Valuation of European gold option in Vietnam Luận văn thạc sĩ

We use the modified Black-Scholes model to price European SJC gold brand option in Vietnam, our focus is to compare the difference between option price derived by modified Black-Scholes

Huynh Ngoc Trang

VALUATION OF EUROPEAN SJC GOLD OPTION

IN VIETNAM

ECONOMICS MASTER THESIS

In Banking Ology code: 60.31.12

Supervisor

Dr Pham Huu Hong Thai

Ho Chi Minh city, 2010

This research project would not have been possible without the support of many people Firstly I wish to express my deep sincere gratitude to my supervisor, Dr Pham Huu Hong Thai for his invaluable advices and helps Without him, this thesis could not have been completed Special thanks to all instructors without whose knowledge and assistance this study would not have been successful

I would like to express my deepest gratitude and honor to my dear parents for not only the love they devote to me but also for the time I took from them which should have been my devotion to them in their aged time

My thanks would also go to all of my classmates, Ms Vu Thi Thu Van and Mr Tran Quoc Trong for all of their friendship and encouragement I also wish to thank

my friends in Eximbank for their great support

Finally, my greatest thanks would go to my husband, Mr Le Minh Nhat who is the greatest inspiration and encouragement for me to overcome all difficulties through the duration of my study.

ABSTRACT

The main idea of this thesis is to find the most suitable approach to the valuation

of SJC gold We use the modified Black-Scholes model to price European SJC gold brand option in Vietnam, our focus is to compare the difference between option price derived by modified Black-Scholes model and option price in TOKYO COMMODITY EXCHANGE (TOCOM) in the period from 1/7/2010 to 15/8/2010 The results show that the option price derived by modified Black-Scholes model is different from the option price in TOCOM Since the two option price is different,

we carry out the ex post and ex ante test to investigate the efficiency of Vietnam gold option market when applying the option price in TOCOM into Vietnam The evidences from these tests provide the rejection of our hypothesis of market efficiency due to the existing of abnormal profit

Key words: SJC gold, option pricing, modified Black-Scholes model, Vietnam gold market

Chapter 2: Review of modified Black- Scholes option pricing models 10

2.4 Testing the market efficiency of K.Shastri and K.Tandon 17

3.1.2 Gold price and option price listed in TOCOM 22

3.2.2 Interest rate of VND and Interest rate SJC gold brand 25

the option price quoted in TOCOM

-

ABBREVIATION

ACB: Asia Commercial Bank

COMEX: NewYork Commodities Exchange

Eximbank: Vietnam Export Import Commercial Joint-Stock Bank PNJ: Phu Nhuan Jewelry Joint Stock Company

SBJ: Sacombank Jewelry Limited Company

SBV: State Bank of Vietnam

SJC: Saigon Jewelry Holding Company

TOCOM: Tokyo Commodity Exchange, Inc

VND: Vietnam Dong

Table 4.3: Excess return from ex post hedging strategy for calls

Table 4.4: Excess return from ex post hedging strategy for puts

Table 4.5: Excess return from ex ante hedging strategy for calls

Table 4.6: Excess return from ex ante hedging strategy for puts

Chapter 1: INTRODUCTION

In Vietnam, the gold markets have developed for more than 7 years, gold is used

as a hedge against inflation, payment for real estate and traded as a currency for speculation Among many gold brand in the market, such as: ACB gold brand, SBJ gold brand, PNJ Gold brand, SJC gold brand, the most popular gold brand traded in the market is SJC gold brand (manufactured by Sai Gon Jewelry holding company) Vietnamese investors trade gold in three ways: spot, forward and option The turn over of spot transaction is largest, about 95% of total, forward 4%, option 1% From 5/2007, a breakthrough of gold market in Vietnam: the opening of first gold floor named Saigon gold exchange run by Asia Commercial Bank, and followed by lots of other gold centers Members of the Gold exchange center are legal entities which have gold trading license and gold traders in Vietnam The Bank is acting as a trading intermediary among the counter-partners, which ensures the settlement capacity and liquidity Margin deposit ratio, transaction fee, and interest rate are regulated by the Bank

At the end of 2009, Vietnam has around 20 gold trading floors where investors could deposit a small fund and then trade 14 times the value of their initial investment Investors can timely grasp their investment opportunities and earn expected profits

On 30 December 2009, the Government Office issued Notice No 369/TB-VPCP

to convey the Prime Minister’s request to all banks to close their gold trading centres and settle all the obligations with customers by 31 March 2010 The decision was an attempt to stabilize the country’s foreign exchange market

On 6 January 2010, SBV issued Circular No.01/2010/TT-NHNN to request all credit institutions in Vietnam to stop their overseas gold margin trading activities, and close overseas margin gold trading accounts by 31 March 2010

Since January 2010, after the issuance of the two circular of the State bank of Vietnam, the domestic gold market become very quiet with investment demand down sharply

However, in Vietnam, investors and hedgers do not usually trade option as a hedge against risk or as an investment in both stock market and currency/gold market At present, only some bank offer option in gold and currency

To calculate the option premium offered for their customer, banks base on the option price of the gold option listed in the TOCOM or COMEX or the premium offered by their counterparts in overseas At the time of the investigation, trading gold in the overseas account has been terminated by the State Bank of Vietnam, moreover it is necessary that the banks base on a model to calculate the option premium base on the characteristic of the domestic gold markets so that the option premium could be acceptable for both the banks and their customer

In the early 1970's, Myron Scholes, Robert Merton, and Fisher Black made an important breakthrough in the pricing of complex financial instruments by developing what has become known as the Black-Scholes model In 1997, the importance of their model was recognized world wide when Myron Scholes and Robert Merton received the Nobel Prize for Economics The Black-Scholes model displayed the importance that mathematics plays in the field of finance It also led to the growth and success of the new field of mathematical finance or financial engineering In this thesis, we use the modified Black-Scholes models for foreign

currency options to calculate the option price of SJC gold in the European style and compare this option price with the option price quoted in the TOCOM

* Objectives and Rationale of the study:

This study is motivated to investigate whether the modified Black-Scholes model should be used in valuation of the SJC gold option price in Vietnam

by examining the difference between the option calculated by modified Scholes model and the gold option price listed in the TOCOM and investigating the efficiency of the gold option market

• The second hypothesis is that the application of TOCOM option price for Vietnam gold option market is still effective The effectiveness of the market is that traders could earn no abnormal profit

We apply the modified Black – Scholes model to price SJC gold option in the period of 1/7/2010 to 15/8/2010 for the options matured on August, October and December The comparison between the SJC option price derived from modified Black – Scholes model and the option price quoted in TOCOM show that these two option price is different Duplicating Kishore Tandon and Kuldeep Shatri, we carry out empirical tests include ex post and ex ante test to test the efficiency of the market when applying the option price quoted in TOCOM into Vietnam market The tests result that the market is not efficient because that the traders can earn abnormal profit

Organization of the thesis: Chapter 1 is an introduction, Chapter 2 is a review on modified Black – Scholes model and on K.Shastri and K.Tandon test of market efficiency Chapter 3 gives the research methodology including Data collection, modified Black – Scholes model and the ex post, ex ante test Chapter 4 is empirical results and discussion The last chapter concludes the thesis

Chapter 2: REVIEW OF MODIFIED BLACK- SCHOLES OPTION PRICING MODELS AND SOME EMPIRICAL EVIDENCES

2.1 Option and boundary conditions:

This means that the value of any American or European call option must be less than or equal to that of the current stock price If the above boundaries were not as they appear, then any arbitrageur would be able to easily make a riskless profit simply by purchasing the stock and then immediately selling the call option

The upper bound for the purposes of both American and European put options is

a little different As for a put option, regardless of the fact that it is an American or a European call option, the option will give the holder the right to sell 1 share of a stock for a certain price (the strike price) However, regardless of how much the stock price falls, the price of the option can never exceed that of the strike price (as this is the price at which you will be able to sell the underlying stock for) Since it is the stock or share price which the option basis its own price on The right can never

be worth the strike price at which you will be selling the underlying stock or share Therefore the upper bound for put option prices is the options strike price This means that the value of any American or European put option must be less than or equal to that of the options strike price

Also, it must be noted that for European options at maturity can not be worth more than the strike price This is because of the fact that the option cannot be worth more than the present value of the strike price today.

This means that the value of any American or European put option must be less than or equal to that of the options strike price multiplied by the natural e, to the power of negative risk-free interest rate multiplied by the options time to expiry

2.1.2 Lower bound

The lower bound for any non-dividend paying call option is:

This means that the lower bound is equal to the current stock price minus the options strike price multiplied by the natural e, to the power of negative risk-free interest rate multiplied by the options time to expiry

The lower bound for the put is the reverse of that of the call It would be the difference between the discounted value of the strike price at the risk free rate against the stock price If this rule is violated the investors can gain profit by borrowing money and buying a put and the stock This would lead to a positive cash flow on maturity

2.2 Review of modified Black- Scholes model

In valuation of the options price, Fisher Black and Myron Scholes (1972) were the first authors to deal with pricing for European style options, the Black-Scholes option pricing model provides the foundation for the modern theory of options valuation In deriving the formula for the value of an option in terms of the price of the stock, they assumed “ideal conditions” in the market for the stock and for the option:

- The stock follows a Geometric Brownian Motion

- There are no penalties for short sales

- Transaction costs and taxes are zero

- The market operates continuously

- The risk-free interest rate is constant

- The stock pays no dividend

The Black – Scholes option pricing formula:

X S XN e T

T r

X S

SN

σ

σ σ

( [ ) /

(1.1)

Where S is the spot price, X is the exercise price, σ2

is the instantaneous variance of the stock’s return, r is the risk free interest rate and N is the cumulative standard normal distribution function, T is the expiration date of the option

Many theoretical studies and empirical tests have provided support for the Black Scholes option pricing formula in pricing stock with no dividend and have some modification in pricing currency option

As Black – Scholes assumes no dividend are paid on the stock during the life of the option, their model cannot be applied to value option on a foreign currency (Nahum Biger and John Hull, 1983) When Merton (1973) and Smith (1976) use the Black-Scholes formula in valuation of the currency options, they found that the Black-Scholes formula can not be applied directly to pricing the currency option if the risk free interest rate can be earned on the foreign currency holding This model

is extended by Merton (1973) and Smith (1976) for continuous dividends Since the interest gained on holding a foreign security is equivalent to a continuously paid dividend on a stock share, the Merton and Smith version of the Black-Scholes can

be applied to foreign security To value currency option, stock price are substituted for exchange rates Merton (1973) and Smith (1976) consider that the dividend yield, δ, is constant, they introduced a formula which called modified Black-Schole formula to calculate the European currency option price:

X S XN e T

T r

X S SN

e

σ

σ δ σ

σ δ

However, often a dividend payment will be scheduled during the life of an option, but the amount of the payment has not yet been announced This is an additional source of uncertainty the Merton model can not reflect

In 1983, using Black- Scholes methodology, Nahum Biger and John Hull assumed that if the risk-free interest rate can be earned on the foreign currency holding, r*, is constant, the dividend yield from an investment in the foreign currency is constant and equal to r*, under the assumption that:

- The price of one unit of foreign currency follows a Geometric Brownian Motion

- The foreign exchange market operates continuously with no transaction cost

r X S XN e T

T r

r X S SN

e

σ

σ σ

( [

) /

*

(1.3)

The variables are redefined as follows:

S: spot price of one unit of the foreign currency

2

: instantaneous variance of the return on the foreign currency holding

X, T: exercise price and date of a European call option to purchase one unit of the foreign currency

R: risk free rate of interest in the home country

In the valuation formulas of option, the forward rate plays a central role (Nahum Biger and John Hull, 1983) Define F as the forward rate on the foreign currency for

a contract with delivery date T, the above formula reduces to:

F XN e T

T X

F FN

e

σ

σ σ

( ) /

(1.4)

Given the premium of a European call option (call C), the premium for a European put option (called P) on the same currency and same exercise price (X) can be derived from put-call parity:

(1.5)

rTe F

X FN e T

T F

X XN

e

σ

σ σ

( ) /

(1.6)

The Black-Scholes model, as the initial assumption, can be used to valuing the European style option price The Black-Schole European option model exhibit systematic mispricing biases when used to value American call and put option (Kuldeep Shastri and Kishore Tandon, 1986), this systematic mispricing is related

to three different factors: the time to maturity of the option, the degree the option is

in or out of the money, and the volatility of the underlying security The probability

of early exercise depends on these three factors

Same as the Black-Scholes model, the model developed by Biger and Hull (1983) is referred to as the European model because it does not account for exercise before the expiration day American options do allow for early exercise while European option does not This extra flexibility of American currency option may justify a higher premium on American currency option than on European currency option with the same characteristics

2.3 Some empirical evidences

When Black and Scholes published their option pricing model in 1973 their study were pioneering Many studies have been published since then some of which

are developments of the Black and Scholes model and some new competing models Many previous studies of the Black and Scholes model show conflicting results andwe will present such results from a couple of authors

The results from the authors who will be presented are: Macbeth and Merville, Merton, Hull and White and finally Byström In 1979 the two researchers Macbeth and Merville tested the Black and Scholes empirically on call options They found that the Black and Scholes model tends to overprice out of the money options and underprice in the money options with a remaining duration

of less than ninety days

Stan Beckers (1982) use a riskless hedging strategy the Black-Scholes call option pricing model was used to test market efficiency and to check for consistent discrepancies between model and market prices Again, no evidence of market inefficiency could be found However, there are indications that the Black-Scholes model is not appropriate to price out-of-the-money gold options

Furthermore, they came to the conclusion that the more in the money an option is the more the model underprices and vice versa for out of the money options Macbeth and Merville relate to the results of Black and Merton in this study and points out the fact that Black on the other hand came to the conclusion that deep out of money options are underpriced by the model while deep in the money options are overpriced by the model

This is not the only conflicting empirical result made by researchers The results

of Merton’s study conflict with the result of the previous mentioned author Macbeth, Merville and Black Merton suggest the Black and Scholes theoretical option prices are lower than market option prices for both deep in the money and deep out the money options

Later in 1987, Hull and White made an empirical study of the Black and Scholes model using random (stochastic) volatility instead of assuming constant volatility This is a wide spread adaptation of the model today, but was new when Hull and

White made their study Their result showed that the theoretical prices of options in the money are underpriced and options out of the money are overpriced These results show that the overpricing increase with the remaining time of duration and also points out the more out of the money the higher the overpricing

A study was made on OMX index call options in 2000 by a Swedish researcher,

H Bystrom He showed that regardless of whether using a constant or a stochastic volatility the Black and Scholes model more accurately prices options at the money and in the money than options out of the money

Marc Chesney and Louis Scott (1989) use the modified Black-Scholes model and a random variance pricing model to study prices of European currency options traded in Geneva They use the modified Black-Scholes model and a random variance model to price calls and puts on the dollar/Swiss franc exchange rate and compare the prices to bid-ask quotes for options traded in Switzerland Data consists of prices on the foreign currency options traded by Credit Suisse First Boston Futures Trading in Geneva They find that the two models produce different theoretical prices and that the Black- Scholes models prices are closer to the bid-ask quote observed in the market

) ( )

( )

,

( S t S e ( )N d1 Xe ( )N d2

C = t −r f T−t − −r d T−t

(1.7) Where:

t T

t T r

r X

S

−

− +

− +

=

σ

2 / 1 (

) /

t T

d

Where N(x) is the distribution function for standard normal random variable T represents maturity and X is the strike price

After calculating the call options, they use put-call parity theorem to obtain the price for European puts:

(1.10)

t t T r t

T r

S e

X e

t S

Black-Corrado C.J and Su T (1998) found that the Black-Scholes model price accurately the at- the- money options but it often misprices deep in- the -money and deep out- of- the money options This due to the model assumption that security log prices follow a constant variance diffusion process

2.4 Testing the market efficiency of Kuldeep Shastri and Kishore Tandon:

Kuldeep Shastri and Kishore Tandon (1986) test the efficiency of the market for foreign currency options using the option pricing model developed by Biger and Hull The test are based on data for four currency options: the British Pound, the German Mark, the Japanese Yen and the Swiss Franc

Kuldeep Shastri and Kishore Tandon examine the ability of hedging strategies to produce excess profits when an option’s market price deviates from its model price The test are in ex post and ex ante form The expost tests assume that the trading strategy can be executed immediately at the market prices that indicate deviations from the model, while in the ex ante tests, the strategy is executed at the price quoted the next day The results of these tests indicate that the ex post hedging strategy yields abnormal profits, but these excess returns disappear if the execution

of the strategy is delay by one day

The results suggest that during the period investigation, if the market participant can duplicate the ex post strategy, the foreign currency market on the Philadenphia

stock exchange inefficient However if they can only duplicate the ex ante strategy, the market would be efficient

Our thesis use the way Kishore Tandon and Kuldeep (1986) test the efficiency

of the market We apply the data of Vietnam market such as: SJC gold price, volatility, interest rate to calculate the option price and apply the gold option price quoted in TOCOM in Vietnam market to test the efficiency of Vietnam market If the market is efficient, trader could not earn abnormal profit when duplicate ex post

or ex ante strategy

2.5 Volatility

One parameter that can not be directly observed is the volatility of the currency price Volatility is an essential element determining the level of option prices, it is a measure of uncertainty about the returns provided by the stock or currency If volatility is high, the premium on the option will be relatively high, and vice versa The higher the volatility, the more likely it is that the underlying price will be above the exercise price before the maturity date and hence the option will then have a higher premium If the volatility of the underlying instrument is expected to

be low then there is less chance that the option will be profitable and will have a lower expected option value

The volatility ( ) can be estimated by using the implied standard deviation (ISD) from observed option prices or by using historical estimates of from changes in the foreign exchange rate (Marc Chesney and Louis Scott, 1989)

The historical volatility is estimated by historical standard deviation (Kuldeep Shastri and Kishore Tandon, 1985) To estimate the volatility of a stock price empirically, the stock price is usually observed at fixed intervals of time (e.g., every day, week, or month)

Define:

n+1 : number of observations

Si : Stock (currency) price at end of ith interval, with i= 0,1,…,n

: number of intervals per annum

And let

1

ln i i

i

S u

1 1

n i i

1

n i i

Where u is the mean of ui

The volatility estimated per annum is:

s

For example: if daily data is used the interval is one trading day and we use

= 252, if the interval is a week = 52 and = 12 for monthly data

Since the volatility of an asset changes over time the measurement of historical volatility is merely an estimate of the future volatility of the asset It is therefore hard to decide on how many historical days to base the calculations

Hull (2003) discusses this issue in his book “Options futures and other derivative s” and he suggests that a good rule of thumb is to set the number of observations to the same amount of days that the volatility is to be applied to In other words when setting the price of an option with 120 days left to expiration on

should base the historical volatility measurement on 120 days alike

The implied volatility is often used in practice These are the volatilities implied

by option prices observed in the market

However the price of deep in-the-money and deep out-of-the-money options are relatively insensitive to volatility Therefore, implied volatilities calculated from these options tend to be unreliable (John C Hull, 2000)

Chapter 3: RESEARCH METHODOLOGY

3.1 Data

3.1.1 SJC gold brand price:

This study uses data from SJC gold price quoted by Vietnam Eximbank

The Vietnam Export Import commercial joint stock bank (Eximbank) was established on May 24, 1989, being one of the first joint-stock commercial banks of Vietnam Trading gold since 2004, Eximbank is one of the first bank which trade gold in Vietnam

Providing 4 methods of gold trading: spot, forward, swap, options Eximbank, along with ACB, SJC and Sacombank, had the largest gold turnover and profit in Vietnam during 2006-2009

Due to the world gold price fluctuation, supply and demand of gold in the domestic market, the gold price quoted by Vietnam Eximbank changed accordingly

in order to keep the price competitively The SJC gold price could represent for the SJC gold price of the domestic market

The gold price quoted at Eximbank used to trade for all branch So, the price quoted have a spread between the buying and selling rate To estimate the volatility accurately, we use the mid point price of buying and selling price The mid point of SJC gold price from 1/1/2010 to 30/6/2010 is used to calculate the historical volatility

The total number of daily observations is 6846 gold price quoted during the time observed, then we to collect the gold price at 11AM, 14PM and daily closing price

at 17PM (Vietnam time)

3.1.2 Gold price and option price listed in TOCOM

The gold price and gold option price listed in TOCOM collected during 01/07/2010 to 15/08/2010

Gold options are listed in TOCOM since May 17, 2004 with 2 types of Trade: Call Options on Gold Futures and Put Options on Gold Futures

Contract Unit: 1 kg / contract

Minimum Price Fluctuation: JPY 1 per gram

Trading Hours : Day Session 9:00 a.m to 3:30 p.m., (JST)

Night Session 5:00 p.m to 11:00 p.m (JST) Trading Method: Computerized Individual Auction

Contract Months: All even months within 6 months (On the day when a New Contract Month is generated, there will be 3 even months starting from the next even month after the month which the said day belongs to)

Last Trading Day : Day session on the last business day of the month preceding

to the underlying futures delivery month

First Trading Day of New Contract Month: Day session on a business day following the First Trading Day of a New Contract Month of the underlying

Strike Price Interval: JPY 50 per gram

Listing of Strike Prices: At the commencement of trading, 5 strikes shall be listed New strikes will be added to maintain 5 strike prices above and below the strike price nearest to the previous day's settlement price of the underlying asset for the current contract month, until the fifth business day prior to the last trading day Exercise Period : The options buyers may exercise their options at any time from the First Trading Day of a New Contract Month to the Last Trading Day (American type)

Limitation on Exercise : No limitation The Exchange may require statement of reason when options are exercised out-of-the Money beyond a certain level

Circuit Breaker Trigger Level: The CB trigger level is to be set everyday at the start of a clearing period (i.e.the start of a night session at 17:00) and is based on the settlement price of the previous clearing periond (or the settlement price of the preceding contract month, in case of a new contract month)

Base Amount of Initial Clearing Margins (sellers only; per contract) 50% of the underlying

Extraordinary Clearing Margin: When the Exchange deems it necessary for market management purposes, Extraordinary Clearing Margins shall be imposed Customer Position Limit : 5,000 contracts (long/short call, long/short put each) Settlement Price: Theoretical price in principle

Assignment of Exercise : Assignment of option contracts starts after 15:45 p.m(JST)

Method of Assignment: TOCOM randomly assigns exercised option contracts to short positions (at the unit of sub account) Within a broker member, allocation will

be made from the oldest outstanding positions with regard to their customer positions (affiliate Members and Member Customer's positions)

We use the gold option price quoted in the TOCOM to compare with the option price derived from the modified Black-Scholed model Although the option in TOCOM is American style In the literature on equity options, it is well known that the use of European model to price American calls represents a major measurement problem only if the dividends on the underlying security are not small In the case

of foreign currency call, the use of a European model would represent a major measurement problem only if the foreign interest rate is large in comparison to the domestic interest rate Kuldeep Shastri and Kishore Tandon show that the early exercise of call on foreign currency will not cause the mispricing of the modified

Black-Scholes model option prices from the market prices if the interest rate paid on the foreign currency option is not comparable The mispricing only occur in the case that the interest rate of foreign currency and domestic currency is comparable For puts, Geske and Johnson have shown that the different between European and American prices is extremely small when the underlying security pays dividend They generate European and American put price for a variety of cases and fine that the different is sufficiently small

In the case of option quoted in TOCOM, the interest rate of Yen and gold is relatively small during the tine of observation, it is 0.1% for Yen and 0% for gold Therefore, we can use the option price of TOCOM to compare with the option price derived by modified Black-Scholes model

Since the SJC gold brand price in Vietnam is not equal to the TOCOM price converted into VND/tael There is usually a gap of world gold price and Vietnam gold price in converted Therefore, we use the price of TOCOM at the same time of the SJC gold price instead of converting the SJC gold price into the gold price in yen/gram and then we collect the option price correspondent to gold price at 11AM, 14PM, 17PM everyday during 01/07/2010 to 15/08/2010

The data of gold option price listed in TOCOM are collect from the internet, website http://www.TOCOM.or.jp/ everyday during 01/07/2010 to 15/08/2010 for the option value on August, October and December

3.2 Modified Black Scholes Model

There is a big interest in option pricing on the market and through the years a number of competing and acknowledged option pricing models have been developed from science research Some of the more renowned options pricing models are Black and Scholes, the Binominal Option Pricing Model more renowned options pricing models are Black and Scholes, the Binominal Option pricing Model used and widely known option pricing model in the financial markets

In this study, we use modified Black – Scholes model developed by John Hull to value call and put options on SJC gold brand:

(2.1)

) ( )

*

d XN e d SN e

c = −r T − −rT

(2.2) )

( )

( d2 Se * N d1N

Xe

p = −rT − − −r T − Where:

(2.3)

T

T r

r X S d

σ

σ / 2 )]

( [

) /

1

+

− +

(2.4)

T d

d2 = 1− σ

3.2.1 Spot rate and exercise price

The option is at-the-money We calculate the option price for the call and put option expired on August, October, December 2010 The spot rate used to calculate the option is spot rate at 11AM, 14PM and 17PM 01/07/2010 to 15/8/2010

3.2.2 Interest rate of VND and Interest rate SJC gold brand

The interest rate of VND collected to value the option is the risk free interest rate of the SBV : 8%

We use interest rate of gold is : 1% (which is the mid point of deposit and loan interest rate of Eximbank at the time of investigation)

3.2.3 Time to expiration

In TOCOM, call and put option for gold is traded in even month, and for 6 next month So to test the diferrence between the option price calculated from the modified Balck-Scholes model, we calculate the call and put option from 1/7/2010

to 15/8/2010 for the value same as the value of gold option traded in TOCOM: August, October, December, the last trading day of these options is on day 30 of each even month

3.2.4 Volatility

We use a daily estimate derived from a sample of daily gold spot price from 1/1/2010 to 30/6/2010 We use the measure of volatility based on spot gold price even though a prefered measure should be the average volatility of forward rates, the average being taken over the maturity of the forward/option contract, since that include information regarding interest rate volatility The measure of volatility on spot price would differ from the volatility of the forward rate only in those periods

in which the foreign/domestic interest rate differential changes substantially In our period of study, the differential is fairly stable for gold and VND, therefore the use

of spot rate volatility do not cause many difference

Although the Black - Scholes Merton assumption of the constant, it is highly possible that the variance is constant over a short time interval but it is time varying over a long time horizon In such a case, the BS is valid model over each of short time intervals This paper assumes that the variance of currency may be constant within a short time interval (one day) but changing from one interval to another, that

is, variance changes on the daily basis but constant within the day

In Vietnam, the derivatives in gold, foreign exchange, stock have not developed The gold option is not traded regularly, therefore the calculation of volatility in method implied volatility can not be carried out in practice In this study, we estimate the volatility based on the historical SJC gold price

To be able to make all the necessary calculations we used MS Excel as a tool for analyzing the data

When calculating historical volatility one must solve for the optimal number of days or observations, n, to base the volatility calculation on Hull (2003) suggests that a good rule of thumb is to set n equal to number of days to which the volatility is to be applied, i.e when valuing an option with 90 days to expiry one should calculate the historical volatility on 90 days backwards We have however added a constraint in that we never calculate the historical volatility on less

than 90 days backwards Our historical volatility is in other words calculated on the same number of days left until expiration unless that value is less than 90 days in which case 90 days is used

We collect daily closing SJC gold price from 01/01/2010 to 30/6/2010 to estimate volatility, since more data generally lead to more accuracy, but volatility change over time and data that are too old may not suitable for predicting the future volatility (John C Hull) and the often used of number of observation is to set equal

to the number of days to which the volatility is to be applied In Vietnam gold market, the investor often have the speculation for the longest interval is one year, usually, they trade gold for 1-6 month

3.3 The ex post and ex ante hedging test

According to Shastri,K and K.Tandon (1986), to test market efficiency, one must investigate whether excess (abnormal) profit opportunities exist in the market

If these opportunities exist and persist over time, it can then be argued that the market is inefficient for the period investigated If profitable opportunities do not appear, then the market is efficient with respect to the trading rule used By definition, an efficient market means that there is no arbitrage profit exists in the market

In this section, the main methodology to be applied to test the market efficiency

is hedging strategy, the modified Black-Scholes model developed by John Hull (presented in Chapter 2) is selected to generate the efficient option prices

We consider two forms of the hedging strategy The first strategy called the ex post strategy, we assume that the strategy can be executed at the market prices that indicate deviations from the corresponding model prices The second form, called the ex ante strategy, assumes that an observed deviation serves only as a signal that triggers a transaction at the next price

It has been argued in the literature that the first test can only indicate the existence of comtemporaneous deviations from equilibrium, while the second test is

a “true” test of market efficiency (K.Shastri and K.Tandon, 1986)

By comparing the model generated option price and the market option price, transaction signals are given to the simulated trader When market price is higher (lower) than the model price, a portfolio composed of written call (put) options and units of bought (borrowing) foreign bonds is established

The domestic currency value at time t of this portfolio is

(3.2) (3.3)

(3.1)

rT T

r t

C Value = + α −* + β −

) (

) (2

1

d XN

d N

=

−

= β α

It can be shown that a long position in a domestic bond is equivalent to a long position in (- / ) foreign bonds and a (1/ ) written (short) position in a call option

If an option is overpriced, a portfolio composed of (1/ ) written calls and (- / ) bought foreign bonds would return a higher return than a long position in a domestic bond If an option is underpriced, a portfolio composed of (1/ ) bought calls and borrowing equivalent to (- / ) foreign bonds would yield a higher return than a short position in a domestic bond

The trading strategy is described as follow:

Step 1: At time t, the equation (2.1) is used to calculate the model price of the call option, Ctmodel

Step 2: if the call is overpriced (Ctmarket >Ctmodel), we form a portfolio composed

of (1/ ) written calls and (- / ) bought foreign bonds The cost of this portfolio is1

t T

(3.4) The portfolio is liquidated at the next trade The excess return on this portfolio is:

T r

T r t

T r

t t

e

Ve C

e S

− +

− +

1

β β

If the call is underpriced, then the portfolio is formed oppositely

The trading strategy for put is similar to those used aboved The strategy is as follows:

Step 1: At time t, the equation (2.2) is used to calculate the model price of the put option, Ptmodel

Step 2: if the put is overpriced (Ptmarket >Ptmodel), we form a portfolio composed

of (-1/ ) written puts and borrowing equivalent to (- / ) foreign bonds The cost of this portfolio is 1

t T

(3.6)

The portfolio is liquidated at the next trade to yield excess returns:

T r

T r t

T r

t t

e

Ve P

e S

− +

− +

1

β β

If the put is underpriced, we follow the opposite strategy

Positive profit (excess return >0) would indicate market inefficiency

1

For the ex ante test, the cost of the portfolio is given by:

11

Equation (3.5) and (3.7) are for the ex post version of the strategy The corresponding equations for the

ex ante strategy are obtained by substituting t+2 for t+1, and t+1 for t in equation (3.5) and (3.7)

Chapter 4: EMPIRICAL RESULTS AND DISCUSSION

4.1 A comparison of SJC gold option price and the option price quoted in TOCOM

We calculate the option price daily from 01/07/2010 to 15/8/2010 for the expiration on August, October, December 2010 Then we compare the option price calculated by the modified Black-Scholes model with the gold option price listed in the TOCOM We devide the call/put option price by the strike price and compare the difference between the two ratio

The call option price derived by modified Black-Scholes model increase when the time to maturity increase The call option of SJC gold derived by modified Black-Scholes model is higher than the call option traded in TOCOM for all call option matured on August, October and December

Table 4.1: A comparison of the difference between the call option price of SJC gold derived by Black Scholes model and call option price of TOCOM

Maturity

month

Average % model call option price / SJC price

Average % call option price / TOCOM price

Table 4.2: A comparison of the difference between the put option price of SJC gold derieved by Black Scholes model and put option price of TOCOM

Maturity

month

Average % model put option price / SJC price

Average % put option price / TOCOM price

Difference

Source: Author’s calculation from appendix 8

The result from table 4.2 indicate that, similar to the case of the call option price

of SJC gold derieved by Black Scholes model and call option price of TOCOM, the put option price derived by the Black – Scholes model is diferrent from the price of

TOCOM

4.2 The result of the Ex post tests

Table 4.3 and 4.4 report the result of the ex post test, the excess returns reported

in table 4.3 and 4.4 indicate that over one half of the portfolio have positive excess (abnormal) returns For all the three maturity month of option investigated, the hedging strategy earn excess return

Table 4.3: Excess return from ex post hedging strategy for calls

Maturity month Mean return t-Value % Positive

December 157.68 6.43 72%

Note: The mean return is VND/tael /day

% positive is the posibility the trader can gain abnormal profit

Soure: Author’s calculation from the appendix 9

Table 4.4: Excess return from ex post hedging strategy for puts

Maturity month Mean return t-Value % Positive

Note: The mean return is VND/tael /day

% positive is the posibility the trader can gain abnormal profit

Soure: Author’s calculation from the appendix 10

The above test indicate that the hedging strategy can yield abnormal profit, this support for the inefficiency of the market (this inline with K.Shastri and K.Tandon

in their test in which excess return more than 0 indicate the market inefficiency)

As mentioned before, the hedging strategy needs to be completed in three steps

at three consequence time: t, t+1, t+2 Shastri and Tandon use the daily closing price for test, this will mean that for the same option, the transaction must be available for three consecutive trading days This will require the simulation to exclude the options which can not be available for three consecutive days If the excluded samples are of significant number, the validity of the result is again questionable

Fortunately, A L Tucker’s article uses intra-daily price This means that the trader do not need to wait until next day for establish a hedge, and liquidate it on the subsequent day We can establish the hedge on 14PM and liquidate the portfolio at 17PM

4.3 The result of the Ex Ante Tests

In performing ex ante test, we assume that the trader will investigate the defference between the market and the model price at time t, then the portfolio will

be formed at time t+1 and liquidate at t+2 There is no guarantee that the prices at t+1 and t+2 still be favorable for the trader because of the market fluctuation

Positive profits in these ex ante test would indicate market inefficiency, while nonpositive profit would support for the market efficiency

Table 4.5: Excess return from ex ante hedging strategy for calls

Maturity month Mean return t-Value % Positive

Note: The mean return is VND/tael /day

% positive is the posibility the trader can gain abnormal profit

Soure: Author’s calculation from the appendix 11

Table 4.6: Excess return from ex ante hedging strategy for puts

Maturity month Mean return t-Value % Positive

Note: The mean return is VND/tael /day

% positive is the posibility the trader can gain abnormal profit

Soure: Author’s calculation from the appendix 12

The result at table 4.5 and table 4.6 indicate that the ex ante strategy which formed and liquilidated at 3 sections of same day (at 11AM, 14PM and 17PM) still gain abnormal profit for trader However, the percentage of positive return is less than that of ex post test For some overprice calls, this delay in execution actually results in a change in sign of excess return from positive to negative

The ex ante results for puts presented in table 4.7 indicate that the ex ante strategy can yield abnormal profit for all three maturity months, and the percentage

of positive excess return is lower than using the ex post test due to the delay in execution

The results from the test in line with K.Shastri and K.Tandon in their test in which excess return more than 0 indicate the market inefficiency

Therefore, this strategy has the high ability of rejecting market efficiency

To sum up, the result from three tests of the SJC gold option price derived by modified Black – Scholes and the price of TOCOM option indicate that the option price derived by modified Black – Scholes is not aproximate to the price of TOCOM option, there is a diferrence between the two option price for maturity month of August, October and December Due to this diferrences between the two option price, we carry out the ex post and ex ante test, the results indicate that trader can earn abnormal profit when using the heding strategy ex post and ex ante Therefore, we can not use the option price quoted by TOCOM to offer for the trader

in Vietnam when trading SJC gold Option This rejects our hypothesis that the TOCOM option price is same as the SJC gold price derived by modified Black- Scholes model Banks can not use the option price of the gold traded in TOCOM to offer for their customer because the trader can earn abnormal profit when execute the ex post and ex ante strategy It is suggested that the modified Black Scholes

model be used in pricing SJC gold price and then the Banks could use delta hedging

to hedge the risk when writing options

Limitation: there is no gold option trading exchange in Vietnam, therefore, we can not collect the implied volatility to calculate the option price Instead, we use the historical volatility (as mentioned in Chapter 3)

Chapter 5: CONCLUSION

In Vietnam, from 2010, trading gold on the overseas account is prohibitted, the hedging of option on TOCOM could not be carried out The ability of banks to hedge the risk of writin options by overseas account is limited, therefore application

of gold option price quoted by foreign counterparts or TOCOM/COMEX could be

no longer compatible Our objective is to investigate whether the modified Scholes model should be used in valuation of European SJC gold option price in Vietnam

Black-The study using two range of data One range is data collected in Vietnam market during 1/1/2010 to 15/8/2010, including SJC gold price, interest rate of VND and gold The other range of data is collected from TOCOM: gold price and gold option price quoted from 1/7/2010 to 15/8/2010 for option maturity on August, October and December

We apply the modified Black Scholes model (developed by Hull and White) to valuate the SJC gold brand option price and use this price to compare with gold option price quoted in TOCOM We find that the TOCOM gold option price different from the option price derived by modified Black-Scholes model

To test the efficiency of Vietnam gold market, we duplicate K.Shastri and K.Tandon (1986) in carrying out the ex post and ex ante test We apply the gold option price quoted in TOCOM into Vietnam gold market and do ex post and ex ante hedging strategy The results of the tests indicate that a trader in the market could make abnormal profit if he execute a hedging strategy at the market price Consequently, the application of gold option price quoted in TOCOM into Vietnam gold market is not compatible due to the existence of trader’s abnormal profit It is suggested that banks use modified Black-Scholes model to value at the money option for SJC gold brand with the European style and hedge the risk using