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 this report is to test the conventional models for covered warrant call option prices and evaluate which model is statistically more reliable along with the suggested statistical inferential modifications in those 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 Option Pricing 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 and Assumptions:...........................................................................................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

List of Appendices Appendix 1 – Binomial Model for CapitaLand Warrant

a) Simple regression output b) Correlogram of simple regression outputc) Serial correlation test for simple regression outputd) Heteroskadacity (White) test for Binomial Model e) Heteroskadacity (ARCH) test for Binomial Model f) Modified Binomial Model for CapitaLand g) VIF test for Modified Binomial modelh) Correlogram for Modified Binomial model i) Static forecasts for Modified Binomial model j) Dynamic Forecasts for Modified Binomial model

Appendix 2 – Black-Scholes Model for CapitaLand


Appendix 3 – Trinomial Model for CapitaLand


I

Appendix 4 – Binomial Model for SembCorp Marine


Appendix 5 – Black-Scholes SembCorp Marine

a. Simple regression output b. Correlogram of simple regression outputc. Serial correlation test for simple regression outputd. Heteroskadacity (White) test for Binomial Model e. Heteroskadacity (ARCH) test for Binomial Model f. Modified Binomial Model for CapitaLand g. VIF test for Modified Binomial modelh. Correlogram for Modified Binomial model i. Static forecasts for Modified Binomial model j. Dynamic Forecasts for Modified Binomial model

Appendix 6 – Trinomial Model for SembCorp Marine


II

Appendix 7 – Binomial Model for Wilmar International Warrant


Appendix 8 – Black-Scholes Wilmar International Warrant


Appendix 9 – Trinomial Model for Wilmar International Warrant


III

Risk Management and Financial Engineering (BAFI2081)

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

prices with least error.

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, heteroskedacity and multicollinearity.

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 and individual warrant price predicted by each of the original model.

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1 – Covered, Normal Equity Warrants and OptionsEquity 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, and options on individual 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

asset allocation, making them desirable to strong investors (Equity Options, 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 for one security (Easley, O’Hara, and Srinivas 1998).

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

obtain a more sophisticated understanding of options and how they are priced.

Beginning with the structure, we can define equity options by the following elements: 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 between call and put options.

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 either before or on a determined date.

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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 option contract.

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

previous position as the holder or writer of the option.

Finally, in order to exercise an option, the holder must advise his broker to submit an exercise notice

to the write, who then has to fulfil their obligations as stated in the contract. 1 A European option allows an option to be exercised any time to the date of expiration.

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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 choices for risk management and investment. Another reason why equity 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

event that the 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

specific pricing models such as the Black Scholes model (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, warrants are often favoured by retail investors who seek less intricate 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

system, ASX Trade, which allows transactions to be certified and cleared through the use of CHESS 2.

2 Clearing House Electronic Subregister System (CHESS)

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Further, using CHESS trading and its settling arrangements provides investors with more familiarity

than trading the options on the ASX, which are usually associated with margin trading.

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 &

Segara R, 2007).

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Table 1: Intraday patterns in the equity warrants

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Figure 1: Equity warrants trading volume

3 – Covered Warrants: 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

expiry date (Chan et al. 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 its structure in meeting what the market demands.

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 – Types of Covered Warrants:There are various types of covered warrants available and in circulation globally, the most prominent

being on stock warrants. Other covered warrants are on:

Baskets; a group of stocks that allows investors to gain exposure in the simplest form just by

purchasing a security in an arrangement of a covered warrant.

Indices; incorporating the overall market of covered warrants globally, indices are the most

heavily traded kind.

Bonds; bonds with option rights

Commodity; gives investors the capability in taking positions of commodities using covered

warrants.

Currency; accessible on a collection of exchange rates.

Barrier; additional terms where if a particular price is hit, the covered warrant will be void.

Terms can also be changed if the barrier price is met dependent upon the initial conditions.

Trigger; triggering a specific matter within the terms that leads to a fixed payout.

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 could be 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 earlier during its lifetime.

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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 obligation in the 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

underlying asset alone therefore decreasing such demand for the particular warrant on 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 extraordinary event

with regards to the underlying asset.

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 warrants in 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

investors also played a major role in the price of warrants in the market.

4 – Option Pricing Models – A comparison of Binomial, Black-Scholes and 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 models introduced 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, in order to discuss the applicability of these models.

European style options: European style option is an option which cannot be exercised

before the expiration date.

American style options: American style option is an option that can be exercised at any time

during the life of that option.

4.1 – Black-Scholes Model: 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

almost all options. Black-Scholes pricing formulas for call options and put options are given below:

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Where:

C = call value

P = put value

S = stock price

K = exercise price of an option

T-t = time between expiration date and the valuation date of an option

r = risk free interest rate

N(d) = standard normal cumulative distribution in point d

∂ = volatility of underlying index

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 higher price it will be for that option.

4.1.1 – Assumptions:

The basic underlying assumptions from the original Black-Sholes model are:

The option is a European style option, which means that it cannot be exercised before the

expiration date

It assumes that the volatility of the underlying stock is constant.

There is no arbitrage due to the efficient markets.

Like volatility it assumes risk free interest rate remains constant over the time.

Return on the underlying stock follows a normal distribution

Markets are always open, giving an opportunity to buy or sell any option at any given time.

No transaction costs involved in buying or selling an option.

Zero dividends are paid during the option life.

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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 down in a certain time period. One step binomial model for option pricing is given below

One Step Binomial Model

Where So is the initial price of stock, p is the probability that value of stock will go up by factor u and

1-p is a probability that stock price will fall down by factor d.

A risk neutral world is assumed over a small period of time, given that the effective return of

binomial mode is equal to risk-free rate.

.

And also the variance of risk-free asset is equal to the variance of an asset in a risk-neutral world

given by the following equation.

The relation between upside factor and a downside factor is given by:

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http://investexcel.net/wp-content/uploads/2011/10/Equation-11.png



From above equations, values for p, u and d have been obtained as follows:

The values of p, u and d given by the Cox, Ross and Rubinstein (CRR) model means that the

underlying initial stock price is symmetric for a multi-step binomial model.

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

with the real life situations. Trinomial tree can be defined as

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http://investexcel.net/wp-content/uploads/2011/10/Cox-Ross-and-Rubenstein-Model1.png


S(t)u with probability pu

S(t + ∆t) =

S(t) with probability 1 − pu − pd

S(t)d with probability pd

According to no arbitrage condition we 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

written as:

1 − pu − pd + puu + pdd = er∆t.

In order to look at upward and downward jump requires an extra constraint that size of upward

jump is a reciprocal of a downward jump i-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 for time interval i is given as

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 , jump sizes of the popular representative of that family are

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And its changing probabilities are:

Stock index So will move up by Su or down by Sd or will remain same as So. Pu is the probability of

upward moment and pd is the probability that stock price will move upward, hence the probability

that the stock remains same will be given by (1-pu-pd).

5 – Comparison between Option Pricing Models & their calculation 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 money and then discounted it back to get the present value 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 and trinomial models where number of steps can be 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 should it 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

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for calculating a large number of option prices very quickly. Binomial model can be much more

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

stationary at each step.

6 – Theoretical modifications in Binomial, Black-Scholes and 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-Scholes Model and Its Alternatives”.

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

prices will provide a closer indication to actual warrant prices.

Secondly, high volatilities tend to result in 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

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basket of stocks that are similar to the underlying stocks (instead of only the warrant’s underlying

stock volatility) to calculate the annualized standard deviation for the models’ inputs.

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 formula to calculate the standard deviation for the model.

7 – Statistical evaluation of the models:The empirical testing of the Binomial, Black-Scholes (BS) and Trinomial was done through running a

simple regression function for the models against the actual warrant prices of the covered warrants.

The structure of the regression was:

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 predicted values of warrants by the selected mathematical 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

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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. 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 compared against the statistical forecasts and dynamic forecasts of the modified models.

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

following subpart classifications format:

a) Simple regression function output with normality tests for residuals

b) Correlogram of the simple regression output

c) Serial Correlation LM Test (Breusch Godfrey)

d) Heteroskedacity – White Test

e) Heteroskedacity – ARCH Test

f) Modified model with ARMA structure

g) Variance Inflation (VIF) test for mluticollinearity

h) Correlogram for the modified model

i) Statistical forecasts for modified models

j) Dynamic forecasts for the modified models

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7.1 – Data collection and Assumptions: In this report, three existing covered warrants for Singaporean companies are considered. The

companies that were selected are:

CapitaLand (Industry: Real Estate)

SembCorp Marine (Industry: Offshore and Marine)

Wilmar International (Industry: Food Processing)

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 European in nature and have the following characteristics:

CapitaLand Covered Warrant – Main features:

Stricke Price 3.2Expiry date 12-Dec-16Risk free rate 1.86%Total days in years 261

Div. Yield 2.82%

SembCorp marine Covered Warrant – Main features:


Div. Yield 2.69%

Wilmar International Covered Warrant – Main features:


Div. Yield 2.48%

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7.1.1 – Assumptions: The assumptions used in this study are the following:

The study is based on the historical data taken from the working days of 1-Jan-2016 to 30-

Jun-2016.

20 nodes are selected for the calculation of the warrant prices in Binomial and Trinomial

models

The risk free rate for the calculations was 1.86% (annualized), which was the yield on the 10

year Singaporean government bond at the time of checking

Time to maturity in years was calculated for each daily calculation. The official number of

working days in 2016 used is according Singapore’s official calendar.

Dividend yields for the warrant price calculation in the Trinomial model were taken from

Bloomberg and Yahoo Finance. These were checked against those listed on Thomson

Reuters and CNBC which were found to be identical

Volatility was the stock’s standard deviation in the selected time period. The volatility was

then annualized for the calculation of theoretical warrant prices

7.2 – Statistical Tests and 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 prices are closer to the unit value 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

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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 (appendices 8 and 9 subparts c).

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 ARCH effects as all

the statistical tests resulted in 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

test was necessary to get all the ARCH effects removed or whether a GARCH would be a better fit. 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 money nature of the option prices which the models calculate.

3 Eviews at RMIT crashed every time whenever an ARCH/GARCH test was run even using the MyDesktop app.

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CapitaLand SembCorp Marine Wilmar InternationalBinomial

ModelBlack-Scholes

ModelTrinomial

ModelBinomial

ModelBlack-Scholes

ModelTrinomial

ModelBinomial

ModelBlack-Scholes

ModelTrinomial

ModelIntercept -0.146 -0.146 -0.113 -0.043 -0.043 -0.034 0.333 0.334 0.308Coefficient (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 0.095Durbin Watson 0.190 0.194 0.189 0.430 0.424 0.438 0.234 0.236 0.234Akaike info. Criterion -4.603 -4.611 -4.919 -7.474 -7.493 -7.880 -1.696 -1.677 -1.757Schwarz Criterion -4.557 -4.566 -4.873 -7.429 -7.447 -7.835 -1.651 -1.630 -1.711Hannan-Quinn Criterion -4.584 -4.592 -4.900 -7.456 -7.474 -7.862 -1.678 -1.658 -1.738

F-stat of regression 316.557 316.824 257.133 620.338 627.319 556.348 12.999 12.387 12.421Prob. Of F-stat of regression 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.001

Jarque-Bera Stats 49.427 47.109 31.536 64.546 61.921 93.909 1.975 1.333 1.222Jarque-Bera Stats (prob) 0.000 0.000 0.000 0.000 0.000 0.000 0.373 0.514 0.543

Table 1 – Statistical Measures Table for the Simple Regression output results for Binomial, Black-Scholes and Trinomial Models vs. Actual Warrant Prices

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CapitaLand SembCorp Marine Wilmar InternationalBinomial

ModelBlack-Scholes

ModelTrinomial

ModelBinomial

ModelBlack-Scholes

ModelTrinomial

ModelBinomial

ModelBlack-Scholes

ModelTrinomial

ModelIntercept -0.212 -0.212 -1.795 -0.053 -0.053 -0.042 0.380 0.337 0.311Coefficient (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 0.946Durbin Watson 1.938 1.919 1.922 1.781 1.902 1.920 2.025 1.836 1.844Akaike info. Criterion -6.831 -6.784 -7.096 -8.510 -8.599 -8.950 -4.405 -4.441 -4.542Schwarz Criterion -6.697 -6.649 -6.962 -8.349 -8.465 -8.815 -4.267 -4.348 -4.450Hannan-Quinn Criterion -6.777 -6.729 -7.042 -8.445 -8.545 -8.895 -4.349 -4.403 -4.505

F-stat of regression 526.303 500.851 449.603 361.552 486.811 428.006 261.545 683.728 699.554Prob. Of F-stat of 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

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7.3 – Modifications of the Binomial, Black-Scholes and 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

performed on the models is as following:

CompaniesModified Binomial

ModelModified Black-Scholes

ModelModified Trinomial

ModelAR Terms MA Terms AR Terms MA Terms AR Terms MA Terms

CapitaLand 1 1 1 1 1 1SembCorp Marine 1 2 (2,3) 1 1(2) 1 1

Wilmar Int. 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 in Appendices 1-9

(subparts g). Since all the coefficients of the modified models had Cantered VIF score of less than 2,

the modified models were deemed to be better than the 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 spikes in the AC and PAC values were

over 15 AR processes and thus, those were left unchanged to keep the parsimony principle in view.

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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 and Black-Scholes model. Table 3 provides a summary of the RMSEs.

CompaniesRoot Mean Squared Errors (RMSE) of Raw Warrant Price Data with Normal

ModelsBinomial Model Black-Scholes Model Trinomial Model

CapitaLand 0.044 0.044 0.028SembCorp Marine 0.033 0.033 0.036

Wilmar Int. 0.318 0.317 0.291

CompaniesRoot Mean Squared Errors (RMSE) of Static Forecasts by Modified ModelsModified Binomial

ModelModified Black-Scholes

ModelModified Trinomial

ModelCapitaLand 0.044 0.044 0.028SembCorp Marine 0.033 0.033 0.036

Wilmar Int. 0.318 0.317 0.291

CompaniesRoot Mean Squared Errors (RMSE) of Dynamic Forecasts by Modified Models

Modified Binomial Model

Modified Black-Scholes Model

Modified Trinomial Model

CapitaLand 0.015 0.015 0.013SembCorp Marine 0.007 0.007 0.006

Wilmar Int. 0.060 0.058 0.056

Table 3 - Summary of Root Mean Square Errors

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8 - ConclusionIn 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 least erroneous as compared to Binomial and Black-Scholes model. Nevertheless, all

3 models could be further improved.

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 on the basis of its least root mean squared error (RMSE) terms.

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9 – References

Aitken, M & Segara, R 2005, ‘Impact of warrant introductions on the behaviour of underlying stocks: Australian evidence’, Accounting and Finance, vol. 45, no. 1, pp.127-144

Black, F & Scholes, M 1973, 'The Pricing of Options and Corporate Liabilities', The Journal of

Political Economy, vol. 81, no. 3, pp. 637-654

Black, F., 1975, “Fact and Fantasy in the Use of Options,” Financial Analysts Journal, vol. 31, pp. 36–72.

Bollen, NPB & Whaley, RE 2004, 'Does Net Buying Pressure Affect the Shape of Implied Volatility Functions?', The Journal of Finance, vol. 59, no. 2, pp. 711-53.

Boyle, PP 1986, 'Option Valuation Using Three-Jump Process', International Options Journal, 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 test of the efficiency of the UK covered warrants market: Stochastic dominance and likelihood ratio test approach’, Journal of Empirical Finance, vol. 19, no. 1. pp. 162-174

Cox, J, Ross, S & Rubinstein, M 1979, 'Option Pricing: A Simplified Approach', Journal of Financial

Economics, vol. 7 pp. 229-263

Easley, D., M. O’Hara, and P. S. Srinivas, 1998, “Option Volume and Stock Prices: Evidence on Where Informed Traders Trade,” Journal of Finance, vol. 53, pp. 431–465.

Fan, K., Shen, Y., Siu T.K., Wang, R., 2013, “Pricing foreign equity options with regime-switching, Economic Modelling, vol. 37, pp. 296-297

Fung, H., Zhang, G. and Zhao, L., 2009, “China's Equity Warrants Market: An Overview and Analysis”,The Chinese Economy, vol. 42, no. 1, pp.86-97.

Horst, J & Veld, C 2008, ‘An Empirical Analysis of the Pricing of Bank Issued Options versus Options Exchange Options’, European Financial Management, vol. 14, no. 2, pp.288-314

Hunt, B. and Terry, C., 2011, “Australian Equity Warrants: Are Retail Investors Getting A Fair Go?”,

The Finsia Journal of Applied Finance, vol. 4, pp.48-64.

Kui, L., 2007, “Do Warrants Lead The Underlying Stocks and Index Futures?”, Singapore: Singapore Management University, p.3.

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Lauterbach, B & Schultz, P 1990, Pricing Warrants: An Empirical Study of the Black-Scholes Model

and Its Alternatives, The Journal of Finance, vol. 45, no. 4, pp. 1181-1209.

Segara L & Segara, R 2007, ‘Intraday trading patterns in equity warrants and equity options markets: Australian evidence’, The Australasian Accounting Business & Finance Journal, vol. 1, no. 2, p. 52

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Appendices:Appendix 1 (a) – Simple regression output for Binomial model for Capital Land

Dependent Variable: BINOMIAL_MODEL_PRICEMethod: Least SquaresDate: 10/17/16 Time: 22:28Sample: 1/04/2016 6/24/2016Included observations: 125

Variable Coefficient Std. Error t-Statistic Prob.

C -0.146451 0.013494 -10.85270 0.0000WARRANT_PRICE 3.625156 0.203751 17.79205 0.0000

R-squared 0.720173 Mean dependent var 0.090576Adjusted R-squared 0.717898 S.D. dependent var 0.045256S.E. of regression 0.024037 Akaike info criterion -4.602565Sum squared resid 0.071067 Schwarz criterion -4.557312Log likelihood 289.6603 Hannan-Quinn criter. -4.584181F-statistic 316.5572 Durbin-Watson stat 0.190229Prob(F-statistic) 0.000000

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Appendix 1 (b) – Correlogram for simple regression output for Capital Land

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Appendix 1 (c) – Serial correlation LM Test for simple regression output for Capital Land

Breusch-Godfrey Serial Correlation LM Test:

F-statistic 63.26895 Prob. F(5,118) 0.0000Obs*R-squared 91.04080 Prob. Chi-Square(5) 0.0000

Test Equation:Dependent Variable: RESIDMethod: Least SquaresDate: 10/17/16 Time: 23:14Sample: 1/04/2016 6/24/2016Included observations: 125Presample missing value lagged residuals set to zero.


C -0.001515 0.007210 -0.210145 0.8339WARRANT_PRICE 0.025677 0.108916 0.235754 0.8140

RESID(-1) 0.851940 0.092193 9.240853 0.0000RESID(-2) -0.026605 0.119736 -0.222193 0.8245RESID(-3) -0.135686 0.119166 -1.138628 0.2572RESID(-4) 0.192246 0.119924 1.603066 0.1116RESID(-5) 5.78E-06 0.092624 6.23E-05 1.0000

R-squared 0.728326 Mean dependent var 5.73E-17Adjusted R-squared 0.714512 S.D. dependent var 0.023940S.E. of regression 0.012791 Akaike info criterion -5.825719Sum squared resid 0.019307 Schwarz criterion -5.667333Log likelihood 371.1074 Hannan-Quinn criter. -5.761375F-statistic 52.72412 Durbin-Watson stat 1.573603Prob(F-statistic) 0.000000

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Appendix 1 (d) – Heteroskadicity (White) test for simple regression output for Capital Land

Heteroskedasticity Test: White

F-statistic 14.86129 Prob. F(2,122) 0.0000Obs*R-squared 24.48760 Prob. Chi-Square(2) 0.0000Scaled explained SS 54.92773 Prob. Chi-Square(2) 0.0000

Test Equation:Dependent Variable: RESID^2Method: Least SquaresDate: 10/17/16 Time: 23:15Sample: 1/04/2016 6/24/2016Included observations: 125


C 0.008402 0.003878 2.166399 0.0322WARRANT_PRICE^2 2.597966 0.928224 2.798856 0.0060

WARRANT_PRICE -0.294102 0.121185 -2.426887 0.0167


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Appendix 1 (e) – Heteroskadacity (ARCH) test for simple regression output for Capital Land

Heteroskedasticity Test: ARCH


Test Equation:Dependent Variable: RESID^2Method: Least SquaresDate: 10/17/16 Time: 23:16Sample (adjusted): 1/18/2016 6/24/2016Included observations: 115 after adjustments


C 5.83E-05 4.74E-05 1.230806 0.2212RESID^2(-1) 0.592820 0.095840 6.185539 0.0000RESID^2(-2) 0.100839 0.110040 0.916384 0.3616RESID^2(-3) 0.242010 0.110015 2.199790 0.0300RESID^2(-4) -0.109843 0.112100 -0.979865 0.3294RESID^2(-5) 0.009521 0.111404 0.085462 0.9321RESID^2(-6) -0.021516 0.110696 -0.194370 0.8463RESID^2(-7) -0.046134 0.100315 -0.459891 0.6466RESID^2(-8) 0.128140 0.078521 1.631915 0.1057RESID^2(-9) -0.169137 0.076309 -2.216474 0.0288

RESID^2(-10) 0.110309 0.051196 2.154653 0.0335

R-squared 0.627148 Mean dependent var 0.000405Adjusted R-squared 0.591297 S.D. dependent var 0.000583S.E. of regression 0.000373 Akaike info criterion -12.85958Sum squared resid 1.45E-05 Schwarz criterion -12.59702Log likelihood 750.4259 Hannan-Quinn criter. -12.75301F-statistic 17.49309 Durbin-Watson stat 2.085835Prob(F-statistic) 0.000000

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Appendix 1 (f) – Modified model for Capital Land Binomial Model

Dependent Variable: BINOMIAL_MODEL_PRICEMethod: ARMA Maximum Likelihood (OPG - BHHH)Date: 10/18/16 Time: 18:41Sample: 2/01/2016 6/10/2016Included observations: 95Convergence achieved after 31 iterationsCoefficient covariance computed using outer product of gradients


C -0.212193 0.026576 -7.984247 0.0000WARRANT_PRICE 4.507190 0.259791 17.34927 0.0000

AR(1) 0.984027 0.019044 51.67061 0.0000MA(1) -0.373227 0.086969 -4.291509 0.0000

SIGMASQ 5.53E-05 4.98E-06 11.11552 0.0000


Inverted AR Roots .98Inverted MA Roots .37

Appendix 1 (g) – VIF Test for Modified Binomial model for Capital Land

Variance Inflation FactorsDate: 10/18/16 Time: 18:41Sample: 2/01/2016 6/10/2016Included observations: 95

Coefficient Uncentered CenteredVariable Variance VIF VIF

C 0.000706 2.703830 NAWARRANT_PRICE 0.067491 2.540167 1.337280

AR(1) 0.000363 2.166351 2.162275MA(1) 0.007564 1.460435 1.427793

SIGMASQ 2.48E-11 2.372646 2.210507

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Appendix 1 (h) – Correlogram of Modified Binomial model for Capital Land

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Appendix 1 (I) – Static forecasts of modified binomial model for Capital Land

Appendix 1 (J) – Dynamic forecasts of modified binomial model for Capital Land

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Appendix 2 (a) – Simple regression output for Black-Scholes (BS) model for Capital Land

Dependent Variable: BS_MODEL_PRICEMethod: Least SquaresDate: 10/17/16 Time: 22:28Sample: 1/04/2016 6/24/2016Included observations: 125


C -0.146008 0.013439 -10.86453 0.0000WARRANT_PRICE 3.611772 0.202914 17.79955 0.0000


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Appendix 2 (b) – Correlogram of Simple regression output for Black-Scholes (BS) model for Capital Land

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Appendix 2 (c) – Serial correlation test of simple regression output of Black-Scholes (BS) model for Capital Land





C -0.001483 0.007242 -0.204719 0.8381WARRANT_PRICE 0.025296 0.109393 0.231238 0.8175

RESID(-1) 0.846586 0.092251 9.176994 0.0000RESID(-2) -0.031336 0.119674 -0.261843 0.7939RESID(-3) -0.109651 0.119283 -0.919255 0.3598RESID(-4) 0.176098 0.119785 1.470118 0.1442RESID(-5) -0.002281 0.092624 -0.024621 0.9804


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Appendix 2 (d) – Heteroskadacity (White) test of Simple regression output for Black-Scholes (BS) model for Capital Land





C 0.008208 0.003805 2.157272 0.0329WARRANT_PRICE^2 2.544777 0.910622 2.794547 0.0060

WARRANT_PRICE -0.287637 0.118887 -2.419417 0.0170


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Appendix 2 (e) – Heteroskadacity (ARCH) test of Simple regression output for Black-Scholes (BS) model for Capital Land






RESID^2(-10) 0.107421 0.052533 2.044844 0.0434


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Appendix 2 (f) – Modified model for Black-Scholes (BS) model for Capital Land Modified BS Model

Dependent Variable: BS_MODEL_PRICEMethod: ARMA Maximum Likelihood (OPG - BHHH)Date: 10/18/16 Time: 18:55Sample: 2/01/2016 6/10/2016Included observations: 95Convergence achieved after 35 iterationsCoefficient covariance computed using outer product of gradients


C -0.212298 0.027382 -7.753102 0.0000WARRANT_PRICE 4.507508 0.279429 16.13112 0.0000

AR(1) 0.982128 0.019722 49.79945 0.0000MA(1) -0.364819 0.082626 -4.415319 0.0000

SIGMASQ 5.81E-05 5.21E-06 11.14643 0.0000



Appendix 2 (g) – VIF test of modified Black-Scholes (BS) model for Capital Land




AR(1) 0.000389 2.219858 2.217503MA(1) 0.006827 1.422085 1.393867

SIGMASQ 2.71E-11 2.475892 2.294333

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Appendix 2 (h) – Correlogram of modified Black-Scholes (BS) model for Capital Land

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Appendix 2 (I) – Static forecasts of modified Black-Scholes (BS) model for Capital Land

Appendix 2 (J) – Dynamic forecasts of modified Black-Scholes (BS) model for Capital Land

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Appendix 3 (a) – Simple regression output for Trinomial model for Capital Land

Dependent Variable: TRINOMIAL_MODEL_PRICEMethod: Least SquaresDate: 10/17/16 Time: 22:29Sample: 1/04/2016 6/24/2016Included observations: 125


C -0.113260 0.011522 -9.830179 0.0000WARRANT_PRICE 2.789592 0.173965 16.03538 0.0000


Appendix 3 (b) – Correlogram of Trinomial model for Capital Land

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Appendix 3 (c) – Serial correlation test for Trinomial model for Capital Land





C -0.001558 0.006090 -0.255813 0.7985WARRANT_PRICE 0.026147 0.092010 0.284172 0.7768

RESID(-1) 0.866331 0.092331 9.382921 0.0000RESID(-2) -0.056403 0.121080 -0.465836 0.6422RESID(-3) -0.088670 0.120933 -0.733218 0.4649RESID(-4) 0.163466 0.121217 1.348543 0.1801RESID(-5) -0.000839 0.092735 -0.009050 0.9928


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Appendix 3 (d) – Heteroskadacity (White) test of Trinomial model for Capital Land





C 0.004847 0.002648 1.830309 0.0696WARRANT_PRICE^2 1.565007 0.633804 2.469229 0.0149

WARRANT_PRICE -0.172787 0.082747 -2.088140 0.0389


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Appendix 3 (e) – Heteroskadacity (ARCH) test of Trinomial model for Capital Land






RESID^2(-10) 0.096549 0.056882 1.697374 0.0926


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Appendix 3 (f) – Modified Trinomial model for Capital Land

Dependent Variable: TRINOMIAL_MODEL_PRICEMethod: ARMA Maximum Likelihood (OPG - BHHH)Date: 10/18/16 Time: 19:13Sample: 2/01/2016 6/10/2016Included observations: 95Convergence achieved after 30 iterationsCoefficient covariance computed using outer product of gradients


C -0.179483 0.023913 -7.505584 0.0000WARRANT_PRICE 3.706089 0.229780 16.12888 0.0000

AR(1) 0.983461 0.020076 48.98651 0.0000MA(1) -0.345738 0.090491 -3.820688 0.0002

SIGMASQ 4.24E-05 3.78E-06 11.23754 0.0000



Appendix 3 (g) – VIF test for modified Trinomial model for Capital Land




AR(1) 0.000403 2.262807 2.262636MA(1) 0.008189 1.371007 1.347798

SIGMASQ 1.43E-11 2.460336 2.288745

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Appendix 3 (H) – Correlogram of modified Trinomial model for Capital Land

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Appendix 3 (I) – Static forecasts for modified Trinomial model for Capital Land

Appendix 3 (J) – Simple regression output for Trinomial model for Capital Land

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Appendix 4 (a) – Simple regression output of Binomial model for Sembcorp Marine



C -0.043047 0.002244 -19.18147 0.0000WARRANT_PRICE 1.230189 0.049392 24.90658 0.0000


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Appendix 4 (b) – Correlogram for simple regression output of Binomial model for Sembcorp Marine

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Appendix 4 (c) – Serial Correlation test for Binomial model for Sembcorp Marine





C -3.51E-05 0.001390 -0.025270 0.9799WARRANT_PRICE 0.001419 0.030580 0.046411 0.9631

RESID(-1) 1.019348 0.092035 11.07569 0.0000RESID(-2) -0.391474 0.131412 -2.978972 0.0035RESID(-3) 0.059469 0.136399 0.435989 0.6636RESID(-4) 0.061421 0.132055 0.465116 0.6427RESID(-5) 0.024109 0.092820 0.259742 0.7955

R-squared 0.635478 Mean dependent var -2.24E-17Adjusted R-squared 0.616943 S.D. dependent var 0.005695S.E. of regression 0.003525 Akaike info criterion -8.403492Sum squared resid 0.001466 Schwarz criterion -8.245106Log likelihood 532.2182 Hannan-Quinn criter. -8.339148F-statistic 34.28530 Durbin-Watson stat 1.911429Prob(F-statistic) 0.000000

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Appendix 4 (d) – Heteroskadacity (White) test for Binomial model for Sembcorp Marine





C 0.000685 7.63E-05 8.979479 0.0000WARRANT_PRICE^2 0.346644 0.034959 9.915622 0.0000

WARRANT_PRICE -0.030938 0.003329 -9.293724 0.0000

R-squared 0.490556 Mean dependent var 3.22E-05Adjusted R-squared 0.482204 S.D. dependent var 6.67E-05S.E. of regression 4.80E-05 Akaike info criterion -17.02819Sum squared resid 2.81E-07 Schwarz criterion -16.96031Log likelihood 1067.262 Hannan-Quinn criter. -17.00062F-statistic 58.73830 Durbin-Watson stat 1.146321Prob(F-statistic) 0.000000

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Appendix 4 (e) – Heteroskadacity (ARCH) test for Binomial model for Sembcorp Marine





C 1.13E-05 6.40E-06 1.768791 0.0799RESID^2(-1) 0.816257 0.098012 8.328171 0.0000RESID^2(-2) -0.383675 0.125771 -3.050577 0.0029RESID^2(-3) 0.124272 0.131298 0.946491 0.3461RESID^2(-4) 0.039004 0.131976 0.295541 0.7682RESID^2(-5) 0.066848 0.132098 0.506048 0.6139RESID^2(-6) -0.018846 0.132143 -0.142619 0.8869RESID^2(-7) -0.054702 0.132065 -0.414206 0.6796RESID^2(-8) 0.022830 0.131570 0.173523 0.8626RESID^2(-9) -0.002918 0.125421 -0.023263 0.9815

RESID^2(-10) -0.014661 0.095331 -0.153787 0.8781


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Appendix 4 (f) – Modified Binomial model for Sembcorp Marine



C -0.052906 0.002917 -18.13432 0.0000WARRANT_PRICE 1.412092 0.052747 26.77128 0.0000

AR(1) 0.913174 0.079114 11.54250 0.0000MA(2) -0.391104 0.137418 -2.846098 0.0055MA(3) -0.352013 0.099698 -3.530798 0.0007

SIGMASQ 1.03E-05 1.78E-06 5.793034 0.0000


Inverted AR Roots .91Inverted MA Roots .89 -.44+.45i -.44-.45i

Appendix 4 (g) – VIF Test for modified Binomial model for Sembcorp Marine



C 8.51E-06 9.343338 NAWARRANT_PRICE 0.002782 10.35827 1.381015

AR(1) 0.006259 5.119722 4.705390MA(2) 0.018884 3.551382 3.422117MA(3) 0.009940 1.822758 1.779146

SIGMASQ 3.16E-12 1.529709 1.463460

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Appendix 4 (h) – Correlogram for modified Binomial model for Sembcorp Marine

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Appendix 4 (i) – Static forecasts for modified model for Sembcorp Marine

Appendix 4 (j) – Dynamic forecasts for modified Binomial model for Sembcorp Marine

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Appendix 5 (a) – Simple regression output of Black-Scholes model for Sembcorp Marine



C -0.042696 0.002224 -19.19906 0.0000WARRANT_PRICE 1.225895 0.048945 25.04633 0.0000


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Appendix 5 (b) – Correlogram of simple regression output of Black-Scholes model for Sembcorp Marine

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Appendix 5 (c) – Serial corerrlation LM test of Black-Scholes model for Sembcorp Marine





C 2.02E-06 0.001365 0.001476 0.9988WARRANT_PRICE 0.000579 0.030041 0.019289 0.9846

RESID(-1) 1.034105 0.091894 11.25323 0.0000RESID(-2) -0.426402 0.132407 -3.220394 0.0017RESID(-3) 0.121896 0.137870 0.884135 0.3784RESID(-4) -0.011467 0.133039 -0.086195 0.9315RESID(-5) 0.060449 0.092669 0.652310 0.5155


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Appendix 5 (d) – Heteroskadacity (White) test for Black-Scholes model for Sembcorp Marine





C 0.000666 7.45E-05 8.937991 0.0000WARRANT_PRICE^2 0.336481 0.034115 9.863117 0.0000

WARRANT_PRICE -0.030033 0.003248 -9.245123 0.0000


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Appendix 5 (e) – Heteroskadacity (ARCH) test of Black-Scholes model for Sembcorp Marine





C 1.08E-05 6.03E-06 1.796662 0.0753RESID^2(-1) 0.849764 0.097959 8.674655 0.0000RESID^2(-2) -0.369647 0.127723 -2.894136 0.0046RESID^2(-3) 0.053860 0.132760 0.405692 0.6858RESID^2(-4) 0.113790 0.132815 0.856754 0.3936RESID^2(-5) 0.000110 0.133325 0.000822 0.9993RESID^2(-6) 0.035107 0.133296 0.263374 0.7928RESID^2(-7) -0.091452 0.132903 -0.688115 0.4929RESID^2(-8) 0.031280 0.132927 0.235318 0.8144RESID^2(-9) 0.022800 0.127023 0.179495 0.8579

RESID^2(-10) -0.040199 0.095062 -0.422874 0.6733


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Appendix 5 (f) – Modified Black-Scholes model for Sembcorp Marine



C -0.053299 0.002625 -20.30663 0.0000WARRANT_PRICE 1.423312 0.048616 29.27654 0.0000

AR(1) 0.985305 0.076036 12.95832 0.0000AR(2) -0.354304 0.071882 -4.928993 0.0000

SIGMASQ 9.61E-06 1.33E-06 7.243185 0.0000


Inverted AR Roots .49-.33i .49+.33i

Appendix 5 (g) – VIF test for Black-Scholes model for Sembcorp Marine




AR(1) 0.005782 1.594611 1.424966AR(2) 0.005167 1.490718 1.455466

SIGMASQ 1.76E-12 1.354383 1.327363

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Appendix 5 (h) – Correlogram of modified Black-Scholes model for Sembcorp Marine

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Appendix 5 (i) – Static forecasts for modified Black-Scholes model for Sembcorp Marine

Appendix 5 (j) – Dynamic forecasts of modified Black-Scholes model for Sembcorp Marine

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Appendix 6 (a) – Simple regression output of Trinomial model for Sembcorp Marine



C -0.033557 0.001832 -18.31805 0.0000WARRANT_PRICE 0.950987 0.040318 23.58704 0.0000


Appendix 6 (b) – Correlogram of simple regression Trinomial model for Sembcorp Marine

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Appendix 6 (c) – Serial correlation LM test for Trinomial model for Sembcorp Marine





C 4.33E-05 0.001122 0.038597 0.9693WARRANT_PRICE -0.000469 0.024689 -0.018993 0.9849

RESID(-1) 1.046455 0.091928 11.38345 0.0000RESID(-2) -0.426625 0.133238 -3.201975 0.0018RESID(-3) 0.084685 0.138904 0.609668 0.5433RESID(-4) 0.012923 0.133839 0.096554 0.9232RESID(-5) 0.054628 0.092721 0.589167 0.5569


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Appendix 6 (d) – Heteroskadacity (White) test for Trinomial model for Sembcorp Marine





C 0.000483 5.46E-05 8.837945 0.0000WARRANT_PRICE^2 0.248886 0.025015 9.949340 0.0000

WARRANT_PRICE -0.022038 0.002382 -9.251941 0.0000


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Appendix 6 (e) – Heteroskadacity (ARCH) test for Trinomial model for Sembcorp Marine





C 7.58E-06 4.20E-06 1.806566 0.0737RESID^2(-1) 0.945942 0.097967 9.655747 0.0000RESID^2(-2) -0.463757 0.134140 -3.457248 0.0008RESID^2(-3) 0.059376 0.141575 0.419395 0.6758RESID^2(-4) 0.163452 0.141606 1.154273 0.2510RESID^2(-5) -0.057433 0.142444 -0.403200 0.6876RESID^2(-6) 0.066209 0.142345 0.465129 0.6428RESID^2(-7) -0.100208 0.141654 -0.707419 0.4809RESID^2(-8) 0.041297 0.141792 0.291247 0.7714RESID^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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Appendix 6 (f) – Modified Trinomial model for Sembcorp Marine



C -0.042378 0.002354 -18.00585 0.0000WARRANT_PRICE 1.118704 0.039248 28.50377 0.0000

AR(1) 0.569424 0.102628 5.548407 0.0000MA(1) 0.462442 0.114595 4.035434 0.0001

SIGMASQ 6.76E-06 1.00E-06 6.750518 0.0000


Inverted AR Roots .57Inverted MA Roots -.46

Appendix 6 (g) – VIF test for modified Trinomial model for Sembcorp Marine




AR(1) 0.010533 2.269428 1.939369MA(1) 0.013132 2.438627 2.188084

SIGMASQ 1.00E-12 1.539229 1.536801

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Appendix 6 (h) – Correlogram for Modified Trinomial model for Sembcorp Marine

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Appendix 6 (i) – Static forecasts for Modified Trinomial model for Sembcorp Marine

Appendix 6 (j) – Dynamic forecasts for Modified Trinomial model for Sembcorp Marine

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Appendix 7 (a) – Simple regression output for Binomial model for Wilmar International



C 0.332709 0.013055 25.48497 0.0000WARRANT_PRICE 0.504644 0.140011 3.604311 0.0005


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Appendix 7 (b) – Correlaogram for Simple regression output for Binomial model for Wilmar International

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Appendix 7 (c) – Serial correlation test for Binomial model for Wilmar International





C 0.024902 0.005019 4.961478 0.0000WARRANT_PRICE -0.387212 0.054788 -7.067498 0.0000

RESID(-1) 0.590192 0.077539 7.611593 0.0000RESID(-2) 0.237470 0.087616 2.710346 0.0077RESID(-3) 0.187075 0.088029 2.125150 0.0357RESID(-4) 0.001597 0.087673 0.018220 0.9855RESID(-5) -0.015496 0.077349 -0.200342 0.8416


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Appendix 7 (d) – Heteroskadacity (White) test for Binomial model for Wilmar International





C 0.008783 0.005738 1.530569 0.1285WARRANT_PRICE^2 0.051178 0.122329 0.418362 0.6764

WARRANT_PRICE 0.017700 0.100361 0.176366 0.8603


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Appendix 7 (e) – Heteroskadacity (ARCH) test for Binomial model for Wilmar International





C 0.001109 0.000971 1.142060 0.2561RESID^2(-1) 0.719418 0.097227 7.399392 0.0000RESID^2(-2) 0.072816 0.120129 0.606147 0.5457RESID^2(-3) 0.019719 0.119336 0.165238 0.8691RESID^2(-4) 0.224577 0.118949 1.888016 0.0618RESID^2(-5) -0.079318 0.120749 -0.656883 0.5127RESID^2(-6) 0.050907 0.119755 0.425089 0.6717RESID^2(-7) -0.098749 0.117312 -0.841761 0.4019RESID^2(-8) -0.150543 0.117672 -1.279343 0.2036RESID^2(-9) -0.005391 0.118190 -0.045613 0.9637

RESID^2(-10) 0.114540 0.095823 1.195330 0.2347


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Appendix 7 (f) – Modified Binomial model for Wilmar International



C 0.380037 0.047176 8.055651 0.0000WARRANT_PRICE 0.022098 0.022871 0.966204 0.3367

AR(1) 0.946363 0.025812 36.66330 0.0000MA(1) 0.209361 0.084102 2.489375 0.0147

SIGMASQ 0.000622 0.000111 5.606801 0.0000


Inverted AR Roots .95Inverted MA Roots -.21

Appendix 7 (g) – VIF test for modified Binomial model for Wilmar International




AR(1) 0.000666 1.175969 1.121351MA(1) 0.007073 1.033015 1.029597

SIGMASQ 1.23E-08 1.194666 1.125863

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Appendix 7 (h) – Correlogram for modified Binomial model for Wilmar International

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Appendix 7 (i) – Static forecasts for modified Binomial model for Wilmar International

Appendix 7 (j) – dynamic forecasts for modified Binomial model for Wilmar International

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Appendix 8 (a) – Simple regression output for Black-Scholes model for Wilmar International



C 0.334196 0.013352 25.03015 0.0000WARRANT_PRICE 0.498048 0.141511 3.519505 0.0006


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Appendix 8 (b) – Correlogram of simple regression output for Black-Scholes model for Wilmar International

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Appendix 8 (c) – Serial correlation test for Black-Scholes model for Wilmar International





C 0.025091 0.005129 4.892365 0.0000WARRANT_PRICE -0.386775 0.055362 -6.986257 0.0000



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Appendix 8 (d) – Heteroskadacity (White) test for Black-Scholes model for Wilmar International





C 0.012130 0.005793 2.093835 0.0384WARRANT_PRICE^2 0.114730 0.122673 0.935246 0.3516

WARRANT_PRICE -0.038323 0.100737 -0.380423 0.7043


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Appendix 8 (e) – Heteroskadacity (ARCH) test for Black-Scholes model for Wilmar International






RESID^2(-10) 0.113341 0.097191 1.166167 0.2463


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Appendix 8 (f) – Modified Black-Scholes model for Wilmar International



C 0.337010 0.074657 4.514121 0.0000WARRANT_PRICE 0.022268 0.025167 0.884820 0.3781

AR(1) 0.969535 0.017881 54.22204 0.0000SIGMASQ 0.000631 9.18E-05 6.873716 0.0000


Inverted AR Roots .97

Appendix 8 (g) – VIF test for Modified Black-Scholes model for Wilmar International




AR(1) 0.000320 1.373864 1.154906SIGMASQ 8.43E-09 1.206686 1.151733

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Appendix 8 (h) – Correlogram for Modified Black-Scholes model for Wilmar International

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Appendix 8 (i) – Static forecast for Modified Black-Scholes model for Wilmar International

Appendix 8 (j) – Dynamic forecasts Modified Black-Scholes model for Wilmar International

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Appendix 9 (a) – Simple regression output for Trinomial model for Wilmar International



C 0.308355 0.012826 24.04175 0.0000WARRANT_PRICE 0.479086 0.135936 3.524339 0.0006


Appendix 9 (b) – Simple regression output for Trinomial model for Wilmar International

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Appendix 9 (c) – Serial correlation test for Trinomial model for Wilmar International





C 0.024095 0.004917 4.900095 0.0000WARRANT_PRICE -0.370186 0.053070 -6.975389 0.0000



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Appendix 9 (d) – Heteroskadacity (White) test for Trinomial model for Wilmar InternationalHeteroskedasticity Test: White




C 0.011600 0.005326 2.177965 0.0314WARRANT_PRICE^2 0.113859 0.112776 1.009605 0.3148

WARRANT_PRICE -0.042513 0.092609 -0.459058 0.6470


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Appendix 9 (e) – Simple regression output for Trinomial model for Wilmar International






RESID^2(-10) 0.120482 0.097187 1.239700 0.2180


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Appendix 9 (f) – Modified Trinomial model for Wilmar International



C 0.311134 0.071083 4.377026 0.0000WARRANT_PRICE 0.022516 0.023435 0.960746 0.3387

AR(1) 0.970092 0.017642 54.98841 0.0000SIGMASQ 0.000570 8.17E-05 6.976230 0.0000


Inverted AR Roots .97

Appendix 9 (g) – Simple regression output for Trinomial model for Wilmar International




AR(1) 0.000311 1.363339 1.147384SIGMASQ 6.68E-09 1.190825 1.143986

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Appendix 9 (h) – Correlogram for Modified Trinomial model for Wilmar International

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Appendix 9 (i) – Static forecasts for Modified Trinomial model for Wilmar International

Appendix 9 (j) – Dynamic forecasts for Modified Trinomial model for Wilmar International

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Risk Management and FInancial Engineering_Final Report

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