Risk Management and FInancial Engineering_Final Report

ROYAL MELBOURNE INSTITUTE OF TECHNOLOGY
Risk Management and
Financial Engineering
Empirical testing of Binomial, Black-Scholes and
Trinomial Models
Henry Santosa (s3583121), Jaimie Comungal (s3454408), Moises Martinez (s3434618), Naeem
Qudeer (s3524294) and Muhammad Usman Khan (s3511699)
Instructor:
Prof. Malick Sy.
Course:
Risk Management and Financial Engineering (BAFI2081)
The aim of thisreport isto testthe conventionalmodelsforcoveredwarrantcall option pricesand
evaluate whichmodel isstatisticallymore reliable alongwiththe suggestedstatistical inferential
modificationsinthose models

Table of Contents
Executive Summary:..................................................................................................................... 1
1 – Covered,Normal Equity Warrants and Options.........................................................................2
2 – Equity Warrants...................................................................................................................... 4
3 – Covered Warrants:.................................................................................................................. 7
3.1 – Types of Covered Warrants: .............................................................................................. 8
4 – Option Pricing Models – A comparison of Binomial, Black-Scholes and Trinomial model:........... 10
4.1 – Black-Scholes Model:...................................................................................................... 10
4.2 – Binomial Model:............................................................................................................. 12
4.3 – Trinomial Model:............................................................................................................ 13
5 – Comparison between OptionPricing Models & their calculation methodologies:...................... 15
6 – Theoretical modifications in Binomial, Black-Scholes and Trinomial:......................................... 16
7 – Statistical evaluation of the models:....................................................................................... 17
7.1 – Data collection andAssumptions:.................................................................................... 19
7.1.1 – Assumptions:........................................................................................................... 20
7.2 – Statistical Tests and findings:........................................................................................... 20
7.3 – Modifications of the Binomial, Black-Scholes and Trinomial models:.................................. 24
8 - Conclusion............................................................................................................................. 26
9 – References............................................................................................................................ 27
Appendices:............................................................................................................................... 29

I
List of Appendices
Appendix 1– Binomial Model forCapitaLand Warrant
a) Simple regressionoutput
b) Correlogramof simple regressionoutput
c) Serial correlationtestforsimple regressionoutput
d) Heteroskadacity(White) testforBinomial Model
e) Heteroskadacity(ARCH) testfor BinomialModel
f) ModifiedBinomial Model for CapitaLand
g) VIFtestfor ModifiedBinomial model
h) CorrelogramforModifiedBinomial model
i) Staticforecastsfor ModifiedBinomial model
j) DynamicForecastsfor Modified Binomial model
Appendix 2– Black-Scholes Model forCapitaLand
c) Serial correlationtestforsimple regression output
Appendix 3– Trinomial Model forCapitaLand
d) Heteroskadacity (White) testforBinomial Model

II
Appendix 4– Binomial Model forSembCorp Marine
Appendix 5– Black-Scholes SembCorp Marine
a. Simple regressionoutput
b. Correlogramof simple regressionoutput
c. Serial correlationtestforsimple regressionoutput
d. Heteroskadacity(White) testforBinomial Model
e. Heteroskadacity(ARCH) testfor BinomialModel
f. ModifiedBinomial Model for CapitaLand
g. VIFtestfor ModifiedBinomial model
h. CorrelogramforModifiedBinomial model
i. Staticforecastsfor ModifiedBinomial model
j. DynamicForecastsfor Modified Binomial model
Appendix 6– Trinomial Model forSembCorp Marine

III
Appendix 7– Binomial Model forWilmarInternationalWarrant
Appendix 8– Black-Scholes WilmarInternational Warrant
Appendix 9– Trinomial Model forWilmarInternationalWarrant

RiskManagement andFinancial Engineering (BAFI2081)
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Executive Summary:
The most widely used models for options and warrants pricing are Binomial model, Black-Scholes
model and Trinomial model. This report defines and tests these models with respect to the covered
warrants (calls) of three companies listed at the Singapore Stock Exchange. The foremost aim is to
test the models to analyze which one of the aforementioned models gives the most reliable warrant
priceswithleasterror.
The report will start with a brief comparison of simple options versus those of warrants and covered
warrants. The focus is both in terms of their functions and how these instruments are used in
present day. A mathematical comparison of all the aforementioned models is also included which
compares the mathematical differences in the models. Since the trinomial model includes an
additional condition of stationary stock price, which Binomial does not include, and Black-Scholes
have long been criticized as inaccurate, it is expected that the Trinomial model would statistically be
more reliable. The tests were performed through simple regressions, followed by tests for serial
correlation,heteroskedacityandmulticollinearity.
Based on a journal article and the results provided by the statistical tests, these models were then
modified using an Autoregressive and Moving Averages (ARMA) structure. Each model was then
fixed as per the nature of the statistical inference that suited the model. Finally for comparison
purposes, the modified models were then used for static and dynamic forecasts to gauge their
reliability. The root mean squared errors (RMSE) of the modified models were compared with that of
the actual warrant price data andindividual warrantprice predictedbyeachof the original model.

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1 – Covered, Normal Equity Warrants and Options
Equity options are known as contract that provide holders the right – though not the obligation – to
buy (call option) or sell (put option) shares of the underlying security at a particular price either on
or before a determined date. Those who grant these rights are known as the seller or the writer of
the contract. Although the sale of options are generally considered unprofitable, some investors still
sell options in order to hedge other positions, and ultimately buy futures (Bollen, NPB & Whaley, RE
2004).
Two of the most commonly accepted forms of equity options are the options on stock indexes or
sub-indexes,andoptionsonindividual shares.
Options are used by a large number of investors as a form of risk reduction in portfolios, to lock in
their target rates of returns, provide crisis insurance, and possibly even enhance their returns from
equity portfolios. Parallel to futures contracts, options also provide low transactions costs for rapid
assetallocation,makingthemdesirabletostronginvestors(EquityOptions,1992).
There are large demands for options in the market. Reasons include, firstly, because the risks built in
the stock markets cannot be hedged against by trading stocks alone. Furthermore, Carr and Wu
(2009) argue that the presence of stochastic volatility causes the market for options to become the
true market to trade volatility risk. In addition, investors may decide to gain exposure to various
shares due to the higher leverage provided by the options (Black, 1975). Informed traders may also
prefer trading options as it allows them to shield themselves through multiple option contracts that
are available forone security(Easley,O’Hara,andSrinivas1998).
All in all, we see that options provide many benefits and are widely used within the investment
world for many purposes. Therefore, we look further into the structure and nature of options to
obtaina more sophisticatedunderstandingof optionsandhow theyare priced.
Beginning with the structure, we can define equity options by the followingelements: their type (call
or put option), their style (American or European), their underlying security, units of trade, exercise
price and their expiration date. Going through each in more detail, we begin distinguishing the
difference betweencall andputoptions.
Call options refer to the right of the holder to buy shares at the exercise price either before or on a
determined date. On the other hand, put options provide the holder the right to sell shares at the
exercise price eitherbefore orona determineddate.

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Underlying security of an option refers to the shares/assets, at which the option will derive their
value hence being categorized as a derivative security. Since equity options often have a ‘unit of
trade’ of 100 shares, we may state that we can buy or sell 100 shares of the underlying security for
one optioncontract.
Next, we consider the exercise price (also known as strike price). This price represents the price at
which the option holder can buy or sell the shares of a company if he decides to exercise his right
against the writer of the contract. This price is often set close to the market price of the share. An
option may be considered in-the-money, out-the-money, or at-the money depending on the
difference between strike and market price of its underlying security. A call option would be in-the-
money if its exercise price is less than the current market price because this allows the holder to buy
the share for less than what it is currently priced at in the market. Similarly, a put option would be
in-the-money if the exercise price were greater than the current market price, as this allows the
holder to sell the share at a price higher than that being currently offered in the market. Therefore,
an option will be out of the money when the strike price is greater than the current market price for
a call option, and when the strike price is less than the current market price for a put option. Lastly,
if the strike price is equivalent to the current market price, we would state that the option is at the
money.
The price of an option is known as the premium which holders/buyers of the options must pay for
the right to buy or sell the underlying security at a locked price in the future. This premium is paid to
the writer of the option in exchange for the seller’s obligation to deliver the underlying security at
the strike price, if the buyer decides to exercise his call option (of to receive delivery if buyer decides
to exercise his put option). The premium is made up of two values: the intrinsic value and the time
value of money. The difference between the option premium and the intrinsic value makes up the
time value, which is ultimately affected by factors such as the interest rates, volatility, time to expiry,
share price,and dividend.
In regards to the style of an option, generally exchange-traded equity options are American-style
options and involve providing the option holder the right to buy or sell his option at any time prior to
the expiration date.1
If the option has not been exercised by this date, the option becomes worthless
and ceases existing as a financial instrument. Once a transaction is closed, this revokes the investor’s
previouspositionasthe holderorwriterof the option.
1 A European option allows an option to be exercised any time to the date of expiration.

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Finally, in order to exercise an option, the holder must advise his broker to submit an exercise notice
to the write,whothenhasto fulfil theirobligationsasstatedinthe contract.
Equity options are favoured by investors for its flexibility in dealing with intricate risks – especially,
risks regarding foreign equity price volatility and the foreign exchanges. In regards to foreign equity
options, there are various types with various payoff functions. This may provide the investors with
more choicesfor risk management and investment. Another reason whyequity options are favoured
is because of exchange trade, meaning that the financial product has more liquidity. On top of that,
regulations allow investors to minimise other risks, e.g. counterparty risk. (Fan, K., Shen, Y., Siu T.K.,
Wang, R.,2013)
2 – Equity Warrants
Now that we understand what equity options are, we now turn our focus to equity warrants, which
are quite similar in the way that they also provide investors with the right to buy a share at a
particular price at a specific date. The difference is that equity warrants and the actual shares can be
issued by the company themselves. Doing this allows the company to raise money and is also
beneficial for the investor as the warrants are often offered at a price lower than equity options.
Investors then profit by trading the warrants before the date of expiration, which is when the
warrant issuer must fulfil the investor’s requirements (Fung, H., Zhang, G. and Zhao, L., 2009).
Through equity warrants, investors can also enjoy them as a hedging tool. Specifically, a put warrant
could operate as a type of insurance policy for an investor’s share portfolio safeguarding them in the
eventthatthe share price falls.
Equity warrants differ further in that the level of supply and demand within the market, or even its
trading volumes, do not determine its price. Instead, issuers of warrants price them according to
specificpricingmodelssuchasthe Black Scholesmodel (Kui,L.,2007).
Another distinction between equity options and equity warrants is that warrants can be traded
continuously and more frequently. They also usually have lower transaction costs, which may be
because the value of the warrants contract may be lower than the value of the underlying asset,
which in turn result in lower brokerage and transactions fees. Further, warrants can be traded
directly through the interest systems provided by various brokers (Hunt, B. and Terry, C., 2011). As a
result,warrantsare oftenfavouredbyretail investors whoseeklessintricate processes.
To trade the warrants, they are often put on a nation’s traditional stock exchange, such as the
Australian Stock Exchange (hereinafter ASX) or the Singapore Stock Exchange (SGX), similar to how
shares are usually traded. For example, in Australia,investors would have to use ASX’s equity trading

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system, ASX Trade, which allows transactions to be certified and cleared through the use of CHESS2
.
Further, using CHESS trading and its settling arrangements provides investors with more familiarity
than tradingthe optionsonthe ASX,whichare usuallyassociatedwithmargintrading.
When we consider their trading volume, we learn that there is an unmistakable pattern during the
trading day. Specifically, at the beginning and at the end of the day, we see that the volume of
equity warrants traded is much higher than any other time of the day. This may be because hedge
traders are more inclined to control positions that they were unable to during the night, as well as,
safeguard their open stock positions which are to be executed during the day. Similarly, speculative
traders may engage in trading during the time in order to exploit the fact that information declared
during the night has not been included in asset prices. We may see this occurring even in Australian
markets through the table and graphs given below. Evidently, we may conclude that during these
times of the trading day, demand for equity warrants may be much more inelastic (Segara, L &
SegaraR, 2007).
2 ClearingHouseElectronic Subregister System (CHESS)

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Table 1: Intraday patterns in the equity warrants

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Figure 1: Equity warrants trading volume
3 – CoveredWarrants:
A Covered Warrant is a listed security that is distributed by a financial organisation, of which
thereafter, is formed to be available for trade on a stock exchange. Covered warrants provide the
holder the right, but not a definite commitment in selling or buying the asset, at a certain amount
whether it was before or during the specific prearranged point in time of the underlying asset’s
expirydate (Chanetal.2012).
For the masses, the similarity between options and covered warrants are significant and can be seen
that both are alike, however, a covered warrant ordinarily comprises of a faster maturity in
comparison to options, and is issued with a larger variety of assets (Aitken & Segara 2005). In
addition, the terms presented in covered warrants vary highly compared to options and to a greater
extent,are more flexible with itsstructure inmeetingwhatthe marketdemands.
On average, covered warrants usually have an ordinary lifespan between six to twelve months while
others may contain greater and/or unrestricted arrangements. The privilege of not being obligated

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in exercising any right whether to buy or sell an underlying asset means any loss is limited to the
initial investment.
3.1 – Typesof CoveredWarrants:
There are various types of coveredwarrants available and in circulation globally, the most prominent
beingon stock warrants.Othercoveredwarrantsare on:
 Baskets; a group of stocks that allows investors to gainexposure in the simplest form just by
purchasinga securityinan arrangementof a coveredwarrant.
 Indices; incorporating the overall market of covered warrants globally, indices are the most
heavilytradedkind.
 Bonds;bondswithoptionrights
 Commodity; gives investors the capability in taking positions of commodities using covered
warrants.
 Currency;accessible onacollectionof exchange rates.
 Barrier; additional terms where if a particular price is hit, the covered warrant will be void.
Termscan alsobe changedif the barrierprice ismetdependentuponthe initial conditions.
 Trigger;triggeringaspecificmatterwithinthe termsthatleadstoa fixedpayout.
The addition of covered warrants as another financial instrument paved way for investors to have a
wider range of options to utilise (Horst & Veld 2008). Similar to options, the existence of covered
warrants can be found in two basic forms; a call and a put. These allow investors of covered
warrants to prevail as a result of either a rise or fall in the market. The maturity of covered warrants
is well established in advance, and dependent on its structure, it usually is the last day for which the
warrant couldbe exercised.
In obtaining the right for the issuance of warrants, it is relatively complex and requires the potential
issuer in meeting a stringent criterion that are set out in order for them to be approved. Over each
underlying security,warrant issuers offer a wide range of strike prices for investors, allowing them to
choose on the basis of how they perceive the market. The exercise style of covered warrants
however is also similar to those of options; an American or European style. In practice, in spite of the
differences between the exercise styles, these have small-scaled influences over the pricing of
covered warrants on the basis that selling the instrument is usually more profitable than exercising
the warrant earlierduringitslifetime.

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Prices of covered warrants have two elements similarly to those of options; intrinsic and time value.
The first relates to the value of the instrument of being exercised instantly while the latter reflects
the time to its maturity date. The time value of a covered warrant decreases rapidly as it approaches
the expiry date, indicating warrants nearing maturity carry a higher risk. The gearing and leverage
offered by a covered warrant are one of the underlying reasons that attract investors. In other words,
covered warrants more than what you initially invested from price movements (leverage) and
exposure (gearing).
There are a number of risks associated with covered warrants ranging from counterparty risk, the
performance of the underlying asset to currency risks. The heavy stringent criterion that exist in
allowing an institution to issue warrants minimises the credit risk involved in negotiating with the
issuer. However, the risk is still present. A covered warrant is seen as a contract between two parties
(issuer and the holder) where the holder bares the risk of the warrant issuer not performing the
contractual obligationinthe instrument.
The fundamental factor in measuring the outcome of success solely relies on the performance of the
underlying asset in question. In cases where a call warrant is present, its success relies on the
underlying assets value to rise and vice versa with put warrants. In a case where the value of the
underlying asset remains steady over time, the losses are limited to the costs incurred in obtaining
the warrant.
Where an antagonistic movement in value of the underlying asset against the desired expectation of
the investor occur, this is known as market risk. There is also a risk in the ability to offload the
warrant in the market for a suitable price. This could be due to a deficiency of liquidity in the actual
underlyingassetalonetherefore decreasingsuchdemandforthe particularwarranton offer.

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Table 2.0 – Covered warrants in circulation (2005-2007)
Covered warrants on overseas indices or foreign currencies expose the investor to currency risks
when an unfavourable movement in the currency occurs with regards to the relevant exchange rate
against a particular currency. Dependent on the terms within the warrant, the issuer may reserve
the right to terminate, withdraw or cancel the warrant on certain triggers like an extraordinaryevent
withregardsto the underlyingasset.
Geng, Qi Ding and Zhang (2013) discuss that a particular warrant’s size on a particular stock and its
level of liquidity will have a substantial impact on the price of the warrant. Their research looks into
the market of warrantsin China and discuss that small & individual investors made up of most of the
warrants. Furthermore, it was also stated that the lack of understanding or lack of financial
knowledge in simply assuming and predicting the market were going to go up or down amongst
investorsalsoplayedamajorrole inthe price of warrantsinthe market.
4 – Option PricingModels – A comparison of Binomial,Black-Scholesand Trinomial model:
There is a high level of complexity in valuing an option as an option contract. It is dependent upon
the number of different variables which may affect the price of underlying assets. Over time there
have been many different types of modelsintroduced to deal with the complexity of valuing options.
The most widely known models to date are Black-Sholes, Binomial and the Trinomial model. It is
necessary to have an idea about the difference between European style options and American style
options,inorderto discussthe applicabilityof these models.
 European style options: European style option is an option which cannot be exercised
before the expirationdate.
 American style options: American style option is an option that can be exercised at any time
duringthe life of thatoption.
4.1 – Black-ScholesModel:
Black-Scholes model was developed by Fischer Black and Myron Scholes in 1973 (Black and Scholes
1973). Black-Scholes model is considered one of the best theoretical models for pricing a European
option and in due time, became one of the most foundation concepts within the realm of modern
financial theory. Its basic principles are used in the formulas found today for the evaluation of
almostall options. Black-Scholespricingformulasforcall optionsandputoptionsare givenbelow:

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Where:
 C = call value
 P = putvalue
 S = stockprice
 K = exercise price of anoption
 T-t = time betweenexpirationdate andthe valuationdate of anoption
 r = riskfree interestrate
 N(d) = standardnormal cumulative distributioninpointd
 ∂ = volatilityof underlyingindex
It can be observed from the above formula that an option with a higher volatility will be more
valuable in comparison to a one that has a lower volatility. Furthermore, the higher the ratio of stock
price to the exercise price is, the higherprice itwill be forthatoption.
4.1.1 – Assumptions:
The basic underlyingassumptionsfromthe original Black-Sholesmodelare:
 The optionisa Europeanstyle option,whichmeansthatitcannotbe exercisedbefore the
expirationdate
 It assumesthatthe volatilityof the underlyingstockisconstant.
 There isno arbitrage due to the efficientmarkets.
 Like volatilityitassumesriskfree interest rate remainsconstantoverthe time.
 Returnon the underlyingstockfollowsanormal distribution
 Markets are alwaysopen,givinganopportunitytobuyor sell anyoptionatany giventime.
 No transactioncostsinvolvedinbuyingorsellinganoption.
 Zerodividendsare paidduringthe optionlife.

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4.2 – Binomial Model:
Binomial option pricing model was invented by Cox-Rubinstein in 1979. He invented it as a tool for
the explanation of Black Scholes model to his students but soon it was found to be a more accurate
model in pricing American style options (Cox and Rubinstein 1979)). American style options are the
options which can be exercised before the date of expiration. Binomial model divides time to
expiration into a large number of short time intervals and produces a tree of prices working forward
from present to expiration step by step. It assumes that the value of current stock will either go up
or downin a certaintime period.One stepbinomial model foroptionpricingisgivenbelow
One Step Binomial Model
Where So isthe initial price of stock, pisthe probabilitythatvalue of stockwill goupbyfactor u and
1-p is a probabilitythatstockprice will fall downbyfactor d.
A riskneutral worldisassumedoverasmall periodof time,given thatthe effective returnof
binomial mode isequal torisk-free rate.
.
Andalsothe variance of risk-free assetisequal tothe variance of anassetin a risk-neutral world
givenbythe followingequation.
The relationbetweenupsidefactoranda downside factorisgivenby:

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From above equations,valuesfor p,uand d have beenobtainedasfollows:
The valuesof p, u and d givenbythe Cox,Rossand Rubinstein(CRR) model meansthatthe
underlyinginitialstockprice issymmetricforamulti-stepbinomialmodel.
Two Steps Binomial Model
4.3 – Trinomial Model:
Trinomial option pricing was proposed by Boyle in 1986 as an extension of the binomial model
(Boyle 1986). Trinomial model is considered to be more of an advanced form of a binomial model as
it gives three possible values that an underlying asset in a certain time period can be greater than,
less than or same as the current value of stock. The Trinomial model contains a third possible value
which assumes a zero change in the value of the stock makes this model more appropriate to deal
withthe real life situations.Trinomial tree canbe definedas

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S(t)u with probability pu
S(t + ∆t) = S(t) with probability 1−pu −pd
S(t)d with probability pd
Accordingto no arbitrage conditionwe have
E[S(ti+1)|S(ti)] = er∆tS(ti) …………………(a)
Var[S(ti+1)|S(ti)] = ∆tS(ti)2σ2 + O(∆t)…..(b)
Assuming that volatility of the underlying stock is constant during time interval t , r is the risk free
rate that the average return from the stock should be equal to the risk-free rate which can be
writtenas:
1− pu − pd +puu +pdd = er∆t.
In orderto lookat upwardand downwardjumprequiresanextraconstraintthatsize of upward
jumpisa reciprocal of a downwardjumpi-e ud = 1…………………(c)
The value of the underlying stock can be find out by using the given knowledge of upward and
downward jump sizes u and d with transition probabilities pu and pd. If Nu, Nd and Nm are the
numbers of upward, downward and middle jumps respectively then the value of underlying stock at
node j and fortime interval i isgivenas
Si,j = uNu dNd S(t0), where Nu + Nd +Nm = n
Three constraints (a), (b) and (c) have been imposed on u, d, pu and pm which results as a family of
trinomial tree models,jumpsizesof the popularrepresentative of thatfamilyare

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Andits changingprobabilitiesare:
Stock index Sowill move upby Su or downby Sd or will remainsame asSo. Pu isthe probabilityof
upwardmomentand pd is the probabilitythatstockprice will move upward,hencethe probability
that the stock remainssame will be givenby (1-pu-pd).
5 – Comparison betweenOptionPricingModels & theircalculation methodologies:
All of above models have an edge on one another according to different circumstances. All of these
models use same inputs; stock price, strike price, time to maturity, risk-free rate and volatility. Black-
Scholes is a continuous time or closed form model while there are discrete steps in binomial and
trinomial models. In binomial model we compute future value of an option at time t by taking into
account the time value of moneyandthendiscounteditbackto getthe presentvalue of the option.
Black-Scholes model is elegant and analytical. It includes the minimum value of the stock price So
minus the discounted strike price ‘ ’ and has added probability functions called standard
normal cumulative distribution functions. Binomial and trinomial models build a map of the future
stock prices with a number of steps and that number can go up to infinity. This means we are
converging to a Black-Scholes model, demonstrating that Black-Scholes model is a special case of
binomial andtrinomial modelswhere numberof steps canbe infinite.
Black-Scholes model is used in a wide range for option pricing, especially for European options.
Binomial model can price an American style options more accurately as it also considers the
possibility of early exercise of an option as it provides an insight of decision at different time
intervals before the expiration date that either an option should be exercised or shouldit be held for
a longer period. On the other hand, Black-Scholes model only considers the possibility of exercising
an option at the expiration date. However it can be implemented on American style options by
considering shorter times for expiration. It is easy to implement a binomial method in a spread
sheet for pricing options giving prices at every step. Black-Scholes model is much more convenient
for calculating a large number of option prices very quickly. Binomial model can be much more

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complex than the Black-Scholes model in doing so. Trinomial model presents a more realistic view of
the behaviour of financial instruments. Trinomial models can sometimes become inconvenient and
inefficient but it crucially considers the third option of possibility of stock value - remaining
stationaryat eachstep.
6 – Theoretical modificationsin Binomial,Black-Scholesand Trinomial:
After calculating theoretical warrant prices using Black-Scholes (BS) Model, Binomial Model and
Trinomial Model, it is apparent that none of the model is particularly close to the actual warrant
prices. Improved versions for all the models are necessary. This section will comment on current
problems of the 3 models that may affect their accuracies in this report, and propose ways to
improve them. Unless otherwise referenced, contents in this section are mainly derived from
Lauterbach and Schultz’s (1990) journal article titled “Pricing Warrants: An Empirical Study of the
Black-ScholesModel andItsAlternatives”.
Firstly, due to the normally long life of a warrant, the variance rate of return on stock and the risk-
free interest rate may be expected to change significantly during its life. To improve BS, Binomial and
Trinomial models, stochastic interest rates and stochastic variance rate of return should be used.
Both factors should not be assumed as constant and their inputs into the model have to be changed
on a daily basis. In particular, as the variance rate of return of a warrant’s underlying stock often
fluctuates and affects warrant prices greatly, the annualized variance rate of return has to be
modelled in a way that will more accurately reflect the market’s expected variance of the underlying
stock. Moreover, dividend should not be assumed as constant, as no/improper dividend adjustments
may lead to inaccurate theoretical model prices. This is because while constant dividends are used to
calculate model prices, it is logical to believe that markets would expect dividends to increase
(decrease) as underlying stock price increase (decrease). In other words, markets will expect the
dividend payout ratio to be the same. To address this problem, improved version of the models can
adjust dividend expectations in a daily basis. By simply adjusting daily expected dividends by a
percentage equivalent to the daily change in the underlying stock prices, the models’ theoretical
priceswill provide acloserindicationtoactual warrantprices.
Secondly, high volatilities tend to resultin extremely high theoretical warrant prices. This meant that
if an individual stock is having a fluctuating year (for example, SGX’s Wilmar from 1 June 2016 to 31
June 2016 as used in this report) theoretical prices tend to be overstated regardless of which model
was used. In order to address such issue, an improved model should utilize equity volatilities of a
basket of stocks that are similar to the underlying stocks (instead of only the warrant’s underlying
stock volatility) tocalculate the annualizedstandarddeviationforthe models’inputs.

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Thirdly, it has been observed that warrant prices are less sensitive to underlying equity values than
the model predicts. As warrants usually trade less frequently than the underlying stock, implied
standard deviation (ISD) in a warrant tend to biased. ISD tend to be downward (upward) biased
when price increases (decreases), which means that actual warrant prices will not increase or
decrease by as much as it should be. In an improved model, sing lagged standard deviation/volatility
will minimize such bias. By taking yesterday’s stochastic annualized volatility to predict today’s prices,
any upward or downward pressure in ISD, and thus theoretical warrant prices, can be neutralized. In
addition, and more importantly, the improved model should allow an inverse relation between
equity volatility and equity value. This means that when equity value increases (decreases), the
model should decrease (increase) equity volatility, which will result in lower (higher) theoretical
prices when warrants are supposed to increase (decrease) in value. This mean movement in
theoretical prices will be less sensitive and more similar to movement in actual warrant prices. The
improved model can, for example, integrate Cox’s Constant Elasticity of Variance (CEV) or Square
Root CEV formulato calculate the standarddeviationforthe model.
7 – Statistical evaluationof the models:
The empirical testingof the Binomial, Black-Scholes(BS) andTrinomial wasdone throughrunninga
simple regressionfunctionforthe modelsagainstthe actual warrantpricesof the coveredwarrants.
The structure of the regressionwas:
Binomial/Black-Scholes/Trinomial Model Price = Intercept + β actual warrant prices
Ideally the regression should produce the coefficient of the warrant prices (β) near to unit value and
the intercept close to zero. The basic framework included testing these regressions as to how
accurately they predict the warrant prices. The foremost statistical measures that were used to
evaluate the models included R-squared measure, Durbin Watson test along with Akaike Info,
Schwartz and Hannan-Quinn criterion. On the basis of the outputs provided by Eviews, further
statistical tests were performed to check for autocorrelation, heteroskedacity and mutlicollinearity.
To check for these effects, Correlogram, Serial correlation LM test, Heteroskedacity White test,
Heteroskedacity ARCH test were performed on the simple regression output of the warrant price
data and predictedvaluesof warrantsbythe selectedmathematical models.
Since coming up with a model that incorporates the complete dynamics of the covered warrants was
out of the scope of the current level of study and of this report, statistical inference approach was
selected. Once the results for these tests were obtained, an Autoregressive Moving Averages (ARMA)
structure was used to make the residuals more structured and make the model statistically fit.

18 | P a g e
Hence the modified Binomial, Black-Scholes and Trinomial models all contained some ARMA (p,q)
structure with improved aforementioned statistical measures. To further validate and compare the
performance of the modified models against that of the raw warrant prices data provided by the
actual models, Root Mean Squared Errors (RMSE) were compared. The RMSE values for raw data
were comparedagainstthe statistical forecasts anddynamicforecasts of the modifiedmodels.
Appendix 1-3 summarise the results for CapitaLand’s Binomial, Black-Scholes and Trinomial models,
Appendix 4-6 contain the same framework for SembCorp Marine and Appendix 7-9 illustrate the
results for Wilmar International. All Appendices summarise the statistical tests and outputs with the
followingsubpartclassificationsformat:
a) Simple regressionfunctionoutputwithnormalitytestsforresiduals
b) Correlogramof the simple regressionoutput
c) Serial CorrelationLMTest(BreuschGodfrey)
d) Heteroskedacity –White Test
e) Heteroskedacity –ARCH Test
f) Modifiedmodel withARMA structure
g) Variance Inflation(VIF) testformluticollinearity
h) Correlogramforthe modifiedmodel
i) Statistical forecastsformodifiedmodels
j) Dynamicforecastsforthe modifiedmodels

19 | P a g e
7.1 – Data collectionand Assumptions:
In this report, three existing covered warrants for Singaporean companies are considered. The
companies thatwere selected are:
 CapitaLand (Industry:Real Estate)
 SembCorpMarine (Industry:Offshore andMarine)
 WilmarInternational (Industry:FoodProcessing)
The selection of these companies was random and the main reason was based on the availability of
data. Data for actual warrant prices were collected from Macquarie Bank’s website. The warrants
are Europeaninnature and have the followingcharacteristics:
CapitaLand CoveredWarrant – Main features:
Stricke Price 3.2
Expirydate 12-Dec-16
Riskfree rate 1.86%
Total daysinyears 261
Div. Yield 2.82%
SembCorpmarine CoveredWarrant – Main features:
Stricke Price 1.95
Riskfree rate 1.86%
Div. Yield 2.69%
WilmarInternational CoveredWarrant – Main features:
Stricke Price 3.3
Riskfree rate 1.86%
Div. Yield 2.48%

20 | P a g e
7.1.1 – Assumptions:
The assumptions usedinthisstudyare the following:
 The study isbasedon the historical datataken fromthe workingdaysof 1-Jan-2016 to 30-
Jun-2016.
 20 nodes are selectedforthe calculationof the warrantpricesinBinomial andTrinomial
models
 The risk free rate for the calculationswas1.86% (annualized), whichwasthe yieldonthe 10
yearSingaporeangovernmentbond atthe time of checking
 Time to maturity inyears was calculatedforeach dailycalculation.The official numberof
workingdaysin2016 usedisaccording Singapore’s officialcalendar.
 Dividendyieldsforthe warrantprice calculationinthe Trinomial model weretakenfrom
BloombergandYahooFinance.These were checkedagainstthose listedonThomson
ReutersandCNBC whichwere foundtobe identical
 Volatilitywasthe stock’sstandarddeviationinthe selectedtime period.The volatilitywas
thenannualized forthe calculationof theoretical warrantprices
7.2 – Statistical Testsand findings:
As shown in Appendices 1-9 (subpart a), the simple regression function output for the models for the
selected sample of companies show a very different result than what was ideally expected. The
values for the intercepts, as shown on Table 1 on the following page, and coefficients were larger
than unit value but remains statistically significant i.e. high f-stat values with less than 0.05 p-values.
The most notable difference was the coefficients for the warrant price variable which pointed out
that Trinomial model are better than the Binomial and Black-Scholes. The values for the actual
warrant price coefficients were as a whole smaller than those of the actual warrant price coefficients
for Binomial and Black-Scholes model. This depicts that less numerical adjustment is needed for the
values given by the Trinomial model to match with that of the actual warrant prices i.e. Trinomial
model pricesare closertothe unitvalue coefficient.
The normality tests for the residuals of simple regression outputs for all three models for CapitaLand
and SembCorp Marine were non-normal as the Jarque-Bera values were high and p-value was less
than 0.05 (resulting in rejection of null of normality as shown in Table 1 on following page). The only
exception was the residuals of the simple regression outputs for the models for Wilmar
International’s warrants. They were all significantly normal following a very close normal distribution
and low JB stat with non-significant p value as shown by Appendix 7a, 8a and 9a. The regression

21 | P a g e
models also had a very low Durbin-Watson stat with the highest value of DW of 0.43 signalling a
severe serial correlation.
The correlogram tests for the simple regression functions for the models, as depicted in Appendices
1-9 (subpart b), showed that these regressions had a very strong AR 1 process with the highest AC
and PAC values at the first lag. On further testing through Breusch-Godfrey LM test with 5 lags, it
became evident that the regressions outputs have strong presence of serial correlation up to lag 2
on average, as the test results were significant and had F-stat. prob < 0.05 (Appendices 1-9 subparts
c). To test for Heteroskedacity, White test was run for all the models for simple regression which
showed a strong statistically significant presence of heteroskedacity for CapitaLand and SembCorp
Marine. In case of simple regression models for Wilmar International, although White test provided
statistically significant results, it was less severe than that of the models for other companies in the
sample (appendices8and9 subpartsc).
The strong presence of heteroskedacity in the residuals indicated a presence of Autoregressive
Conditional Heteroskedacity (ARCH) in the regressions. To confirm that, a heteroskedacity (ARCH)
test was run (appendices 1-9 subparts d). The tests proved significant presence of ARCHeffects as all
the statistical tests resultedin the rejection of null of ARCH. Due to the limitation of the functionality
of Eviews, exact ARCH tests couldn’t be performed to further evaluate whether which level of ARCH
testwas necessarytogetall the ARCHeffectsremovedorwhetheraGARCHwouldbe a betterfit. 3
In addition, an observation of the raw data (i.e. comparison of the actual warrant prices and prices
predicted by the selected models for the sample companies) showed a trend towards more accurate
prediction whenever the stock price moved closer to exercise price. The pattern was present in all
three models for all the companies, and was strongest for the trinomial model. This effect could be
the outcome of the in the moneynature of the optionpriceswhichthe modelscalculate.
3 Eviews at RMIT crashed every time whenever an ARCH/GARCH test was run even usingthe MyDesktop app.

22 | P a g e
CapitaLand SembCorp Marine Wilmar International
Binomial
Model
Black-Scholes
Model
Trinomial
Model
Binomial
Model
Black-Scholes
Model
Trinomial
Model
Binomial
Model
Black-Scholes
Model
Trinomial
Model
Intercept -0.146 -0.146 -0.113 -0.043 -0.043 -0.034 0.333 0.334 0.308
Coefficient(β) of
Warrant Variable
3.625 3.612 2.790 1.230 1.226 0.951 0.505 0.498 0.479
R-squared 0.720 0.720 0.676 0.835 0.836 0.819 0.096 0.094 0.095
DurbinWatson 0.190 0.194 0.189 0.430 0.424 0.438 0.234 0.236 0.234
Akaike info.Criterion -4.603 -4.611 -4.919 -7.474 -7.493 -7.880 -1.696 -1.677 -1.757
SchwarzCriterion -4.557 -4.566 -4.873 -7.429 -7.447 -7.835 -1.651 -1.630 -1.711
Hannan-QuinnCriterion -4.584 -4.592 -4.900 -7.456 -7.474 -7.862 -1.678 -1.658 -1.738
F-statof regression 316.557 316.824 257.133 620.338 627.319 556.348 12.999 12.387 12.421
Prob.Of F-statof
regression
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.001
Jarque-BeraStats 49.427 47.109 31.536 64.546 61.921 93.909 1.975 1.333 1.222
Jarque-BeraStats
(prob)
0.000 0.000 0.000 0.000 0.000 0.000 0.373 0.514 0.543
Table 1 – Statistical MeasuresTable for the Simple Regressionoutput results for Binomial, Black-Scholes and Trinomial Models vs. Actual Warrant Prices

23 | P a g e
CapitaLand SembCorp Marine Wilmar International
Binomial
Model
Black-Scholes
Model
Trinomial
Model
Binomial
Model
Black-Scholes
Model
Trinomial
Model
Binomial
Model
Black-Scholes
Model
Trinomial
Model
Intercept -0.212 -0.212 -1.795 -0.053 -0.053 -0.042 0.380 0.337 0.311
Coefficient(β) of
Warrant Variable
4.507 4.508 3.706 1.412 1.423 1.119 0.022 0.022 0.023
R-squared 0.957 95.701 0.952 0.953 0.956 0.950 0.924 0.946 0.946
DurbinWatson 1.938 1.919 1.922 1.781 1.902 1.920 2.025 1.836 1.844
Akaike info.Criterion -6.831 -6.784 -7.096 -8.510 -8.599 -8.950 -4.405 -4.441 -4.542
SchwarzCriterion -6.697 -6.649 -6.962 -8.349 -8.465 -8.815 -4.267 -4.348 -4.450
Hannan-QuinnCriterion -6.777 -6.729 -7.042 -8.445 -8.545 -8.895 -4.349 -4.403 -4.505
F-statof regression 526.303 500.851 449.603 361.552 486.811 428.006 261.545 683.728 699.554
Prob.Of F-statof
regression
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
Table 2 – Statistical Measures Table for the Modified (ARMA – structure) Binomial, Black-Scholes and Trinomial Models

24 | P a g e
7.3 – Modificationsofthe Binomial,Black-Scholesand Trinomial models:
As discussed in the previous section, the regression outputs provide strong evidences of serial
correlation and heteroskedacity. These conditions lead to the modification of the models as per the
Autoregressive Moving Averages (ARMA) structure. A summary of the modifications that were
performedonthe modelsis asfollowing:
Companies
Modified Binomial
Model
Modified Black-Scholes
Model
Modified Trinomial
Model
AR Terms MA Terms AR Terms MA Terms AR Terms MA Terms
CapitaLand 1 1 1 1 1 1
SembCorp
Marine
1 2 (2,3) 1 1(2) 1 1
WilmarInt. 1 1 1 0 1 0
Table 2 - ARMA Structure of the Modified Models for sample companies
After adjusting the models to the above ARMA structures, some notable differences were observed
in the models. First, the R-squared measure increased whereas the Durbin Watson measure became
much closer to 2. This suggests that although the modifications have taken care of serial correlation,
some effects still remain although is not as severe as before. Like previously, the coefficient for the
warrant price variable in the equation was the least for the Trinomial model here. This is consistent
with the original regression function. The results of the modified regression outputs are shown in
Appendices 1-9 (subsection f). The inverted AR Roots were also less than 1 (unit measure) and
Eviews did not give any warning related to the non-stationary relationship of the added ARMA terms.
The ARMA terms were statistically significant i.e. having large t-stat values with p-values < 0.05. The
modification also had a collective impact over the Akaike Info, Schwarz and Hannan-Quinn criterion
which became lesser than the ones for simple regression output as shown in Table 1 and Table 2 on
the previous page. To check for the presence of multicollinearity of the added variables and the
coefficients, Variance Inflation Factor (VIF) test was run for all the ARMA structure models which
showed that mutlicollinearity is not a problem with the modified models as shown inAppendices 1-9
(subparts g). Since all the coefficients of the modified models had Cantered VIF score of less than 2,
the modifiedmodelsweredeemedtobe betterthanthe original regressions.
The correlograms of the modified models, as shown in the Appendices 1-9 (subparts h), also showed
that the serial correlation problem is not as severe anymore. The AC and PAC values were within the
limits and any significant spike was not present. The concerned spikesin the AC and PAC values were
over15 AR processesandthus,those were leftunchangedtokeep the parsimonyprincipleinview.

25 | P a g e
The final step to evaluate the modified models was to run a static forecast through the end portion
of the sample data points. This was accompanied by more robust dynamic forecasts of the modified
models, then having their Root Mean Squared Errors (RMSE) compared with RMSE of the raw data
that was obtained from the original Binomial, Black-Scholes and Trinomial models. As shown in the
table below the modified models contained the lowest RMSE scores for the models. In addition,
even in the modified models, Trinomial model clearly has the least RMSE scores as compared to
Binomial andBlack-Scholesmodel. Table 3providesasummaryof the RMSEs.
Companies
Root Mean Squared Errors (RMSE) of Raw Warrant Price Data with Normal
Models
Binomial Model Black-Scholes Model Trinomial Model
CapitaLand 0.044 0.044 0.028
SembCorp
Marine
0.033 0.033 0.036
WilmarInt. 0.318 0.317 0.291
Companies
Root Mean Squared Errors (RMSE) of Static Forecasts by Modified Models
Modified Binomial
Model
Model
Modified Trinomial
Model
CapitaLand 0.044 0.044 0.028
SembCorp
Marine
0.033 0.033 0.036
WilmarInt. 0.318 0.317 0.291
Companies
Root Mean Squared Errors (RMSE) of Dynamic Forecasts by Modified Models
Modified Binomial
Model
Model
Modified Trinomial
Model
CapitaLand 0.015 0.015 0.013
SembCorp
Marine
0.007 0.007 0.006
WilmarInt. 0.060 0.058 0.056
Table 3 - Summary of Root Mean Square Errors

26 | P a g e
8 - Conclusion
In conclusion, the report has looked into the most widely used options and warrants pricing models,
namely the Binomial, Black-Scholes and Trinomial model. These models were empirically tested
using covered warrants for three selected Singaporean companies. The main objective was to
analyze the predictability and accuracy of the models in estimating call warrant prices. Although all
the models proved to be somewhat weak predictors of the covered warrant prices, Trinomial model
seems to be the leasterroneous as compared to Binomial and Black-Scholes model. Nevertheless, all
3 modelscouldbe furtherimproved.
These models were tested through simple regression models which were further tested for
statistical nature and error patterns. Based on the statistical tests, these models were then modified
to reign in the nature of the errors of which these models have in predicting the warrant values. It
was concluded that even in the modified models Trinomial models stands out as the more reliable
measure onthe basisof itsleastroot meansquarederror(RMSE) terms.

27 | P a g e
9 – References
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stocks:Australianevidence’, Accounting and Finance,vol.45,no.1, pp.127-144
Black,F & Scholes,M1973, 'The Pricingof Optionsand Corporate Liabilities', The Journal of
Political Economy,vol.81,no.3, pp.637-654
Black,F., 1975, “Fact andFantasy inthe Use of Options,” FinancialAnalystsJournal,vol.31,pp.
36–72.
Bollen,NPB&Whaley,RE 2004, 'Does NetBuyingPressure Affectthe Shape of ImpliedVolatility
Functions?', TheJournalof Finance,vol.59, no. 2, pp. 711-53.
Boyle,PP1986, 'OptionValuationUsingThree-JumpProcess', International OptionsJournal,vol.
3, pp. 7-12
Carr, P., and L. Wu, 2009, “Variance Risk Premiums,” Review of Financial Studies, vol. 22, no. 3, pp.
1311– 1341.
Chan,CY, Peretti,CD,Qiao,Z & Wong,WK 2012, ‘Empirical testof the efficiencyof the UK
coveredwarrantsmarket:Stochasticdominance andlikelihoodratiotestapproach’, Journal
of Empirical Finance,vol.19, no. 1. pp. 162-174
Cox,J, Ross,S & Rubinstein,M1979, 'OptionPricing:A SimplifiedApproach', Journal of Financial
Economics,vol.7 pp. 229-263
Easley,D.,M. O’Hara,and P. S.Srinivas,1998, “OptionVolume andStockPrices:Evidenceon
Where InformedTradersTrade,” Journalof Finance,vol.53, pp.431–465.
Fan,K., Shen,Y.,SiuT.K.,Wang, R., 2013, “Pricingforeignequityoptionswithregime-switching,
EconomicModelling,vol.37, pp.296-297
Fung,H., Zhang,G. and Zhao,L., 2009, “China'sEquityWarrants Market: AnOverview and
Analysis”,TheChineseEconomy,vol.42,no.1, pp.86-97.
Horst, J & Veld,C2008, ‘AnEmpirical Analysisof the Pricingof BankIssuedOptionsversus
OptionsExchange Options’, European FinancialManagement,vol.14,no.2, pp.288-314
Hunt,B. and Terry, C.,2011, “AustralianEquityWarrants:Are Retail InvestorsGettingA FairGo?”,
The Finsia Journalof Applied Finance,vol.4, pp.48-64.
Kui,L., 2007, “Do WarrantsLead The Underlying Stocksand Index Futures?”, Singapore:Singapore
ManagementUniversity,p.3.

28 | P a g e
Lauterbach,B & Schultz,P 1990, PricingWarrants:An Empirical Studyof the Black-ScholesModel
and ItsAlternatives, TheJournalof Finance, vol. 45, no.4, pp. 1181-1209.
SegaraL & Segara,R 2007, ‘Intradaytradingpatternsinequitywarrantsand equityoptions
markets:Australianevidence’, TheAustralasian Accounting Business&FinanceJournal,vol.
1, no. 2, p. 52

29 | P a g e
Appendices:
Appendix1 (a) – Simple regressionoutputfor Binomial model for Capital Land
DependentVariable:BINOMIAL_MODEL_PRICE
Method:Least Squares
Date: 10/17/16 Time:22:28
Sample:1/04/2016 6/24/2016
Includedobservations:125
Variable Coefficient Std.Error t-Statistic Prob.
C -0.146451 0.013494 -10.85270 0.0000
WARRANT_PRICE 3.625156 0.203751 17.79205 0.0000
R-squared 0.720173 Mean dependentvar 0.090576
AdjustedR-squared 0.717898 S.D. dependent var 0.045256
S.E. of regression 0.024037 Akaike infocriterion -4.602565
Sumsquaredresid 0.071067 Schwarz criterion -4.557312
Log likelihood 289.6603 Hannan-Quinncriter. -4.584181
F-statistic 316.5572 Durbin-Watsonstat 0.190229
Prob(F-statistic) 0.000000
0
5
10
15
20
25
30
-0.04 -0.02 0.00 0.02 0.04 0.06 0.08
Series: Residuals
Sample 1/04/2016 6/24/2016
Observations 125
Mean 5.73e-17
Median 0.003068
Maximum 0.090313
Minimum -0.042560
Std. Dev. 0.023940
Skewness 0.799383
Kurtosis 5.633244
Jarque-Bera 49.42723
Probability 0.000000

30 | P a g e
Appendix1 (b) – Correlogram for simple regressionoutputfor Capital Land

31 | P a g e
Appendix1 (c) – Serial correlationLM Testfor simple regressionoutputfor Capital Land
Breusch-GodfreySerial CorrelationLMTest:
F-statistic 63.26895 Prob.F(5,118) 0.0000
Obs*R-squared 91.04080 Prob.Chi-Square(5) 0.0000
TestEquation:
DependentVariable:RESID
Date: 10/17/16 Time:23:14
Sample:1/04/2016 6/24/2016
Presample missingvaluelaggedresidualssettozero.
C -0.001515 0.007210 -0.210145 0.8339
WARRANT_PRICE 0.025677 0.108916 0.235754 0.8140
RESID(-1) 0.851940 0.092193 9.240853 0.0000
RESID(-2) -0.026605 0.119736 -0.222193 0.8245
RESID(-3) -0.135686 0.119166 -1.138628 0.2572
RESID(-4) 0.192246 0.119924 1.603066 0.1116
RESID(-5) 5.78E-06 0.092624 6.23E-05 1.0000
R-squared 0.728326 Mean dependentvar 5.73E-17
AdjustedR-squared 0.714512 S.D. dependentvar 0.023940

32 | P a g e
Appendix1 (d) – Heteroskadicity(White) testfor simple regressionoutputfor Capital Land
HeteroskedasticityTest:White
ScaledexplainedSS 54.92773 Prob.Chi-Square(2) 0.0000
TestEquation:
DependentVariable:RESID^2
Date: 10/17/16 Time:23:15
Sample:1/04/2016 6/24/2016
C 0.008402 0.003878 2.166399 0.0322
WARRANT_PRICE^2 2.597966 0.928224 2.798856 0.0060
WARRANT_PRICE -0.294102 0.121185 -2.426887 0.0167
R-squared 0.195901 Mean dependent var 0.000569

33 | P a g e
Appendix1 (e) – Heteroskadacity(ARCH) test for simple regressionoutputfor Capital Land
HeteroskedasticityTest:ARCH
TestEquation:
Date: 10/17/16 Time:23:16
Sample (adjusted):1/18/2016 6/24/2016
Includedobservations:115 afteradjustments
C 5.83E-05 4.74E-05 1.230806 0.2212
RESID^2(-1) 0.592820 0.095840 6.185539 0.0000
RESID^2(-2) 0.100839 0.110040 0.916384 0.3616
RESID^2(-3) 0.242010 0.110015 2.199790 0.0300
RESID^2(-4) -0.109843 0.112100 -0.979865 0.3294
RESID^2(-5) 0.009521 0.111404 0.085462 0.9321
RESID^2(-6) -0.021516 0.110696 -0.194370 0.8463
RESID^2(-7) -0.046134 0.100315 -0.459891 0.6466
RESID^2(-8) 0.128140 0.078521 1.631915 0.1057
RESID^2(-9) -0.169137 0.076309 -2.216474 0.0288
RESID^2(-10) 0.110309 0.051196 2.154653 0.0335
Sumsquaredresid 1.45E-05 Schwarz criterion -12.59702

34 | P a g e
Appendix1 (f) – Modifiedmodel forCapital Land Binomial Model
Method:ARMA MaximumLikelihood(OPG- BHHH)
Date: 10/18/16 Time:18:41
Sample:2/01/2016 6/10/2016
Convergence achievedafter31 iterations
Coefficientcovariance computedusingouterproductof gradients
C -0.212193 0.026576 -7.984247 0.0000
WARRANT_PRICE 4.507190 0.259791 17.34927 0.0000
AR(1) 0.984027 0.019044 51.67061 0.0000
MA(1) -0.373227 0.086969 -4.291509 0.0000
SIGMASQ 5.53E-05 4.98E-06 11.11552 0.0000
InvertedARRoots .98
InvertedMA Roots .37
Appendix1 (g) – VIF Test for ModifiedBinomial model forCapital Land
Variance InflationFactors
Date: 10/18/16 Time:18:41
Sample:2/01/2016 6/10/2016
Coefficient Uncentered Centered
Variable Variance VIF VIF
C 0.000706 2.703830 NA
WARRANT_PRICE 0.067491 2.540167 1.337280
AR(1) 0.000363 2.166351 2.162275
MA(1) 0.007564 1.460435 1.427793
SIGMASQ 2.48E-11 2.372646 2.210507

35 | P a g e
Appendix1 (h) – Correlogram of ModifiedBinomial model forCapital Land

36 | P a g e
Appendix1 (I) – Static forecastsof modifiedbinomial model forCapital Land
-.02
.00
.02
.04
.06
.08
.10
.12
10 13 14 15 16 17 20 21 22 23 24
2016m6
MODBIN_STF ± 2 S.E.
Forecast: MODBIN_STF
Actual: BINOMIAL_MODEL_PRICE
Forecast sample: 6/10/2016 6/24/2016
Included observations: 11
Root Mean Squared Error 0.005033
Mean Absolute Error 0.004332
Mean Abs. Percent Error 12.58233
Theil Inequality Coefficient 0.056820
Bias Proportion 0.300654
Variance Proportion 0.123759
Covariance Proportion 0.575587
Appendix1 (J) – Dynamic forecasts of modifiedbinomial model forCapital Land
-.04
-.02
.00
.02
.04
.06
.08
.10
10 13 14 15 16 17 20 21 22 23 24
2016m6
MODBIN_DYF ± 2 S.E.
Forecast: MODBIN_DYF
Forecast sample: 6/10/2016 6/24/2016

37 | P a g e
Appendix2 (a) – Simple regressionoutputfor Black-Scholes(BS) model for Capital Land
DependentVariable:BS_MODEL_PRICE
Date: 10/17/16 Time:22:28
Sample:1/04/2016 6/24/2016
C -0.146008 0.013439 -10.86453 0.0000
WARRANT_PRICE 3.611772 0.202914 17.79955 0.0000
0
4
8
12
16
20
24
28
32
-0.04 -0.02 0.00 0.02 0.04 0.06 0.08
Series: Residuals
Sample 1/04/2016 6/24/2016
Observations 125
Mean 2.86e-17
Median 0.002584
Maximum 0.090007
Minimum -0.043039
Std. Dev. 0.023842
Skewness 0.802003
Kurtosis 5.544027

38 | P a g e
Appendix2 (b) – Correlogram of Simple regressionoutputfor Black-Scholes(BS) model for Capital
Land

39 | P a g e
Appendix2 (c) – Serial correlationtest of simple regressionoutputof Black-Scholes(BS) model for
Capital Land
TestEquation:
Date: 10/18/16 Time:18:49
Sample:1/04/2016 6/24/2016
C -0.001483 0.007242 -0.204719 0.8381
WARRANT_PRICE 0.025296 0.109393 0.231238 0.8175
RESID(-1) 0.846586 0.092251 9.176994 0.0000
RESID(-2) -0.031336 0.119674 -0.261843 0.7939
RESID(-3) -0.109651 0.119283 -0.919255 0.3598
RESID(-4) 0.176098 0.119785 1.470118 0.1442
RESID(-5) -0.002281 0.092624 -0.024621 0.9804

40 | P a g e
Appendix2 (d) – Heteroskadacity (White) testofSimple regressionoutputfor Black-Scholes(BS)
model for Capital Land
TestEquation:
Date: 10/18/16 Time:18:50
Sample:1/04/2016 6/24/2016
C 0.008208 0.003805 2.157272 0.0329
WARRANT_PRICE^2 2.544777 0.910622 2.794547 0.0060
WARRANT_PRICE -0.287637 0.118887 -2.419417 0.0170

41 | P a g e
Appendix2 (e) – Heteroskadacity(ARCH) test ofSimple regressionoutput for Black-Scholes(BS)
model for Capital Land
TestEquation:
Date: 10/18/16 Time:18:51
Sample (adjusted):1/18/2016 6/24/2016
C 6.08E-05 4.77E-05 1.272659 0.2060
RESID^2(-1) 0.618070 0.096025 6.436547 0.0000
RESID^2(-2) 0.068679 0.111514 0.615879 0.5393
RESID^2(-3) 0.254685 0.111404 2.286148 0.0243
RESID^2(-4) -0.139011 0.113850 -1.221001 0.2248
RESID^2(-5) 0.031538 0.113222 0.278555 0.7811
RESID^2(-6) -0.046684 0.112998 -0.413135 0.6804
RESID^2(-7) -0.006233 0.101233 -0.061571 0.9510
RESID^2(-8) 0.107083 0.080283 1.333813 0.1852
RESID^2(-9) -0.163184 0.078214 -2.086381 0.0394
RESID^2(-10) 0.107421 0.052533 2.044844 0.0434

42 | P a g e
Appendix2 (f) – Modifiedmodel forBlack-Scholes(BS) model for Capital Land ModifiedBSModel
Date: 10/18/16 Time:18:55
Sample:2/01/2016 6/10/2016
Included observations:95
C -0.212298 0.027382 -7.753102 0.0000
WARRANT_PRICE 4.507508 0.279429 16.13112 0.0000
AR(1) 0.982128 0.019722 49.79945 0.0000
MA(1) -0.364819 0.082626 -4.415319 0.0000
SIGMASQ 5.81E-05 5.21E-06 11.14643 0.0000
Adjusted R-squared 0.955097 S.D. dependentvar 0.036945
InvertedARRoots .98
Appendix2 (g) – VIF test ofmodifiedBlack-Scholes(BS) model for Capital Land
Date: 10/18/16 Time:18:55
Sample:2/01/2016 6/10/2016
C 0.000750 3.250658 NA
WARRANT_PRICE 0.078081 2.992788 1.372972
AR(1) 0.000389 2.219858 2.217503
MA(1) 0.006827 1.422085 1.393867
SIGMASQ 2.71E-11 2.475892 2.294333

43 | P a g e
Appendix2 (h) – Correlogram of modifiedBlack-Scholes(BS) model forCapital Land

44 | P a g e
Appendix2 (I) – Static forecastsof modifiedBlack-Scholes(BS) model for Capital Land
-.02
.00
.02
.04
.06
.08
.10
.12
10 13 14 15 16 17 20 21 22 23 24
2016m6
MODBS_STF ± 2 S.E.
Forecast: MODBS_STF
Actual: BS_MODEL_PRICE
Forecast sample: 6/10/2016 6/24/2016
Appendix2 (J) – Dynamic forecasts of modified Black-Scholes(BS) model for Capital Land
-.04
-.02
.00
.02
.04
.06
.08
.10
10 13 14 15 16 17 20 21 22 23 24
2016m6
MODBS_DYF ± 2 S.E.
Forecast: MODBS_DYF
Forecast sample: 6/10/2016 6/24/2016

45 | P a g e
Appendix3 (a) – Simple regressionoutputfor Trinomial model for Capital Land
DependentVariable:TRINOMIAL_MODEL_PRICE
Date: 10/17/16 Time:22:29
Sample:1/04/2016 6/24/2016
C -0.113260 0.011522 -9.830179 0.0000
WARRANT_PRICE 2.789592 0.173965 16.03538 0.0000
AdjustedR-squared 0.673799 S.D. dependent var 0.035934
0
5
10
15
20
25
30
35
-0.04 -0.02 0.00 0.02 0.04 0.06
Series: Residuals
Sample 1/04/2016 6/24/2016
Observations 125
Mean 4.30e-17
Median 0.002553
Maximum 0.073848
Minimum -0.037055
Std. Dev. 0.020440
Skewness 0.679445
Kurtosis 5.051431

46 | P a g e
Appendix3 (b) – Correlogram of Trinomial model for Capital Land

47 | P a g e
Appendix3 (c) – Serial correlationtest for Trinomial model for Capital Land
TestEquation:
DependentVariable: RESID
Date: 10/18/16 Time:19:03
Sample:1/04/2016 6/24/2016
C -0.001558 0.006090 -0.255813 0.7985
WARRANT_PRICE 0.026147 0.092010 0.284172 0.7768
RESID(-1) 0.866331 0.092331 9.382921 0.0000
RESID(-2) -0.056403 0.121080 -0.465836 0.6422
RESID(-3) -0.088670 0.120933 -0.733218 0.4649
RESID(-4) 0.163466 0.121217 1.348543 0.1801
RESID(-5) -0.000839 0.092735 -0.009050 0.9928

48 | P a g e
Appendix3 (d) – Heteroskadacity (White) testofTrinomial model for Capital Land
TestEquation:
Date: 10/18/16 Time:19:03
Sample:1/04/2016 6/24/2016
C 0.004847 0.002648 1.830309 0.0696
WARRANT_PRICE^2 1.565007 0.633804 2.469229 0.0149
WARRANT_PRICE -0.172787 0.082747 -2.088140 0.0389

49 | P a g e
Appendix3 (e) – Heteroskadacity(ARCH) test ofTrinomial model for Capital Land
TestEquation:
Date: 10/18/16 Time:19:04
Sample (adjusted): 1/18/2016 6/24/2016
C 5.02E-05 3.71E-05 1.355171 0.1783
RESID^2(-1) 0.622102 0.096796 6.426907 0.0000
RESID^2(-2) 0.073477 0.112808 0.651339 0.5163
RESID^2(-3) 0.249660 0.112679 2.215679 0.0289
RESID^2(-4) -0.144028 0.115045 -1.251929 0.2134
RESID^2(-5) 0.043006 0.114808 0.374590 0.7087
RESID^2(-6) -0.058190 0.114678 -0.507418 0.6129
RESID^2(-7) -0.012578 0.103354 -0.121699 0.9034
RESID^2(-8) 0.099355 0.085893 1.156724 0.2500
RESID^2(-9) -0.146181 0.083482 -1.751042 0.0829
RESID^2(-10) 0.096549 0.056882 1.697374 0.0926

50 | P a g e
Appendix3 (f) – ModifiedTrinomial model for Capital Land
Date: 10/18/16 Time:19:13
Sample:2/01/2016 6/10/2016
C -0.179483 0.023913 -7.505584 0.0000
WARRANT_PRICE 3.706089 0.229780 16.12888 0.0000
AR(1) 0.983461 0.020076 48.98651 0.0000
MA(1) -0.345738 0.090491 -3.820688 0.0002
SIGMASQ 4.24E-05 3.78E-06 11.23754 0.0000
InvertedARRoots .98
Appendix3 (g) – VIF test for modifiedTrinomial model forCapital Land
Date: 10/18/16 Time:19:14
Sample:2/01/2016 6/10/2016
C 0.000572 2.856546 NA
WARRANT_PRICE 0.052799 2.582642 1.336992
AR(1) 0.000403 2.262807 2.262636
MA(1) 0.008189 1.371007 1.347798
SIGMASQ 1.43E-11 2.460336 2.288745

51 | P a g e
Appendix3 (H) – Correlogramof modifiedTrinomial model for Capital Land

52 | P a g e
Appendix3 (I) – Static forecastsfor modifiedTrinomial model for Capital Land
-.02
.00
.02
.04
.06
.08
.10
10 13 14 15 16 17 20 21 22 23 24
2016m6
MODTRI_STF ± 2 S.E.
Forecast: MODTRI_STF
Actual: TRINOMIAL_MODEL_PRICE
Forecast sample: 6/10/2016 6/24/2016
Appendix3 (J) – Simple regressionoutputfor Trinomial model for Capital Land
-.04
-.02
.00
.02
.04
.06
.08
10 13 14 15 16 17 20 21 22 23 24
2016m6
MODTRI_DYF ± 2 S.E.
Forecast: MODTRI_DYF
Actual: TRINOMIAL_MODEL_PRICE
Forecast sample: 6/10/2016 6/24/2016

53 | P a g e
Appendix4 (a) – Simple regressionoutputof Binomial model for Sembcorp Marine
Date: 10/17/16 Time:22:36
Sample:1/04/2016 6/24/2016
C -0.043047 0.002244 -19.18147 0.0000
WARRANT_PRICE 1.230189 0.049392 24.90658 0.0000
0
2
4
6
8
10
12
14
16
-0.005 0.000 0.005 0.010 0.015 0.020
Series: Residuals
Sample 1/04/2016 6/24/2016
Observations 125
Mean -2.24e-17
Median -0.000953
Maximum 0.022579
Minimum -0.008564
Std. Dev. 0.005695
Skewness 1.350464
Kurtosis 5.257852

54 | P a g e
Appendix4 (b) – Correlogram for simple regressionoutputof Binomial model for Sembcorp
Marine

55 | P a g e
Appendix4 (c) – Serial Correlationtest for Binomial model for Sembcorp Marine
F-statistic 41.14236 Prob. F(5,118) 0.0000
TestEquation:
Date: 10/18/16 Time:19:53
Sample:1/04/2016 6/24/2016
C -3.51E-05 0.001390 -0.025270 0.9799
WARRANT_PRICE 0.001419 0.030580 0.046411 0.9631
RESID(-1) 1.019348 0.092035 11.07569 0.0000
RESID(-2) -0.391474 0.131412 -2.978972 0.0035
RESID(-3) 0.059469 0.136399 0.435989 0.6636
RESID(-4) 0.061421 0.132055 0.465116 0.6427
RESID(-5) 0.024109 0.092820 0.259742 0.7955
R-squared 0.635478 Mean dependentvar -2.24E-17

56 | P a g e
Appendix4 (d) – Heteroskadacity (White) testforBinomial model for SembcorpMarine
Heteroskedasticity Test:White
TestEquation:
Date: 10/18/16 Time:19:54
Sample:1/04/2016 6/24/2016
C 0.000685 7.63E-05 8.979479 0.0000
WARRANT_PRICE^2 0.346644 0.034959 9.915622 0.0000
WARRANT_PRICE -0.030938 0.003329 -9.293724 0.0000
AdjustedR-squared 0.482204 S.D. dependentvar 6.67E-05
S.E. of regression 4.80E-05 Akaike infocriterion -17.02819

57 | P a g e
Appendix4 (e) – Heteroskadacity(ARCH) test for Binomial model for Sembcorp Marine
TestEquation:
Date: 10/18/16 Time:19:54
Sample (adjusted):1/18/2016 6/24/2016
C 1.13E-05 6.40E-06 1.768791 0.0799
RESID^2(-1) 0.816257 0.098012 8.328171 0.0000
RESID^2(-2) -0.383675 0.125771 -3.050577 0.0029
RESID^2(-3) 0.124272 0.131298 0.946491 0.3461
RESID^2(-4) 0.039004 0.131976 0.295541 0.7682
RESID^2(-5) 0.066848 0.132098 0.506048 0.6139
RESID^2(-6) -0.018846 0.132143 -0.142619 0.8869
RESID^2(-7) -0.054702 0.132065 -0.414206 0.6796
RESID^2(-8) 0.022830 0.131570 0.173523 0.8626
RESID^2(-9) -0.002918 0.125421 -0.023263 0.9815
RESID^2(-10) -0.014661 0.095331 -0.153787 0.8781

58 | P a g e
Appendix4 (f) – ModifiedBinomial model forSembcorp Marine
Date: 10/18/16 Time:20:00
Sample:2/01/2016 6/10/2016
C -0.052906 0.002917 -18.13432 0.0000
WARRANT_PRICE 1.412092 0.052747 26.77128 0.0000
AR(1) 0.913174 0.079114 11.54250 0.0000
MA(2) -0.391104 0.137418 -2.846098 0.0055
MA(3) -0.352013 0.099698 -3.530798 0.0007
SIGMASQ 1.03E-05 1.78E-06 5.793034 0.0000
InvertedARRoots .91
InvertedMA Roots .89 -.44+.45i -.44-.45i
Appendix4 (g) – VIF Test for modifiedBinomial model forSembcorp Marine
Date: 10/18/16 Time:20:01
Sample:2/01/2016 6/10/2016
C 8.51E-06 9.343338 NA
WARRANT_PRICE 0.002782 10.35827 1.381015
AR(1) 0.006259 5.119722 4.705390
MA(2) 0.018884 3.551382 3.422117
MA(3) 0.009940 1.822758 1.779146
SIGMASQ 3.16E-12 1.529709 1.463460

59 | P a g e
Appendix4 (h) – Correlogram for modified Binomial model forSembcorp Marine

60 | P a g e
Appendix4 (i) – Static forecasts for modifiedmodel forSembcorp Marine
-.015
-.010
-.005
.000
.005
.010
.015
10 13 14 15 16 17 20 21 22 23 24
2016m6
MODBIN_STF ± 2 S.E.
Forecast: MODBIN_STF
Forecastsample:6/10/20166/24/2016
Theil InequalityCoefficient 0.833371
Appendix4 (j) – Dynamic forecasts for modifiedBinomial model for SembcorpMarine
-.025
-.020
-.015
-.010
-.005
.000
.005
.010
10 13 14 15 16 17 20 21 22 23 24
2016m6
MODBIN_DYF ± 2 S.E.
Forecast: MODBIN_DYF

61 | P a g e
Appendix5 (a) – Simple regressionoutputof Black-Scholesmodel forSembcorp Marine
Date: 10/17/16 Time:22:37
Sample:1/04/2016 6/24/2016
C -0.042696 0.002224 -19.19906 0.0000
WARRANT_PRICE 1.225895 0.048945 25.04633 0.0000
0
2
4
6
8
10
12
14
16
-0.005 0.000 0.005 0.010 0.015 0.020
Series: Residuals
Sample 1/04/2016 6/24/2016
Observations 125
Mean -2.51e-17
Median -0.000714
Maximum 0.021734
Minimum -0.008560
Std. Dev. 0.005644
Skewness 1.334747
Kurtosis 5.182328

62 | P a g e
Appendix5 (b) – Correlogram of simple regressionoutputof Black-Scholesmodel forSembcorp
Marine

63 | P a g e
Appendix5 (c) – Serial corerrlationLM testof Black-Scholesmodel for Sembcorp Marine
TestEquation:
Date: 10/18/16 Time:20:09
Sample:1/04/2016 6/24/2016
C 2.02E-06 0.001365 0.001476 0.9988
WARRANT_PRICE 0.000579 0.030041 0.019289 0.9846
RESID(-1) 1.034105 0.091894 11.25323 0.0000
RESID(-2) -0.426402 0.132407 -3.220394 0.0017
RESID(-3) 0.121896 0.137870 0.884135 0.3784
RESID(-4) -0.011467 0.133039 -0.086195 0.9315
RESID(-5) 0.060449 0.092669 0.652310 0.5155

64 | P a g e
Appendix5 (d) – Heteroskadacity (White) testforBlack-Scholesmodel for Sembcorp Marine
TestEquation:
Date: 10/18/16 Time:20:10
Sample:1/04/2016 6/24/2016
C 0.000666 7.45E-05 8.937991 0.0000
WARRANT_PRICE^2 0.336481 0.034115 9.863117 0.0000
WARRANT_PRICE -0.030033 0.003248 -9.245123 0.0000
AdjustedR-squared 0.479414 S.D. dependent var 6.49E-05

65 | P a g e
Appendix5 (e) – Heteroskadacity(ARCH) test ofBlack-Scholesmodel for Sembcorp Marine
TestEquation:
Date: 10/18/16 Time:20:10
Sample (adjusted):1/18/2016 6/24/2016
Includedobservations:115 after adjustments
C 1.08E-05 6.03E-06 1.796662 0.0753
RESID^2(-1) 0.849764 0.097959 8.674655 0.0000
RESID^2(-2) -0.369647 0.127723 -2.894136 0.0046
RESID^2(-3) 0.053860 0.132760 0.405692 0.6858
RESID^2(-4) 0.113790 0.132815 0.856754 0.3936
RESID^2(-5) 0.000110 0.133325 0.000822 0.9993
RESID^2(-6) 0.035107 0.133296 0.263374 0.7928
RESID^2(-7) -0.091452 0.132903 -0.688115 0.4929
RESID^2(-8) 0.031280 0.132927 0.235318 0.8144
RESID^2(-9) 0.022800 0.127023 0.179495 0.8579
RESID^2(-10) -0.040199 0.095062 -0.422874 0.6733

66 | P a g e
Appendix5 (f) – ModifiedBlack-Scholesmodel forSembcorpMarine
Date: 10/18/16 Time:20:17
Sample:2/01/2016 6/10/2016
C -0.053299 0.002625 -20.30663 0.0000
WARRANT_PRICE 1.423312 0.048616 29.27654 0.0000
AR(1) 0.985305 0.076036 12.95832 0.0000
AR(2) -0.354304 0.071882 -4.928993 0.0000
SIGMASQ 9.61E-06 1.33E-06 7.243185 0.0000
InvertedARRoots .49-.33i .49+.33i
Appendix5 (g) – VIF test for Black-Scholesmodel forSembcorp Marine
Date: 10/18/16 Time:20:17
Sample:2/01/2016 6/10/2016
C 6.89E-06 9.316912 NA
WARRANT_PRICE 0.002364 9.892869 1.329405
AR(1) 0.005782 1.594611 1.424966
AR(2) 0.005167 1.490718 1.455466
SIGMASQ 1.76E-12 1.354383 1.327363

67 | P a g e
Appendix5 (h) – Correlogram of modifiedBlack-Scholesmodel forSembcorpMarine

68 | P a g e
Appendix5 (i) – Static forecasts for modifiedBlack-Scholesmodel forSembcorp Marine
-.020
-.015
-.010
-.005
.000
.005
.010
.015
10 13 14 15 16 17 20 21 22 23 24
2016m6
MODBS_STF ± 2 S.E.
Forecast: MODBS_STF
Appendix5 (j) – Dynamic forecasts of modifiedBlack-Scholesmodel forSembcorpMarine
-.025
-.020
-.015
-.010
-.005
.000
.005
.010
.015
10 13 14 15 16 17 20 21 22 23 24
2016m6
MODBS_DYF ± 2 S.E.
Forecast: MODBS_DYF

69 | P a g e
Appendix6 (a) – Simple regressionoutputof Trinomial model for Sembcorp Marine
Date: 10/17/16 Time:22:37
Sample:1/04/2016 6/24/2016
C -0.033557 0.001832 -18.31805 0.0000
WARRANT_PRICE 0.950987 0.040318 23.58704 0.0000
0
4
8
12
16
20
-0.005 0.000 0.005 0.010 0.015 0.020
Series: Residuals
Sample 1/04/2016 6/24/2016
Observations 125
Mean -1.52e-17
Median -0.000534
Maximum 0.019125
Minimum -0.007344
Std. Dev. 0.004649
Skewness 1.482325
Kurtosis 6.039967

70 | P a g e
Appendix6 (b) – Correlogram of simple regressionTrinomial model forSembcorp Marine

71 | P a g e
Appendix6 (c) – Serial correlationLM testfor Trinomial model for Sembcorp Marine
TestEquation:
Date: 10/18/16 Time:20:26
Sample:1/04/2016 6/24/2016
C 4.33E-05 0.001122 0.038597 0.9693
WARRANT_PRICE -0.000469 0.024689 -0.018993 0.9849
RESID(-1) 1.046455 0.091928 11.38345 0.0000
RESID(-2) -0.426625 0.133238 -3.201975 0.0018
RESID(-3) 0.084685 0.138904 0.609668 0.5433
RESID(-4) 0.012923 0.133839 0.096554 0.9232
RESID(-5) 0.054628 0.092721 0.589167 0.5569
Adjusted R-squared 0.625858 S.D. dependentvar 0.004649

72 | P a g e
Appendix6 (d) – Heteroskadacity (White) testforTrinomial model for SembcorpMarine
TestEquation:
Date: 10/18/16 Time:20:27
Sample:1/04/2016 6/24/2016
C 0.000483 5.46E-05 8.837945 0.0000
WARRANT_PRICE^2 0.248886 0.025015 9.949340 0.0000
WARRANT_PRICE -0.022038 0.002382 -9.251941 0.0000

73 | P a g e
Appendix6 (e) – Heteroskadacity(ARCH) test for Trinomial model for Sembcorp Marine
TestEquation:
Date: 10/18/16 Time:20:27
Sample (adjusted):1/18/2016 6/24/2016
C 7.58E-06 4.20E-06 1.806566 0.0737
RESID^2(-1) 0.945942 0.097967 9.655747 0.0000
RESID^2(-2) -0.463757 0.134140 -3.457248 0.0008
RESID^2(-3) 0.059376 0.141575 0.419395 0.6758
RESID^2(-4) 0.163452 0.141606 1.154273 0.2510
RESID^2(-5) -0.057433 0.142444 -0.403200 0.6876
RESID^2(-6) 0.066209 0.142345 0.465129 0.6428
RESID^2(-7) -0.100208 0.141654 -0.707419 0.4809
RESID^2(-8) 0.041297 0.141792 0.291247 0.7714
RESID^2(-9) 0.006904 0.133835 0.051583 0.9590
RESID^2(-10) -0.046210 0.096272 -0.479993 0.6322

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