Expected return on Put Option

In the example in the attached material, why do you think the put option has a negative expected return (-16.67%), while the call option has a very high positive expected return (42.86%)?

Compare the payoff patterns of these two options. When do they pay? Note that one pays when the economy is good, and the other pays when the economy is bad.

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1.1 Different Instruments in Derivative Markets Derivatives, or derivative securities, are financial arrangements whose payments are derived from other securities or assets. In other words, derivative markets are markets for contractual instruments whose performance is determined by how other securities or assets perform. Two major types of derivatives include option-type contracts and forwardtype contracts.Options are the right, but not the obligation, to buy or sell an asset at a pre-set price (=exercise price or striking price) over a specified period. Forward-type contracts are the commitment to buy or sell a given asset at a set price on a future date. Forwards, futures, and swaps are included in this type. Read pages 2-4 of the textbook for more discussion of these types of derivative securities. Now in the rest of this lecture, we'll go over two concepts: (1) risk and risk aversion and (2) arbitrage. 1.2. Risk and Risk Aversion We all know that the expected return on a security is determined by the risk of the security, i.e., the higher the risk, the higher the expected return. Risk-averse investors ask for a premium for the risk they are taking. To understand the concept of risk-aversion, let's consider an imaginary economy where uncertainty is represented by the two states (good and bad) of the economy in the next period. In this simple economy, the good and the bad, economy has the same probabilities of 50%-50%. Let us further assume that the next three securities exist in an imaginary economy: T-bill Common stock 10 Call options written on the same common stock with the strike price (X) of 100 T-bill is the risk-free asset and will provide certain cash flow of $100 whether economy is good or bad. (See the table below.) The common stock pays off $120 and $80 in good and bad economy, respectively. The call option is the right (not an obligation) to buy the common stock for $100 at the end of the period. If the economy is good, you exercise your call option and make $20 (=120 -100) profit per option. Therefore, the payoff to the 10 call options in the good economy = 10 x (120 - 100) = 200. In the bad economy, you will not exercise the call options, and its payoff is 10 x (0) = 0. These three securities are designed so that their expected payoffs are the same as shown in the table. For example the expected payoff of common stock can be calculated as follows: 0.5 x 120 + 0.5 x 80 = 100 probability x payoff + probability x payoff = Expected payoff Suppose, for the time being, that these securities are all free. If you are to choose one out of these three securities, which one would you choose, and why? Some of you may choose the T-bill because it is sure money. Some others may choose 10 call options because there is nothing to lose. Still some others may say common stock because it is between the two extremes. Now, let me show you why you should prefer the T-bill to common stock, and common stock to 10 call options; in other words, why you should be risk-averse. Now consider new securities. (See the next table.) Bond #1 is created from T-bill by adding $1 to the good economy payoff and subtracting $1 from bad economy payoff. Likewise Bond #2 - #19 are created the same way: You add $1 to the good economy and subtract $1 from the bad economy. The expected payoffs remain the same at $100. Now ask yourself, which $1 is more valuable, +$1 or -$1? Of course they are the same $1. However, one dollar becomes less valuable when you get richer and richer. If we exaggerate a bit, the value of one dollar when you are starved to death is greater than the value of one dollar when you are a millionaire. Do you agree? Economists use the term Diminishing Marginal Utility to describe this property. In other words, the $1 you lose is more important than the $1 you gain. Therefore, every time you add one dollar and subtract one dollar to move to the next step, the value of the security decreases. In other words, the dollar you lose is more valuable than the dollar you gain. Therefore, the T-bill is more valuable than Bond #1, Bond #1 is more valuable than Bond #2, ... , Bond #19 is more valuable than common stock, and common stock is more valuable than the 10 call options. We can conclude, based on this example, that investors are riskaverse because of the diminishing marginal utility. Suppose that these three securities (T-bill, common stock, and call option) are not free any more. The next question is, how much would you pay for these securities? Let's assume the payoffs will be made in one year. One obvious thing is that you will pay less for common stock than for the T-bill, and less for the 10 call options than for common stock. Right? Why? Because you said so. You prefer the T-bill to common stock, and common stock to 10 call options. Suppose you pay $95 for the T-bill. Make sure that it is a reasonable price because it makes the rate of return on this investment 5.26% (= 100 / 95 - 1). How much would you pay for the common stock? More or less? Of course you will pay less than $95 for common stock because it is not as attractive as the Tbill. Say you pay $90 for common stock. How about 10 call options? The price of $75 for the 10 call options would be a reasonable price. Then, based on these prices you can calculate the expected rate of return (= Expected payoff / Price 1) as follows: T-Bill Common Stock 10 Call Option with X =100 Expected Payoff $100 $100 $100 Price $95 $90 $75 Expected rate of return 100 / 95 - 1 = 5.26% 100 / 90 - 1 = 11.11% 100 / 75 - 1 = 33.33% Based on these returns, we now can see that the rate of return on a riskier security is higher because risk-averse investors pay lower prices for the riskier security. Risk-averse investors require compensation (=risk premium) for the extra risk they are taking. In the above example, the risk premium on the common stock is 5.85% (=11.11% - 5.26%) and the risk premium on the call option is 28.07%. (=33.33% - 5.26%). 1.3. Arbitrage Arbitrage is a type of transaction in which an investor seeks to profit when the same good sells for two different prices. For example, if a computer monitoring markets notices that ABC stock can be bought on a New York exchange for $10 a share and sold on a London exchange at $10.12, the arbitrageur or a special program can simultaneously purchase ABC stock in New York while selling the same amount it in London, thus pocketing the difference. Since derivative securities are securities whose payoffs are derived from other securities, their prices also depend upon the underlying assets' prices. Therefore, certain relationships should be satisfied between the prices of derivative securities and the prices of the underlying securities. If the relationships are not satisfied, you can engage in an arbitrage transaction to make profits. We have some examples of arbitrage transactions below. 1.3.1. Value of a Call Option Example: Arbitrage Profit: Call Options - Right to buy Continue with the same simple economy we considered above. Remember there are two possible states, good and bad economy. If the market price of call option with X = 100 were $7.50 when stock price = 90 and T-bill interest rate = 5.26%, how can you make arbitrage profit? To answer this question, let's first create a portfolio of shares of common stock and T-bills to replicate a call option. In this example, it is possible to replicate the call option with shares and T-bills because there are only two possible states (good and bad economy). Compare the payoffs from the two positions in the next table: {Buy 5 shares + Sell 4 T-bills} and {Buy 10 call options}. Since the payoffs from these two positions are the same, their prices (or values) must be the same. Let us assume the common stock and T-bill are fairly priced, then the price of five shares purchased and four T-bills sold is 5 x $90 - 4 x $95 = $70, and the price of 10 call options should be $70. Positions Payoff at t=1 Buy 5 shares of stock + Sell 4 T-bills Buy 10 Call options good econ. 600 + 400 200 bad econ. 400 - 400 0 -(5 x $90 - 4 x $95) = $-70 This must be $-70. CF(0) Based on the table above, the equilibrium price of one call option is $7. Then, the market price of $7.5 for one call option we assumed above was too high. In other words, the call option is overvalued. The next table shows how we can take advantage of the overvalued call option. We sell 10 call options at the price of $75, and at the same time buy the replicating portfolio (i.e., buy 5 shares and sell 4 T-bills) and pay $70. Since {Buy 5 shares + Sell 4 T-bills} is essentially the same as selling 10 call options, the future payoffs will be all covered and you can make $5 profit now. (See the table below.) It is risk-free because you will be covered regardless of the state of the economy. We call it an arbitrage profit. CF (t = 1) Positions CF (t = 0) Good economy (Stock Price = $120) Bad economy (Stock Price = $80) 75 -(120-100) x 10 = -200 0 x 10 = 0 Buy 5 shares of stock -450 120 x 5 = 600 80 x 5 = 400 Sell 4 T-bills 380 -100 x 4 = -400 -100 x 4 = -400 5 0 0 Sell 10 call options (X = 100) Total Therefore, the price of the call option cannot stay at $7.50 because of the arbitrageurs. It will soon go down to the equilibrium price of $7. 1.3.2. Value of a Put Option Now let's consider a put option written on the same common stock with the exercise price of $100 in the next example. Example: Arbitrage Proft: Put Options - Right to Sell Suppose that the market price of put option with X = 100 were $11 when stock price = 90 and T-bill interest rate = 5.26%. How can you make arbitrage profit? The next table shows that you can generate the arbitrage profits of $10 by {Buying 10 put options} and {Buying 5 shares and Selling 6 T-bills}. CF (t=1) Positions CF (t=0) Good economy (Stock Price = $120) Bad economy (Stock Price = $80) Buy 10 put options (X=100) -11 x 10 = -110 0 x 10 = 0 (100-80) x 10 = 200 Buy 5 shares -90 x 5 = -450 120 x 5 = 600 80 x 5 = 400 Sell 6 T-bills 95 x 6 = 570 -100 x 6 = -600 -100 x 6 = -600 10 0 0 Total Let us assume, in the above example, that the common stock and T-bill are fairly priced. Then, we have shown that the put option is undervalue by $1 per option because you could make $10 profit by buying 10 put options. The next table shows the same fact. Positions Sell 5 shares of stock + Buy 6 T-bills Buy 10 Put options Payoff at t=1 good econ. -600 + 600 0 bad econ. -400 + 600 200 5 x $90 - 6 x $95 = $-120 This must be $-120. CF(0) The equilibrium price of 10 put options should be $120, or $12 per put option with X=100. The market price of $11 is lower than $12. That is why we could make the arbitrage profits. The above table is created based on the fact that the 10 put options can be replicated by the portfolio of {Buy 6 T-bills + Sell 5 shares of common stock}. If this is the case, the right price of the 10 put options should be 95 x 6 - 70 x 5 = 120. In other words, the put options are undervalued. That is why we are buying 10 put options and selling the replicating portfolio. Since we buy 10 put options and sell the replicating portfolio, the future cash flows should be zero regardless of the state of the economy. Since the current cash flows are positive, it is an arbitrage opportunity. This kind of opportunity would not stay long before it is taken advantage of and disappears. One question you may ask is: Where do we get the number of shares and T-bills to create the replicating portfolio? The answer to this question will be provided when we discuss the hedge ratio of an option later in Chapter 4. For the time being, let us just assume these replicating portfolios are given. 1.3.3. Expected Return and Risk Again Remember we have seen the relationship between the risk and the expected return. The risk-free asset has no risk but the expected return is low (5.26%); the stock has medium risk and has the medium expected return (11.11%); the call option has the highest risk and the highest expected return (42.86%). In this calculation, I used $70, the equilibrium price, not the original price of $75. How about the put option's expected return? The next table shows the calculations: T-Bill Common Stock 10 Call Option with X =100 10 Put Option with X =100 Expected Payoff $100 120x0.5 + 80x0.5 =$100 200x0.5 + 0x0.5 =$100 0x0.5 + 200x0.5 =$100 Price $95 $90 $70 $120 Expected rate of return 100 / 95 - 1 = 5.26% 100 / 90 - 1 = 11.11% 100 / 70 - 1 = 42.86% 100 / 120 - 1 = -16.67% It is shown in the above table that the put option has a negative expected return of -16.67%. The negative return is due to the price ($120) investors would pay for this option when the expected payoff is $100. Remember the call option's expected return is 42.86%. This high expected return is due to the low price of $70. Go to the Discussion Board and answer the following question: Discussion Board In the above example, why do you think the put option has a negative expected return (-16.67%), while the call option has a very high positive expected return (42.86%)? Compare the payoff patterns of these two options. When do they pay? Note that one pays when the economy is good, and the other pays when the economy is bad. 1.3.4. Relationship among Stock, Call, Put and Risk-free Asset The arbitrage argument can be used to derive the relationship among stock, call, put, and risk-free asset. Let's assume the same imaginary economy to see the relationship. In the next table, compare the payoffs to two portfolios: (1) stock + put, (2) Call + T-bill with face value same as exercise price. Since the payoffs to these two portfolios are the same in these two states, we may conclude that their values are the same. If their values were not the same, we should be able to take advantage of the arbitrage opportunities by buying the lower valued portfolio and selling the higher-valued one. Based on this arbitrage argument, we may state the following relationship called Put-Call parity relationship. (A more rigorous proof will be given later in Chapter 2.) S + P = C + PV(X) We can confirm the put-call parity relationship in the above example as follows: S+P C + PV(X) 90 + 12 = 102 7 + 100/1.0525 = 7 + 95 = 102