Arbitrage With Options

8/4/2019 Arbitrage With Options

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ARBITRAGE IN THE MARKETS

FOR CALL AND PUT OPTIONS

Dallas BrozikP f f Fi


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P f f Fi

ARBITRAGE IN THE MARKETS

FOR CALL AND PUT OPTIONS

ABSTRACT

Option pricing theory encompasses two distinct contracts, thecall option and the put option. Existing option pricing modelstreat the two contracts as opposites that use the same relevant

pricing factors in similar manners. These models also assumethat the option market is efficient and unbiased. This paperexamines the option market and finds it to be inefficient andbiased in the pricing of call and put options. This difference inpricing behavior gives rise to profitable arbitrage opportunitiesby combining call and put options with an underlying stock.

INTRODUCTION

Option pricing is one of the most researched areas of finance. Several different

option pricing models have been developed, each with its own strengths and weaknesses.

One common characteristic of these models is that call options and put options are treated

as opposites by the pricing model. While this might be intuitively appealing, there is no a

priori reason to believe that market participants price these contracts in an identical but


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showed evidence of systematic mispricing, symmetric pricing models would not be

possible.

LITERATURE REVIEW

The Black-Scholes [1973] option pricing model provided a basis for a better

understanding of the pricing of options and contingent claims securities. When applied to

stock options, the model is able to explain much about how option prices are set, but there

have been numerous studies that have found discrepancies between actual and predicted

prices. Many researchers have attempted to explain these anomalies by changing certain

characteristics of the model, like the interest rate specification process, and other

researchers have focused on market factors like dividend payments to the underlying stock

(for example, Black [1975], Black and Scholes [1973], Geske and Roll [1984], Gultekin,

Rogalski, and Tinic [1982], MacBeth and Merville [1989], and Rubinstein [1985]). It should

be noted that all of these researchers restricted their data to call options. Even the review

of various option pricing models conducted by Bakshi, Cao, and Chen [1997] used only call


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Resnick [1979], Stoll [1969]), it is accepted as true, and this acceptance results in the

assumed equal-but-opposite nature of call and put option prices.

While all this work is indeed impressive and indicative of a similarity between the

pricing of calls and puts, it has not yet been proven that the pricing models for these two

different contracts are the same. The lack of testing of option pricing models using data

from put options is no reason for the assumption of symmetry. Theoretical constructs

notwithstanding, it is possible that call and put options are priced differently in their relevant

markets.

METHODOLOGYAND DATA

One of the difficulties in comparing call and put options is identifying comparable

data points. Existing option pricing theory implies that there are several factors involved

in setting the price of the contract such as time to maturity and the volatility of the

underlying stock, and it can be hard to find data points that can be compared directly

without having to make adjustments for specific contractual differences. There is, however,


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would have the same relevant volatility, time to maturity, and implied cost of capital. In a

perfect market with symmetric pricing, all call/put combinations with matching maturities

would have identical prices. In a less than perfect market with symmetric pricing, there

could be differences in the call and put prices, but the differences should be randomly

distributed, and there should be a similar number of occasions when the call price exceeds

the put price as when the put price exceeds the call price. The less efficient the market,

the noisier the market and the more price mismatches, but in a market with symmetric

prices the price spread distribution would be balanced.

The comparison of the prices of the call and put options in each observed price pair

provides evidence of the symmetry or asymmetry of the call option and put option pricing

processes. If the pricing processes are really symmetric, the prices of the call and put

options will be the same, though some noise could occur due to market inefficiencies. This

comparison also provides a direct test of the put/call parity theory. One specification of the

theory is:


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put options. The question examined is one of market efficiency in pricing and does not

provide evidence for or against any particular pricing model.

The data used to examine the pricing of call and put options is from 2006. All

options listed on the CBOE and traded during 2006 were used in this study. On those

dates when the stock closed at an option strike price, the closing price of the call and put

options with strike prices the same as the stock closing price were collected whenever

there was a maturity-matched call/put pair. For example, if stock XYZ closed at 50 and

there were both a call and put option with a strike price of 50 for a given maturity, the two

option closing prices would comprise one observation. Since it is possible to have different

maturity dates with the same strike price, it was possible to get more than one observation

on each date.

Not all contracts trade on all days, even if they are offered. On such days, contracts

may have a bid and ask price, but the lack of market activity does not validate this price.

Similarly, if only a few contracts are traded, the prices recorded for those contracts might

reflect mispricing in a thin market. One further filter was placed on the data set in order to


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Price Relationship Number of Occurrences Percent of Sample

Call Price > Put Price 1079 68.29%

Call Price = Put Price 208 13.16%

Call Price < Put Price 293 18.55%

Totals 1580 100.00%

In 68% of the observations, the call price exceeded the put price while the put price

exceeded the call price about 19% of the time. The relative number of mispricings

indicates that the market is not symmetric. If the price differences are small, they could be

considered noise in the trading process, but if the differences are large, they would be

a sign of inefficiency. Put/call parity could be supported if the call prices were either zero

or greater, but the frequency of occurrences when the put price exceed the call price

cannot be dismissed simply as noise. Put/call parity does not appear to hold, but efficiency

or inefficiency is not yet established.

Exhibit 1 examines the pricing structure of call and put options. The number of


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that on average a call option is worth roughly a quarter of a cent per day to maturity more

than a put option. Recall that this observation does not test any underlying option pricing

theory, but the nature of option prices that actually occur in the market does not support

the concept of put/call parity.

The existence of systematic mispricing between the markets for call and put options

creates the possibility of arbitrage opportunities if the mispricing events are significant. If

the differences are not significant, the markets could still be regarded as efficient if not

symmetrical. There are two possible ways to exploit this mispricing. If the call is priced

higher than the put, a hedged position can be formed by buying the stock, selling a call and

buying a put. The profit/loss diagram for this hedge is shown in Exhibit 3. If the put is

priced higher than the call, a hedge can be formed by short-selling the stock, buying a call

and selling a put; this profit/los diagram is shown in Exhibit 4.

The question of whether or not arbitrage is profitable depends on the spread

between the call and put option prices, transactions costs, and the time value of money.

If a stock is purchased, the time value of the investment should be considered. It is


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commission schedules indicate that the appropriate stock and option trades for 1,000

shares could be done for around $0.03 per share. Exhibit 5 shows the number and

proportion of profitable arbitrage opportunities that existed when the price of the call option

was greater than the price of the put option. At a 6% cost of capital, there were 8 profitable

arbitrage opportunities at a trading cost of $0.40 per share, but at a trading cost of $0.10

per share there were 85 profitable trades. Even at a cost of capital of 12%, there are 26

profitable trades at a trading cost of $0.10 per share.

Exhibit 6 reports the number and proportion of profitable trades when put prices

were greater than call prices, and the pattern is similar. What is interesting to note here

is that though there were fewer arbitrage opportunities, there were usually as many or more

profitable trades and a higher proportion of profitable trades than for instances when call

prices exceeded put prices. This is in some degree caused by the fact that the short sale

of stock only requires a 50% margin, and so the amount of borrowed cash was less.

The number of profitable trades reported in Exhibits 5 and 6 were actually a very

thin slice of the arbitrage available. These trades were restricted to situations with


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possible. The number and magnitude of arbitrage opportunities cannot be regarded as

random noise in the data.

CONCLUSIONS

The development of any security pricing model often assumes that the market from

which pricing data is obtained is efficient. If the market is not efficient or shows evidence

of bias, any model which does not make allowance for this may be of limited value. It

makes no difference how theoretically elegant a pricing model may be if it does not

accurately reflect the market conditions.

Existing option pricing models implicitly assume that calls and puts are opposite

contracts and that the only difference between them is that one is a call and the other a

put. Pricing models derived for call options are modified with properly placed negative

signs and presented as pricing models for puts. This approach can only be valid if the

option market is efficient and unbiased.

This paper presents evidence that the market for call and put options is not efficient.


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these different markets. Further research is necessary to identify option pricing models

which take into account such differences in behavior.

REFERENCES

Bakshi, R., C. Cao, and Z. Chen. 1997. Empirical Performance of Alternative OptionPricing Models, The Journal of Finance (December), 2003-2049.

Black, F. 1975. "Fact and Fantasy in the Use of Options," Financial Analysts Journal(July/August), 36-41+.

Black, F., and M. Scholes. 1973. "The Pricing of Options and Corporate Liabilities," Journalof Political Economy(May/June), 637-54.

Brenner, M. and D. Galai. 1986. Implied Interest Rates. Journal of Business 59: 493-507.

Frankfurter, G.M. and W.K. Leung. 1991. Further Analysis of the Put-Call Parity ImpliedRisk-Free Interest Rate. The Journal of Financial Research 14 (Fall): 217-232.

Geske, R., and R. Roll. 1984. On Valuing American Call Options with the Black-ScholesEuropean Formula, The Journal of Finance (June), 443-455.

Gultekin, N., R. Rogalski, and S. Tinic. 1982. "Option Pricing Model Estimates: SomeEmpirical Results," Financial Management(Spring), 58-69.

Klemkosky, R.C. and B.G. Resnick. 1979. Put-Call Parity and Market Efficiency. The


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EXHIBIT 1


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EXHIBIT 2

Call/Put Option Price Spread


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EXHIBIT 3

Arbitrage When (Call Price) > (Put Price)


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EXHIBIT 5

Call Price > Put Price

Number of Profitable Arbitrage Opportunities at Various Costs of Capital and Price Spreads

Price Spread Cost of Capital

(Call - Put) 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12%0.00 1053 981 841 639 446 326 250 204 169 147 130 119

0.03 999 860 685 493 318 225 174 139 120 105 97 85

0.05 846 744 551 389 234 164 127 96 85 74 67 60

0.10 664 532 376 217 124 85 58 53 42 35 29 26

0.15 526 394 265 151 78 51 39 30 24 23 21 18

0.20 429 313 198 94 50 37 27 23 18 16 14 12

0.25 349 240 133 65 40 21 18 13 12 11 8 7

0.30 284 195 104 52 25 16 9 8 8 8 7 6

0.35 238 154 76 33 15 9 8 8 7 7 6 6

0.40 200 127 64 22 11 8 7 7 7 6 6 6

Relative Proportion of Profitable Arbitrage Opportunities at Various Costs of Capital and Price Spreads


(Call - Put) 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12%

0.00 0.9759 0.9092 0.7794 0.5922 0.4133 0.3021 0.2317 0.1891 0.1566 0.1362 0.1205 0.1103

0.03 0.9259 0.7970 0.6348 0.4569 0.2947 0.2085 0.1613 0.1288 0.1112 0.0973 0.0899 0.0788

0.05 0.7841 0.6895 0.5107 0.3605 0.2169 0.1520 0.1177 0.0890 0.0788 0.0686 0.0621 0.0556

0.10 0.6154 0.4930 0.3485 0.2011 0.1149 0.0788 0.0538 0.0491 0.0389 0.0324 0.0269 0.0241

0.15 0.4875 0.3652 0.2456 0.1399 0.0723 0.0473 0.0361 0.0278 0.0222 0.0213 0.0195 0.0167

0.20 0.3976 0.2901 0.1835 0.0871 0.0463 0.0343 0.0250 0.0213 0.0167 0.0148 0.0130 0.0111

0.25 0.3234 0.2224 0.1233 0.0602 0.0371 0.0195 0.0167 0.0120 0.0111 0.0102 0.0074 0.0065

0.30 0.2632 0.1807 0.0964 0.0482 0.0232 0.0148 0.0083 0.0074 0.0074 0.0074 0.0065 0.0056

0.35 0.2206 0.1427 0.0704 0.0306 0.0139 0.0083 0.0074 0.0074 0.0065 0.0065 0.0056 0.0056

0.40 0.1854 0.1177 0.0593 0.0204 0.0102 0.0074 0.0065 0.0065 0.0065 0.0056 0.0056 0.0056


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EXHIBIT 6

Put Price > Call Price

Number of Profitable Arbitrage Opportunities at Various Costs of Capital and Price Spreads


(Put - Call) 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12%0.00 291 283 273 260 252 237 229 217 206 195 184 177

0.03 281 262 236 222 208 199 188 177 165 154 150 142

0.05 186 176 169 160 154 145 139 129 121 118 111 106

0.10 109 104 100 93 88 84 79 74 69 63 61 58

0.15 81 75 72 69 64 60 51 49 46 45 41 39

0.20 57 55 52 48 44 43 40 37 33 32 31 27

0.25 45 44 41 38 36 33 29 29 27 26 23 22

0.30 39 38 34 32 31 28 26 24 23 20 19 19

0.35 33 32 28 28 25 24 23 21 19 18 18 18

0.40 27 25 25 22 21 21 21 19 17 15 15 15

Relative Proportion of Profitable Arbitrage Opportunities at Various Costs of Capital and Price Spreads


(Put - Call) 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12%

0.00 0.9932 0.9659 0.9317 0.8874 0.8601 0.8089 0.7816 0.7406 0.7031 0.6655 0.6280 0.6041

0.03 0.9590 0.8942 0.8157 0.7577 0.7099 0.6792 0.6416 0.6041 0.5631 0.5256 0.5119 0.4846

0.05 0.6348 0.6007 0.5768 0.5461 0.5256 0.4949 0.4710 0.4403 0.4130 0.4027 0.3788 0.3618

0.10 0.3720 0.3549 0.3413 0.3174 0.3003 0.2867 0.2696 0.2526 0.2365 0.2150 0.2082 0.1980

0.15 0.2765 0.2560 0.2457 0.2355 0.2184 0.2048 0.1741 0.1672 0.1570 0.1536 0.1399 0.1331

0.20 0.1945 0.1877 0.1775 0.1638 0.1502 0.1468 0.1365 0.1263 0.1126 0.1092 0.1058 0.0922

0.25 0.1536 0.1502 0.1399 0.1297 0.1229 0.1126 0.0990 0.0990 0.0922 0.0887 0.0785 0.0751

0.30 0.1331 0.1297 0.1160 0.1092 0.1058 0.0956 0.0887 0.0819 0.0785 0.0683 0.0648 0.0648

0.35 0.1126 0.1092 0.0956 0.0956 0.0853 0.0819 0.0785 0.0717 0.0648 0.0614 0.0614 0.0614

0.40 0.0922 0.0853 0.0853 0.0751 0.0717 0.0717 0.0717 0.0648 0.0580 0.0512 0.0512 0.0512

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Arbitrage With Options