4.3 Trading Strategies

4.3.1 Portfolios of Calls and Puts with the Same Maturity Date

The payoffs from puts and calls can be interpreted as payoffs that were generated from the stock by stripping it to its basic Footnote 8 components: cash and a put and a call with the same strike price. Let us look at the pattern of the payoff if we put together a portfolio of a short put and a long call with the same strike price (try to think about it as already suggested -- mechanically putting together the pieces of the graph of the long call and the short put).

Assume that we create a portfolio of a call with a strike price of $60 and a short put with the same strike price. Thus, we are looking at a portfolio whose payoff in state s will be

Call (s,60)- Put (s,60).

For each s in the region [0, ), we sum the value of the put and the call. MAPLE is capable of graphing the result. The resultant cash flow is shown in figure 4.8.

> plot(Call(s,60)-Put(s,60),s=0.01..100,title=`Figure 4.8: Payoff from a Long Position in a Call and a Short Position in a Put`,titlefont =[TIMES,BOLD,10],thickness=2,labels=[`Stock Price`,`Value`], scaling=constrained);

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To highlight the building blocks of the above position, as in Figure 4.9, one can issue the following command (note that instead of plotting the sum of the Call and the negative of the Put we simply plot them together; thus the "," instead of the "-" in Figure 4.8):

> plot({Call(s,60),-Put(s,60)},s=0.01..100,title=`Fig. 4.9: Payoff from a Long Position in a Call and a Short Position in a Put`,titlefont = [TIMES,BOLD,8],thickness=2,labels=[`Stock Price`,`Value`],scaling=constrained);

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Remembering that the graph of the payoff from a stock is a line emanating from the origin at a angle, we see that we just need to "lift up'' the line in Figure 4.8 by $60 to get the payoff from the underlying stock. This can be accomplished by investing an amount of money in a risk-free asset that will grow to $60 by the maturity date of the call and the put. Thus, the payoff of the call and the put at maturity will be as above, plus $60. The reader can graph this payoff by executing the command (Figure 4.10)

> plot(Call(s,60)-Put(s,60)+60,s=0.01..100,thickness=2,title=`Figure 4.10: A Payoff from a Long Call and a Short Put (with an exercise price of $60) + $60`,titlefont=[TIMES,BOLD,8],labels=[`Stock Price`,`Value`],scaling=constrained);

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This is indeed the payoff from the stock as confirmed by MAPLE. Footnote 9

> assume(s,positive);

> about(s);

Originally s, renamed s~:

is assumed to be: RealRange(Open(0),infinity)

> is(Call(s,60)-Put(s,60)+60=s);

>

It is not surprising that combining the stock with options on the same stock can either reinforce or reduce the risk inherent in the stock. The payoff from the stock in certain states of nature can be neutralized by taking an offsetting short position, and can be enhanced in other states of nature by taking a long position. Options allow us to give up future possible payoff from the stock in exchange for a decrease in the current price of the portfolio. They can facilitate risk reduction at the cost of an increase in the price of the portfolio. They also allow creation of a position for which the investor is paid today for buying risk out of somebody else's hands.

The analogy of holding options as a side bet between two investors on the outcome of the stock value should be clearer now. It is like a zero sum game between the writer and the holder of an option. If the writer profits, the holder loses and vice versa. The availability of options in the market allow buying and selling "parts of the risk'' inherent in the stock. It introduces a market for risk. Options can be used to reduce risks assumed by certain positions.

Consider a portfolio which is only a long stock. There is a risk inherent in such a portfolio from the possibility of a decrease in the price of the stock. An investor can guarantee that the payoff from the portfolio will never fall below a certain value. This can be done by including a put in the portfolio. Let us first look at the graph, Figure 4.11, of the cash flow from a portfolio composed of a long stock and a long put with an exercise price of $60.

> plot(s+Put(s,60),s=0.01..100,y=0..100, title=`Figure 4.11: A Payoff from a Long Stock and a Long Put`, titlefont=[TIMES,BOLD,8],thickness=2,labels=[`Stock Price`,`Value`],scaling=constrained);

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Figure 4.11 clearly demonstrates that holding a stock long and a put long ensures a payoff above the strike price of the put. Such a position is called a protective put . The larger the strike price is, the larger is the minimum payoff. This is demonstrated via three-dimensional illustration in Figure 4.12, where the minimum payoff increases as the exercise price K increases.

> plot3d(s+Put(s,K),s=0..100,K=20..90, title=`Figure 4.12: The Payoff from a Protective Put`,titlefont=[TIMES,BOLD,8],orientation=[52,38],axes=framed,labels=["Stock Price","Exercise Price","Value"]);

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Effectively, holding the (long) put protects the investor from price decreases of the stock. When the stock price decreases below the strike price of the put, the put becomes valuable and has a payoff equal to the difference between the strike price and the price of the stock. Thus, the put introduces a "floor'' -- a lower bound on the payoff from the portfolio. The investor purchased the put as an "insurance policy'' which would guarantee that the payoff from the portfolio would not fall below the strike price. The writer of the put took on this risk from the hands of the holder of the stock (for a price -- the premium). The writer was willing to receive a certain amount of money today (the price of the put, or the premium) and in return guaranteed payment in the future if the price of the stock fell below the strike price of the put. This insurance-type argument can also work for a short position.

Consider a short position in the stock. The potential payoff from such a position can be very negative -- it can potentially be any negative number. There is no limit to the loss one can suffer from such a position. If the price of the stock increases to a huge number, the holder of the short position will have to pay it when closing the position. However, it is possible to truncate this potentially very negative payoff by taking a long position in a call. Let us first graph, Figure 4.13, the payoff from shorting a stock and longing a call at an exercise price of $60 on the same plane.

> plot({-s,Call(s,60)},s=0..199, title=`Figure 4.13: Building Blocks of Payoff from Shorting a Stock and Longing a Call`, titlefont=[TIMES,BOLD,8],thickness=2,axesfont=[TIMES,BOLD,6],labels=[`Stock Price`,`Value`],scaling=constrained);

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It is evident that the higher the stock price, the larger the amount the investor who shorts the stock must pay. At the same time, it can be seen that a position in a call can offset the decrease in the stock price. Figure 4.14 demonstrates what happens when the short stock and the call are combined in a portfolio.

> plot(-s+Call(s,60),s=0..199,title=`Figure 4.14: A Payoff from a Short Position in a Stock and a Long Position in a Call`,titlefont =[TIMES,BOLD,8],thickness=2,labels=[`Stock Price`,`Value`]);

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The loss from the short position is bounded by the negative of the strike price. Whenever the stock price goes above the strike price, the call becomes valuable and produces a payoff equal to the difference between the stock price and the strike price. Thus, the payoff from the portfolio of a short call and a stock cannot be more negative than the strike price.

Writing a call also exposes the investor to a potentially unbounded loss, in the same way that shorting a stock does. However, in the case of shorting a call, the payoff can be negative only if the price of the stock increases above the strike price. This can also be offset by holding a long position in the stock. Such a position is called writing a covered call . Let us first graph the payoff from a short call position and a long stock in the same plane in Figure 4.15.

> plot({s,-Call(s,60)},s=0..199,title=`Figure 4.15: A Payoff from a Long Position in a Stock and a Short Call Position`,titlefont=[TIMES,BOLD,8],thickness=2,labels=[`Stock Price`,`Value`],scaling=constrained);

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Again it is seen that if the stock and the short call are combined in a portfolio, the potential payoff is bounded below. Figure 4.16 demonstrates that the payoff from such a position is positive for every state of nature.

> plot(s-Call(s,60),s=0..199, title=`Figure 4.16: The Payoff from a Long Position in the Stock and a Short Position in the Call`,titlefont =[TIMES,BOLD,8],thickness=2,axesfont=[TIMES,BOLD,8],labels=[`Stock Price`,`Value`] );

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We can also investigate the effect of changes in the strike price of the option on such a position. Figure 4.17 demonstrates the three-dimensional graph of the payoff of this portfolio as a function of the state of nature, s , and the strike price, K . The reader should be convinced, even before seeing the graph, that the higher the strike price, the larger the minimum payoff from such a portfolio.

> plot3d(s-Call(s,K),s=0..199,K=20..80,title=`Figure 4.17: The Payoff from a Long Position in a Stock Plus a Short Position in a Call as a Function of s and K`, axes=frame,titlefont=[TIMES,BOLD,8], orientation=[83,38],axes=framed,labels=["Stock Price","Exercise Price","Value"]);

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In writing a covered call, a stock is held long against every short call position. It is also possible to cover the call, but not "fully". The ratio of the number of written calls to the number of long positions in the stock, referred to as a hedging ratio , is not one to one. In such cases, the payoff can be negative (indeed it is unbounded). It is, though, for every state of nature, less negative than the payoff from a simple short position in the call.

Shorting (writing) a call without holding the stock is called a naked position . The best way to illustrate it is to graph an example of such a payoff. Consider the payoff from shorting three calls and holding two units long in the stock. Figure 4.18 illustrates these two graphs in the same plane.

> plot({-3*Call(s,60),2*s-3*Call(s,60)},s=0..200, title=`Figure 4.18: A Payoff from Writing Three Calls, and a Payoff from Two Long Positions in the Stock Plus Writing Three Calls`, titlefont =[TIMES,BOLD,8],thickness=2,axesfont=[TIMES,BOLD,6],labels=[`Stock Price`,`Value`]);

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While the payoff can be potentially very negative when the call is not fully covered, it is still less negative, for every state of nature, than that of a naked position. The payoff decrease per dollar increase in the stock price in the partially covered position is less than in the case of the naked call. This is evident from the slopes of the graph: shorting the three calls has a steeper slope (for a stock price greater than $60) than that of the portfolio. The reader is encouraged to investigate how the payoff changes with changes in the ratio of written calls to holdings of the stock.

Consider a portfolio composed of a call with a strike price of $50 and a put with a strike price of $50. the cash flow in state s will be the sum of the function Call (s,50) and the function Put ( ). (Try to visualize the two pieces put together -- the call like a hockey stick with a handle from zero to the strike price and the put a hockey stick with an infinite handle starting at the strike price and continuing to infinity.)

This portfolio will increase in value if the price of the stock moves away from $50. If the stock price is $50 the value of this portfolio is zero (both the put and the call are worthless at this value). At any other value of the stock the portfolio is worth something. If the price of the stock either increases from $50 by $10 or decreases from $50 by $10 it will have the same effect on the cash flow. This position is called a purchased straddle . The result is depicted in Figure 4.19.

> plot(Call(s,50)+Put(s,50),s=0.01..100,title=`Figure 4.19: Straddle - Call Plus Put with the Same Exercise Price K`, titlefont=[TIMES,BOLD,8],thickness=2,labels=[`Stock Price`,`Value`],scaling=constrained);

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To highlight the building blocks of the above position one can replace the "," with the "+" in Figure 4.19, as shown in the on-line version of the book. Can you graph the payoff from a written straddle (i.e., instead of holding the put and call long, hold them short)?

> plot({Call(s,50),Put(s,50)},s=0.01..100,title=`The building blocks of a straddle - Call Plus Put with the Same Exercise Price K`, titlefont=[TIMES,BOLD,8],thickness=2,labels=[`Stock Price`,`Value`],scaling=constrained);

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Imagine a situation in which we would like to be able to fix today, in spite of any uncertainty, the price of a commodity or a stock which we know we will need in the future. Suppose we entered into an agreement in which we obligated ourselves to buy a stock (or more commonly, a certain commodity) at some time T in the future for a certain price. Suppose we had committed to buy a stock for $100 at time T . Note that indeed the price of the transaction which will take place in the future has been fixed now, and we are obliged to buy the stock for $100. However, no cash changes hands now. This type of an agreement is of course a forward contract Footnote 10, which has been introduced in Section 2.4.1, albeit here we have a continuum of states of nature.

At the current time it is not known what the price of the stock will be at time T . If the price is above $100, we achieve a positive cash flow which is the difference between the price in the future and $100 (buy the stock for $100 and sell it in the market for its market price). If the price of the stock at time T is less than $100, we achieve a negative cash flow of the price difference. Thus the cash flow at time T , as a result of this commitment, is stochastic (or random): it is a function of the state of nature s as shown in Figure 4.20.

> plot(s-100, s=0..200, title=`Figure 4.20: Time T cash flow resultant from commitment to buy a stock for $100`, titlefont=[TIMES,BOLD,8], thickness=2,labels=[`Stock Price`,`Value`],scaling=constrained);

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There is a way to make sure that regardless of the state of nature at time T , our commitment will be for exactly $100. The graph above, Figure 4.20, nearly spells out the answer. We need to devise a portfolio which will have exactly the opposite cash flow to the above commitment. This portfolio should be worth $1 if the price of the stock is $99 and should be worth -$ if the price of the stock is $101. If it is possible to devise such a position, it would lock in the value of the commitment for $100 regardless of the price of the stock (state of nature) in the next period. Graphically, we are attempting to generate a line intersecting the x -axis at $100 which makes a angle with the line in Figure 4.20. Can we generate such a profile from puts, calls, and the underlying stock? It is actually easy to see what type of pieces we will need from our "Lego'' box. Here it is in Figure 4.21:

> plot(Put(s,100)-Call(s,100),s=0.01..180,title=`Figure 4.21: Put minus Call at strike of $100`,titlefont =[TIMES,BOLD,9],thickness=2,labels=[`Stock Price`,`Value`],scaling=constrained);

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If you would like to be convinced that indeed these two positions exactly offset each other, you can ask MAPLE to graph it. This is done by issuing the command which generates Figure 4.22

> plot(Put(s,100)-Call(s,100)+(s-100),s=0.01..180,title=`Figure 4.22: Sum of the Two Offsetting Positions`,titlefont =[TIMES,BOLD,9],thickness=2,labels=[`Stock Price`,`Value`]);

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or to evaluating it:

> assume(s,positive);

> is(Put(s,100)-Call(s,100)+(s-100)=0);

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The reader is encouraged to look at some combinations defined in the appendix to this Chapter and to graph the corresponding payoff diagrams. Many other payoffs are possible, which can be tailored to your needs. This is done by combining different assets. Taking a look at our "Lego'' pieces and at those trading strategies we have managed to devise, the reader should now be convinced that we can generate any payoff which can be represented as a piecewise linear function. This is indeed the case and the last section of this Chapter pertains to this issue.

Before we leave this section, we provide a "practical'' demonstration by summing together many puts and calls whose exercise prices are close to one another so that we can approximate a smooth curve. Two examples are shown here and are termed the "smile'' (Figure 4.23), and the "frown'' (Figure 4.24):

> plot(sum(Put(s,i)+Call(s,i),i=1..50),s=0.01..70, title =`Figure 4.23: The Smile`, titlefont = [TIMES,BOLD,10],labels=[`Stock Price`,`Value`]);

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Footnote 8

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Footnote 9

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Footnote 10

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