Experiences with Nikkei put warrants

Market efficiency: Experiences with Nikkei put warrants

Wei, Jason Z

In the past several years, many exchanges in North America and Europe have introduced options on foreign stock indexes. The most popular are Nikkei put warrants (hereafter, NPWs). In Canada, six NPWs have been listed on the Toronto Stock Exchange (see the Appendix for details). The mushrooming of NPWs around the turn of the decade was mainly due to the belief that the Japanese stocks were seriously overvalued. That belief was subsequently confirmed by the movement of the Nikkei index: Over the last three years, it has dropped by about 50%. As a result, most of the NPWs have been exercised. Although the creation of NPWs was mainly motivated by an assessment of the market movement, many similar products have been introduced for other reasons.(1) One important rationale is the ability of investors to participate in the foreign stock market without incurring otherwise prohibitive transaction costs. In fact, some of the foreign index warrants allow investors to participate in the foreign stock index movements without incurring exchange rate risk. Given the globalization of the world financial markets, it is logical to foresee more development and creation of cross-currency instruments such as NPWs.

Despite the popularity of these innovative products, the literature has been lacking in formal treatments of the new instruments. Although some authors have examined the theoretical pricing of cross-currency products, very few have examined the empirical evidence in this market.(2) There are two exceptions. Chen, Sears, and Shahrokhi (1992) studied the price behaviour of NPWs traded on the American Stock Exchange. Wei (1992a) empirically tested pricing models, using NPWs traded on the Toronto Stock Exchange. The question of market efficiency of the NPW market is largely unanswered.

The purpose of this paper is to investigate efficiency of the NPW market. Specifically, NPWs traded on the Toronto Stock Exchange were used to evaluate two trading strategies designed to make arbitrage profits. In the remainder of the paper, we will present the NPWs and their pricing models, describe the data and testing methodologies, and report empirical results.

NIKKEI PUT WARRANTS AND THEIR VALUATION

An NPW is an American put option written on the Nikkei 225 index. Although the index the underlying asset is denominated in Japanese yen, the warrants are traded in dollars. Therefore, it matters as to how the Japanese yen payoff is converted into dollars. Existing NPWs can be classified into three different categories, according to conversion scheme. We will use the following notation.

S = Nikkei 225 index level in yen;

X = current Cdn$/JapY exchange rate;

X sub 0 = pre-specified, fixed exchange rate;

K = exercise price in yen;

r = Canadian risk-free interest rate;

r sub f = Japanese risk-free interest rate;

sigma sub s , = volatility of the Nikkei index;

sigma sub x , = volatility of the exchange rate;

rho = correlation coefficient between the index and the exchange rate;

q = continuous dividend yield on the Nikkei index;

t = current time;

T = maturity date.

In terms of payoffs, the three categories can be characterised as follows.

Category–Payoff upon exercise in dollars

Category I NPWs–Max [0, X sub T (K – S sub T )]

Category II NPWs–Max [0, X sub 0 (K – S sub T )]

Category III NPWs–Max [0, X sub 0 K – X sub T S sub T ]

It is seen that the exchange rate enters the payoff function in different ways. For category I NPWs, the exchange rate simply serves as a currency conversion factor. Therefore, holding a category I NPW is exactly like holding an ordinary Japanese index put option. This is because the spot exchange rate at the time of exercise is always used to convert the yen payoff. It is, however, not the case with a category III NPW. Here, the exercise price is converted upfront, at exchange rate X sub 0 , which is constant throughout the life of the warrant, but the Nikkei index is converted at the spot exchange rate at the time of exercise. Finally, for a category II NPW, the yen payoff is converted at a pre-fixed exchange rate X sub 0 (which is also constant). Therefore, a category II NPW essentially allows a Canadian investor to participate in the Nikkei index movements without incurring exchange rate risk. It should be pointed out that all NPWs are American, in that they can be exercised any time before maturity. Moreover, all NPWs were issued with a three-year time to maturity.

Dravid, Richardson, and Sun (1991), Reiner (1992), and Wei (1992c) showed that NPWs could be priced using a one state variable numerical procedure, despite the fact that there are two underlying variables (i.e., the index price and the exchange rate). Specifically, Wei (1992c) demonstrated that a binomial (Cox, Ross, & Rubinstein, 1979) or trinomial (Boyle, 1988) lattice framework could be used to price NPWs. In this framework, one builds the lattice in exactly the same way as in the case for ordinary options. The only difference is that the model inputs take different forms. We summarize the model inputs as shown in the box below.

For example, to price a category III NPW, we can build a binomial tree for a (fictitious) state variable whose current value is SX. The drift rate and variance for the variable are r and sigma sub s sup 2 + 2 rho sigma sub s sigma sub x + sigma sub x sup 2 respectively. The dividend yield is q. Once we get the tree, we work backwards and price an American put option on this variable with exercise price KX sub 0 . The sub-period discounting in the backward deduction is done using risk-free rate r. The final price so obtained is the value of a category III NPW.(3)

There is a unique problem in the exercise settlement of an NPW. Due to the time zone difference, a concurrent quote for the Nikkei index is not available when the NPWs are traded on the Toronto Stock Exchange. To overcome this problem, issuers of Nikkei put warrants, upon receiving an exercise notice, would wait for one or two business days and use the subsequent Nikkei index quote to settle the exercise. Among the six warrants, only Trilon Financial Corporation NPWs have a two business day delay. The exercise delay is one day for other warrants. It is obvious that investors incur exercise risk due to this settlement delay. To mitigate this risk, issuers grant a free “limit option,” which protects investors from adverse index movements during the waiting period. All NPWs have an index movement limit of 500 points. Specifically, at the time of exercise, if a warrant holder chooses this limit option,(4) and if the index moves up by more than 500 points during the waiting period, then the warrant holder can (and must) get the warrant back. Otherwise, it is regularly settled. Wei (1992c) showed that the effects of these two twists on the warrant prices were negligible. Therefore, we will ignore these two twists when calculating the model prices.

DATA AND METHODOLOGY

DATA

This study used the same data set as in Wei (1992a). The daily warrant prices (high, low, and close) were obtained from RBC Dominion Securities.(5) The Nikkei index daily close data and the exchange rate data were retrieved from the Nikkei Telecom–Japan News & Retrieval. For the risk-free interest rates, ideally we should have used long-term discount bond yields. However, they are not available in Canada and Japan. As a proxy, we used yields on government coupon bonds which had similar maturities to the NPWs. Japanese bond prices were obtained from the Nikkei Telecom–Japan News & Retrieval. Canadian government bond prices were obtained from The Financial Post. Yields were calculated from bond prices and then converted to continuous compounding rates. The calculated Canadian (Japanese) bond yield was then be used as a proxy for Canadian (Japanese) risk-free interest rate. Finally, monthly dividend yields, starting in January 1980 and ending in December 1990, were supplied by Nomura Canada. A simple moving-average forecast of 0.43% p.a. was used as the constant dividend yield during the testing period.

METHODOLOGIES

Many trading strategies can be used to explore arbitrage profits. We identified two strategies: trading based on boundary condition violations, and forming hedging portfolios based on mispricing. The tests associated with the two strategies will be referred to as the boundary condition test and the hedging test, respectively.

BOUNDARY CONDITION TESTS

The boundary condition test is independent of distributional assumptions and is a pure test of market efficiency, because no specific pricing models are used. To examine the NPW market efficiency, we tested the following simple boundary conditions for each warrant:

Category I NPW: P sub a (I) >= (K – S) X

Category II NPW: P sub a (II) >= (K – S) X sub 0

Category III NPW: P sub a (III) >= (KX sub 0 – SX),

where P sub a (i) (i = I, II, III) is the ask price for the NPWs in each category and other notation is defined as before.(6) Note that e input for the index S is the closing quote from the previous trading session on the Tokyo Stock Exchange. We call the terms on the right hand side of the above inequalities the “intrinsic values” of the NPWs.(7) The specific testing procedures for each NPW were as follows.

a) On each trading day, the “intrinsic value” was calculated and subtracted from the ask price P sub a (i). Let epsilon sub t (i) (i = I, II, III) denote the resultant deviation. We then obtained a time series of epsilon sub t (i) (i = I, II, III).

b) The negative deviations were identified: epsilon sub t , (i) (i = I, II, III).

c) For those trading days on which the deviations were negative, the following trading strategy was employed: Buy the warrant on that day (based on the ask price) and deliver an exercise notice immediately. The exercise will be settled on the next business day (for Trilon Financial NPWs, the exercise will be settled on the business day after the next business day). EXP (i) was used to denote the actual exercise value of the warrant, which could be zero if the limit option was not chosen.

d) The average of EXP(i) – P sub a (i), denoted by delta (i) (i = I, II, II), was calculated. (We called delta (i) the average arbitrage profit.)

e) Finally, the null hypothesis, H sub 0 : delta (i) = 0, was tested.

The above strategy is based on the so-called “options rational pricing,” namely, that if an NPW is rationally priced, its current price should be at least as high as its intrinsic value, the value of the warrant exercised immediately. Otherwise, arbitrage opportunities exist and the market is not efficient. Specifically, if the arbitrage profit delta (i) is not significantly different from zero, then the market is believed to be efficient; if, on the other hand, the arbitrage profit is positive, we still cannot claim that the market is inefficient, because we have omitted the costs of executing the trades. To accommodate transaction costs, we repeated the above testing procedures with the amount EXP (i) – P sub a (i) in step d) being calculated as EXP (i) – P sub a (i) – c. “c” is the commission fee paid on each warrant for a buy or sell transaction.(8) Finally, to see whether the limit option made a significant difference, we did the above tests for two cases: one with the limit option and one without. (“With limit option” means the limit option is chosen when delivering the exercise notice.)

HEDGING TESTS

Our hedging tests were based on the classical neutral hedge argument of Black and Scholes (1973). Namely, if the model is correct, then a hedge portfolio consisting of an option and a certain number of shares should be riskless over a short time interval, and should earn the risk-free interest rate.(9) A violation of this condition implies market inefficiency. Since we need model prices to carry out the hedging tests, we have to identify models that can price American put options. In this study, we used the numerical framework summarized in the previous section. Specifically, a trinomial lattice(Boyle, 1988) with 100 steps was used. The next problem is to estimate the unobservable parameters sigma sub s , sigma sub x , and rho. We used two sets of estimates: historical and implied. For each trading day, historical estimates for sigma sub s , sigma sub x , and rho were calculated, using the most recent 250 observations for the index and the exchange rate. Implied estimates were calculated differently for each category of warrants. For category I, sigma sub s , was imputed. For category II, a, was imputed by inputting historical sigma sub x and rho.(10) For category III, sigma sub s sup 2 + 2 rho sigma sub s sigma sub x + sigma sub x sup 2 was imputed as a single parameter. We took two measures to avoid spurious model prices. First, for each day, we calculated a weighted average of the implied sigma sub s (from categories I and II), with the weight being the degree to which the warrant is in-the-money.(11) We then used this aggregate estimate as the index volatility input for categories I and II NPWs. Second, for all NPWs, we used yesterday’s implied estimates as today’s model inputs.

The potential profits in the boundary condition tests are not risk-free. As a result, it is hard to argue against market efficiency, even if positive profits are actually found. A hedging test will overcome this difficulty. If persistent abnormal profits can be earned using a trading strategy based on riskless hedging, then the market is deemed not to be efficient. If, however, the profitable opportunities do not materialize, then the market is efficient at least with respect to the trading strategy being used.

Generally, there are two alternatives for forming a riskless hedge. The first involves taking positions in an option and the underlying security. For example, in a Black-Scholes world, one long call and Delta units of the underlying security short compose a riskless hedge. Over a very short period of time, the hedge portfolio should earn the risk-free interest rate. The second alternative involves taking positions in two or more options written on the same underlying security. Strike price spread or time-to-maturity spread are examples of this type of hedge.

From the pricing results reviewed in the previous section, we know that in order to hedge categories I and III NPWs, we must take positions in both the index and some currency instruments, since the exchange rate directly enters the pricing models. Indeed, it is also true for a category II NPW. This may sound peculiar, as the value of a category II NPW is clearly independent of the exchange rate X. The key lies in the underlying variable S, the Nikkei index level denominated in yen. As dictated by Black-Scholes risk neutral hedge argument, we could hedge a category II NPW by taking a position in the index. However, once we take a position in the index, we are exposed to the exchange rate risk, which in turn needs to be hedged.

Two factors prevent us from taking the first hedging alternative: the lack of data and the complexity involved in carrying out the hedge. Regarding the latter, suppose we want to form a hedge portfolio with a category I NPW. Then we need to take positions in three instruments: the warrant, the Nikkei index futures, and the Japanese yen futures. Neither of the last two instruments is traded in Canada. To get around all these difficulties we developed hedging schemes based on the second alternative.

To develop the idea better, let’s first examine a simple case. Suppose we have two European put options, A and B, written on there stock (that pays a continuous dividend yield q) but with differing time-to-maturities and/or exercise prices. We use Delta sub A and Delta sub B to denote the deltas of the two puts. By riskless hedge arguments, one put A plus -Delta sub A units of the stock make up a riskless hedge, and so will one put B plus -Delta sub B units of the stock. To put it another way, one stock and -1/Delta sub A units of put A comprise a riskless hedge, and so will one stock and -1/Delta sub B units of put B. Since the one unit of stock is common to both hedges, we can infer that 1/Delta sub A units of put A and -1/Delta sub B units of put B will also comprise a riskless hedge. In other words, one unit of put A and -Delta sub A /Delta sub B , units of put B will make up a riskless hedge. (We will call Delta sub A /Delta sub B the hedge ratio.) Therefore, we have achieved a riskless hedge with two puts, without taking positions in the underlying stock.

By generalizing the above arguments to category II NPWs, we can see that any two NPWs can be combined in proper proportions to form a riskless hedge portfolio. The only complication is that the deltas of the warrants must be calculated numerically.(12) Once the deltas and the hedge ratio are calculated, we can take advantage of relative mispricing among warrants simply by buying the relatively underpriced warrant and selling the relatively overpriced warrant. As Galai (1977) has pointed out, two options could be correctly priced relative to each other, yet both prices be nonequilibrium ones. But if two options are not even correctly priced relative to each other, then at least one option is not correctly priced.

Following Galai (1977), we will detect the relative mispricing between warrants A and B by comparing the ratio of model prices P sup mod sub At / P sup mod sub Bt with the ratio of market prices P sup mod sub At /P sup mod sub Bt . The decision rule is: Buy warrant A and sell warrant B if the model price ratio is greater than the market price ratio (and se warrant A and buy warrant B if the relationship between the ratios is the reverse). Since we have three category II NPWs, we can form three different pairs. Specifically, we execute the following so-called ex post trading strategy.

a) For each of the three pairs, calculate and compare the model price ratio and the market price ratio. Long the underpriced warrant and short the overpriced warrant based on the market prices on day t. (For all calculations we use the midpoint average of the bid and ask prices as the market price.)

b) For each pair, liquidate the portfolio based on the market prices on day t + 1.

c) Repeat a) and b) for the sample period. We get three time series of trading profits or daily returns. For example, the daily return for a pair consisting of warrants A and B on day t’s trading (if A is believed to be underpriced) is given by

(Equation 1 omitted)

where P sup mkt sub At , P sup mkt sub A(t + 1) , P sup mkt sub B(t + 1) are market prices of warrants A and B on day t and day (t + 1), respectively. Delta sub At and Delta sub Bt are the deltas of warrants A and B calculated based on day t’s model inputs.

d) Calculate average daily returns (in dollars) for each pair. If the market is efficient, the average daily returns should be close to zero.(13)

The above trading strategy was executed for two scenarios, one based on implied volatilities (when calculating the model prices and deltas) and another based on historical volatilities.

EMPIRICAL RESULTS AND MARKET EFFICIENCY

BOUNDARY CONDITION TESTS AND MARKET EFFICIENCY

The results of the boundary condition tests are reported in Table 1. (Table 1 omitted) There are several observations. First, except for Bankers Trust Bank of Canada (BT B of C) Series IV NPWs, there are a number of boundary condition violations for each warrant. The average violations range from $0.203 to $0.394 per warrant and are all statistically significant, as reflected in the first column of the table. (“Same day profit” denotes the difference between the ask price and the “intrinsic value.”) Second, when the limit option is not chosen, the trading strategies generate positive profits (except in Panel D), with or without transaction costs. However, only three warrants (Panels B, C, and E) exhibit persistent positive profits (in b the sense that the t-vales are significant). Third, when the limit option is chosen, the trading profits are significantly reduced. Only BT B of C Series III NPW (Panel C) is a constant winner. The fact that the limit option reduces trading profits is apparently counterintuitive, since the limit option is a “protector” against adverse index movements. However, we should realize that it is an option with “obligations.” This is so because whenever the index moves up by more than 500 points, you must get the warrant back. It is possible that the warrant is so deeply mispriced that even a 500-point index movement could not completely correct the mispricing, however. In this case, it is still desirable to exercise the warrant. A further examination of the data confirms this point. Around the days of “big” index movements, the warrants are most severely mispriced. This point is also reflected in panel C. The BT B of C Series III NPW can sustain profits even when the limit option is chosen, because this warrant has the most severe mispricing, as reflected by the high average trading profits. (The severe mispricing in turn is due to the relatively high exercise price. See the Appendix, for a comparison of exercise prices.) (Appendix omitted)

Note that the above tests are in a sense ex post tests, since the warrant ask price is not necessarily the one at which we could complete the transaction. (See Galai (1978) for a good discussion of the distinction between ex post and ex ante tests.) In light of this, we also carried out an ex ante test which was similar to the above, except that we evaluated the boundary conditions on day t, bought the warrant on day t + 1, and settled the exercise on day t + 2. The results are shown in Table 2. (Table 2 omitted) It can be seen that the average arbitrage profits uniformly declined when ex ante trading strategies were used. Specifically, when the limit option was not chosen, only Trilon Financial NPWs registered a statistically positive profit, which became insignificant when transaction costs were considered. For all other warrants, we obtained either negative or statistically insignificant positive profits: indicates that the market does correct itself over time terms of mispricing. Overall, the NPW market seems to be reasonably efficient.

HEDGING TESTS AND MARKET EFFICIENCY

Panel A, Table 3, contains the results for the hedging tests. (Table 3 omitted) It can be seen that the average profit per pair of warrants is positive for all cases, and the t-values are significant for almost all cases. The highest average per-pair (per trade) profit is 17c (achieved with the first pair, using historical volatilities). The highest per trade profit is $2.394, achieved with pair A, using implied volatilities.

Note that the above test is only an ex post one, since we assumed that we could buy and sell the warrants at the prices on day t. An ex ante (and more realistic) test can be carried out as follows: Calculate the price ratios and determine relative mispricing based on the prices of day t, execute the trade based on the prices of day t + 1, and liquidate the hedge based on the prices of day t + 2. In this case, the trading profit for a pair consisting of warrants A and B (if A is believed to be underpriced) is given by

(Equation 2 omitted)

where the variables are defined in the same way as in (1). Panel B of Table 3 contains the results of this ex ante test. The average profit per pair declines for all cases, although all averages are still positive. Only the second pair maintains significant t-values. The highest average profit per pair (per trade) is also associated with the second pair, which is about 9e.

Although positive profits seem to be available, based on the above tests, we still cannot draw a conclusion on market efficiency. An important factor that must be considered is the transaction cost. It might be the case that the trading profits are not sizable enough to offset necessary transaction costs. To determine the profitability of the hedging strategies, we repeated the above tests, both ex post and ex ante, by incorporating a one-way transaction cost of 2.5c per warrant. The testing results are reported in Table 4. (Table 4 omitted) It can be seen from Panel B that the potential profits are completely wiped out when an ex ante trading strategy is employed. (All averages are negative.) An ex post trading strategy (in Panel A) did manage to produce some positive averages, but the numbers are very small and their t-values are not significant.(14) This leads us to conclude that the NPW market is fairly efficient with respect to riskless hedging schemes.

CONCLUSIONS

Nikkei put warrants are recently created, innovative cross-currency products. They allow investors outside Japan to participate in the index movements without incurring otherwise prohibitive transaction costs. Certain types of Nikkei put warrants (e.g., category II NPWs) also allow investors outside Japan to participate in the index movements without incurring exchange rate risk. Given the globalization of the world financial markets, it is likely that more cross-currency instruments similar to Nikkei put warrants will prevail.

Although Nikkei put warrants have been available on the Toronto Stock Exchange for more than three years, no formal research has been done to examine the efficiency of this newly created market. This study is the first to fill this gap.

Two market efficiency tests were performed: a boundary condition test and a hedging test. In the first test, frequent boundary condition violations were detected. Simple, naive trading strategies designed to take advantage of the violations did generate significant positive profits. But after modifying the strategies in light of trading complications (such as exercise delay) and transaction costs, positive profits either disappeared or became insignificant. Similar findings were obtained with the hedging test. Although relative mispricing could be identified frequently (with model prices based on either historical volatilities or implied volatilities), consistent risk-free profits could not be made using hedging strategies after transaction costs. The empirical results as a whole indicate that the Nikkei put warrant market is relatively efficient. This is somewhat surprising, since newly created markets tend to be less efficient in their early stages.

REFERENCES

Bailey, W., & Ziemba, W.T. (1990). An introduction to Japanese stock index options. Working Paper, Johnson Graduate School of Management, Cornell University, Ithaca, NY.

Black, F. (1975). Fact and fantasy in the use of options. Financial Analyst Journal, 31, 36-72.

Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637-659.

Boyle, P. (1988). A lattice framework for option pricing with two state variables. Journal of Financial and Quantitative Analysis, 13, 1-12.

Chen, K.C., Sears, R.S., & Shahrokhi, M. (1992). Pricing Nikkei put warrants: Some empirical evidence. Journal of Financial Research, XV, 231-251.

Cox, J.C., Ross, S.A., & Rubinstein, M. (1979). Option pricing: A simplified approach. Journal of Financial Economics, 7, 229-264.

Derman, E., Karasinski, P., & Wecker, J.S. (1990). Understanding guaranteed exchange-rate contracts in foreign stock investments. International equity strategies. Unpublished manuscript. NY: Goldman Sachs.

Dravid, A., Richardson, M., & Sun, T. (1991). Options on two risky assets: Nikkei index warrants. Weiss Center for International Financial Research, The Wharton School, University of Pennsylvania, Philadelphia, PA.

Galai, D. (1977). Tests of market efficiency of the Chicago Board Options Exchange. Journal of Business, 50, 167-197.

Galai, D. (1978). Empirical tests of boundary conditions for CBOE options. Journal of Financial Economics, 6, 187-211.

Galai, D. (1983). A survey of empirical tests of option-pricing models. In M. Brenner (Ed.), Option pricing theory and applications (pp. 45-80). Lexington, MA: D.C. Heath.

Gruca, E., & Ritchken, P. (1991). Exchange traded foreign warrants. Working Paper, Weatherhead School of Management, Case Western Reserve University, Cleveland, OH.

MacBeth, J.D., & Merville, L.J. (1979). An empirical examination of the Black-Scholes call option pricing model. Journal of Finance, 34, 1173-1186.

MacBeth, J.D., & Merville, L.J. (1980). Tests of Black-Scholes and Cox call option valuation models. Journal of Finance, 35, 285-303.

Nelson, D., & Ramaswamy, K. (1990). Simple binomial processes as diffusion approximations in financial models. Review of Financial Studies, 3, 393-430.

Reiner, E. (1992, March). Quanto mechanics. Risk, pp. 59-63.

Rumsey, J. (1991). Pricing cross-currency options. The Journal of Futures Markets, 11, 89-93.

Sterk, W.E. (1983). Comparative performance of the Black-Scholes and Roll-Geske-Whaley option pricing models. Journal of Financial and Quantitative Analysis, 18, 345-354.

Wei, J.Z. (1992a). Empirical tests of the pricing of Nikkei put warrants. Working Paper, College of Commerce, University of Saskatchewan, Saskatoon, SK.

Wei, J.Z. (1992b). Pricing forward contracts and options on foreign assets. Working Paper, College of Commerce, University of Saskatchewan, Saskatoon, SK.

Wei, J.Z. (1992c). Pricing Nikkei put warrants. Journal of Multinational Financial Management. 2, 45-75.

NOTES

1 For example, the American Stock Exchange has listed warrants on such foreign indexes as CAC40 and FT-SE 100.

2 A list of theoretical papers includes: Bailey and Ziemba (1990), Derman, Karasinski, and Wecker (1990), Dravid, Richardson, and Sun (1991), Gruca and Ritchken (1991), Reiner (1992), Rumsey (1991), Wei (1992b), and Wei (1992c).

3 When building a lattice (binomial or trinomial) to p options, we chose the size and probability of the jump step such that the first two moments of the discrete process converged to those of the original, continuous process. Our model, together with the parameters given above, satisfied the conditions required for the convergence. See Boyle (1988), Cox, Ross, and Rubinstein (1979), and Nelson and Ramaswamy (1990) for details.

4 A warrant holder can forego this limit option.

5 The AB Svensk Exp. Corp. warrant was excluded from the study because of its thin trading. See the Appendix for the sample period of each NPW.

6 We used the ask price to make the test a weak one. Also, the use of the ask price makes more Sense when we consider taking advantage of the boundary condition violations.

7 For NPWs there is no such a thing as intrinsic value, since the two markets (i.e., Toronto Stock Exchange and Tokyo Stock Exchange) are never open at the same time.

8 Conversations with traders reveal that c ranges from two to five cents. We chose a conservative transaction cost: five cents per warrant. As pointed out by an anonymous referee, the bid-ask spread is also a component of transactions costs. The use of the ask price (rather than the closing price) in executing the trading strategies implicitly accounts for the bid-ask component of transaction costs.

9 For a call option, the number of shares is given by “delta,” the partial derivative of the call option price with respect to the underlying stock’s price. In what follows, we use “Delta” to denote delta.

10 We feel that this is warranted, since sigma sub x and rho enter the model as a product, which is a relatively small number.

11 This is based on the empirical evidence that at the money options tend to give the least biased implied volatility estimate. See, for example, Black (1975), MacBeth and Merville (1979), MacBeth and Merville (1980), and Sterk (1983).

12 The same lattice used to calculate warrant prices (i.e., a trinominal lattice with 100 steps) was used to calculate warrant deltas. Specifically, we let the lattice start one step early (so we had a 101 step lattice) and used the index prices and the warrant prices on the three nodes of the second step to calculate the delta. We used this approximating procedure because an American put option’s delta could not be obtained analytically. However, simulation results (not reported here; but available from the author upon request) showed that the procedure had excellent convergence properties. In other words, the calculated approximate deltas are sufficiently close to the true deltas.

13 Theoretically we should have included interest expenses/incomes when calculating daily returns. Given that the daily interest rate is very small, we simply ignored the cash flows associated with borrowing/lending. For a given hedge portfolio, the total cash position can be positive or negative. Therefore the total effects of borrowing and lending over time tend to be self-cancelling. Also, the short holding period in the trading scheme is essentially dictated by the need to maintain a riskless portfolio. As correctly pointed out by a referee, theoretically the position could be held for a longer term, which requires proper re-balancing. The more frequent re-balancing will bring in higher transaction costs.

14 A five cent one-way transaction cost would reduce the potential trading profits to an even larger extent.

This paper is extracted from my doctoral dissertation, completed at the University of Toronto. I would like to thank the following individuals for their help and support: Guy Bellemare, Laurence Booth, John Hull, Paul Potvin, Marlene Puffer, and Alan White. I also thank two anonymous referees for useful suggestions.

Address all correspondence to Jason Z. Wei, Department of Finance and Management Science, College of Commerce, University of Saskatchewan, Saskatoon, SK, Canada, S7N 0W0.

Copyright Administrative Sciences Association of Canada Mar 1994

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