that allow volatility to vary deterrninistically with the asset price or time. For
example, Rubinstein (1994) and Jackwerth and Rubinstein (1996) used implied
binominal trees to estimate the underlying binomial probability model assumed to be
driving the asset price. However, many of these methods involve complex numerical
simulation and optimization routines which are somewhat costly because they are
difficult to implement and can take a long time to converge. Some researchers have
essentially ignored the evolution of daily volatilities over time and began with an
assumption about the future distribution of the underlying asset to directly recover the
parameters of that distribution (e.g. Fackler and King 1990). Among these future
distributional assumptions, mixtures of lognormal distributions have been applied to
different markets.

The mixture of lognormals distribution density is simply tractable. This method
has a natural economic interpretation — that of multiple alternative regimes. It has been
shown to be appropriate for stock (Ritchey, 1990), crude oil futures (Melick and Tomas
1997), and foreign currency (Leahy and Thomas 1996). In Mclick and Thomas (1997),
for example, three different lognormal distributions for the price of oil correspond to
various outcomes of the 1991 Gulf War. In Mclick and Thomas’s work, they suggested
“this methodology should be useful to researchers who wish to impose a minimum of
structure and are i) examining other markets during unsettled times, or ii) investigating

asset price distribution that are not adequately described by the lognormal distribt’rtion

(e. g. a distribution leptokurtosis).” Although this method has been used to derive implied
volatility for many markets, the performance of this model has not yet been tested for
estimating distributions from commodity futures options. Commodity price series tend to
' exhibit stochastic volatility and non-stationarity features. Furthermore, commodity prices
tend to have significantly “peaked” and “fat-tailed” distributions relative to the Gaussian
density. The mixture of lognormals distribution could give a flexible shape to the
distribution of commodity prices at maturity. The mixture assumption could also
accommodate large price changes and be capable of capturing other observed features of

commodity prices.

Section 3: Option Pricing and Implied Probability Distributions

In this section, an option pricing formula for commodity futures option is
developed to compute the implied mixture of lognormal distribution for the underlying
commodity futures prices. First, Black’s option pricing model for commodity futures is
presented based on the standard lognormal distribution assumption for the underlying
futures prices. Second, a risk-neutral option pricing formula for European options will be
developed for the case when the underlying futures price can be described by a mixture
of two lognormal distributions (MLN). This functional form of the terminal distribution
can easily accommodate a wide variety of shapes for the terminal futures price
distributions, giving it the advantage of flexibility, parsimony, and generality. Finally, the
methods to derive implied distributions of commodity prices from both Black’s model

and mixture of lognormals method will be discussed.

3.1 Black’s Commodity Option Pricing Model
Black’s extension of the Black-Scholes formula can be used to value European

options5 on commodity futures contracts (Black 1976). The assumption behind Black’s

 

5 A “European option’ is one that can be exercised only on the maturity date and not earlier. An “American
option” is one that can be exercised at any time up to the date the option expires. The price that is paid for
the asset when the option is exercised is called the “exercise price” or “strike price.” The last day on which
the option may be exercised is called the “expiration date” or “maturity date.” Most options on
commodities are American options, meaning that they can be exercised at any date on or before maturity or
at any time within a specific period (e.g. one month) before maturity. Thus, an option’s value will depend
on the entire stochastic process for future prices, not just the distribution of future prices at the option’s
expiration. In this paper, a model is developed to value European options, which involve a less costly and
simple solution technique. Some studies have found that the early exercise feature of American options on
futures contracts are generally small and there is little loss in pricing efficiency by ignoring the early
exercise feature (Sgastri and Tandon 1980, Ramaswamy and Sunderesan 1985). Such results suggest that
European pricing models can serve as a useful approximation.

10

model is that the underlying commodity price follows a Geometric Brownian process. It
can be expressed as

d?F=Cdt+0dZ (3.1)

Here F is the futures price; C is the instantaneous expected relative change in the
futures price; 0' is the instantaneous standard deviation; t is time and Z is a Wiener
process. Both 4’ and 0’ are assumed to be constant. Here dZ =£Jdt, where 8 is a

standard .variate. Thus the implicit assumption is that percentage price changes dF/F, over
the interval dt are independently drawn from a stationary normal distribution.

Black’s futures option pricing model is based on an arbitrage relationship between
the risk-free rate of return and the return on a portfolio containing the option. A hedge
portfolio is adjusted continuously such that the resultant portfolio is riskless and has a
return that replicates a risk-free bond. The number of options hedged against the
underlying commodity depends on the strike price, the current price of the underlying
commodity, the time to expiration, the interest rate on a risk-free bond, and the variance
of proportional price changes of the underlying commodity price. When these factors are
known, the proper hedging portfolio is known. Black succeeded in solving the partial
differential equation to obtain exact formulas for the prices of European call and put
options. In the case of futures the solution to the partial differential equation doesn’t
include the expected return. Thus, the differential equation for option pricing does not
involve parameters that are affected by risk preferences. Therefore, Black’s equilibrium
option premium is invariant to risk preference and to expected change in the underlying

commodity price.

11

A simpler approach to obtain the same solution is risk neutral valuation (Cox and
Ross 1976). Cox and Ross developed a general theory of option pricing which relies on
the minimal assumption that no arbitrage opportunities exist. They show that this
condition is equivalent to the existence of an artificial probability distribution such that
the asset price equals the stream of expected returns discounted at the risk-free rate. The
idea of an artificial distribution is widely applied in the finance literature. It is called the
risk neutral valuation measUre. In a world where investors are risk neutral, the expected
return on all securities is the risk-free rate of interest, r.

Hull (1993) says, “When we move from a risk-neutral world to a risk-averse
world, two things happen. The expected growth rate in the underlying asset changes and
the discount rate that must be used for any payoffs from the derivative security changes.
It happens that these two effects always offset each other exactly.”

In a risk neutral world, two important restrictions hold. The first restriction is that
the option would be priced according to its risk neutral expected value at maturity,
discounted back to the current period at the risk-free rate of interest. For commodity.

options on futures contracts, this implies,

P = (“T-”Emerita, — K,O)]

call

P = e"(T")E[max(K — F, ,0)] (3.2)

put
where E(-) denotes expected value in a risk-neutral world; Pam is the price of call
options; Pp“, is the price of put options; K is the strike price; F r is price of future contract
at maturity; T is expiration date; and r is the risk free interest rate.
The second restriction is that futures prices are unbiased. Current futures price is

the expected value of the futures price at maturity. When using the risk-neutral approach

12

for Black’s futures option model, the geometric mean, ln(F,+A,/F,), where F, is the

future price at time t, is zero. This zero geometric mean of the log-price return comes
from Black’s assumption of a zero holding cost for a futures contract. It is shown in
Black (1976) that a futures price can be treated in the same ”way as a security paying a
continuous dividend yield at rate r. Thus, the expected growth rate in a futures price in a
risk neutral world is zero. If the two restrictions mentioned above did not hold, there
would be unexploited profit opportunities.

Using the lognormal distribution and risk neutral valuation, Black’s formula for

European call and put options written on futures are:

P :e-r(T-!)[F'N(dl)-K'N(d2)]

(all

P :-e—r(T_')[F-N(-dl)"K'N(—d2)]

put

where

2

[lnF-an+g—(T—t)]
d = 2
‘ oJ(T—z)

d, =d, —-o.,/(T—z) . (3.3)

where N (-) is cumulative normal density function.

 

The only parameter in the pricing formula that can not be observed is 6. When the
market option price and the four parameters, K, F, r, T are known, the only unknown
parameter 0' can be backed of the forrnula. The volatility implied by an option price
observed in the market is called “implied volatility” and completely describes the
lognormal distribution at maturity that is implied by the option premium. Very often,

several implied volatilities are obtained simultaneously from different options on the

13

same stock and a composite implied volatility for the price is then calculated by taking a
suitable weighted average of the individual implied volatilities.

Black’s work provides the foundation for the theory of futures option valuation.
The model for valuing futures options leads to a closed form solution which implies only
nonnegative prices. In actual applications, however, the model has certain well-known

deficiencies as discussed in section 2.

3.2 A Mixture of Lognormal Approach

In order to overcome the deficiencies in the Black model for valuing commodity
futures options, an alternative method is proposed here. In choosing a functional form for
the terminal distribution, one should try to balance flexibility, parsimony, and ease of '
interpretation. Here the futures price at the option’s expiration is specified to be drawn
from a mixture of two lognormal distributions. The benefits of this technique arise from
its flexibility, generality, and directness. This functional form for the terminal distribution
can easily accommodate a wide variety of shapes. This approach leads to a probability
density function (PDF) clearly distinct from the lognormal benchmark, and typically
characterized by skewness and leptokurtosis.

With European style options, the value of the call (put) option is simply the risk
neutral expected payoff discounted back to the present using an appropriate interest rate.

At expiration the call and put option values must be max(FT — K ,0) and max(K — FT ,0)

respectively. Assuming risk neutral valuation, the option price for calls and puts can be

expressed as

14

P

(all

z (“T") f (F, — K) - g(F,)dF,
K

put

P = em") I (K — F, ) - g(F, )dF, (3.4)
0 .

where g( F r) is the risk-neutral probability density function for the commodity futures at
time T, which is assumed to be a mixture of two lognormal distributions here.

In the mixture of two lognormals method, the change of log futures price at the
option’s expiration is specified to be drawn from a mixture of two normal distributions.

That is,
rnF, —1nF, ~ AN[p,,o,2]+(1—/1)N[u,,o,2] (3.5)
Where N (u,- ,0',-2 ) is a normal distribution with mean it,- and standard deviation 0",; and x1

and (1 —/1) are the weights on each normal distribution respectively, where O S A S 1.

Thus, the probability density function of log future prices, G(ln FT) , is given by:

1 (1" FT "#1) l (1" F'r‘llz)
G(1nF,)=A———-e 2"! +(1—A)———e 202 (3.6)

427w, 427w,

Because ln F, follows a mixture of normal distributions, and Fr follows a mixture of

lognormal distributions. Thus, the density function of F T is,

(In Fr —#1) 1 (In FT—F‘Z)
e 20' +(1-A)———e “’2 ]

4% o,

g<FT>=iu

FT 5°01
(3.7)
=481(FT)+(1-4)82(FT)

where g ,. (F, ) is the single lognormal distribution

_(ln FT —/“r )2

,
20,

 

1
,(F )= e
g T JZfl-a,-F,

15

Here g(FT) represents the PDF of mixture of two lognormal distributions, and

g ,(F,) represents the PDF of a single lognormal. There are five unknown parameters A,

in, 01, 11.2, 0'2 in this equation.
The option pricing formula is derived as follows where the futures prices follows
a mixture of two lognormal distributions. First, consider the call option pricing formula

for European options. Substituting (3.7) into (3.4) yields

Pm” = (“T-”(A 1 (F, - 10g, (F, )dF, + (1 —/t) I (F, — K)g,(F, )dF,] (3.8)
K K
Thus Pm” = 1PM, + (1 - A)Pm,,2 (3.9)
Where Pa, = MT") I (F, — 10g, (F, )dF, (3.10)
K

The call option pricing formula for European option with mixture of two lognormal

distributions is then (using the standard Black model)”,

 

__,. _, (F‘|+a—')
[11:11:16 (T ’[e 22.N(d11)_K'N(d12)] (3.11)
+ (1 — Mex” [e(p2+T-) -N(d21 ) — K - Mar22 )1
2
Where d,1 = an+#’ +0’ , 61,, 2M i=l,2
a, 0',
By using the same approach for put,
-r —r (””07”)
Pp... =46 ‘T ’[e 'N(_d11)-K'N(—d12)] (3.12)

1

—(1—,t)e"<T"’[e”"+ 2 ’ -N(—d2, )— K . N(—d,,)]

 

6 The derivation of option pricing formula 3.11 and 3.12 refer to appendix 1.

l6

Under the risk neutral assumption, current futures price is an unbiased predictor of the
futures price at maturity. That is the expected value of futures price at maturity equals the

current futures price:
E(F,) = F, (3.13)
Where E(F,) is the risk neutral expectation of F, conditional on information available

at t. In the case of mixture of two lognormal distributions,

2
0'1

E(F,) = 2 1““? + (1—3) - eW—é— (3.14)
This constraint can be used to substitute out one of the five unknown parameters

in the option pricing formula.

3.3 Using Market Traded Option Prices to Generate Implied Distribution
Forecasts

The mixture of lognormal formula is an appealing choice, most importantly
because it is sufficiently flexible to allow a wide range of stochastic price behavior. Thus,
the implied distribution derived from a mixture of lognormals model should be more
realistic than that derived from Black’s model. i

In order to derive the implied distribution, one typically defines a loss function
expressed as the sum of squared deviations of the estimated option prices and the actual

option prices. In the case of Black’s model, there is only one unknown parameter in the
option pricing formulas, e. Let F, (K ,T, r, F , 0) denote the estimated price at time t using
Black’s option pricing model for a call or put that has an exercise price K and maturity at
T. Then if the option pricing model is correct, 13,(K ,T, r, F ,0') is expected to .be close to

the observed market option price P.

17

Therefore, (I can be estimated by solving

min 2 Q,[P, — 13,,(K,,T, r, F;a)]2
mianJPp —13p,(K,,T,r,F,0')]2 (3.15)

Where pc is the price for call and PP is the price for put. i is the number of call and put
options at time t. Equation (3.15) is weighted sum of squared deviations between realized
market options and those estimated via Black option pricing formula. Q, is weight of each
option. The summation i is over call and put Options with identical maturities but

different strike prices.

In the mixture of two lognormals method, let F,(K,,T,r,F,/I,u,,,u,,0',,0'2)
denote the price of the, option (call ‘or put) at time t with strike price of K,- and maturity
date of T by using the mixture of lognormal option pricing model conditional on
parameters (Luv/1251,02). This suggests that the mixture of lognormal parameters

implied by options can be estimated by solving:

minEQrP. —é,(K,,T,r,F,/1,u..u..o,.0211:

minEQJP, — 13,,(K-,T,r.F,/1,u,,u2,0..02)? (3.16)

which is weighted sum of squared deviations between realized market option prices and
those estimated via the mixture of normal option pricing model.

In this paper, options with the same maturity date and different strike prices from
the same day will be employed to derive the implied mixture of lognormal parameters. In
literature, there are different methods to derive the weights Q. One approach in Black’s

model is to put all the weights on a single contract for which the price of the options is

18

more Sensitive to changes in the volatility of the underlying commodity. Usually, this is
the at-the-money option, but occasionally it is the near-the-money option7. The reasons to
use at-the-money options are as follows: First, at—the-money option price is sensitive to
volatility, it should therefore return the most accurate measure of volatility. Second, at-
the-money options have high volume, which indicate they contain more information
about underlying asset’s volatility than thinly traded options that are well out of money.
However, this method ignores information about volatility that is available in the
contracts that are not at-the-money. In order to account for all the information in the
option market, a method is chosen in this research that will contain the information of all
options. This method recognizes that the true distribution is the same for all the options
on a given future contract with the same maturity date, but different strike price, and
chooses the single estimate of distribution which is closest to satisfying the option pricing
equation for all exercise price. “Closeness” is measured by the mean square deviation
between the observed and the theoretical prices, aggregated over all contracts with a

given maturity and weighted by Q..

The weight is used to influence the relative contribution of each option in the
determination of the implied mixture of lognormal model. The weight Q, may be
assumed to be equal. But a more reasonable assumption is to let each option be weighted
by some measure of its liquidity such as volume. This weighting mechanism seems
reasonable because options that are far from the money have prices that are not as
sensitive to volatility, and should therefore get less weight in the forecast than the more

sensitive near-the-money options. However, in this research, due to the lack of

 

7 This is a loosely defined term that means the strike price and spot price are not too far.

19

information of the trading volume of option with different strike price, we will give equal

weight to each option.

20

Section 4: Application to Corn, Soybean and Wheat~ Futures Prices

4.1 Data Sources

In this section, the mixture of lognormals method is applied to corn, soybean, and
wheat futures options traded on the Chicago Board of Trade. Soybean futures contracts
are available for January, March, May, July, August, September, and November
expiration dates; corn futures contracts are available for March, May, July, September,
and December expiration dates; and wheat futures contracts are available for March,
May, July, September, and December expiration dates. Options are written on the futures
prices for each commodity with an expiration date on the last Friday preceding the first
notice days of the corresponding futures contract by at least five business days. Each of
these. grains has a “storage contract” which expires around the end of usual planting
period or the beginning of the usual harvest period, and a “harvest contract” which
expires around the end of usual harvesting period. The storage contracts for corn,
soybean, and wheat futures are the July contract, July contract, and March contract,
respectively; and the harvest contracts for corn, soybean and wheat are December
contract, November contract, and July contract, respectively. In this section, the implied
distributions for each contract are presented at three points in time. For the harvest
contracts, implied distributions are presented for a single day around the most active
planting date, a single day near the maturity date of the contract, and a single day in the
middle of these two daysg. For the storage contracts, implied distributions are generated

for a single day right after the maturity of the harvest contract, a single day near the

 

8 Notice day is the seventh business day preceding the last business day of the delivery month.
9 The reason for this data selection method is that the pattern of changes of distribution can be observed as
we get closer to the maturity date and market uncertainty decreases.

21

maturity date of the contract, and one day in the middle of these two days. That is, data
selection starts at the earliest date and moves closer to the maturity date. Data for market
option premiums on futures and the corresponding futures prices were taken from the

Wall Street Journal for each contract on the dates specified in Table 4.1.

Table 4.1: Data source

 

Soybean November 2000 contract — harvest contract

 

 

 

 

 

Date: 05/24/00 08/ 16/00 10/ 1 1/00
Time to maturity: 149 65 9
Soybean J uly2000 contract — storage contract

Date: 10/ 13/99 03/15/00 06/14/00
Time to maturity: 248 97 6

Corn December 2000 contract - harvest contract .
Date: 05/10/00 08/16/00 1 1/15/00
Time to maturity: 194 96 5

Corn July 2000 contract — storage contract

Date: 1 1/17/00 03/15/00 06/14/00
Time to maturity: 216 97 6

Wheat July 2000 contract — harvest contract

Date: 10/ 13/99 02/ 1 6/00 06/ 14/00
Time to maturity: 251 125 6

Wheat March2000 contract — storage contract

Date: 06/14/00 1 1/29/00 02/14/01
Time to maturity: 251 83 6

 

For each day, closing option premiums for 6 separate strike prices were quoted in
the Wall Street Journal for both call and put options. This provides a total of 12 closing
option premiums (6 calls and 6 puts)”. In addition, the corresponding closing futures
price was collected. There are a total of 6 contracts, each priced at 3 different dates for a
total of 18 implied distributions estimated.

The only other data needed to estimate the implied distributions is the risk-free

interest rate, which is estimated using data collected from the Wall Street Journal on the

 

'0 In some days, quotations of call or put for certain strike price are not available.

22

yields to maturity on Treasury bills with the same maturity date as the options used in the
estimation.

For each day, the implied distributions at the option maturity date were estimated
for the MLN model and the Black model. A total of 18 implied distribution were derived
for each model varying by commodity, storage versus harvest contract, and time to

maturity.

4.2 Estimating the Implied Distribution

Traded option premia on selected days are employed to generate implied
distributions for MLN and Black methods. The optimization of the objective function
(3.16) was performed with a trust-region reflective Newton algorithm using MATLAB
function fmincon to find the minimum of a constrained nonlinear multivariable objective

function.11

 

H The MATLAB code can be run under any system that has installed MATLAB with the version equal or
higher than 5.3.1 R1 1.1. Bounds for the parameters were set so that, lnF,-10*r<u,< lnF,+10*r,
0.001 <a,~<0.8. According to Black’s model, u=lnF,-oz /2, thus a reasonable searching bound for u in MLN
should center around lnF. Here Mr is chosen because this range is big enough in all our calculation cases.
After trying other larger bounds, such as 15r or 20r, the optimization results are the same. This means the
chosen bound has already included the minimum points. Similar arguments go to O'i. I have tried other
larger upper bounds for 0,, and the optimization results remain the same. The chosen bounds here are a
good tradeoff between calculation cost and accuracy. The chosen bound for this program ensures accurate
optimization results and also make it possible to finish the calculation within several minutes using a PC.
Because this is a constrained nonlinear optimization problem, in some cases, the fitting parameters are local
optimization points. The results are sensitive to the initial searching points selected. Therefore the
parameter searching ranges were divided into multiple smaller lattices and the center of each individual
lattice was chosen as the initial points. After this kind of global search, the global minimum was chosen
from the minimal local minimum. For the chosen searching bounds, each dimension was divided into three
segments, the optimization results would converge and further separation will give the same results.

23

Table 4.2: Estimated parameters for all contracts/days by MLN methods12

 

 

A - 111 112 01 02
Soybean November 2000 contract — harvest contract
05/24/00 0.08585 2.16043 1.64139 0.16473 0.13922
08/16/00 0.00689 1.85784 1.54968 0.59303 0.07689
10/11/00 0.40717 1.58125 1.55150 0.05289 0.02212
Soybean July 2000 contract — storage contract
10/13/99 0.07779 2.03517 1.61374 0.16767 0.15402
03/15/00 0.42548 1.74122 0.61329 0.17482 0.08058
06/14/00 0.15659 1.60771 1.63390 0.00148 0.04529
Corn December 2000 contract — harvest contract
05/10/00 0.17366 1.31442 0.85136 0.04287 0.15358
08/16/00 0.04498 0.62658 0.64106 0.36855 0.08662
11/15/00 0.00298 0.44647 0.73009 0.12726 0.01484
Com July 2000 contract — storage contract
11/] 7/99 0.43659 0.85537 0.69627 0.20277 0.10594
03/15/00 0.44671 0.96236 0.81482 0.19403 0.07584
06/14/00 0.01928 0.47976 , 0.70982 0.54835 0.04340
Wheat July 2000 contract — harvest contract
10/13/99 ‘ 0.07973 1.03267 1.03866 0.65019 0.15930
02/16/00 0.07436 1.0881 1 1.02844 0.66529 0.11043
06/14/00 0.00419 1.18298 0.95861 0.05150 0.03578
Wheat March 2000 contract — storage contract
06/14/00 0.17352 1.33423 1.02885 0.27324 0.14286
11/29/00 0.41174 1.03822 0.95037 0.11549 0.07028
02/14/01 0.00552 0.89180 0.98443 0.04961 0.00376

 

Table 4.2 presents the implied mixture of lognormal estimated parameters (2,111,112, 01,02)

for each contract on each set of data.

In the following three sections, the graphs of distributions and option pricing errors of

each commodity will be presented. The comparison of graphs of different commodities

and different days to maturity will be given. The contract selected is harvest contract for

 

'2 A, [1,, [12, 0'). 0'; are the five parameters in the mixture of lognormal density function defined in equation

(3.7)

24

each commodity. The corresponding graphs of storage contract of these commodities are

attached in appendix.

25

Section 5: Soybean Estimation Results

Figure 5.1 and Figure 5.2 depict probability density functions and cumulative
density functions taken from both Black and MLN models using data of May 24, 2000,
August 16, 2000, and October 11, 2000 respectively. The top panel uses estimates from
1 the November contract of soybean (harvest contract) on May 24, 2000 (149 days to
maturity). The middle panel uses estimates of the same contract on August 16, 2000 (65
days to maturity). The bottom panel uses estimates of the same contract on October 11,
2000 (9 days to maturity). Figure 5.3 plots the difference between estimated option prices
and actual prices for calls and puts of these three days.

In Figure 5.1 for the data that are far from maturity date, the PDFs are flat and fat,
which indicate a bigger variance compared with those from the data that are near the
maturity. The PDFs from data that are near the maturity are tall and thin. This is the same
for PDFs from both Black and MLN methods. Compared with single lognormal
distribution, PDFs of MLN model are always high-peaked and fat-tailed compared with
PDFs of Black model. I

As shown in Figure 5.2, on the right side, the CDF from MLN model crosses
Black CDF at certain point. For Example, in Figure 5.2.0 the CDF from MLN model
cross the Black CDF at $5, about 4.60% above the current future price ($4.78). To the
right of this point, the derived CDF of MLN method lies below that from Black model,
indicating greater probability attributed to realization above this level.

Figure 5.3 depicts call and put option pricing errors. It is clear from the graph that
Black model tend to underprice in-the-money call and out-of-the—money put options, and

overprice the out-of—the-money call and in-the-money put options, which is consistent

26

with the findings from literature. In Figure 5.3, the option pricing errors from MLN
method are in the order of $1*10'3, while the errors from Black model is in the order of

$1*10'2. This indicates the MLN method gives better fitting results than Black model.

27

Figure 5.1: Estimated PDFs -- November (harvest) contract of soybean

a. November Option Maturity as of May 24, 2000

0.5 -

- - - Black
[14 — — Mixture of lognormals

0.3 —
0.2 -

0.1—

 

0.0 — -

 

 

futures price ($)

b. November Option Maturity as of August 16, 2000

   
 
   

 

 

1.0 a
- - - Black
0.8 — —— Mixture of lognormals
0.6 —
0.4 ~—
02 -
0.0 I I l T 1
3 4 5 6 7

futures price ($)

c. November Option Maturity as of October 11, 2000

2.5 -
- - - Black

— Mixture oflognormals
2.0 -

1.5—
1.0-

0.5 -—1

 

0.0 — _

 

futures price ($)

28

Figure 5.2: Estimated CDFs — November (harvest) contract of soybean

a. November Option Maturity as of May 24, 2000

1.0—

08 '-

05 ——

0.4 —

0.2 —‘

 

0.0 -—

  

   

- - - Black
— Mixture oflognormals

 

futures price (5)

b. November Option Maturity as of August 16, 2000

1.0—

0.8 —

0.6 -

0.4 -

0.2 —

 

0.0 —

 

    

- - - Black
— Mixture of lognormals

 

l
3

I I

4 5 8 7
futures price (5)

‘

c. November Option Maturity as of October 11, 2000

 

 

  

 

1.0 - - -
0.8 — - - - Black
— Mixture oflognormals
0.6 —
0.4 -1
0.2 —
0'0 _i I l I l
3 4 5 6 7

futures price ($)

29

Figure 5.3: Option pricing errors -- November (harvest) contract of soybean

a. Call Pricing Errors of May 24, 2000

predicted call premium - real value (5)

predicted put premium - real value (5)

C.

predicted call premium - real value (5)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

4g — - 0- Black
.- ______ a. -A— Mixture of lognormals
20 — I ’ * ~ .
‘~ ‘ -
0 '—‘1_3r 5 fil— ‘ ~ Le ‘ ; a
' -20 — " s.
-40x10'3 ~ ~ ‘
C
l I l I I I I
4.8 5.0 5.2 5.4 5.6 5.8 6.0
_ striking price (5)
b. Put Pricing Errors of May 24, 2000
“U T. - 0- Black
...... a. .
e . ‘ -rs- MIxture of lognormals
20 —* 7 ~ “
o ; ’h— ;_ I “ n. in —..
-20 — I s.
410x1133 — ‘ I ~
O
I I I I I I l
4.8 5.0 5.2 5.4 5.6 5.8 6.0
striking price (5)
Call Pricing Errors of August 16, 2000
15 _ - 0- Black
10 -r_':- Mixture of lognormals
a ' - ‘b - ‘ 9
' F b h .\ f
It. _ s ‘ __...--""'
s . ‘ ~
~ ‘ ‘.
45x10" —
I I I I I I I
4.0 4.2 4.4 4.6 4.8 5.0 5.2

striking price (5)

30

(I. Put Pricing Errors of August 16, 2000

 

 

 

 

 

 

 

 

 

a 15 m
g - 0— Black
T,“ 10 " , A, —A- Mixture of lognormals
g 5 _. ‘ fl 4' - ' I. T T \ ~
g ‘ ’ ‘ 1..
E 0 W
9 '7 ~
Q ‘
S -S— 7*:
a. ‘ s
a 1 ~ .
g -10 a
'D
9 -3
9' -15x10
I I I J I I
4.0 4.2 4.4 4.6 4.8 5.0 5.2
striking price (5)
e. Call Pricing Errors of October 11, 2000
Q 20 —
g —r£~.- Mixture oflognormals
(U
3 - 0' Black
g 10 "1
é 4- " i.‘
.2 if ' o " p A \ l
g 0 —F- — -—- «:5 — 5 .- .-
rF-Qo- \ x _ .. - - . ' ' ' ’
w- " '

3 -10 -
(I)
.§
13
9 -3
0— -20x10 —

I I I I l I

4.4 4.6 4.8 5.0 5.2 5.4

striking price (5)
f. Put pricing Errors of October 11, 2000

20 -
qr Mixture oflognormals
- l- Black

10—

-10—

 

Predicted put premium - real value (
C]
B
[I
I
I
I
I
I
t

-2ox1o'3 4

4.4 4.6 4.8 5.0 5.2 5.4
striking price (5)

31

Section 6: Corn Estimation Results

Figure 6.1 and Figure 6.2 depict probability density functions and cumulative
density functions taken from both Black and MLN models by. using data of May 10,
2000, August 16, 2000, and November 11, 2000 respectively. The top panel uses
estimates from the November contract of corn (harvest contract) on May 10, 2000 (194
days to maturity). The middle panel uses estimates of the same contract on August 16,
2000 (96 days to maturity). The bottom panel uses estimates of the same contract on
November 15, 2000 (5 days to maturity). Figure 6.3 plots the difference between
estimated option prices and actual prices for calls and puts of these three days.

In these graphs, MLN captures key deviations from Black such as a fatter right-
hand-side tail, and a mode shifted slightly to the left. A lot of empirical test of commodity
price show that variance of commodity price distribution is not constant and commodity
price distributions are characterized by hi gh-peak and fat-tail. Thus, the derived
distribution from MLN is consistent with the literature.

Figure 6.3 depicts call and put option pricing errors. The MLN method tends to
give much smaller option pricing errors when the date is far from maturity. When the
date is near maturity, there is no significant difference between the option pricing errors
from the two methods. When the date is far from maturity, there is more uncertainty in
the market. The MLN has a more flexible shape to accommodate the market participants’

expectations.

32

Figure 6.1: Estimated PDFs -- December (harvest) contract of com

a. December Option Maturity as of May 10, 2000

     
 
   
   

 

 

0.8 — -— Mixture oflognormals
- - - Black
0.6 -
0.4 —
0.2 —
U 0 T I 7 l I i ’ Iii-T—

futures price (5)

b. December Option Maturity as of August 16, 2000

 

 

 

2'0 — - - - Black
— Mixture oflognormals
1.5 -
1.0 —
0.5 -
0.0

0.5 1.0 1.5 2.0 2.5 3.0 3.5
futures price

c. December Option Maturity as of November 15, 2000

14m

12 -1 — Mixture oflognormals

10 _ - - - Black

8%
6—4
4...

2—

 

 

U—I I I I I I I

1.8 1.9 2.0 2.1 2.2 2.3 2.4
futures price (5)

33

Figure 6.2: Estimated CDFs - December (harvest) contract of com

a. December Option Maturity as of May 10, 2000

   
   

 

 

1.0 _
0 3 ‘ — Mixture oflognormals
- - - Black
0.5 —
0.4 -
0.2 "
0.0 — I A I I I I
1 2 3 4 5

futures price (5)

b. December Option Maturity as of August 16, 2000

 

 

 

1.0 — _____ A
0.8 - - - - Black

—— Mixture oflognormals
0.6 —
0.4 -
0.2 -
0'0 I l _ I l l l

0.5 1.0 1.5 2.0 2.5 3.0

futures price

c. December Option Maturity as of November 15, 2000

 

 

 

 

1.0 a
0.8 - Mixture of lognormals
- - - Black

0.6 —

0.4 -

0.2 —

0'0 TI 1 I I I I I
1.8 1.9 2.0 2.1 2.2 2.3 2.4

futures price (5)

34

Figure 6.3: Option pricing errors -- December (harvest) contract of corn

a. Call Pricing Errors of May 10, 2000

Predicted call premium - real value (5)

Predicted put premium - real value (5)

Predicted call premium - real value (5)

 

 

 

 

 

 

 

 

 

 

 

30 -
—rh- Mixture of lognormals
20 - - 0- Black
c. _ ._ _
10 "1 F - ‘ —._ ‘
U " h a. m “ A W
u u u ‘ co _ .‘ ~ —:§—
-10 —1 ‘ " - . g
o .. .. ., _
' - -e
-20 .—
-30x10 -*
I l I I I
2.4 2.5 2.6 2.7 2.8 2.9
striking price (5)
Put Pricing Errors of May 10, 2000
30 -
-1!x- Mixture oflognormals
20“-~_ -0- Black
10 - "‘ ~ ~ .. _
~ - ‘- .-
0 E ; " f-‘fi—‘b
0.. g _
‘10 _" ~ ‘ ‘ . . ‘ Q- _
T " “-9
-20 -
-3ox10'3 —
I l I I I I
2.4 2.5 2.6 2.7 2.8 2.9
striking price (5)
Call Pricing Errors of August 16, 2000
10 —
--a— Mixture of lognormals
5 _ - 0- Black
---- “ Q-
.- "' - - t ‘ ~ ___,__a--—"¢
0 ,3—_. r ‘ A C. .1 h ‘ «‘H
. ' + ‘- .. ~ -
* t - u
T ‘ ' -
_5 _
-10x10'3 -
I I I I I l
1.7 1.8 1.9 2.0 2.1 2.2

Striking Price (5)

35

(I. Put Pricing Errors of August 16, 2000

10—

-5—

 

-10x10-3 — -
I I I I

Predicted put premium - real value (5)
D
'1
I

  

1— Mixture of lognormals
,\ - 0- Black

 

 

 

1.7 1.8 1.9 2.0 2.1 2.2
Striking Price (5)
e. Call Pricing Errors of November 15, 2000
e 2 -
3 - 0- Black
2 —:h— Mixture of lognormals
To 1 A
9."
é
.2 0
E $ - * """ . ------ ‘ \ s
Q.
T:
'D ’1 -
2
.2
'U
E .3
ll -2x10 -
l I I I I I
1.8 1.9 2.0 2.1 2.2 2.3

striking price (5)

f. Put pricing Errors of November 15, 2000

 

 

 

a 2 —
T; -.e:— Mixture of lognormals
To 1 —
9.’
l
g A 1’
_ 0 - Y
Q ‘, J
s -a-"
:- 'I - . ------- .- ------- r
.93.
9
‘c
2 -3
0- -2x10 -—
I r I I l l
1.8 1.9 2.0 2.1 2.2 23

striking price (5)

36

The PDF in Figure 6.1.a turns out as a bi-modal distribution, while Black is
unimodal all the time due to the single lognormal assumption. As the data near the
maturity date, there is little qualitative difference in the two estimates, while during
periods far away from maturity; the estimate from single lognormal method cannot easily
accommodate the significant probability mass on the right tail. The bimodal shape could
be the result of uncertainty about supply and demand.

To investigate further, Figure 6.5 and Figure 6.6 show three consecutive week’s
PDFs and CDFs of corn December (2000) contract in the month of May. A trend can be
observed when time goes on. The right tail becomes smaller and smaller. The pattern of
the graphs could reflect a change in the market situation. When the date is far away from
maturity, there is a lot of uncertainty in the market. There might be different expectations
of outcomes in the market. The mixture lognormal distribution can accommodate
different expectation of market participants. It can also easily accommodatea single
lognormal distribution if that would best fit the data. We expect as news hit the market,
the relative weighting of the two lognormals might change, as well as the parameters of
each of the two lognormals.

A hypothesis for the following patterns is that there was some news about weather
situation, demand or supply conditions. For example, there was an expectation of drought
in the beginning of May 2000; thus, there was an expectation of high corn price at
maturity if the drought was realized. As May progressed, there was rain and expectation

of high corn price became small.

37

I Figure 6.4 Estimated PDFs — December (harvest) contract of corn — 2000

a. December Option Maturity as of May 3, 2000

0.8 -

0.6 —

0.4 —

0.2 —

 

0.0

— Mixture of lognormals
- - - Black

    

futures price (5)

b. December Option Maturity as of May 10, 2000

0.8 -1

0.6 -

0.4 —

0.2 .4

   

— Mixture of lognormals
- - - Black

 

 

 

0.0

 

 

futures price (5)

0. December Option Maturity as of May 24, 2000

0.8 —‘

0.6 —

0.4 '-

02 —

 

0.0 —

—— Mixture of lognormals
- - - Black

 

 

futures price (5)

38

Figure 6.5 Estimated CDFs — December (harvest) contract of corn — 2000

a. December Option Maturity as of May 3, 2000

 

   
  
  

 

 

1.0 T _______
0.8 —
—- Mixture of lognormals

0.6 — - - - Black
0.4 -1
0.2 _
0‘0 I _ I I I I

1 2 3 4 5

futures price (5)

b. December Option Maturity as of May 10, 2000

 

1.0— _______

 

0.8 -
— Mixture oflognormals

- - - Black
0.6 —

0.4 -

0.2 '-

 

0.0 — I ‘ I l

1 2 3 4 5
futures price (5)

c. December Option maturity as of May 23, 2000

 

—
—

1.0— ____. ..... 4._

— Mixture oflognormals
- - - Black

0.8 -—

0.6 —

0.4 '-

02 -

 

 

0.0 —

—t

l I I I

1 2 3 4 5
futures price (5)

39

Figures 6.7- 6.8 and Figures 6.9- 6.10 show three consecutive week’s PDFs and
CDFs of December contract of corn in the year of 1998 and year 1999 respectively.
Figure 6.7 and 6.8 depict probability density functions and cumulative density function
taken from both Black and MLN models by using data of May 6, 1998, May 13, 1998
and May 20, 1998 respectively. Figure 6.9 and 6.10 depict probability density functions
and cumulative density function taken from both Black and MLN models by using data
of May 5, 1999, May 12, 1999 and May 19, 1999 respectively.

The PDF in Figure 6.7.a turns out as a bi-modal distribution, while this bi-modal
pattern doesn’t apply to the other graphs in the year of 1998 and year 1999. The results
suggest that the market perceptions of the harvest futures price distribution are quite
different during the three weeks in the year 1998 and 1999. Again, this may be due to the
news about weather or supply and demand situations during that period. More analysis is

needed to test these hypothesis.

4O

Figure 6.6 Estimated PDFs - December (harvest) contract of corn -- 1998

a. December Option Maturity as of May 6, 1998

 
 
  

 

 

 

1.0 —
- - - Black
0.8 — — Mixture of lognormal
0.6 -<
0.4 —
0.2 —
0.0 F - I l I - l
1 2 3 4 5

futures price (5)

b. December Option Maturity as of May 13, 1998

1.0—

- - - Black
0.8 — -— Mixture of lognormal

0.6 —
0.4 —

0.2 "l

 

0.0 — -

 

 

1.0 -
- - - Black

[18 -l — Mixture oflognormal

0.6 *
0.4 -

0.2 —

 

0.0

 

 

futures price ($)

41

Figure 6.7 Estimated CDFs - December (harvest) contract of corn -- 1998

a. December Option Maturity as of May 6, 1998

 

1.0.“

   

0.8 *

   

- - - Black
— Mixture of lognormal

0.8 —~

0.4 -l

0.2 —

 

0.0 - -

 

futures price (5)
b. December Option Maturity as of May 13, 1998

1.0— .......

 

'08 d

   
  

- - - Black
— Mixture oflognormal

0.6 -

0.4 -

0.2 —

 

0.0 — -.

 

1.0 —
—— Mixture of lognormal
0.5 —

0.4 '-

02 —

 

0.0

 

_

I I I l

1 2 3 4 5
futures price ($)

42

Figure 6.8: Estimated PDFs — December (harvest) contract of corn -- 1999

a. December Option Maturity as of May 5, 1999

1.0-1

- - - Black
— Mixture of lognormal

0.8 '3

0.5 -

0.4 '-

02 -

 

0.0

 

 

futures price (5)

b. December Option Maturity as of May 12, 1999

 
 
  
  
  

0.8 d : Enliiftlire of lognormal
0.6 —

0.4 "l

0.2 -‘

0.0 f j r

 

 

 

futures price($)

c. December Option Maturity as of May 19, 1999

1.0 -
- - - Black

0.8 — — Mixture oflognormal

0.8 -
0.4 d

0.2 -

 

 

0.0 —

 

futures price ($)

43

Figure 6.9: Estimated CDFs - December (harvest) contract of corn -- 1999

a. December Option Maturity as of May 5, 1999

b.

1.0— ___________

 

0.8 _ - - - Black
— Mixture of lognormal

0.6 —
0.4 -

0.2 '3

 

 

0.0 —

 

-

I I I

1 2 3 4
~ futures price ($)

December Option Maturity as of May 12, 1999

1.0—

  

0.8 —

   
  

- - - Black
-— Mixture of lognormal

0.6 -—

0.4 -I

0.2 —

 

0.0 —

 

I I I I

1 2 3 4
futures price($)

December Option Maturity as of May 19, 1999

 

1.0— __

DB '3 - - - Black

Mixture of lognormal

 

0.5 —

0.4 -

0.2 —

 

0.0

 

—

I T I

1 2 3 4
futures price (5)

Section 7: Further Comparisons of Enlarged graphs of Black and MLN Models

From Figure 6.1, it is not easy to tell the difference of tails intuitively, especially
for those distributions that are near maturity. Figure 7.11 is an enlarged PDF for the
December contract of corn on August 16, 2000. It gives intuitive sense of the difference
of tails between MLN and Black methods. The numerical integration13 results in Table
6.4 give a better explanation about the difference of Black and MLN methods.

The reason that MLN’s PDF has a larger standard error can be illustrated in
Figure 7.11 From the figures, it is obvious that around 1.4 and 2.6, the two PDFs have the
same density value. Numerical integration can be done region by region to examine their
individual contribution.

From Table 7.4, the major reason why MLN has a larger standard error is that it
has a longer and thicker tail (from 2.6 to 38). This can also be shown by the kurtosis
values. MLN has a much larger kurtosis (0.56127) than that of the Black’s (0.262851).
This means MLN has a much larger tail and this affects the standard error in a larger

degree than what is seen intuitively from'the graph of PDF.

 

13 In order to calculate (l) and (2) numerically, the approximate equations for expected value and Variance

are respectively: 7t «- Exi -g(x,.)-Ax,. ; Var z 20‘.- - if -g(x,.)~Ax,.

where x is the futures price and g(x) is the PDF of the futures price derived from both MLN and Black
model. Ax: is the small step used to divide the integration region, and X, is the value of x at grid point i.

Numerical integration have been done from 0.0001 to 37.5787 with the step size of 0.00037578. Here
numerical integration range is not from 0 to positive infinity, because outside of the calculation region, the
PDF’s value is almost zero and that contribution to the integration can be neglected. For MLN’s PDF, the
calculated expected value is 1.91, and the standard error is 0.22972. For Black’s PDF, the calculated
expected value is 1.91, and the standard error is 0.19696. This is almost the same as those from the
analytical results presented in Table 4.3.

45

Figure 7 .1: Enlarged PDF of Black and MLN models - December contract of corn

a. December Option maturity as of August 16, 2000

2.0 ~

- - - Black

-— Mixture of lognormals

 

 

 

1.5 2.0 2.5
futures price

b. Enlarged PDF of the same PDF as above

- - - Black
— Mixtlure oflognormals

$
I
I
I
I
I
I
I
I
1.

 

 

   

1.0 1.5 2.0

futures price

46

 

 

 

 

 

Table 7 .1: Contribution to variance by different range of futures price

— by Black and MLN models

Integration Integration Integration Integration Total Standard

from .0001 from 1.4 from 1.91 from 2.6 to Variance Error

to 1.4 to 1.91 to 2.6 38
MLN 0.0059713 0.013848 0.015066 0.017886 0.0527713 0.22972
Black 0.00045408 0.017353 0.020308 0.00067893 0.03879401 0.19696
MLN- 0.00551722 -0.003 505 0005242 0.01720707 0.01397729 0.03276
Black

 

 

 

 

 

 

 

47

 

Section 8: Wheat Estimation Results

Figure 8.1 and Figure 8.2 depict probability density functions and cumulative
density functions taken from both Black and MLN models by using data of October 13,
1999, February 16, 2000, and June 14, 2000 respectively. The top panel uses estimates
from the November contract of soybean (harvest contract) on October 13, 1999 (251 days
to maturity). The middle panel uses estimates of the same contract on February 16, 2000
(125 days to maturity). The bottom panel uses estimates of the same contract on June 14,
2000 (6 days to maturity). Figure 8.3 plots the difference between estimated option prices
and actual prices for calls and puts of these three days.

The derived distributions of wheat futures prices have the same characteristics as
those of soybean and corn. PDFs for wheat are characterized by high peak and fat-tail.
The option pricing errors from MLN methods are much smaller than that from Black
model. The differences of these two models are obvious for those days that are far from

maturity.

48

Figure 8.1: Estimated PDFs -- July (harvest) contract of wheat

a. July Option Maturity as of October 13, 1999

0.8
T - - - Black
— Mixture oflognormals

 

 

 

futures price (5)

b. July Option Maturity as of February 16, 2000

   
  
  

1.2-
- - - Black

1.0 — — Mixture oflognormals

0.8 —
0.6 4
0.4 —

0.2 —

 

no 'r“'*‘

futures price (5)

0. July Option Maturity as of June 14, 2000

 

 

 

4 —I
- - - Black
— Mixture of lognormals
3 —
2 _I
1 _.
U “I I I I I I I I
2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4

futures price (S)

49

Figure 8.2: Estimated CDFs — July (harvest) contract of wheat

a. July Option Maturity as of October 13, 2000

1.0- .4 ________

   
  

0.8 —
- - - Black

[15 ._ — Mixture of lognormals

0.4 -

0.2 -‘

 

0.0 —————

 

futures price (5)
b. July Option Maturity as of February 16, 2000

1.0— --------

   

- - - Black
—— Mixture of lognormals

0.8 d

0.5 —

0.4 -

0.2 -*

 

0.0 _ -v$

 

futures price ($)

0. July Option Maturity as of June 14, 2000

 

 

 

1.0 - -
0.8 — - - - Black
—— Mixture oflognormals
0.6 —
0.4 —
0.2 J
0'0 _I I I I T I I r
2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4

futures price ($)

50

Figure 8.3: Option pricing errors — July (harvest) contract of wheat

a. Call Pricing Errors of October 13, 2000

Predicted call premium - real value (5)

Predicted put premium - real value (5)

Predicted call premium - real value (5)

20*

10—

- O- Black
--3- Mixture oflognormals

 

 

 

 

 

 

 

 

 

 

 

 

 

 

‘ . ‘
-10 —I ‘ q. ~ ‘
~ ‘ ‘O
-3
-20x10 -1
I I I I I
3.0 3.1 3.2 3.3 3.4
striking price (5)
Put Pricing Errors of October 13, 2000
10 —
l‘ _ - 0- Black
___________ -:£r- Mixture of lognormals
5 — ____________
3 ..... 9
0 r 4,
-5 _
-10x10'3 —
I I I I I I
3.00 3.02 3.04 3.05 3.08 3.10
striking price ($)
Call Pricing Errors of February 16, 2000
20 —
- 0- Black
10 -fi- Mixture oflognormals
o-----"'"‘~--_
h h - _.‘ ~ - ~ -
0 A
L; W— ‘ ‘ a .—
\. ‘
-10 — s ‘
‘ ‘.
-20x1r3'3 —
I I I I l I
2.5 2.7 2.8 2.9 3.0 3.1

striking price ($)

51

(1. Put Pricing Errors of February 16, 2000

 

 

 

’5 20 —
g - 0- Black
7;: ' -¢- Mixture oflognormals
E 10 _._ ______ .g _ _ _ ..
h § .- *‘ ‘
g D g, A " ~ . 9
E w ‘ ~. ‘7';
a. “ ‘
S ‘ .
3' -10 — ‘ ~ , ~
0) Q.
.22 ‘ ‘~
'5
9 -3
‘1- -20x10 —
I I I I I I
2.5 2.7 2.8 2.9 3.0 3.1

striking price ($)

e. Call Pricing Errors of June 14, 2000

 

 

 

 

 

 

 

a 2 —
(1'5 - .- Black
.2 -a- Mixture of lognormals
E 1 3 _
g I """ d d‘ - - - ‘5 - a -‘ /_——‘
g 0 E :. :- ‘Z i
‘5. ‘- ‘ '3
<=3 ‘ ,
_ fl -
g 1 --1 ” s .. ‘ ~ + ------ _.
.13 -
“O
9 -3
CL -2x10 — .
I I I I I I
2.4 2.5 2.6 2.7 2.8 2.9

striking price ($)
f. Put pricing Errors of June 14, 2000

2 _
I ~ - C- Black
-z:- Mixture of lognormais

 

 

Predicted put premium real value (5)
’{
‘u

 

 

I I I I I

2.4 2.5 2.8 2.7 2.8 _ 2.9
striking price (5)

52

Section 9: Conclusion

9.1 Comparison of the two models

Compared with the single lognormal distribution, the mixture of two lognormal
distributions clearly reflects some additional information embedded in option data. The
empirical results show that MLN model has a stronger ability to account for the market
information of soybean, corn and wheat futures prices.

Table 9.3 reports the mean, standard deviation, skewness, and kurtosis of corn,
soybean, wheat prices and mean square deviation from objective function. The mean of
the risk-neutral distribution always equals the current futures price. From the estimated
results above, the standard errors, skewness and kurtosis from MLN methods are always
greater than that from Black model.

Skewness reflects simply a distribution’s asymmetry. Positive skewness indicates
that the right tail is “fatter” (high probability) and usually extends out further than the left
tail; large positive realizations are more likely than large negative realizations. Kurtosis is
a measure of the thickness of the tails of the distribution. Table 9.3 further suggests the
skewness from commodity price distribution is often systematically one-sided. And the
bigger kurtosis of distribution of MLN is an indication of fatter tail than the Black’s.

Mean square deviation is the rrrinimum value of the objective function defined in
Section 3. For those days that are far from maturity, the differences between these two

methods are large.

53

Table 9.1: Moments of the implied distribution and mean square deviation of
objective function for all contracts/days"

 

 

Expected Standard Skewness Kurtosis Mean Square
Value Error Deviation
Soybean November contract—harvest contract
05/24/00
Black 5.520 1.099 0.666 1.520 8.067e-004
MLN 5.520 1.294 2.754 2.250 2.696e-006
08/16/00
Black 4.744 0.404 0.103 0.537 3.659e-005
MLN 4.744 0.599 9.781 3.032 7.164e-006
10/1 1/00
Black 4.780 0.188 0.022 0.248 2.071e-005
MLN 4.780 0.197 0.187 0.294 1.735e-005
Soybean July contract —storage contract
10/13/99
Black 5.290 1.006 0.581 1.386 1.742e-004
MLN 5.290 1.105 1.720 1.807 1.686e-006
03/15/00
Black 5.3575 0.781 0.344 1.056 5.304e-004
MLN 5.3575 0.823 1.110 1.277 2.916e-006
06/14/00 '
Black 5.1075 0.213 0.027 0.281 1.413e-005
MLN 5.1075 0.219 0.086 0.297 1.292e-006
Corn December contract—harvest contract
05/10/00
Black 2.606 0.635 0.474 0.899 1.284e-004
MLN 2.606 0.615 0.498 0.764 2.697e-006
08/16/00
Black 1.910 0.197 0.061 0.263 7.909e-006
MLN 1.910 0.230 0.610 0.561 2.230e-006
11/15/00
Black 2.074 0.031 1.39e-003 0.041 1.028e-006
MLN 2.074 0.043 -0.296 0.138 1.025e-006

 

 

'4 The third moment of the distribution, or its skewness is defined mathematically as E[ ( F r103 I, where u is
the mean of the distribution and F r is the futures price at maturity. The fourth moment of the distribution,

or its kurtosis is defined as E[( F 7104].

54

 

Corn July contract -storage contract

11/17/99

Black
MLN
03/15/00
Black
MLN
06/14/00
Black
MLN

2.185
2.185

2.445
2.445

2.0325
2.0325

0.396
0.409

0.391
0.422

0.113
0.179

Wheat July contract—harvest contract

10/13/00

Black
MLN
02/16/00
Black
MLN
06/14/00
Black
MLN

Wheat March contract —storage contract

06/14/00

Black
MLN
1 1/29/00
Black
MLN
02/14/01
Black
MLN

2.91
2.91

2.88
2.88

2.6125
2.6125

3.020
3.020

2.696
2.696

2.675
2.675

0.628
0.852

0.478
0.844

0.098
0.103

0.640
0.723

0.272
0.282

2.40e-003
0.022

0.218
0.545

0.193
0.633

0.019
1.302

0.413
5.687

0.241
7.864

0.011
0.135

0.413
1.622

0.083
0.255

6.36e-006
-0.258

0.543
0.633

0.538
0.666

0.150
0.732

0.875
2.797

0.652
3.104

0.129
0.191

0.890
1.326

0.363
0.408

3.13e-003
0.082

7.891e-005
8.579e-007

1 .719e-004
1 . 1596-006

3.984e-005
2.338e-005

5.420e-005
8.456e-007

7.7456-005
1 .063e-006

9.718e-007
4. 1916-007

2.51 le-004
6.924e-007

2.605e-005
2. 139e-006

4.4706-005
4.466e-005

 

55

9.2 Conclusion

This paper develops a method for using options prices to estimate the market’s
probability distribution for the underlying asset’s price. The mixture of two lognormal
method developed in this paper overcomes the deficiencies in standard Black option
pricing model for commodity futures option. Black’s model assumes proportional
changes in the underlying futures price follow identical lognormal distribution. Yet
empirical evidence suggests that variance of commodity futures price are not constant
and the tails of empirical price distributions appear to be much fatter than the lognormal
benchmark, indicating excess kurtosis in commodity price changes.

Since deriving price distributions has a strong motivation from both theoretical
and practical perspective, it is very important to derive a option pricing formula with a
more realistic distribution assumption which can be used to back out the implied
distribution of commodity prices.

The mixture of lognormals method is quite general, allowing the standard
lognormal distribution to be replaced by any distribution from within a wide class. As the
focus is only on the asset’s distribution, minimum structure is placed on the stochastic
process governing movements in the asset price over time. This lack of structure is
appealing since we generally have little prioriinformation about the stochastic process
that market participants have assumes.

In the application to the soybean, corn and wheat futures market, MLN model is
able to capture more information embedded in option data. Empirical results show that

MLN model accounts for the true properties of commodity price distribution, which

56

characterized by peaked, fat-tailed, and frequently positively skewed patten relative to a
Gusssian density.

The mean square deviation generated from MLN is always smaller than those
from Black’s model, which indicates smaller option pricing errors by MLN method.
Black model frequently overprices in—the-money call and out-of—the-money put options,
and underprice out-of-the-money call and in-the-money puts options. MLN formula
overcomes this deficiency by reducing the levels of errors in option pricing. The in—
. sample option pricing errors from MLN are usually at a level of $1’l‘10'3 or even smaller,
which has a better performance than that from Black model.

Extension of this research could go in a number of directions. First, statistical
comparison of Black’s model and mixture of lognormal distribution could be conducted
to test whether the results from MLN is statistically significant. Second, out of sample
forecast of option prices could be conducted by using the estimated parameters to check

whether the results form MLN are stable.

57

Appendix I: Derivation of formulas for MLN method

1.1 Derivation of the density function for lognormal distribution

Theorem: G(ln FT )d(ln FT) = g(FT )dFT (1.1.1)
Here G(-) is the probability density function of In F, ; g(-) is the probability density
function of FT.

Thus, a relationship equation can be derived,
G(ln FT )d(ln FT) = g(F, )dFT

=> Fl—Gfln FT)dFT = g(FT)dFT (1.12)

T

:9 7:1—G(ln FT) = g(FT)

T
This relationship equation is applied in the case of normal and lognormal
distributions. Assume In FT follows normal distribution; and F, follows lognormal

distribution. (2.1.1) can be expanded as

G(ln FT)d 1n FT = N(u,0)d In F, = N(u,a)?l-dFT

T

 

1 1 _(ln F,~—p)2
: __ . e 2‘72 dFT (1.1.3)
. FT 27w '
= g(FT )dFT_

Thus, (F)=—- 2“: (1.1.4)
g T F, 27w

 

This is the density function for single lognormal distribution. Thus, the density

function for mixture of two lognormal distributions is

58

1 1 , 1 ,
(F )= —(/1——6 “°‘ (1 -l)————e ””2 )
g T' FT JZn-a, JZn-o,
(1.1.5)
= 181(FT)+(1_A)32(FT)
1.2 Derivation of call option pricing formula for mixture of two lognormal
distributions
The option prices for call is
P... = e"""’ I (F. — K>g(F.>dP.
,. K .. (1.2.1)
= (“T-"[2 j (FT — K)g,(FT )dFT + (1 — A) I (F, — K)g,(F, )dFT]
K K
Thus Pm” = 3PM”l + (1 — )1)Pm,,2 (1.2.2)
pm,“ = e-“T‘” j (F, — K)g,(F, )dFT (1.2.3)
K ,
The following steps derive the formula for Pm,“
Pmu = e‘rU—I) J(FT _ K)8(FT )dFT
K (1.2.4)

= e"”"’[ j F.g<P.>dP. — K j g(P.)dF.I
K K

_(lnF-,—;1)2
., 2
e -0

where g(FT)=J2710 F
. . T

 

It contains two integrations. The calculation will be done one by one.

59

{FTg(FT)dFT

(111 FT—p)2

e 20’ dFT (1.2.5)

 

+ 1
IF Fug?

Let x = lnFT, then exdx = dFT

Thus, (1.2.5) is equal to

 

 

+oo (_x- _14'_)
20
e 6 dx
1,, K (Ix/1275
(n+0 ) -u +oo I _(Jr—(u+cr2))2
_ 203 20' -
— e e dx (1.2.6)
1,, K m/Zn

 

_ 2
Let y = x (”+0 ), then (1.2.6) is equal to
0'

 

(#32:) +0” 1 -§
= e I e - dy
an-(p+02)
0'

E

-ln K+(p+a‘2)

 

 

_ (11+?) 1 ‘%
—e I We dy (1.2.7)
:eIw—Z—i ~cdfi1(_an+”+0)

Next go to calculate the second integration.

60

KTg<F.>dF.

+oo 1 l _(lnF,:211)
— e 2" dF
KF, m/zn T (1.2.8)

 

1 (In __F-_, 14):

=KI————e 3" dlnFT
27:

First use x = lnFT, then (1.2.8) is equal to

 

 

e
M (hf—2,; . (1.2.9)

 

 

 

e 2 dy (1.2.10)

Therefore, the expected call option price under single lognormal distribution

could be written as

0.2
fipwgwyx —InK+u+a

cdfn(

Pmu = e

 

)-K°chfi(

inK—“f-n (1.2.11)
0'

Thus, the option pricing formula for mixture of two lognormal distributions is

61

—r -r (”1+ 1 )
[:01le (T )[e " 'Cdfi1(dri)_K'Cdfi1(d21)] (1.2.12)

.’
A.

+(1—A)e*"T-"Ie‘”” 2 ’ -cdfi2<d..>—K-cdfiz<d..>1

 

2
Where d“ = IHK‘H‘. +0’ , d2, =—an—+f—l—‘ i=1,2
0'. o.

I l

1.3 Derivation of put option pricing formula for mixture of two lognormal
distributions

put = APP“,| +(1- 1)!)me (1.31)
K ,
where PM = e-“T-P j (K — F, )g,(F, )dF, (1.3.2)
0

The following steps derive PM

put

K
P = e"‘T"’j (K — F, )g(F,)dF,
0 (1.3.3)

K +oo
= e"‘T"’[Kj g(F,)dF, — I F, g(F, )dF,]
0 K

_(1n F, ~14)2
2c;2

 

where g(F,)=J§;10 F e
. . T

So it contains two integrations. The calculation Will be done one by one.

62

KIg(FT)dFTl

_(In FT :2”)

202 dFT

 

(1.3.4)

 

During the integration, first let x = In F, , then let y = x _ I"

 

to get the derivation.

The second integration could be calculated as:

K
j F.g(F.)dP.

:1

 

NH

q.

§
“a

5' O‘—-.>c O‘—-.>:
H
E
i’
t

 

2 Q

._.N
:1

 

i

q
S,
«*1

(11+az’2—u2 111K 1 _(x-(#+02))
: e 202 I 8 202 dx
27w
nix—(11ml)
(In?) 1 -%h
1 -—er dy
_. 27:
a2 2
<u+—> In K — — a
= e 2 -cdfi2( ”

=6

 

) (1.3.5)

63

- . . . —- + 2
During the rntegrauon, we frrst let x = In F, , then let y = x (l1 G )
0'

 

to get the

derivation.
Therefore, the expected put option price under single lognormal distribution could

be written as

an_#)—em+gzi)-cdfn(an—#-02

 

P = e"‘T"’[K - cdfiz(

pur

)] (1.3.6)

Thus, the option pricing formula for mixture of two lognormal distributions is

2
0'1

P... = flew—”12"” 2 ’ ~cdfi2(-d,, ) — K -cdfi1(—d,,)]

 

(1.3.7)
- <1— 4)e"‘T"’ rem”? -cdfi2(—d..) — K - cdfizr—d. >1
2
Where (1,, = “”1 +0, , d, =M i=1,2
0', 0',
1.4 Derivation of the relationship between [11, u; and F r
Expected future price F r can be calculated as
E(F,) = jF,g(F,)dF,)
= F (11 (F )+(l-l) .(F ))dF
I T g! T g- T T (1.4.1)

= AjF,g,(F,)dF, + (1—A)jF,g,(F,)dF,
0 0
=AE,(F,)+(1—A)E,(F,)
where E,(F,) is the expected value of future price with a single lognormal

distributions in the risk neutral world. (In order to make it easy, I will not use the

subscript in the derivation).

64

E103): IFT81(FT)dFT

 

 

 

+... 1_(lnF,--;p)l
= I FT. f?“ OJ— 6 20- dFT
0
T 1 _(lnFr ”)2
= e ”2 dF
0 0127: T (1.4.2)
T l_-‘(‘ #____)_2
= e e 3" dx
.. 0J2?
(n+0 ) -y .. l _(x-(rt+,03))2
: e 20' I e 20' dx

 

During this integration, let x = In F, .
Thus, 133(F,) = n - e’“? + (1 —7r) - X“? (1.4.3)
1.5 Derivation of formulas for Expected Value, Variance, Skewness, and Kurtosis of

mixture of two lognormal distributions

The single lognormal distribution can be expressed as

 

 

g(F.) = 1 60—221 (151)
Jib-F, . H
Thus
.E(F ")=+j°°F "-g(F )dF =+I°F "- 1 612123413” (1.5.2)
T o T T T 0 T 5.0+} .T
Let lnF, =X,then F, =ex,and dX =d—IfT—.
T

65

E(F,")= jF," -g(F,)dF, = IF,"
0

x_#)3

=1???— Ue dX= —:J:‘/2_7:.O_e

 

 

 

 

 

 

+1» 1 ()(-(;I+no'))2 —2n/.40‘ 2~rrzcr4
= [2? Ce 20' dX
6'7“" 1 e_(X-(:l+2na))’
fij .. dX
4.4/er -ecr

 

Thus the following relationship exist.

E(F,)=e” 2 (n=1)

E(F~T2):621H-20’2 (11:2)

902

E(F.3)=e3” 2 (n=3)

E(FT4) : edfl+86Z (n: 4)

l _<1n F. ~11 )2

 

- e 2": dF
JZn-mF, T
_(X _~_#_)2+
’0 .de

(1.5.4)

(1.5.5)

(1.5.6)

(1.5.7)

The above formula are for single lognormal distribution. In mixture of two

lognormal distributions, E(F,") is a linear combination of E,(F,") and E,(F,") with

the weight of A. and 1-2» respectively. That is,

El (F,") = AE1(F,") + (1-l1)E2(F,")

(1.5.8)

The definitions of expected value, variance, skewness and kurtosis are as follows:

El (F,) is expected value.

For variance:

66

Var(F, ) = E[(F, - E(F,))2] = FIF,2 - 2 - F, -E(F, ) + E(F, )2]
= Exp(F,2)—2-E(F,)-E(F,)+E(F,)2 (1.5.9)
= E(F.2)— E11512

For skewness:

E[(F, — E(F,))3] = E[F,3 —3F,2E(F, ) + 3F,E(F,)2 — E(F, )3]
= E(F,3)-3E(F,2)E(F,) + 2(E(F,))3 (1.5.10)

For kurtosis:

EI(F, - E(Fr D4]

= EIF,‘ + 4F,"'(E(F,))2 + (HE D4

— 4F,3 (E(F, )) + 2F,2(E(F, ))2 - 4F. (E(F. ))3]

= E(F,‘) + 6E(F,2)(E(F, ))2 — 4E(F,3)(E(FT )) - 3(E(F. ))‘

(1.5.11)

67

Appendix II: PDFs, CDFs and option pricing errors of storage contracts of

soybean, corn and wheat

2.1 Estimated PDFs - July (storage) contract of soybean

a. July Option Maturity as of October 13, 1999

0.4 -—

0.3 '3

0.2 —‘

0.1—4

 

0.0 —

- - - Black
—— Mixture of lognormals

 

 

futures price ($)

b. July Option Maturity as of March 15, 2000

0.7 -—
0.6 -
0.5 4
0.4 —1
0.3 —i
0 2 —*

- - - Black
Mixture oflognormals

  

 

 

0.1 7
0.0

 

futures price (5)

c. July Option Maturity as of June 14, 2000

15..

1.0—

0.5 —

 

0.0 -—

' - - - Black

  

— Mixture of lognormals

 

—

futures price ($)

68

2.2 Estimated CDFs - July (storage) contract of soybean

a. July Option Maturity as of October 13, 1999

 

     
   
   

 

 

1.0 - ________
0.8 —
- - - Black
— Mixture of lognormals
0.6 —
0.4 -
0.2 -
2 4 6 8 10

futures price (5)

b. July Option Maturity as of March 15, 2000

 

 

 

1.0 — ----- 4
0.8 — - - - Black
—— Mixture oflognormals
0.6 —
0.4 -
0.2 -
2 4 6 8 10

futures price (5)

c. July Option Maturity as of June 14, 2000

 

 

 

1.0 m
0.8 m '
- - - Black
— Mixture oflognormals
0.6 -
0.4 —
0.2 -n
0.0 _ . I 17 I I
3 4 5 B 7

futures price ($)

69

2.3 Option pricing errors - July (storage) contract of soybean

a. Call Pricing Errors of October 13, 1999

 

 

 

 

 

 

 

 

 

g 30 —
3 - C- Black
E 20 4 -tr- Mixture of lognormals
'5
5 10—‘-"'“"---g~
E ~ .. ~
.2 0 . ~ .. _ . A
c2. ‘ ~ .. ~
77:? ~10 - ‘ ~ ~ . . ‘
.8 ~ ‘ §
*5 -20 — I ‘ -
E '7'
°-—’ -3
0- -30x10 "J
I I I I I
4.8 5.0 5.2 5.4 5.5
striking price ($)
b. Put Pricing Errors of October 13, 1999
g 30 —
3 - .- Black
E 20 T. ________ _‘ ~ -¢- Mixture of lognormals
(u ‘ - - _ -
P 10 — ‘ - .- ‘
E - . g
a A : A -. " u
E 0 u are . 1.
(D s
a ‘ '- ~
5 -10 -— ~ - g
o. ‘9
3
E '20 ‘-
“C
“-3 .3
0- -30x10 —
I I I I I
4.5 4.8 5.0 5.2 5.4

striking price (5)

c. Call pricing Errors of March 15, 2000

40
T - 0- Black
1— Mixture of lognormals

 

 

Predicted call premium - real value (5)
O
|
D
.fi

 

I I I I l I I

4.8 5.0 5.2 5.4 5.5 5.8 5.0
striking price ($)

70

(:1. Put Pricing Errors of March 15, 2000

Predicted put premium - real value ($)

 

 

 

4U —
- o- Black
‘ """" "- ., 1- Mixture oflognormals
2U — g ‘
N ‘~
0 ér 3; M
§ .‘
-20 — ‘ 7 C .‘
- I ‘9
40.1103 — -
I I I I I I I
4.8 5.0 5.2 5.4 5.5 5.8 5.0

striking price (5)

e. Call Pricing Errors of June 14, 2000

predicted call premium - real value ($)

 

 

 

15 —
- 0- Black .
10 — -121- Mixture of lognormals
5 - ' o l.‘
a ‘ ’ ‘ ‘
0 -W
A ‘ M
\ - _ o . ‘ - " - - '-
-5 _ 9' ' ' "
-10 —
-15x10'3 -
I I I I I I
4.5 4.8 5.0 5.2 5.4 5.5

striking price (5)

f. Put Pricing Errors of June 14, 2000

predicted put premium - real value ($)

15—

10—

 

- 0- Black
-A- Mixture of lognormals

 

 

\ _’ 8"
-5 _ ~ ' ¢ '
y ' d
-10 —
-15x10-3 —-
I I T I I F
4.5 4.8 5.0 5.2 5.4 5.5

striking price ($)

71

2.4 Estimated .PDFs - July (storage) contract of corn

3. July Option Maturity as of November 17, 1999

1.2 — - - - Black
— Mixture oflognormals

1.0 m
0.8 -l
0.5 -I
0.4 -
02 —

 

 

0.0 —

 

futures price (5)

b. July Option Maturity as of March 15, 2000

1.6 _ — Mixture oflognormals
1-4 T - - - Black

1.2 —
1.0 —
0.8 -
0.8 -
0.4 —
0.2 -‘

0.0

 

 

 

futures price (5)

c. July Option Maturity as of June 14, 2000

   

 

 

4 - — Mixture of lognormals
- - - Black
3 _
2 _
1 —-1
1.0 1.5 2.0 2.5 3.0

futures price ($)

72

2.5 Estimated CDFs — July (storage) contract of com

a. July Option Maturity as of November 17, 1999

 

1.0— ,_-_

   
  
   

0.8 - - - - Black
— Mixture oflognormals

0.5 —

0.4 T

0.2 —

 

0.0 —

 

futures price (5)

b. July Option Maturity as of March 15, 2000

 

1.0— ------

0.8 - — Mixture oflognormals

- - - Black
0.6 e

0.4 -

0.2 —

 

0.0 —

 

I I I I

1 2 3 4
futures price (5)

c. July Option Maturity as of June 14, 2000

 

 

 

 

1.0 -
0.8 — -— Mixture of lognormals
- - - Black
0.5 -
0.4 -
0.2 -
0'0 ‘1 ' ' ' ‘1‘ I r I
1.0 1.5 2.0 2.5 3.0

futures price (5)

73

2.6 Option Pricing errors - July (storage) contract of com

a. Call pricing errors of November 17, 1999

 

 

 

 

 

 

 

 

 

 

 

 

 

g 20 —
3
T; - 0- Black
7., 10 —‘ - ~ - _ . -tr- Mixture of lognormals
e ‘ "~ . .
S ‘ t~ ‘ - __¢:_‘——¢
g 0 — ! _. a _ ‘_ .. .r
a .-_‘
C—U . u. .- . . ‘ -
3 ~10 -" — 7 ‘ ‘ -.
2
.2
"O
9 -3
0— -20x10 '—
I I I I I I
2.0 2.1 2.2 2.3 2.4 2.5
striking price ($)
b. Put pricing errors of November 17, 1999
g 20 —
E - 0- Black
9 " - - _ -a— Mixture of lognormals
E 10 _‘ ~ ‘ “ t‘.
e ‘ ~ .
E I ‘ v.
3 L *
g 0 A tfifl .3 .
a) ‘ 1
a ‘ t- g
E ~-.~
.2- -10 —I ‘ fi . - ‘-
cu ~ .. .
73 "
‘5
9 -3
‘3- -20x10 —
I I I I I I
2.0 2.1 2.2 2.3 2.4 2.5
striking price (5)
c. Call pricing errors of March 15, 2000
a 20 —
§ ‘ ------- .. ‘ 1— Mixture of lognormals
T: 10_ N..- -0- Black
2 ‘ 1b.,
E ‘ .
.2 0 ‘A
M'— s: . -
°- to
«=3 ‘ ‘ ~
,3 -10 - " ~ .
f3 " ~ -
e ‘ ~ ~ ~ -
i -20x10'3 — '
l I I I I I
2.2 2.3 2.4 2.5 2.5 2.7

Striking Price (5)

74

(1. Put pricing errors of March 15, 2000

Predicted put premium - real value (5)

20—

 

-20x10'3 -]

 

 

c» . q ‘
" ~ - m ‘ -z:— Mixture oflognormals
10‘ ~.‘~ -o- Black
~ “
é— ‘ s ‘ ‘ ‘ ‘___ _.A
\ .‘ Cl
‘ \
\-
‘10 — ~ g
\ . ‘ ‘I

Q 5 h ‘.
I I I l I
2.2 2.3 2.4 2.5 2.6 2.7

Striking Price ($)

e. Call pricing errors of June 14, 2000

Predicted call premium - real value (5)

2U—

1U—

- 0- Black
—a— Mixture oflognormals

 

 

  

 

I I I I I

1.8 1.9 2.0 2.1 2.2 2.3
Striking Price ($)

f. Put pricing errors of June 14, 2000

Predicted put premium - real value (5)

207

1D—

- 0- Black
-t:- Mixture of lognormals

 

 

 

 

I I I I l

1.8 1.9 2.0 2.1 2.2 2.3
Striking Price ($)

75

2.7 Estimated PDFs - March (storage) contract of wheat

a. March Option Maturity as of June 14, 2000

  
  
   

 

0.8 -
- - - Black
— Mixture of lognormals
0.8 —
0.4 —
0.2 -
0'0 - I ‘ I I I I ..... r7

 

 

futures price ($)

b. March option Maturity as of November 29, 2000

1.8 —
1.4 —*
1.2 d
1.0 -
0.8 -
0.8—
0.4 —
0.2 —

- - - Black
— Mixture of lognormals

 

 

 

futures price ($)

c. March Option Maturity as of February 14, 2000

183d
140—
120-
1CD—
80d
80-
40-
20—
D

- - - Black
— Mixture of lognormals

 

 

 

I l I I I

2.80 2.82 2.84 2.88 2.88 2.70 2.72 2.74
futures price (5)

76

 

2.8 Estimated CDFs - March (storage) contract of wheat

a. March option maturity as of June 14, 2000

1.0

0.8

0.8

0.4

0.2

0.0

fl

‘

—

 

-----
-P-

     

- - - Black
— Mixture oflognormals

futures price ($)

b. March Option Maturity as of November 29, 2000

1.0
0.8
0.8
0.4
0.2

0.0

—

—I

—1

—-I

 

 

- - - Black
— Mixture oflognormals

 

I I I

1 2 3 4 5
futures price ($)

-
-

c. March Option Maturity as of February 14, 2000

1.0

0.8

0.8

0.4

0.2

0.0

 

 

 

 

_ - - - Black
— Mixture of lognormals
r j I I I
2.80 2.82 2.84 2.88 2.88 2.70 2.72 2.74

futures price (5)

77

2.9 Option pricing errors - March (storage) contract of wheat

a. Call pricing errors of June 14, 2000

 

 

 

 

g 30 —
g
T; 20 .1 - 6‘ Black
:3 ‘ . —a-— Mixture of lognormals
‘ 10 - 7 ‘ ~ - ‘
E ‘ ~ ~ .
a ‘ ‘ _‘
E 0 A— u
2 ‘ ~ .
Q- ‘ h
e 40 a ‘ h t .. _
(J - b
a ‘ ‘ ~ ~ -
§ -20 'fi ~ ~ "'
”O
9 -3
0- -30x10 —
I I I I l I
2.9 3.0 3.1 3.2 3.3 3.4

striking price ($)
b. Put Pricing Errors of June 14, 200015

20—

 

 

 

5 .
g 0 Black
T; A Mixture oflognormals
75 10—
2
é
3
E 0 2
0.)
‘5.
‘3
u -10_‘
(D
.‘Q
“U
93 -3
CL -20x10 - 1
2.80

striking price (5)
0. Call Pricing Error of November 29, 2000

10—

- 0- Black
-a- Mixture of lognormals

 

 

 

Predicted call premium real value (5)
O
\
4
I
I
I
[b

I I I I I

2.4 2.5 2.8 2.7 2.8 2.9
striking price (ii)

 

'5 There is only one set of data for put options on June 14, 2000.

78

(1. Put Pricing Errors of November 29, 2000

 

 

 

5 10 -
0)
% “Jo-JR. —9- Black
.2 o ‘ ’ ‘ . 1- Mixture oflognormals
(U 5 — ‘ h.‘
.02 ‘ ~ ..
2 0 s - . —¢
E ~ .
Q- s
5 fl
'3- '5 d x ‘ - w ‘
O.) ‘ .
.§ "
3
~— -3
CL -10x10 —
I I I I I I
2.4 2.5 2.8 2.7 2.8 _ 2.9

striking price (5)

e. Call Pricing Errors of February 14, 2000

 

 

 

 

1.3;. 10 —
3. - 0- Black
2 -ie:- Mixture of lognormals
To 5 -
2
S
E o _.a-::':‘—'_'____'%\
2 \ a a
Q. \
(=0 \
Q
B '5 —
(1)
§
'0
3 -10x10-3 -—
I I T I I I
2.5 2.8 2.7 2.8 2.9 3.0

striking price ($)

f. Put Pricing Errors of February 14, 2000

    

.4
20 - 0- Black

1- Mixture of lognormals

 

 

 

E
(D
2
(U
3
E 10 e
E
3
E 0 — .t a
9
D.
25; -10 —
"D
2’.
.2 -3
3 -20x10 -
i
I I I I l I
2.5 2.8 2.7 2.8 2.9 3.0

striking price ($)

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81

 

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