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 Abstract Financial Specialist, Dubai, Saudi Arabia 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 BlackScholes 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 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 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 A warrant is worth $10 in this day given stock price of $70 (Max 70 – 60) $70 $50 A warrant is worth nothing because current market price is less than the strike price $40 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 H* 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* − ∗ 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 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 Calculate Pu: Pu = Pu = 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)= 1P(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(t) t=0 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)] = ∆t (t) (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) = ( P(d) = ( 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 = [ ∗ ( )+( − − )∗ ( )+ ∗ ( )] ∗ − ∗ 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: − ∗ move P warrant =[ ∗ ( )+( −− ) ∗ ( ) + ∗ ( )] ∗ C(u) : The option (warrant) price of an up 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 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 CBJW CBHW CBRW - 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 Mean value in months Average of Difference T test P value Conclusion 54 SINGAPORE_CBHW Mean value in months Average of Difference T test P value Conclusion SINGAPORE_CBRW Mean value in months Average of Difference T test P value Conclusion 3.2 Hong Kong market Warrant Ticker 16179 20104 22249 - The risk-free rate is the HIBOR in months The price of 16179, 20104 and 22249 are evaluated based on three models including BlackScholes, Binomial and Trinomial as below: HONGKONG_161179 55 Mean Average of Difference T test P value Conclusion HONGKONG_21014 Mean value Average of Difference T test P value Conclusion HONGKONG_22249 Mean Average of Difference T test P value Conclusion Table 2: The results of warrant prices from models 56 3.3 Australia market Warrant Ticker ANZWOB BHPWOA CBAWOA - 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 Mean value in months Average of Difference T test P value Conclusion AUSTRALIA_BHPWOA Mean value in months Average of Difference T test P value Conclusion AUSTRALIA_CBAWOA Mean value in months Average of Difference T test P value Conclusion 3.4 The United Kingdom market Warrant Ticker CWN8144B2665 CWN8143L3076 CWN8141J6687 The risk-free rate is LIBOR in months The prices of CWN8144B2665, CWN8143L3076 and CWN8141J6687 are calculated based on three models including Black-Scholes, Binomial and Trinomial as below THE UK_CWN8144B2665 Mean value in months Average of Difference T test P value Conclusion THE UK_CWN8143L3076 58 Mean value in months Average of Difference T test P value Conclusion THE UK_CWN8141J6687 Mean value in months Average of Difference T test P value Conclusion Table 4: The results of warrant prices from models 3.5 Summary Empirical Results Between thin markets (Singapore and Hong Kong) and developed markets (Australia and the UK), there are some differences in the selection of models to calculate the price of covered warrants Towards the Singapore market, the results from models indicate that the Binomial model may be the best solution to calculate the value of covered warrants in this market This is because the theoretical prices (mean value) from the Binomial model of two covered warrants (CBJW and CBRW) are closer to the market prices than those of the Black-Scholes and Trinomial model Similar to the Singapore market, regarding the Hong Kong market, the Binomial model may be the best selection because of some reasons The results of the Binomial model (mean value) are nearly similar to the market price In addition, since the returns of underlying assets such as China mobile or AIA not follow the log-normal distribution (based on Kolmogorov-Smirnov Test), the Black-Scholes model may not be suitable to apply for this market In terms of developed markets (Australian and the UK market), the Black-Scholes may be the best model to calculate the price of covered warrants This is due to the fact that the prices (mean values) from the Black-Scholes model may be less different from the market price compared to the Binominal and Trinomial model Additionally, as the most returns of underlying assets in such markets follow the log-normal distribution, the 59 Black-Scholes model can be applied to calculate the value of covered warrants ((based on Kolmogorov-Smirnov Test) The Black-Scholes Comments Black-Scholes (BS) can be applied for pricing the value of covered warrants in developed markets such as Australia and the UK as the model is saving time BS can calculate a huge number of warrant prices in the short period of time (Kilic 2015) However, BS has many limitations that prevent it from forecasting accurately Firstly, this model assumes that the distribution of underlying price is lognormal Unfortunately, in the reality, the distribution is very different from lognormal such as having fatter left and right tails (Kilic 2015) For instance, thin market has very few buyers and sellers that lead to low transactions which lead to price more volatile and illiquidity (Rostek & Weretka 2008) This causes the distribution of underlying price movement is moved far from lognormal Thus, Black- Scholes model is no longer applied correctly Moreover, the high volatility in price also relates to second huge limitation of BS model which is assuming volatility constant Unfortunately, in the reality, the volatility is very hard to forecast and change randomly which often is referred to the term “stochastic volatility” (Lorig & Sircar 2014) Black-Scholes model has failed to take this issue into account; hence, it can lead to incorrect predictions As a consequence, to modify BS model to enhance its result, volatility should be modelled stochastically (Fouque et al 2000) It means that underlying price volatility will be random variables which could increase the precision of forecasts Similar to the volatility, another drawback of Black-Scholes is that interest rate is remained constant This model only uses the same risk-free rate in calculation for the whole period However, in practice, this rate is subjected to change every day (Spas’ka & Sheychenko n.d.); hence, it violates the Black-Scholes assumptions and makes the empirical results less accurate The fourth limitation of Black-Scholes is that it assumes there is no dividend payment throughout option’s life (Jonsson et al 2013) Meanwhile, most companies pay dividend to their shareholders; therefore, it is a huge limitation of Black-Scholes because dividend yield can impact investors on deciding which underlying has better value According to Spas’ka and Sheychenko (n.d.), they suggest that the common way to improve Black-Scholes model in this aspect is subtracting the discounted value of future dividend from stock price As a result, dividend payment will be taken into account and make the results more accurately Some other limitations that make Black-Scholes model be less accurate is that when it comes to price warrants with American-style exercise, Black-Scholes model cannot calculate and forecast correctly (Kilic 2015) The reason is that the Black-Scholes model only calculate price options at expiration, while in American-style, options can be exercised at any time and even before expiration Spas’ka and Sheychenko (n.d.) argue that there are some methods to adjust Black-Scholes price to make it calculating American option prices more precisely such as Fischer Black Pseudo American method but it is not really effective In addition, Black-Scholes model also ignores commissions (Spas’ka & Sheychenko n.d.) In fact, investors always have to pay commission when buying and selling options This may be small amount of money but it still has impact on output Conclusion It can permit that there are three approaches, used to price the value of covered warrants including Binomial, Trinomial and Black-Scholes model Between thin markets and developed markets, the selection of pricing models may be different While the Binomial model may be the best method to calculate the price of covered warrants of markets namely Singapore and Hong Kong, the Black-Scholes should be employed to evaluate the value of covered warrants in advanced markets including Australia and the UK However, it is 60 important to notice that the Black-Scholes may have some limitations due to its assumptions and BS may not be applied for warrants with American-style exercise Towards the Vietnamese market, investors and issuing originations may consider applying the Binomial model to evaluate the value of covered warrants REFERENCES Besley, S 2016, ‘Corporate finance’, Cengage Learning Britton, A & Waterston, C 2013, ‘Financial accounting’, Harlow: Financial Times Prentice Hall Duguid, C 2017, ‘Stock exchange’, Routledge Fouque, P, Papanicolaou, G & Sircar, R 2000, ‘Stochastic Volatility Correction to Black-Scholes’, Stanford Univeristy Fard, F 2014, ‘Introduction to Mathematics and Statistics of Financial Markets’, Corpus Education Publications Hull, J 2015, ‘Option, Futures and Other Derivatives (9 th ed)’, Pearson Publications Helbæk, M, Lindset, S & McLellan, B 2010, ‘Corporate finance’, Maidenhead, Berkshire: Open University Press, McGrawHill Education Haahtela, T2010, ‘Recombining Trinomial Tree for Real Option Valuation with Changing Volatility ‘, Helsinki University of Technology Jonsson, R, Karlen, A & Weigardh, A 2013, ‘THE BLACK-SCHOLES MODEL: When the underlying pays dividends’, School of Education, Culture and Communication, Malardalen University Kilic, E 2005, ‘A Comparison of Option Pricing Models’, Finecus, 11 January, viewed 20 May 2017, < http://www.finecus.com/docs/option.pdf> Lorig, M & Sircar, R 2014, ‘Stochastic Volatility: Modeling and Asymptotic Approaches to Option Pricing & Portfolio Selection’, Princeton London Stock Exchange 2017, ‘Covered warrants – an in depth guide’, London Stock Exchange, viewed 29 April 2017, Moon, K 2013, ‘An adaptive averaging binomial method for option valuation’, Operations Research Letters, vol.1, no.3 Powers, M & Needles, B 2012, ‘Financial accounting’, Cengage Learning Rostek, M & Weretka, M 2008, ‘Thin markets’, The New Palgrave Dictionary of Economics, viewed 20 May 2017, < http://www.ssc.wisc.edu/~mrostek/PalgraveRW.pdf> Rahman, N 2015, ‘Corporate Finance’, North Ryde: McGraw-Hill Australia Spas’ka, N & Sheychenko, O n.d., ‘Black-Scholes Limitations’, Department of Mathematics and Physics Sy, M 2017, ‘ Equity risk management’, RMIT University Welch, I 2014, ‘Corporate finance’, Los Angeles: Ivo Welch Zaboronski, P 2008, ‘Pricing Options Using Trinomial Trees’, viewed 24 May 2017, 61 ... Between thin markets and developed markets, the selection of pricing models may be different While the Binomial model may be the best method to calculate the price of covered warrants of markets. .. 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. .. (Australia and the UK), there are some differences in the selection of models to calculate the price of covered warrants Towards the Singapore market, the results from models indicate that the Binomial