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

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 months of 12 stocks in 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 47 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 Equity warrants Equity options 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 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 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 48 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 Beginning of the period Ending of the period $70 A warrant is worth $10 in this day given stock price of $70 (Max 70 – 60) $50 $40 A warrant is worth nothing because current market price is less than the strike price 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 49 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 Return of the investment Value of investment H*(Stock price) – Warrant price H*70 – (70 – 60) H*(Stock price) – Warrant price H*(Stock price) – Warrant price H*70 – 10 = H*40 – H = 1/3 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 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 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 = – 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: 50 Beginning of period Ending of the period Pu (Possibility of stock price increasing) $70 $50 Pd = – Pu (Possibility of stock price decreasing) $40 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 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 50 = [Pu ∗ 70 + (1 − Pu) ∗ 40] ∗ 𝑒 −𝑅𝑓∗𝑇 Calculate Pu: Warrant Price = [Pu ∗ 10 + (1 − Pu) ∗ 0] ∗ e−Rf∗T Pu = Pu = 50∗𝑒 −𝑅𝑓∗𝑇 −40 Warrant price = (0.326*$10)* e−0.05∗0.0833 70−40 50∗𝑒 −0.05∗0.0833 −40 Warrant price = $3.25 70−40 Pu = 0.326 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 51 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) S(u) S(t) S(t) 1-P(u)-P(d) S(d) t=0 t = E[S(t+∆t)|S(t)] = Exp(rf * ∆t) * S(t) (1.a) 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 Var[S((t + ∆t)|S(t)] = ∆t𝑆 (t)𝜎 (t)+ O(∆t) (1.b) For a given ∆t time, the combination of equation (1.a) and (1b) is created as follow: – P(u) – P(d)+ u*P(u) + d*P(d) = Exp(rf * ∆t) Another assumption is established allowing the tree of trinomial complexity to grow only polynomial in order to make the calculation easier P(u)*P(d) =1 The size of jump is assumed now d= Exp(-σ* √ *∆t) u= Exp(σ* √ *∆t) The probability of lower and upper job of every intermediate note is obtained after solving these equations 52 P(u) = ( ∆𝒕 (𝒓𝒇∗ ) 𝟐 − 𝒆 ∆𝒕 ( ) 𝒆 𝟐 − P(d) = ( 𝒆 ∆𝒕 )) 𝟐 𝒆 ∆𝒕 (−𝛔∗𝐒𝐐𝐑𝐓( )) 𝟐 𝒆 (−𝛔∗𝐒𝐐𝐑𝐓( ∆𝒕 (𝒓𝒇∗ ) 𝟐 𝒆 ∆𝒕 (−𝛔∗𝐒𝐐𝐑𝐓( )) 𝟐 𝒆 (𝛔∗𝐒𝐐𝐑𝐓( 𝒆 ( ∆𝒕 ) 𝟐 − ∆𝒕 )) 𝟐 − 𝟐 ) 𝟐 ) Valuation of option (warrant) P(d) + P(u) + P(m) = 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 53 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 and the paying’s present value of the expiration exercise price is demonstrated in part Empirical Study 3.1 Singapore market Warrant Ticker Underlying Security Ticker Issuer Strike Price Expiry Date Style Option type CBJW DBS Macquarie Bank Ltd 19.5 10/07/2017 European Call CBHW OCBC Macquarie 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 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 Mean value in months 0.0945 0.1203 0.1199 0.2062 0.0258 0.0254 0.1116 Average of Difference Null hypothesis: H0 =Market Price = Price given by Model T test H1: Market Price ≠ Price given by Model Alpha = 0.05 (95% confidence interval) P value 0.0982 0.1167 3.48E-08 P Value > 0.05 P Value > 0.05 P Value < 0.05 Cannot reject H0 Cannot reject H0 Reject H0 Conclusion 54 SINGAPORE_CBHW Market price Black-Scholes Binomial Trinomial Mean value in months 0.1102 0.2079 0.2933 0.2595 0.0977 0.1831 0.1492 Average of Difference Null hypothesis: H0 =Market Price = Price given by Model T test H1: Market Price ≠ Price given by Model Alpha = 0.05 (95% confidence interval) P value Conclusion 2.95E-15 < 2.2E-16 < 2.2E-16 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 Mean value in months 0.1098 0.1305 0.1287 0.1918 0.0207 0.0188 0.0819 Average of Difference Null hypothesis: H0 =Market Price = Price given by Model T test H1: Market Price ≠ Price given by Model Alpha = 0.05 (95% confidence interval) P value 0.1076 0.1574 3.04E-06 P Value > 0.05 P Value > 0.05 P Value < 0.05 Cannot reject H0 Cannot reject H0 Reject H0 Conclusion Table 1: The results of warrant prices from models 3.2 Hong Kong market Warrant Ticker Underlying Security Ticker Issuer Strike Price Expiry Date Style Option type 16179 China mobile CS 90.93 19/06/2017 European Call 20104 AIA MB 47.28 02/08/2017 European Call 22249 SMIC UB 10.88 17/07/2017 European Call - The risk-free rate is the HIBOR in 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 55 Binomial Trinomial Mean 0.1902 Average of Difference 0.2396 0.233 0.3951 0.0494 0.0428 0.2049 Null hypothesis: H0=Market Price = Price given by Model T test H1: Market Price ≠ Price given by Model Alpha = 0.05 (95% confidence interval) P value 0.149 0.2174 2.97E-06 P Value > 0.05 P Value > 0.05 P Value < 0.05 Cannot reject H0 Cannot H0 reject H0 Conclusion reject HONGKONG_21014 Market price Black-Scholes Binomial Trinomial Mean value 0.3762 2.4893 2.4906 3.0327 2.113 2.1144 2.6565 Average of Difference Null hypothesis: H0 =Market Price = Price given by Model T test H1: Market Price ≠ Price given by Model Alpha = 0.05 (95% confidence interval) P value Conclusion < 2.2E-16 < 2.2E-16 < 2.2E-16 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 Mean 0.1114 0.204 0.2065 0.243 0.0926 0.0951 0.1316 Average of Difference Null hypothesis: H0= Market Price = Price given by Model T test H1: Market Price ≠ Price given by Model Alpha = 0.05 (95% confidence interval) P value Conclusion 0.0005 0.0004 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 models 56 3.3 Australia market Warrant Ticker Underlying Security Ticker Strike Price (AUD) Expiry Date Style Option type ANZWOB ANZ 30 29/07/2017 European Call BHPWOA BHP 24 29/07/2017 European Call CBAWOA CBA 74 29/07/2017 European Call - The risk-free rate is the yield of 10 Year Australian Bond The price of ANZWOB, BHPWOA and CBAWOA are evaluated based on three models including BlackScholes, Binomial and Trinomial as below: AUSTRALIA_ANZWOB Market price Black-Scholes Binomial Trinomial Mean value in months 0.4854 1.2569 1.2583 1.5266 0.7715 0.7729 1.0412 Average of Difference Null hypothesis: H0 =Market Price = Price given by Model T test H1: Market Price ≠ Price given by Model Alpha = 0.05 (95% confidence interval) P value Conclusion