The Pricing Models of Covered Warrants and Empirical Study in Thin Markets and Developed Markets

This paper aims to present three models including Binomial, Trinomial and Black-Scholes model, used to price the value of covered warrants, based on the value of their underlying assets.

The Pricing Models of Covered Warrants and Empirical Study

in Thin Markets and Developed Markets

Thi Kieu Hoa Phan

University of Economics and Law, Vietnam National University, Vietnam

Thi Tham Dinh

The State Bank of Vietnam, Vietnam

Quy Thai Duong Nguyen

Accountant, Melbourne, Australia

Nam Thai Nguyen

Ministry of Finance, Vietnam

Nada Thabet S Alluhaibi

Financial Specialist, Dubai, Saudi Arabia

Abstract

The covered warrant is a financial product which is widely applied in many markets not only the thin markets such as Hong Kong, Singapore and Korea but also developed markets namely Germany, Australia, the UK and the USA Towards the Vietnamese market, the covered warrant is a new type of investment, assisting investors in hedging and diversifying their portfolios The occurrence of covered warrants in the next few months may increase the ranking of the Vietnamese financial market However, in the reality, it is not easy for investors to value the price of covered warrants to make decisions This paper aims to present three models including Binomial, Trinomial and Black-Scholes model, used to price the value of covered warrants, based on the value of their underlying assets Moreover, based on these valuation models, the paper conducts the empirical study in thin markets (Hong Kong and Singapore) and developed markets (Australia and the UK) to select which is the best model for each type of market The research is involved in 1,275 input data (the daily price in 6 months of 12 stocks in 4 markets) and then pricing the value of their covered warrants respectively The research also applies the Kolmogorov-Smirnov method to test whether the stock price of underlying assets follows the log-normal distribution or not In addition, the P-value approach is conducted

to test the hypothesises The research results indicate that the Binomial model may apply to price the value of covered warrants in thin markets, while developed markets should employ the Black-Scholes Model Thus, this result may be a suggestion for Vietnamese investors as well as issuing organisations in selecting an approach to price the value of covered warrants Nevertheless, the further research is required to apply for pricing covered warrants with the American-style exercise

Keywords: Covered Warrants, Underlying Asset, Thin Market, Developed Market, Binomial, Trinomial, Black-Scholes

1 Introduction

A new type of financial instrument in the year 1998, was introduced to the market of Italy known as covered warrants The covered warrant is referred to a type of financial products that allows the holders to sell (put) or purchase (call) an underlying asset at a predetermined price (strike price) on or before a fixed day (exercised day) in the future (London Stock Exchange 2009) The investors have joined the world of derivatives and purchasing the covered warrants which are the financial instruments that derives their values from the performances of other assets such as stock exchanges indices, shares, interest rates, currencies, and commodities

Distinctions between covered warrants, equity warrants, equity options

The covered warrant allows holders to sell (put) or purchases (call) underlying assets from third party at

a predetermined price on or before a specific date in the future

The equity warrant is the instruments issued by the issuing company that give rights to holders to purchase stocks at a predetermined price on or before a certain day in the future The buyers of the equity warrant need

to make the upfront payment to gain rights to the warrants issuer

The equity option is a type of derivatives, issued by Stock Exchanges allow investors to buy or sell stocks

at a predetermined price on or before a specific date in the future The equity option has the powers to protect, diversify or develop share portfolios since it can be used regardless of conditions of the share market (Britton and Waterston 2013)

Covered warrants are issued

by the third party

Underlying assets: Equities,

Currency, Index or other

assets

Covered warrants work in a

similar way to the options

contracts that allows the

private investors to carry out

trade in the prices of assets

such as currencies, global

indices, equities, and

commodities

The covered warrants generate

a return if the prices of the

underlying assets have

increased above the strike

price before the maturity date

Equity warrants issued by companies

Underlying asset: The stocks of the issuing company

The warrants are issued by the way

of the preferential allotment to institutional investors, promoters and other investors

Equity warrants protect the participants of the market from the defaulting counterparties and provide the hedging opportunities

The equity warrants are not the equity shares because they do not carry any voting or dividend rights

It is issued by the organization during the period of financing in order to purchase the security

Equity options issued by stock exchanges

Underlying asset: Equities on the market

The equity options are commonly used by the market participants which includes investors seeking exposure to movements of the shares for a fraction of the cost of

an actual share

Traders and brokers can access options, listed on the stock exchange through a technology platform that offers dual options structure of the market

The equity option allows the regulators to gain the exposure to the share price movements for less than the actual share cost

2 Literature Review And Methodology

2.1 Binomial Derivative Pricing Models

2.1.1 Overview

An options valuation method so-called Binominal option is developed by Cox, Rubinstein and Ross in 1979

In order to price options (or warrants), an iterative procedure is used by Binomial option pricing Although

Binomial option has an additional relationship with Black-Scholes model, when comparing with Black-Scholes model, it not only has simpler formal deduction but also is more concretely to demonstrate valuation of option’s concept (Hull 2015)

The binomial option pricing model has an assumption that the stock’s margin and probability will not change as well as the movement of price fluctuations is only possible between up and down during the whole period of observation (Fard 2014) In some phases of the duration, the entire potential development path of basic stock is simulated by using the share price’s historical volatility and handling in order to evaluate the exercise profits, the guaranteed price on each path and single node is calculated by discounted method

In general, the Binomial option pricing model’s basic hypothesis is that the price of share could only fluctuate two orientations in every phases including down and up The pricing strategy of Binomial option depends on using stocks portfolio in order to simulate an option’s market value, before creating a risk free hedge; otherwise, between different options, a cheaper one could be bought by the investor then sell the higher one to attain profit on risk free, if it exists the opportunities for arbitrage However, these opportunities for arbitrage only happen in very short time period

The call option’s pricing method is given by the main function of this stock portfolio The option hedging must be constantly adjusted until the date of expiration, different from futures, once created; the future hedging could not change Generally, there are two approaches in order to use the Binomial pricing model including the Risk Neutral Approach and the Risk Less Hedge Approach which will indicate the similar consequences when estimate the warrant price with the different basic approach (Moon 2013)

2.1.2 The Riskless Hedge Approach

In term of the riskless hedge approach, the warrant price is calculated by using hedge ratio which could be found in a portfolio in each period between warrants and stocks Therefore, finding a general hedge ratio for every single period is the first step of using riskless hedge approach In this situation, a portfolio is established

by an assumption including selling call warrants and buying shares Assuming that, a number of shares are bought at $50 and the strike price is $60 if the call warrant is sold immediately Under the situations, when the price of share rises to $70, the price of warrant is $10 ($70-$60) However, if the price of share falls to $40, the warrant shall be valueless

The initial portfolio investment is created by using the cost of shares after eliminating the profit from the sales of warrant At the end of period, if the price of underlying asset increases to $70, the return will consist

of all value stocks as well as the warrant exercise’s loss However, the total portfolio return will be equivalent

to the current price of stock if the price of stock decreases In the below example, H is the hedge ratio

$50

$40

$70

A warrant is worth $10 in this day given stock price of $70 (Max 70 – 60)

A warrant is worth nothing because current market price is less than the strike price

Beginning of the period

Ending of the period

Assuming that portfolio is invested without risk, resulting in an equivalent in the investment return Otherwise, the hedge ratio also could be derived by using the equation

The return value could be checked when the hedge ratio is gotten from the equation In this situation, the portfolio value will equal $13.33 = (70*1/3 – 10) if the price of share has been risen This result will be similar

to the price of share when it decreases = $13.33 = 40*1/3 The beginning portfolio value shall be discounted by the ending period return with the discount rate equivalent to risk free rate resulting in the present value of end period return Therefore, the warrant price also could be calculated by using the present value Assuming that, risk free rate equal to 5% with monthly period, every period will be equivalent to 1/12 = 0.0833

As a result, warrant price is $325

2.1.3 The Risk–Neutral Approach

In term of the risk neutral approach, this model could apply for all the above assumptions In order to comparing with the risk free hedge approach, there is an assumption that arbitrage is eliminated in the risk neutral approach Generally, the return and risk always exchanges For an illustration, if we are going to invest

in an investment with high return, the money loss also must be proposal in case of failure As a consequence,

if the opportunity of arbitrate is ignored, there is not exist the potentiality of risk and return When the probabilities of the stock price’s trend are found by using this model, it is called the risk neutral probability When warrant price is calculated by using the risk neutral approach, the probability of the increase in stock price is pointed out by using Pu, and Pd indicates the probability of the fall in the stock price with Pd = 1 – Pu

is the association between Pd and Pu Assuming that, risk free rate equal to 5% with monthly period, every period will be equivalent to 1/12 = 0.0833 The same price is used to estimate as below:

H*50 – warrant price = 13.33*𝑒−𝑅𝑓∗𝑇 1/3*50 – warrant price = 13.33*e-0.05*0.0833 Warrant price = 16.67 – 13.42

Warrant price = $3.25

Value of investment

H*(Stock price) – Warrant price

H*(Stock price) – Warrant price H*70 – (70 – 60)

H*(Stock price) – Warrant price Return of the investment

H*70 – 10 = H*40 – 0

H = 1/3

All ending period possible warrant price must be discounted in order to calculate the warrant price Based

on the result above we have Pu (Possibility of the increase of stock price) as well as the ending period of warrant price is calculated by using the riskless hedge approach As a result, when the price of stock has been risen, the warrant price will be equivalent to $10 On the other hand, the warrant price will be equivalent to 0

if the price of stock has been fallen

The results of two calculation methods demonstrate that the result of warrant price can be attained from two different approaches It can be concluded that the warrant price only could be calculated in one period by these approaches, if the underlying assets have many stages, we must calculate the warrant price for every stage (using Excel)

2.1.4 Restrictions of Binominal Model

Assuming for the price of future is the main limitation of the Binomial model In fact, the stochastic performance is one characteristic of the stock market resulting in the most accurate number could not be obtained by the investor Therefore, the prediction could lead to failure meaning that the results deriving from Binomial model cannot match the price of the market at all time

2.2 Trinomial Option Pricing Model

2.2.1 Overview

Binomial option pricing model was one of the first and most popular models for option pricing basing on straightforwardness and it’s easy to understand However, this model also reveals a number of restrictions in the time of application resulting in emerging another model with more promising, so-called trinomial option pricing model

Comparing to binomial model, the trinomial model adds a state which can be considered the main feature

of this model An assumption of binomial model is the option price in each step might be go up or might be

$50

$40

$70 Pu (Possibility of

stock price increasing)

Pd = 1 – Pu (Possibility of stock price decreasing)

Beginning of period

Ending of the period

50 = [Pu ∗ 70 + (1 − Pu) ∗ 40] ∗ 𝑒−𝑅𝑓∗𝑇

Calculate Pu:

Pu = 50∗𝑒−𝑅𝑓∗𝑇−40

70−40

Pu = 50∗𝑒

−0.05∗0.0833 −40 70−40

Pu = 0.326

Warrant Price = [Pu ∗ 10 + (1 − Pu) ∗ 0] ∗ e−Rf∗T Warrant price = (0.326*$10)* e−0.05∗0.0833

Warrant price = $3.25

go down while in the assumption of trinomial model, the option price could remain stable This hypothesis is not only more similar to reality but also pricing better than the binomial model

Assuming that, S(t) is the price of stock at time t In the future, at time t+∆t, we have three values of S(t+∆t) including S(u) = S(t)*u, S(d) = S(t)*d and S(t) with P(u), P(d) and P(m)= 1-P(u)-P(d) are their probabilities respectively

2.2.2 Assumptions of the model

The price distribution over time t+∆t is provided on the section above The section above provides a distribution of price over time t+∆t There are a number of assumptions before reaching a price in order to use this model

No Arbitrage Pricing / Risk Neutrality: This assumption indicates that only one fix return could be reached for any given risk level meaning there is only one return is reached in a situation of risk free, so-called risk free return (rf)

The price of stock is assumed following the Brownian motion geometric with constant variance σ This assumption will be equivalent to the log normal distribution’s returns As a result, we have the following equation

For a given ∆t time, the combination of equation (1.a) and (1b) is created as follow:

Another assumption is established allowing the tree of trinomial complexity to grow only polynomial in order to make the calculation easier

The size of jump is assumed now

The probability of lower and upper job of every intermediate note is obtained after solving these equations

E[S(t+∆t)|S(t)] = Exp(rf * ∆t) * S(t) (1.a)

Var[S((t + ∆t)|S(t)] = ∆t𝑆2(t)𝜎2(t)+ O(∆t) (1.b)

1 – P(u) – P(d)+ u*P(u) + d*P(d) = Exp(rf * ∆t)

P(u)*P(d) =1

d= Exp(-σ* √ 2 *∆t) u= Exp(σ* √ 2 *∆t)

S(t)

S(d)

S(t)

S(u)

1-P(u)-P(d)

=

Valuation of option (warrant)

2.2.3 Valuation of options (warrants)

The option valuation is analysed as below after we have the results of P(d), P(u), P(m), u, d and m (m=1)

a The factors including S(u) = u*S(t), S(d)= d*S(t) and S(t)= m*S(t) is used to assign the price of underlying asset on t+∆t ,

Sn = [𝐏𝐮 ∗ 𝐒(𝐮) + (𝟏 − 𝐏𝐮 − 𝐏𝐝) ∗ 𝐒(𝐭) + 𝑷𝒅 ∗ 𝑺(𝒅)] ∗ 𝒆−𝑹𝒇∗𝑻

b The price of call option (warrant) or price of put option (warrant) is calculated respectively its intrinsic

value situation

In term of Call Option: Max[(Sn – K),0]

In term of Put Option: Max[( K- Sn),0]

c Thus, the formula of final warrant price by using the Risk-Neutral method as below:

P warrant = [𝐏𝐮 ∗ 𝐂(𝐮) + (𝟏 − 𝐏𝐮 − 𝐏𝐝) ∗ 𝐂(𝐦) + 𝑷𝒅 ∗ 𝑪(𝒅)] ∗ 𝒆−𝑹𝒇∗𝑻

C(u) : The option (warrant) price of an up move

C(m): The option (warrant) price of the middle move (remain constant)

C(d): The option (warrant) price of a down

Rf: Risk free rate of return, T: Time to maturity

2.3 Black-Scholes Option Pricing Model

2.3.1 Overview

The Black-Scholes Option Pricing Model was established and developed by Merton and Scholes who won the 29th economics Nobel Prize in 1997 The bonds, stocks as well as numerous of new derivative financial instruments are prised reasonably by this model

2.3.2 Assumptions

Similar to other models, Black-Scholes model has a number of assumptions

a Constant volatility: The movement of a stock in short term (constant over time) could be measured by

volatility which is one of the most important assumptions The market will never change in long term although

it experiences a fluctuation in very short term and relatively stable

b Efficient markets: The trend of individual stocks or market could not be consistently predicted by the

Black-Scholes model in this hypothesis, instead of this, the stock is assumed walk randomly to move

c No dividends: Generally, the dividends are paid to the shareholders by most companies However, the

Black-Scholes model assumes that the shareholders will not receive dividends from basic stocks

d Lognormal distribution: The returns of underlying assets follow normal distribution in Black-Scholes

model This means that the price of underlying assets is lognormal distribution

e European-style options: European-style options are assumed

P(u) = ( 𝒆(𝒓𝒇∗

∆𝒕

𝟐)− 𝒆(−𝛔∗𝐒𝐐𝐑𝐓(

∆𝒕

𝟐))

𝒆(

∆𝒕

𝟐)− 𝒆(−𝛔∗𝐒𝐐𝐑𝐓(

∆𝒕

𝟐))

)

𝟐

P(d) = ( 𝒆(𝛔∗𝐒𝐐𝐑𝐓(

∆𝒕

𝟐))− 𝒆(𝒓𝒇∗

∆𝒕

𝟐)

𝒆(

∆𝒕

𝟐)− 𝒆(−𝛔∗𝐒𝐐𝐑𝐓(

∆𝒕

𝟐))

)

𝟐

P(d) + P(u) + P(m) = 1

f Interest rates constant and known: The interest rates is assumed invariably

g No commissions and transaction costs:

There are not existed any trade barriers or extra costs in selling or buying options

h Liquidity: Assuming that every

transaction takes place at any time and the markets are full liquidity

2.3.3 Black-Scholes model’s Function

This model consists of two parts with the expected return is absolutely basis of the purchase in part 1 and the paying’s present value of the expiration exercise price is demonstrated in part 2

3 Empirical Study

3.1 Singapore market

Warrant

Ticker

Underlying Security Ticker Issuer

Strike Price Expiry Date Style

Option type

Bank Ltd 19.5 10/07/2017 European Call

Bank Ltd 9.5 02/10/2017 European Call CBRW UOB Macquarie Bank Ltd 22.5 01/11/2017 European Call

- The risk-free rate is the SIBOR in 6 months

The covered warrant price of CBJW, CBHW and CBRW is evaluated based on three models including

Black-Scholes, Binomial and Trinomial as below:

SINGAPORE_CBJW Market price Black-Scholes Binomial Trinomial

T test

Null hypothesis: H0 =Market Price = Price given by Model

H1: Market Price ≠ Price given by Model Alpha = 0.05 (95% confidence interval)

Conclusion

P Value > 0.05 P Value > 0.05 P Value < 0.05 Cannot reject H0 Cannot reject H0 Reject H0

SINGAPORE_CBHW Market price Black-Scholes Binomial Trinomial

T test

Null hypothesis: H0 =Market Price = Price given by Model

H1: Market Price ≠ Price given by Model Alpha = 0.05 (95% confidence interval)

Conclusion P Value < 0.05 P Value < 0.05 P Value < 0.05

Reject H0 Reject H0 Reject H0 SINGAPORE_CBRW Market price Black-Scholes Binomial Trinomial

T test

Null hypothesis: H0 =Market Price = Price given by Model

H1: Market Price ≠ Price given by Model Alpha = 0.05 (95% confidence interval)

Conclusion

P Value > 0.05 P Value > 0.05 P Value < 0.05 Cannot reject H0 Cannot reject H0 Reject H0

Table 1: The results of warrant prices from 3 models

3.2 Hong Kong market

Warrant

Ticker

Underlying Security Ticker Issuer Strike Price Expiry Date Style Option type

- The risk-free rate is the HIBOR in 6 months

The price of 16179, 20104 and 22249 are evaluated based on three models including Black-Scholes, Binomial and Trinomial as below:

HONGKONG_161179 Market price Black-Scholes Binomial Trinomial

Mean 0.1902 0.2396 0.233 0.3951

T test

Null hypothesis: H0=Market Price = Price given by Model H1: Market Price ≠ Price given by Model

Alpha = 0.05 (95% confidence interval)

P Value > 0.05 P Value > 0.05 P Value < 0.05 Cannot reject H0 Cannot reject H0 reject H0 HONGKONG_21014 Market price Black-Scholes Binomial Trinomial

T test

Null hypothesis: H0 =Market Price = Price given by Model H1: Market Price ≠ Price given by Model

Alpha = 0.05 (95% confidence interval)

Conclusion

P Value < 0.05 P Value < 0.05 P Value < 0.05 Reject H0 Reject H0 Reject H0 HONGKONG_22249 Market price Black-Scholes Binomial Trinomial

T test

Null hypothesis: H0= Market Price = Price given by Model H1: Market Price ≠ Price given by Model

Alpha = 0.05 (95% confidence interval)

Conclusion

P Value < 0.05 P Value < 0.05 P Value < 0.05 Reject H0 Reject H0 Reject H0

Table 2: The results of warrant prices from 3 models