An Introduction to the Mathematics of Finance. Basu, Goodman, Stampfli

Transcription

1 An Introduction to the Mathematics of Finance Basu, Goodman, Stampfli 1998

2 Click here to see Chapter One.

3 Chapter 2 Binomial Trees, Replicating Portfolios, and Arbitrage 2.1 Pricing an Option A Special Case A Call option on a stock is the right, but not the obligation, to purchase the stock on a certain day. This day is termed the strike date or date of expiration. The purchase price is agreed on in advance, it is termed the strike price or exercise price. A Put option on a stock is a similar right, but not the obligation to sell the stock on a certain date. It is understood that one does not need to own a share of stock to exercise a put. Example A Call Option. Ford 27 1/2 Dec / /8 27 1/2 Mar /8 30 Nov / /8 30 Dec 84 3/4 28 5/8 30 Mar /8 28 5/8 32 1/2 Dec 59 1/4 28 5/8 32 1/2 Mar / /8 35 Mar /16 The Wall Street Journal We have a stock presently priced at $100. In exactly one year the stock price will be either $90 or $120. We are not given probabilities. The current interest rate is 5% (a dollar invested today is worth $1.05 in one year). What is a fair price for the option on the stock with a strike of $105 and expiring in one year? 3

4 4 CHAPTER 2. REPLICATION AND ARBITRAGE We will present two methods which answer this question. One method could be called the probabilistic (expected value) approach and the other the financial (game theory, arbitrage) approach. Some readers may feel we do not have enough information to answer the question. They are right in that we will have to make some assumptions. The assumptions are all part of the approach or solution to the problem and we feel no concern at adding them as we go along Financial Method Let V = price of the option, and let S = price of the stock. We will construct a portfolio as follows: We buy a shares of the option and b shares of the stock. The numbers a or b may be negative. If b, for example, is negative, it indicates that we are short selling the stock. Let Π 0 = the value of the portfolio at time t =0. Π 0 = av + bs. At this point we do not know a or b. We will next value the portfolio at time t =1. Since there are two possible states (scenarios) at t = 1, we take them separately. (Up State, S = $120) Π 1 = a( ) + b120. Why does the term appear? We convert the option into stock and sell the stock. The cost of the conversion is $105 per share and the sale price is $120 per share. (Down State, S = $90) Π 1 = a 0+b90. We seem to have made little progress but now for a brilliant stroke. We cannot control the stock performance. So we use a trick from game theory. Eliminate Uncertainty. We will choose a and b so that Π 1 does not depend on the outcome. Thus we set a( ) + b120 = a 0+b90.

5 2.1. PRICING AN OPTION A SPECIAL CASE 5 This gives the equation 15a = 30b and we make the choice b = +1 and a = 2. This strategy tells us we should sell two options and buy one share of stock. If we do so, the outcome is deterministic; we can ignore the real world. Valuing the Option. Note that Π 0 = 2V and Π 1 = (+1)120 = 90. We could have taken the funds Π 0 and invested them at 5%. So the value of that investment should equal the value Π 1 of the stock-option trade. In other words 1.05Π 0 =1.05(100 2V )=Π 1 =90. Thus, 100 2V = and so V =7.14. Arbitrage. Suppose V > Then we could make a deterministic, risk free profit by borrowing money at 5% to buy the stock and sell the options. If, on the other hand, V < 7.14, then we could make a deterministic, risk free profit by buying two options and selling one share of stock and investing the net proceeds at 5%. Note that arbitrage forces or drives the price to $7.14. Let s see this arbitrage in action. We will put concrete numbers into the scenario. Assume the price of the option is V = $8. Our analysis tells us the options are overpriced. So, we will buy one share of stock and sell two shares of the option. We will borrow the money necessary to fund this position. The cost at time 0 is = 84. We borrow the $84 (none of our money is at stake). After one year our position is worth $90 (independent of the stock s up down movement). We repay the loan, handing over This gives us a profit of = = $1.80. Note that our profit was a certainty, independent of events. And, we might have scaled up our purchases, perhaps, purchased a thousand shares of stock and sold two thousand options. The reader may wish to work out the details when the options are underpriced at $6 per share (exercise 5).

6 6 CHAPTER 2. REPLICATION AND ARBITRAGE The Probabilistic Approach. Stock Price Tree 120 p p 90 Let s begin by thinking about this situation using one characteristic of real markets. We know the stock price is $100, the up price is $120, and the down price is $90. Suppose we were viewing real market behavior over a one year period. A reasonable choice of p is one where the expected return of the stock is on the order of 15%. This return is much larger than if we invested the $100 in a secure bank account, and the following calculations suggest why this is so. A p value that roughly matches this expected return is p =.90. This seems to produce an attractive situation. The expected payoff is given by E(P )=.9( ) +.1(90 100) = $17. Notice that the expected return each year is 17%. But there is still some uncertainty. Since only a probability of success is involved, the situation here is that 90% of the time you make $20 and 10% of the time you lose $10. Many investors would purchase stock under these circumstances. The healthy price increase of the stock would offset the occasional loss so that this investment choice is attractive to those who can afford the risk of some loss. However, each investor is different. How do we decide what is a reasonable amount of risk and reward for this stock? To circumvent this impossible task we introduce a hypothetical investor, hereafter designated by HI, who has the following characteristics: I. The HI is risk neutral; this is the oposite of a bird in the hand outlook. A risk neutral investor is risk indifferent; a certain dollar is no

7 2.1. PRICING AN OPTION A SPECIAL CASE 7 more preferable than an expected dollar. Most people are not risk neutral. The insurance industry is based on this fact. II. Our HI thus has no preference between the stock introduced above and a risk free investment. Given these assumptions, what value of p in our stock model will make the stock and the risk free return (.05) equally attractive to the investor? p 1-p 90 If we form a portfolio, Π, consisting of one share of stock then Π 0 = $100 and E[Π 1 ]=p120 + (1 p)90 =30p +90 after one year. If we had simply invested the $100 at the risk free rate, then the value of that selection would be $105 at the end of one year. Being risk neutral, our HI sees these investments as equal. That is, 30p + 90 = 105 which implies p =0.5. Important Caveat. The value we have just found does not necessarily correspond to investor sentiment, nor to some real but unknown probability associated with the stock. It is simply the probability which produces a stock return equivalent (in the hypothetical value sense) to the risk free return. Now, take this p value and use it to compute the expected value for a call option on the stock. Let C be our call price. Then E[C] =p( ) + +(1 p)(90 105) +

8 8 CHAPTER 2. REPLICATION AND ARBITRAGE = =$7.50. However, we pay for the call today and receive our payoff in one year, so we should discount the expected return on the call. We then arrive at the value E[C]/1.05 = $7.50/1.05 = $7.14. Astonishingly, this is exactly the price we calculated using the financial (arbitrage, replicating portfolio) method. 2.2 Replicating Portfolios We have encountered several powerful techniques in Section 2.1. We wish to try them out in a slightly more general environment. The Context. As before, we have a stock where S 0 is the stock price at t = 0 and the stock can achieve one of two possible values at time t = τ. S 0 S u S d V U D We also have a derivative security, V, whose value at t = τ will depend on the performance of S. IfS goes up, V will be worth U. IfS goes down, V will be worth D. What is a fair price for V today? At first glance this may seem to be an impossible question. We know almost nothing about V. It could be an option, but it could be some complicated combination of options, futures, swaps, or derived values. To price V, we will employ an ingenious device known as a replicating portfolio A Portfolio Match We need to fix one more item. The risk-free interest rate will be denoted by r and we make use of the short-term lending and borrowing possible at this rate. Let s record such amounts of money in terms of a bond by assuming

9 2.2. REPLICATING PORTFOLIOS 9 the bond is intially worth $1. Then the value of such a bond at time t is e rt. Our portfolio, Π, will consist of a units of the stock and b units of the bond. The value, Π 0, of the portfolio at time t = 0 is just Π 0 = as 0 + b. Let s compute the value of Π at t = τ. Our stock model gives two future values for the portfolio: Up State Down State Π τ = as u + be rτ Π τ = as d + be rτ We now set as u + be rτ = U as d + be rτ = D (2.1) so that the value of our portfolio, Π, is identical to the value of the derivative security. The portfolio replicates V. Since the portfolio and the derivative have the same value at t = τ, they should have the same value today. After all, they are indistinguishable at the next future date. We conclude that V 0 = as 0 + b. This expression for V 0 is rather remarkable when we solve for a and b using the equations (??). We have two linear equations whose solution is a = U D (2.2) and b = [U U D ] S u e rτ. Although these expressions look complicated, they produce simpler expressions for portfolio values. We focus on the expression for V 0 (which is just as 0 + b). V 0 = U D S 0 + [U U D ] S u e rτ, and we separate the U terms and the D terms to get [ S 0 V 0 = U + e rτ S u e rτ] [ S 0 + D + S u e rτ]

10 10 CHAPTER 2. REPLICATION AND ARBITRAGE [ e = e rτ rτ S 0 U S ] [ d + e rτ S u D erτ S ] 0. There is something very special here. The U coefficient, ignoring the exponential term, is q = erτ S 0 S d and the D coefficient, S u e rτ S 0, is just 1 q. So, our portfolio value does simplify to V 0 = e rτ [qu +(1 q)d]. (2.3) Expected Value Pricing Approach. It appears from equation (??) that the present value of a portfolio is obtained by discounting (e rτ is referred to as a discount factor) an expected value of the future portfolio values. Indeed, the formula for q would serve as some sort of probability if we could check that the condition 0 q 1 holds. Let s look at the value for q once more: q = erτ S 0 S d. (2.4) If q were negative, this stock would be a great buy; its worst value, S d,at the future date would exceed the return we would get by initially investing $ S 0 in the bond. Note that the bond return would be $ e rτ S 0. This is an example of a sure money-making scheme known as arbitrage, and we believe this is too good to be true in the real world. Equally unrealistic is the case that 1 q is negative. We see from the expression just before equation (??) that 1 q = S u e rτ S 0, and, if this were negative, then the stock would be a dud. In this case, the best future value of the stock, S u, is not as large as the return on an initial bond investment of $ S 0. There would be no reason to buy such a stock. Again, a real market would not support such stock behavior.

11 2.2. REPLICATING PORTFOLIOS 11 as This motivates us to rewrite our simple portfolio value in equation (??) V 0 = e rτ E q [V 1 ]. (2.5) The subscript in equation (??) will indicate that we are using the specially computed no-arbitrage pricing probability, q, given by (??) The No-arbitrage Pricing Probability We wish to emphasize that a good way to remember the q-value in the context of this section is to consider the most simple, one period portfolio to own: a single share of the stock. This corresponds to the tree S 0 S u q 1-q S d and thus (after discounting) e rτ S 0 = qs u +(1 q)s d. (2.6) When the U and D values are the (two) future stock values, equation (??) includes, as a special case, equation (??). The reader should take a few moments to solve this equation for q in terms of S 0, S u, and S d.heor she will appreciate the fact that equation (??) determines the no-arbitrage pricing probability, q. It will be important to remember one other quantity. In this section we have focussed on a portfolio that matches or replicates the results of some other equity. The key idea is to hold the correct number of shares of the stock. Our formula for this number of shares, equation (??), is just # of shares = U D (2.7) Notice that this ratio compares the change in the equity being replicated to the change in the stock price. This ratio is termed the delta quantity when it determines an investment process. You will see the pricing probability, q,

12 12 CHAPTER 2. REPLICATION AND ARBITRAGE and the delta quantity,, appear in many calculations of portfolio behavior later on. The techniques and results of this section are so elegant and powerful, one naturally wonders whether they can be extended to a more general situation. Suppose our stock took on just three values: up, middle, and down (a Goldilock s stock). Alas, there is no way to apply the foregoing approach without making truely ridiculous assumptions. We present the case in the next section. 2.3 Limits of the Binomial Arbitrage Method We have just seen that the replicating portfolio method, combined with the absence of arbitrage, exactly determines an option price. It converts what appears to be a stochastic situation into a deterministic one, in the binomial context. However, a stock that takes only one of two values isn t very realistic. Can we extend this approach to a stock that takes on three values? The answer is, no (unless we make unreasonable assumptions). Let s see why. S u S 0 S m S d V U M D Stock Tree Our portfolio will (as before) consist of a units of the stock S Option Tree b units of a bond (price is $1 at an interest rate r). the value of the portfolio at t =0is Π 0 = a S + b We now set the portfolio value at t = 1 equal to the option value for the three scenarios:

13 2.3. LIMITS OF THE BINOMIAL ARBITRAGE METHOD 13 Up Case Middle Case Down Case as u + be r = U as m + be r = M as d + be r = D We wish to solve for a and b. But, in general that is impossible since we have more equations to satisfy (three) than unknowns (two). To put it another way, we need a three dimensional solution space, and yet, what we have is two dimensional. However, all is not lost. Our observation suggests we should add another financial instrument. Let s begin by adding a bond with a different interest rate, R. Thus, our new portfolio consists of a units of the stock S b units of Bond One (price is $1 at an interest rate r). c units of Bond Two (price is $1 at an interest rate R). Again, we set up the portfolio value at t = 1 in the three scenarios: Up Case Middle Case Down Case as u + be r + ce R = U as m + be r + ce R = M as d + be r + ce R = D Now we have three equations and three unknowns; but, look at the second and third column on the left hand side. They are identical. So, again we cannot solve for a, b, c in general. Again, our solution space is only two dimensional (or the space of the column vector is two dimensional). And, we need three dimensions. We still have one more strategy. Let s add another stock, P, which also takes exactly three values, V u, V, and V d at t = 1. So, our new portfolio consists of a units of the stock b units of the bond (price is $1 at an interest rate r) c units of the stock P. We set up the portfolio value at t = 1 and we make the following:

14 14 CHAPTER 2. REPLICATION AND ARBITRAGE Enormous Assumption 1. When S S u ; P P u 2. when S S m ; P P m 3. When S S d ; P P d. Thus, the portfolio values in the three cases become as u + be r + cp u = U as m + be r + cp m = M as d + be r + cp d = D We now have three independent equations and three unknowns so we can solve for a, b, c. Having done so, we can determine the initial portfolio value. Indeed, Π 0 = a S + b + cp But look back at our Enormous Assumption. We have won a hollow victory. First, it doesn t seem reasonable that if we wish to price an option on Ford, we should be using the additional stock International Paper. But, second, it is even more unreasonable to assume that Ford and International Paper move up, down, or sideways in tandem. We don t have to assume that P u, P m, and P d occur in any particular order, but the linked, coupled, or tandem movement is just too much to swallow. 2.4 Repeated Binomial Trees and Arbitrage In the next chapter we look at repeated binomial trees in detail. But, let s look at the two step case before moving on. We take our binomial tree when S 1 = XS 0 and X is a random variable and takes either the values u or d. So, our tree becomes

15 2.4. REPEATED BINOMIAL TREES AND ARBITRAGE 15 S 0 us 0 ds 0 u 2 S 0 uds 0 dus 0 d 2 S 0 Since u d = d u, we can simplify our tree by representing it as: u 2 S 0 us 0 S 0 uds 0 ds 0 Time d 2 S 0 Can we use a replicating portfolio to price an option by arbitrage in this two step process? More precisely, can we make the outcome deterministic? Can we protect the seller of the option from any loss? The answer is no, but since this limitation is so important, let s look at this in detail. If we disregard the intermediate time step we have a trinomial (three outcome) situation, and the comments from the previous discussion apply. But, here we have a lot more information via the intermediate step. Can that work to our advantage?

16 16 CHAPTER 2. REPLICATION AND ARBITRAGE Step 1 We first use the replicating portfolio method (RPM) to obtain a sure-thing deterministic price for the portfolio at time t = 1. Step 2 We would like to use the RPM again to price the option at t =2. But there is a problem. Are we now in the up state or the down state? Since the state at t = 1 is the result of a random process, we don t know the answer at t = 1. Moreover, the price we should assign to the option depends on the state at t = 1. let s illustrate this with a concrete example based on our earlier example. Let The tree for the process is: S 0 = 100 u =1.2 d =.9 r =.05 (discrete) Strike Price for Option = 110 Option Payoff at t = Time Using RPM or the Expectation Method, we arrive at the deterministic price of $4.76 for the portfolio at t =1. Step 2 We wish to price the option (deterministically if possible) from t =1 to t = 2. But look what happens. If we are in the Down state the option value or price is 0. it can never rise above the strike price.

17 2.4. REPEATED BINOMIAL TREES AND ARBITRAGE 17 If we are in the Up state, the option clearly has some value, and if we work through the numbers, we find that is $ The $16.19 is unimportant. What is important is the fact that we don t know in advance just which state we will be in at t = 1. There is no possible way to determine our state at t = 1 ahead of the event. We can take some sort of a weighted average over the up state and the down state, and that is precisely what we will do in the next chapter. However, the main point of this section declares that we can not deterministically price even a binomial option in the two step case.