aswath_damodaran-investment_philosophies_2012

E x e r c i s e A r b i t r a g e The easiest arbitrage opportunities in the option market exist when options violate simple pricing bounds. No option, for instance, should sell for less than its exercise value. Thus, the minimal arbitrage conditions are as follows:

For instance, a call option with a strike price of $30 on a stock that is currently trading at $40 should never sell for less than $10. If it did, you could make an immediate profit by buying the call for less than $10 and exercising right away to make $10.

In fact, you can tighten these bounds for call options if you are willing to create a portfolio of the underlying asset and the option and hold it through the option’s expiration. The bounds then become:

With a call option: Value of call Value of underlying asset – Present value of strike price

With a put option: Value of put Present value of strike price – Value of underlying asset

To see why, consider the call option in the previous example. Assume that you have one year to expiration and that the riskless interest rate is 10 percent.

Present value of strike price = $30/1.10 = $27.27

Lower bound on call value = $40 − $27.27 = $12.73

The call has to trade for more than $12.73. What would happen if it traded for less, say $12? You would sell short a share of stock for $40, buy the call for $12, and invest the remaining proceeds from the short sale of $28 ($40 – $12) at the riskless rate of 10 percent. Consider what happens a year from now:

If the stock price strike price ($30): You first collect the proceeds from the riskless investment [$28(1.10) = $30.80], exercise the option (buy the share at $30), and cover your short sale. You will then get to keep the difference of $0.80.

If the stock price strike price ($30): You collect the proceeds from the riskless investment ($30.80), buy a share in the open market for the prevailing price then (which is less than $30), and keep the difference.

In other words, you invest nothing today and are guaranteed a positive payoff in the future. You could construct a similar example with puts.

The arbitrage bounds work best for non-dividend-paying stocks and for options that can be exercised only at expiration (European options). Most options in the real world can be exercised prior to expiration (American options) and are on stocks that pay dividends. Even with these options, though, you should not see short-term options trading violating these bounds by large margins, partly because exercise is so rare even with listed American options and dividends tend to be small. As options become long-term and dividends become larger and more uncertain, you may very well find options that violate these pricing bounds, but you may not be able to profit off them.

R e p l i c a t i n g P o r t f o l i o One of the key insights that Fischer Black and Myron Scholes had about options in the 1970s that revolutionized option pricing was that a portfolio composed of the underlying asset and the riskless asset could be constructed to have exactly the same cash flows as a call or put option.3 This portfolio is called the replicating portfolio. In fact, Black and Scholes used the arbitrage argument to derive their option pricing model by noting that since the replicating portfolio and the traded option have the same cash flows, they would have to sell at the same price.

To understand how replication works, let us consider a very simple model for stock prices where prices can jump to one of two points in each time period. This model, which is called a binomial model, allows us to model the replicating portfolio fairly easily. In the diagram, we have the binomial distribution of a stock, currently trading at $50, for the next two time periods. Note that in two time periods, this stock can be trading for as much as $100 or as little as $25. Assume that the objective is to value a call with a strike price of $50, which is expected to expire in two time periods.

3F. Black and M. Scholes, “The Valuation of Option Contracts and a Test of Market Efficiency,” Journal of Finance 27 (1972): 399–417.

Now assume that the interest rate is 11 percent. In addition, define:

= Number of shares in the replicating portfolio

B = Dollars of borrowing in replicating portfolio

The objective is to combine shares of stock and B dollars of borrowing to replicate the cash flows from the call with a strike price of $50. Since we know the cash flows on the option with certainty at expiration, it is best to start with the last period and work backward through the binomial tree.

Step 1: Start with the end nodes and work backward. Note that the call option expires at t = 2, and the gross payoff on the option will be the difference between the stock price and the exercise price if the stock price exercise price, and zero if the stock price exercise price.

The objective is to construct a portfolio of shares of stock and B in borrowing at t = 1, when the stock price is $70, that will have the same cash flows at t = 2 as the call option with a strike price of $50. Consider what the portfolio will generate in cash flows under each of the two stock price scenarios after you pay back the borrowing with interest (11 percent per period) and set the cash flows equal to the cash flows you would have received on the call.

If stock price = $100: Portfolio value = 100 − 1.11B = 50

If stock price = $50: Portfolio value = 50 − 1.11B = 0

We can solve for both the number of shares of stock you will need to buy (1) and the amount you will need to borrow ($45) at t = 1. Thus, if the stock price is $70 at t = 1, borrowing $45 and buying one share of the stock will give the same cash flows as buying the call. To prevent arbitrage,

the value of the call at t = 1, if the stock price is $70, has to be equal to the cost (to you as an investor) of setting up the replicating position:

Value of call = Cost of replicating position = 70 − B = (70)(1) − 45 = $25

Considering the other leg of the binomial tree at t = 1,

Solving for and B: = 0; B = 0

If the stock price is 35 at t = 1, then the call is worth nothing.

Step 2: Now that we know how much the call will be worth at t = 1 ($25 if the stock price goes to $70 and $0 if it goes down to $35), we can move backward to the earlier time period and create a replicating portfolio that will provide the values that the option will provide.

In other words, borrowing $22.50 and buying 57 of a share today will provide the same cash flows as a call with a strike price of $50. The value of the call therefore has to be the same as the cost of creating this position.

Value of call = Cost of replicating position

=57 (Current stock price) − 22.5

=57 (50) − 22.5 = $13.21

Consider for the moment the possibilities for arbitrage if the call traded at less than $13.21, say $13.00. You would buy the call for $13.00, sell the replicating portfolio for $13.21, and claim the difference of $0.21. Since the cash flows on the two positions are identical, you would be exposed to no risk and you make a certain profit. If the call traded for more than $13.21, say $13.50, you would buy the replicating portfolio, sell the call, and claim the $0.29 difference. Again, you would not have been exposed to any risk. You could construct a similar example using puts. The replicating portfolio in that case would be created by selling short on the underlying stock and lending the money at the riskless rate. Again, if puts are priced at a value different from the replicating portfolio, you could capture the difference and be exposed to no risk.

What are the assumptions that underlie this arbitrage? The first is that both the underlying asset and the option are traded, that you can be long or short on either, and that you can trade simultaneously in both markets, thus locking in your profits. However, the very assumption that you can constantly or dynamically replicate the cash flows on the option by using the underlying asset and riskless asset has been vigorously contested by many, thus breaching a fundamental requirement for arbitrage.4 The second assumption is that there are no (or at least very low) transaction costs. If transaction costs are large, prices will have to move outside the band created by these costs for arbitrage to be feasible. The third assumption is that you can borrow at the riskless rate and sell short, if necessary. If you cannot, arbitrage may no longer be feasible.

A r b i t r a g e A c r o s s O p t i o n s When you have multiple options listed on the same asset, you may be able to take advantage of relative mispricing—how one option is priced relative to another—and lock in riskless profits. We will look first at the pricing of calls relative to puts and then consider how options with different exercise prices and maturities should be priced relative to each other. We will close the section by looking at returns from buying versus selling options.

Put-Call Parity When you have a put and a call option with the same exercise price and the same maturity, you can create a riskless position by selling the call, buying the put, and buying the underlying asset at the same time. To see why, consider selling a call and buying a put with exercise price K and expiration date t, and simultaneously buying the underlying asset

4E. Derman and N. N. Taleb, “The Illusions of Dynamic Replication” (SSRN Working Paper 720581, 2005).

at the current price S. The payoff from this position is riskless and always yields a cash flow of K at expiration t. To see this, assume that the stock price at expiration is S*. The payoff on each of the positions in the portfolio can be written as follows:

Since this position yields K with certainty, the cost of creating this position must be equal to the present value of K at the riskless rate (K e−rt).

S + P − C = Ke−rt

C − P = S − Ke−rt

This relationship between put and call prices is called put-call parity. If it is violated, you have arbitrage.

If C – P S – K−rt, you would sell the call, buy the put, and buy the stock. You would earn more than the riskless rate on a riskless investment.

If C – P S – K−rt, you would buy the call, sell the put, and sell short the stock. You would then invest the proceeds at the riskless rate and end up with a riskless profit at maturity.

Note that violations of put-call parity create arbitrage opportunities only for options that can be exercised only at maturity (European options) and may not hold if options can be exercised early (American options).

Does put-call parity hold up in practice, or are there arbitrage opportunities? An early study examined option pricing data from the Chicago Board Options Exchange from 1977 to 1978 and found potential arbitrage opportunities in a few cases.5 However, the arbitrage opportunities were small and persisted only for short periods. Furthermore, the options examined were American options, where arbitrage may not be feasible even if put-call parity is violated. A study by Kamara and Miller of options on the S&P 500 (which are European options) between 1986 and 1989 found that

5R. C. Klemkosky and B. G. Resnick, “Put-Call Parity and Market Efficiency,” Journal of Finance 34 (1979): 1141–1155.

there were few violations of put-call parity, and that the deviations tend to be small.6

Mispricing across Strike Prices and Maturities A spread is a combination of two or more options of the same type (call or put) on the same underlying asset. You can combine two options with the same maturity but different exercise prices (bull and bear spreads), two options with the same strike price but different maturities (calendar spreads), two options with different exercise prices and maturities (diagonal spreads), and more than two options (butterfly spreads). You may be able to use spreads to take advantage of relative mispricing of options on the same underlying stock.

Strike prices. A call with a lower strike price should never sell for less than a call with a higher strike price, assuming that they both have the same maturity. If it did, you could buy the lower strike price call and sell the higher strike price call, and lock in a riskless profit. Similarly, a put with a lower strike price should never sell for more than a put with a higher strike price and the same maturity. If it did, you could buy the higher strike price put, sell the lower strike price put, and make an arbitrage profit.

Maturity. A call (put) with a shorter time to expiration should never sell for more than a call (put) with the same strike price with a long time to expiration. If it did, you would buy the call (put) with the shorter maturity and sell the call (put) with the longer maturity (i.e., create a calendar spread) and lock in a profit today. When the first call expires, you will either exercise the second call (and have no cash flows) or sell it (and make a further profit).

Even a casual perusal of the option prices listed in the newspaper or on the Internet each day should make it clear that it is very unlikely that pricing violations that are this egregious will exist in a liquid options market.

There are riskier strategies that do not require the blatant mispricing that pure arbitrage does but do require relative mispricing, either across options or over time. For instance, there is evidence that investors are not rational in their behavior just before ex-dividend days and often choose not to exercise call options when they should. In particular, rational exercise would require the exercise of options when the time value on the option is less than the expected dividend, and Hao, Kalay, and Mayhew find that

6A. Kamara and T. W. Miller, “Daily and Intradaily Tests of European Put-Call Parity,” Journal of Financial and Quantitative Analysis 30, no. 4 (1995): 519–541.

about 30 percent of call options that should have been exercised are not, leading to windfall profits for those who sold these options.7

Long versus Short Positions Studies that have looked at the profit opportunities from trading on options have uncovered an irregularity. In general, strategies that require selling options seem to offer much better returns, given their risk, than equivalent strategies that require buying options. There are, however, three potential roadblocks to using this strategy to generate arbitrage profits in the long term. The first is the transaction costs of these strategies tend to be high, wiping out a large portion of the profits over time. The second is that the margin requirements (and the resultant margin calls) if the stock price moves in the wrong direction can limit how much an investor can invest in a strategy and also sometimes force the investor to close down positions prematurely. Finally, these strategies also suffer from exposure to extreme events: risks that occur infrequently but when they do cause major and even catastrophic losses.

V O L A T I L I T Y V I E W S A N D O P T I O N S

Option prices are determined by expected volatility, and there are strategies that are designed to profit from views on the future direction of volatility or from mispricing of volatility across options on the same underlying asset. These strategies are not riskless and thus don’t fit into the arbitrage spectrum. They can, however, still be profitable.

Taking the volatility timing strategies, there are patterns to volatility over time that investors may be able to exploit. In particular, just

as prices and P/E ratios often revert back to historic norms, there is evidence that volatility does as well. A strategy of buying options (straddles or other positions that minimize exposure to stock price direction) when the implied volatility in option prices is lower than the historic realized volatility generates significant excess returns for investors, with option positions in the highest decile (in terms of difference between historic realized volatility and implied volatility) earning 0.21 percent a month more than expected. However, allowing

(continued)

7J. Hao, A. Kalay, and S. Mayhew, “Ex-Dividend Arbitrage in Options Markets,”

Review of Financial Studies 23 (2010): 271–303.

for transaction costs lowers that excess return to just under 0.08 percent a month. A strategy of selling options when the implied volatility is much higher than the realized volatility generates excess returns of about 0.16 percent a month, but incorporating transaction costs wipes out most of those returns.8

There is clear evidence that the implied volatilities in option prices are different for different options on the same underlying asset at the same time; this is clearly not consistent with conventional option pricing models. In particular, there seems to be evidence of a volatility smile, where options at-the-money for a given underlying asset and

option maturity have lower implied volatilities than options that are deep-in-the-money or deep-out-of-the-money. To the extent that this consistently shows up in different option markets and across time, most analysts incorporate the smile into their option pricing models, using higher volatilities when valuing deep-in-the-money or deep-out- of-the-money options. However, to the extent that a smile is exaggerated (the difference in implied volatilities between at-the-money and deep-in-the-money or deep-out-of-the money options is too high), selling straddles can be profitable, assuming the smile will revert back to normal levels.

F i x e d I n c o m e A r b i t r a g e

Fixed income securities lend themselves to arbitrage more easily than equities because they have finite lives and contractually specified cash flows. This is especially so when you have default-free bonds, where the contractual cash flows are also guaranteed. Consider one very simple example. You could replicate a 10-year Treasury bond’s cash flows by buying zero coupon Treasuries with expirations matching those of the coupon payment dates on the Treasury bond. For instance, if you invest $100 million in a 10-year Treasury bond with an 8 percent coupon rate, you can expect to get cash flows of $4 million every six months for the next 10 years and $100 million at the end of the tenth year. You could have obtained exactly the same cash flows by buying zero coupon Treasuries with face values of $4 million, expiring every six months for the next 10 years, and an additional 10-year

8A. Goyal and A. Saretto, “Option Returns and Volatility Mispricing,” Journal of Financial Economics 94 (2009): 310–326.