ESTM 60202: Financial Mathematics Alex Himonas 03 Lecture Notes 1 October 7, 2009 1 Introduction to Option Pricing We begin by defining the needed finance terms. Stock is a certificate of ownership of a company. A share is a unit of stock. At any given time, t = 0, the stock value,, is given by = Market value of company Number of shares of company Example 1. If a company today has 40 million shares and the value of each share is $50, then the market value of the company is $50 40 million or $2 billion. European call option gives the owner the right but not the obligation to buy a share of stock at a specified price K, which is exercised only on the future date T stated in the contract. The time T, at which the option can be exercised, is called the maturity date. The predetermined price K is called the strike price. Example 2. Today the value of a share of a given company is = $50. Buying one European call option with maturity date T = 1 and strike price K = $70 means that a year later you have the right to buy a share of stock. If S 1 > 70 then you would exercise the option. While, if S 1 70 then you would let the option expire. Your profits would be [S 1 70] + = { S1 70 if S 1 70 > 0, 0 if S 1 70 0. The Question: What is a fair price (not strike price) of a European call option? (1.1) 1.1 The one-period binomial option pricing model Imagine the following situation: You live in an (economic) world which consists of stocks (all of the same kind) and money (dollars) deposited in a bank or borrowed from a bank. The bank charges you the interest rate r when you borrow money from the bank and pays you the same interest rate r when you deposit money in the bank. Question. Think about how the development below changes in the more realistic case in which the lending rate is above the deposit rate. Let us assume that at the time t = 0 the value of a share of the stock is (say $50 as in Example 2). One period later (say 1 year) the stock can take one of the following values: S 1 (H) or S 1 (T ) (say S 1 (H) = 100 or S 1 (T ) = 25) as is indicated in the following diagram: 1 Based on joint notes with Tom Cosimano 1
p S 1 (H) 1 - p S 1 (T) You can imagine that at time t = 1 the price of the stock is determined by a weighted coin toss. That is, this coin is tossed and if the outcome is H then the price is S 1 (H), while if the outcome is T then the price is S 1 (T ). (In probabilistic terms S 1 is a random variable.) Let us denote this by We can always assume Otherwise, we just change notation. p = P (H) and 1 p = P (T ). 0 < S 1 (T ) < S 1 (H). Question. Why do we assume that the stock price is always positive? Since S 1 (H) is the up stock value and S 1 (T ) is the down stock value it is helpful to introduce the terminology Since u = S 1(H) = up factor, and d = S 1(T ) = down factor. 0 < S 1 (T ) < S 1 (H) it follows that 0 < d < u. The interest rate for this period in the economy is denoted by r (say r = 0.05 or 5%). Therefore, we have that $1 deposited in the bank at time t = 0 becomes 1 + r at t = 1. In other words we have the following diagram: p 1 + r $1 1 - p 1 + r We now introduce a fundamental idea in finance. First define the following: Arbitrage is a trading strategy which: 1. Begins with zero money. 2
2. Has a zero probability of losing money. 3. Has positive probability of making money. No Arbitrage Axiom (NA). It is impossible to have a trading strategy which can turn nothing into something without running the risk of a loss. That is, there is no arbitrage. Lemma 1.1 If there is no arbitrage (NA) then d < 1 + r < u. (1.2) Proof. (By contradiction) First we prove that d < 1 + r. If the opposite is true, that is d 1 + r, then at t = 0 we could borrow from the bank dollars to buy one share of stock. Then no matter what is the outcome of the coin toss at time t = 1 our stock will be worth S 1 d, while we will need (1 + r) to pay off the loan from the bank. Therefore with a certain probability of 1 we will be making a profit of S 1 (1 + r) d (1 + r) = [d (1 + r)] 0, since d 1 + r. Also, with probability p > 0 we will make a profit equal to (1.3) S 1 (H) (1 + r) = u (1 + r) () > 0, since u > d. Thus, we have produced a strategy which begins with zero money, has a zero probability of losing money, and positive probability of making money. This violates the NA axiom. Therefore, inequality d 1 + r is false. Next we prove that 1 + r < u, again by contradiction. If 1 + r u, then at t = 0 we could borrow a share of stock (this is called short selling a stock), sell it for dollars, and invest the proceeds in a bank paying interest r. At t = 1 this deposit at the bank would grow to (1 + r). At this time to replace the stock borrowed at time t = 0, in the worst situation it would cost u. Therefore, with a certain probability of 1 this strategy produces a gain of (1.4) (1 + r) S 1 (1 + r) u = [(1 + r) u] 0, since 1 + r u. Also, with probability 1 p > 0 we will make a profit equal to (1.5) (1 + r) S 1 (T ) = (1 + r) d () > 0, since u > d. Thus, again we have produced a strategy which violates the NA axiom. Remark. The reason why the NA axiom comes about follows from this proof. If you had the inequality d 1 + r, then companies would find it profitable to buy lots of stock at time 0 with price. As this occurs the price would be bid up. As a result, both u and d would go down. Thus, the profit motive or the fact that people and corporations are greedy leads to the NA axiom in finance. 3 (1.6)
1.2 Computing the price of a European call option Here we assume the 1-period binomial model satisfying the NA axiom 0 < d < 1 + r < u. (1.7) In this case it is natural to require that the strike price K satisfies the condition S 1 (T ) < K < S 1 (H). (1.8) If we denote by V 0 the unknown price of the option at time t = 0 and V 1 (H) and V 1 (T ) the two possible payoffs for the European call option at time t = 1, then we have the following diagram for this option. p V 1 (H) = S 1 (H) - K V 0? 1 - p V 1 (T) = 0 Now we are ready to state our main result which is the Black-Scholes option pricing formulas in our one period two state world. Theorem 1.2 [Binomial Black-Scholes formula] The price V 0 one-period binomial model is given by of a European call option in the where (1 + r) d p =, and q = and V 1 (H) and V 1 (T ) are the two possible payoffs. V 0 = 1 1 + r [ p V 1(H) + q V 1 (T )], (1.9) Remark 1. The probability p does not appear explicitly in the formula. u (1 + r), (1.10) Exercise 1. Can you think of how p enters into the pricing of the call option? Remark 2. Note that both p and q are positive and that p + q = 1. They look like probabilities and are called risk-neutral probabilities. Exercise 2. Think why p and q are called risk-neutral probabilities. Also, explain why p and q = 1 p must satisfy the inequality < 1 [ ] p S 1 (H) + q S 1 (T ). (1.11) 1 + r 4
Example 3. If = 50, S 1 (H) = 100, S 1 (T ) = 25, K = 70 and r = 0.05, then we have The possible payoffs are Also, and u = S 1(H) = 100 50 = 2, and d = S 1(T ) = 25 50 = 1 2 = 0.5. V 1 (H) = S 1 (H) K = 100 70 = 30, and V 1 (T ) = 0. p = q = Therefore, applying formula (1.9) gives V 0 = 1 1 + 0.05 1 + 0.05 0.5 2 0.5 2 (1 + 0.05) 2 0.5 [ 55 95 30 + 150 150 0 = 0.55 1.50 = 55 150, = 0.95 1.50 = 95 150. ] = 11 1.05 10.48 dollars. Next, we provide the terminology needed in the proof of Theorem 1.2. Portfolio is a combination of assets held and liabilities incurred. Option Replication: Finding the correct initial wealth X 0, in money, and the number 0 of shares of stock that we need to buy in order to replicate a European call option. Proof of Theorem 1.2. Let us assume that we begin with a wealth of X 0 dollars and that we buy 0 shares of stock. Then at time t = 0 we hold a portfolio with: 1. Cash position deposited in the bank X 0 0. 2. 0 shares of stocks. The value of this portfolio at time t = 1 is equal to X 1 = 0 S 1 + (1 + r)(x 0 0 ). Therefore, we have the diagram of payoffs from the wealth invested in the portfolio. p X 1 (H) X 0? 1 - p X 1 (T) 5
Replicating the option would create the same payoffs as the option. As a result, we want to choose X 0 and 0 such that X 1 (H) = V 1 (H) and X 1 (T ) = V 1 (T ). Using the formula for X 1, these two equations take the form 0 S 1 (H) + (1 + r)(x 0 0 ) = V 1 (H), 0 S 1 (T ) + (1 + r)(x 0 0 ) = V 1 (T ). After separating the X 0 from the 0 terms, we have X 0 + [ 1 S ] 1+r 1(H) 0 = 1 V 1+r 1(H), X 0 + [ 1 1+r S 1(T ) ] 0 = 1 1+r V 1(T ). Subtracting the second from the first equation gives 1 1 + r [S 1(H) S 1 (T )] 0 = 1 1 + r [V 1(H) V 1 (T )]. Solving for 0 we obtain the so called delta-hedging formula: (1.12) 0 = V 1(H) V 1 (T ) S 1 (H) S 1 (T ). (1.13) Next, substituting in the first equation of system (1.12) formula (1.13) gives [ ] 1 X 0 + 1 + r S 1(H) V1(H) V 1 (T ) S 1 (H) S 1 (T ) = 1 1 + r V 1(H). Then, letting S 1 (H) = u and S 1 (T ) = d we obtain [ ] 1 X 0 + 1 + r u 1 V1(H) V 1 (T ) () = 1 1 + r V 1(H). Eliminating and solving for X 0 gives X 0 = 1 [ 1 + r V 1(H) + 1 1 ] 1 + r u V1(H) V 1 (T ) = 1 [ V 1 (H) + (1 + r u) V1(H) ] V 1 (T ) 1 + r = 1 [( 1 + 1 + r u ) ( ) ] 1 + r u V 1 (H) V 1 (T ), 1 + r or X 0 = 1 [ ] (1 + r) d u (1 + r) V 1 (H) + V 1 (T ). (1.14) 1 + r This value of X 0 must be equal to V 0, the desired price of the European option. This completes the proof of the theorem. 6
Example 3. (Continued) In the case of Example 3, using delta-hedging formula ( 1.13), we find that 0 = 30 0 100 25 = 2 5. That is, to replicate this option we must buy 2/5 shares of stock. This will cost 0 = (2/5) 50 = 20 dollars. Therefore, our cash position is X 0 0 = (11/1.05) 20 9.52, which means we need to borrow 20 (11/1.05) dollars from the bank at the interest rate of r = 0.05. Exercise 3. Find the price of a European option for a stoch whose value today is $40 a share, its up value is $60 and its down value is $30, its strike price a year from now is $50, the annual interest rate is 4% and the probability of the good state is 0.6. Exercise 4. For a stock in a binomial world assume that = 100, S 1 (H) = 140, S 1 (T ) = 75, and r = 0.08. What is the price of a novel call option for this stock which at t = 1 has payoffs V 1 (H) = 25 and V 1 (T ) = 5. Exercise 5. Derive a formula for the price of a European put option in the case of the one-period binomial option pricing model. Exercise 6. Find the price of a European put option for a stock whose value today is $100 a share, its up value is $150 and its down value is $75, its strike price a year from now is $125, and the annual interest rate is 5%. Unresolved Issues 1. In the Theorem the formula for the option price is independent of the probabilities of the ups and down movements in stock. This result is puzzling since the purpose of the option is to insure the purchaser against events they do not want to occur. We will see that the probabilities are related to the variables u = S 1(H) and d = S 1(T ). The reason for this is that the stock price today is dependent on the probability of the good and the bad state. For example, if there is a 100% chance of the high price next period, then the stock price today would be much higher relative to a 100% chance of the low price next period. Consequently, taking the stock price today as given implicitly assumes a given probability of an up and down move. In future classes we will show this explicitly. 2. We have not discussed why someone would want to buy or sell a European call option. A buyer would want to use the call option to insure against a particular change in prices. For example suppose you run an Airline in which your cost of operation is dependent on oil prices since jet fuel prices go up whenever the price of oil goes up. Now suppose the call option the airline bought pays off when the price of oil goes up. For example it could be a call option on BP stock. Now when the price of oil goes up, your airline would face higher jet fuel cost. On the other hand they make a bigger payoff on the option written on BP. Thus, the call option reduces fluctuations in the profits of the airline company when the price of oil goes up. If the airlines do not like uncertainty, then they are made better off. This opens the question as to why someone would sell a call option. It turns out that an oil drilling company may also want to avoid fluctuations in oil prices since they make their money mainly from producing oil and they wish to avoid uncertainty. As a result, the oil drilling company (say BP) would sell a call option on its oil. When the price of oil goes down below a certain level, the call option on BP stock is not exercised and so BP make a profit V 0 per share. As a result, the 7
decreased value of oil for the oil drilling company is canceled (partially?) by the profit of the call options sold. Thus, the call option also insures the oil drilling company against fluctuations in the price of oil. Why this works will be discussed in more detail as we go through the semester.... 8