Expected payoff = 1 2 0 + 1 20 = 10.

Chapter 2 Options 1 European Call Options To consolidate our concept on European call options, let us consider how one can calculate the price of an option under very simple assumptions. Recall that the terminal payoff of a call option is c(t ) = max(s(t ) E, 0) = (S(T ) E) +. Example 1. Say, today s date is January 11, and on May 11, the holder of a European call option may purchase one XYZ share for $250 by exercising his call option. Assume that there are only two possible situations that might occur on the expiry date, May 11, four months in the future; namely S(T ) = 270 or S(T ) = 230. If the XYZ share price is $270 on May 11, then the holder will be able to buy the asset for only $250. This exercising of the option yields an immediate payoff of $270 $250 = $20. On the other hand, if the XYZ share price is only $230 on May 11, then it would not be sensible to exercise the option. Suppose with equal probabilities the XYZ share price takes the values $230 or $270 on May 11. The expected payoff is given by: Expected payoff = 1 2 0 + 1 20 = 10. 2 Forget about the interest rate for the moment. It seems reasonable that the value of the option should be around $10. From this simple example, we can see why someone would prefer to buy an option rather than the underlying asset. Suppose that the holder does pay $10 premium for this option, and if the share price rises to $270 at T, then he has made a net profit of $10 ($20 payoff on exercising less $10 the cost of the option). Observe that this net profit of $10 is 100% of the up-front premium. However, the downside of this speculation is that if the share price is less than $250 at expiry date T, the loss is also $10, which is 100% of what he invested in the option. Say, if the investor purchases instead the share for $250 on January 11, the corresponding profit or loss of $20 will be only 20/250 = 8% of the original investment. Option prices thus respond in an exaggerated way to changes in the underlying asset price. This effect is called gearing or leverage. The price of an option is made up of two components: its intrinsic value and its time value. The intrinsic value is the payoff that would be received if the underlying asset is at its current level when the option expires. That is, the intrinsic value of a

14 MAT4210 Notes by R. Chan call option is defined as max(s(t) E, 0). Time value is any value that the option has above its intrinsic value, i.e. max(c(t) max(s(t) E, 0), 0). The uncertainty surrounding the future value of the underlying asset means that the option value is generally different from the intrinsic value. Options are referred to as in the money, at the money, or out of the money. An in-the-money option is one that would lead to a positive cash flow to the holder if it were exercised immediately, i.e., an option is in the money when it has a positive intrinsic value. Similarly, an at-the-money option would lead to zero cash flow if it were exercised immediately, and an out-of-the-money option would lead to a negative cash flow if it were exercised immediately. For a call option, it is in the money when S(t) > E, at the money when S(t) = E, and out of the money when S(t) < E. 2 Put Options The option to buy an asset is known as a call option. The right to sell an asset is known as a put option, and its payoff properties are opposite to those of a call. In other words, a put option gives the holder a right to sell a share of stock in the future for a guaranteed price the strick price. A European put option allows its holder to sell the asset only on the exercise date at a prescribed strike price if the holder chooses to exercise the option. The writer is then obliged to buy the asset at the prescribed strick price. More precisely, the buyer of the option pays the seller of the option a fee called the premium. On the expiration date, the holder of this contract may give the writer a share of stock or, equivalently, the market price of a share of stock. In other words, the holder of this put option does have a choice. When the stock price on the expiration date is higher than the exercise price, the holder of this contract can do nothing. He or she does not have to give up a share of stock. In contrast, if the holder of a put see a lower stock price at expiration, then he or she selects to obtain the fee (i.e., the exercise price) from the writer and use some of it to obtain a share of stock from the market. The rest is profit. If the writer of the contract receives the share or its price from the holder, the writer must pay the exercise price to the holder on the expiration date. Obviously, the holder of a call option wants the asset price to rise the higher the asset price at expiry the greater the profit; whereas the holder of a put option wants the asset price to fall as low as possible. Again, a put option has some value. We denote the price of a put option by p(t) = p(s(t), E, τ, r,...), (1) where τ = T t is the remaining time of the option and r is the interest rate. Example 2. Refer to Example 1 again, but change the call into the put. What is the payoff of a put option at May 11, the expiry date T? Note that E = 250. If S(T ) = 230, then the holder of this put option is allowed to sell the share at the price E = 250, which is higher than S(T ) = 230. The payoff (ignoring other factors such as interest rate, tax, transaction costs, etc.) is E S(T ) = 20. Suppose we do not own the share. What can we do? Here is the details: at T once we see S(T ) = 230 < E, we buy 1 share at the market price S(T ) = 230 and sell it at E = 250. So we make a profit of $20. Suppose that at T the share price rises to $270. Then we, the holder of the put option, can choose to do nothing, i.e., not exercising the put option!

Options 15 From this simple example, we arrive at the following conclusion: at the expiry date T, the value of a European put option is p(t ) = max (E S(T ), 0) = (E S(T )) + { E S(T ), if S(T ) < E, = 0, if S(T ) E. (2) In comparison, the payoff for a call option at expiry date is given by c(t ) = max (S(T ) E, 0) = (S(T ) E) + { S(T ) E, if E < S(T ), = 0, if S(T ) E. (3) A put option can serve as a protective insurance. For illustration, see the following example. Example 3. Suppose that an investor owns a substantial amount of telecommunication stocks of XYZ company, which is now trading at $50 per share. The investor believes that the stock price is likely to vary widely in the months ahead, and hence he wishes to begin selling this stock soon. But since he has a substantial amount of the stocks, by selling them, the stock price is likely to plumbet. What can the investor do? Well, he can begin a program of buying puts of approximately three months to expiration, say, with strike price $45. He has to pay an option premium, which, for such a case, costs $2.80 for each put. This strategy protects the investor in the following way: a future low stock price will cause the puts to be exercised, and hence it will allow the investor to obtain at least $45 for each share called away by a put. If the stock price stays above the $45 threshold, the put expires worthless. So, in any case, the investor is guaranteed a minimum price for shares of the stock he sells, as long as he owns puts for these shares. The $2.80 expense for each put may be regarded as an insurance premium that guarantees the ability to sell some of the investor s stock at or above the level $45. Computing or estimating the price of a put is as difficult as that for a call. As in the case of a call price, the computed price will depend on which mathematical model we wish to use to describe the underlying stock behavior. Before 1973 all option contracts were what is now called over-the-counter (OTC). That is, they were individually negotiated by a broker on behalf of two clients, one being the buyer and the other the seller. Trading on an official exchange began in 1973 on the Chicago Board Options Exchange (CBOE), with trading initially only in call options on some of the most heavily traded stocks. As increased competition followed the listing of options on an exchange, the cost of the setting up an option contract deceased significantly. Options are now traded on all of the world s major exchanges. They are no longer restricted to equity options but include options on indices, futures, government bonds, commodities, currencies etc. The OTC market still exists, and options are written by institutions to meet a client s needs. This is where exotic option contracts are created; they are very rarely quoted on an exchange. Many options are registered and settled via a clearing house. This central body is also responsible for the collection of the margin from the writers of options. Recall

16 MAT4210 Notes by R. Chan that margin is a sum of money which is held by the clearing house on behalf of the writer. It is a guarantee that he is able to meet his obligation should the asset price move against him. 3 Long and Short The term short sell means that we borrow shares and sell them out to the market immediately, and in the future we buy back the same amount of shares and return them. This is a trading strategy that yields a profit when the price of a security goes down and a loss when it goes up. To explain the mechanics of short selling, we suppose that an investor contacts a broker to short 500 IBM shares. The broker immediately borrows 500 IBM shares from another client and sells them in the open market in the usual way, depositing the sale proceeds to the investor s account. Providing there are shares that can be borrowed, the investor can continue to maintain the short position for as long as desired. At some stage, however, the investor will choose to instruct the broker to close out the position. The broker then uses funds in the investor s account to purchase 500 IBM shares and replaces them in the account of the client from which the shares were borrowed. This investor makes a profit if the stock price has declined and a loss if it has risen. If at any time while the contract is open, the broker runs out of shares to borrow, the investor is what is known as short-squeezed and must close out the position immediately even though he or she may not be ready to do so. In short, here are some important procedures in a short sale: One borrows a concrete, specific number of shares of a stock (usually from a broker) and sells these shares today. The date on which the borrowed shares must be replaced is not specified. If the owner of the borrowed shares decides to sell, then the short seller must borrow other shares and replace the first borrowed shares. Regulators currently only allows shares to be sold on an uptick, i.e., when the most recent movement in the price of the security was an increase. A broker requires significant initial margins from clients with short positions. An investor with a short position must pay to his or her broker any income, such as dividends or interest, that would normally be received on the securities that have been shorted. The broker will transfer this to the account of the client from whom the securities have been borrowed. Example 4. Consider the position of an investor who shorts 500 IBM shares in July when the price per share is $120 and closes out his or her position by buying them back in October when the price per share is $100. Suppose that a dividend of $4 per share is paid in August. The investor receives 500 $120 = $60, 000 in July when the short position is initiated. The dividend leads to a payment by the investor of 500 $4 = $2, 000 in August. The investor also pays 500 $100 = $50, 000 when the position is closed out in October. The net gain is, therefore, $60, 000 $2, 000 $50, 000 = $8, 000. Observe that, using short sell, an investor can make profits when the share price falls. But the shortcoming of short sell is that the risk is unlimited. Say, the asset price is S(t) at time t when an investor is selling short this asset. If the asset price

Options 17 rises to S(T ) > S(t) at time T, the investor makes a loss of S(T ) S(t), which may possibly go to. In this sense, we say the risk is unlimited. Note that the opposite of a short position is a long position. It is equivalent to the simpler and frequently used word buying. There are two sides to every option contract. On one side is the trader who has taken the long position (i.e., has bought the option). On the other side is the trader who has taken a short position (i.e., has sold or written the option). The writer of an option receives cash up front but has potential liabilities later. The writer s profit or loss is the reverse of that for the purchaser of the option. Four basic option positions are possible: (i) a long position in a call option, (ii) a long position in a put option, (iii) a short position in a call option, and (iv) a short position in a put option. The payoff functions of the options are given in Figure 1. Note that the premia of the options are not considered in the figures. Payoff Payoff Payoff Short Call Payoff Short Put E S E S E S E S Long Call Long Put Figure 1. Payoff functions of options at expiry date. I should emphasize that selling an option is equivalent to shorting an option, so we can say without ambiguity that we sell, short or short sell an option. An intuitive reason is that selling an option binds one for a future obligation, like shorting a stock. To see that clearly suppose Mr. Seller has sold an option with exercise price E and expiry T to Mr. Buyer. Then Mr. Buyer in principle will have a certificate stating that he can exercise the option with Mr. Seller for one stock at the price of E dollar at expiry T. Now suppose I would like to short sell an option to Mr. Trustme. Since I don t have any option in hand, I borrow the option (or certificate) from Mr. Buyer and sell it to Mr. Trustme for c dollars say. Notice that with the certificate, if Mr. Trustme wants to exercise the option at T, he can claim that with Mr. Seller and not with me (because that is what the certificate has stated he can exercise it with Mr. Seller). There is no deals between me and Mr. Trustme whatsoever. Yes, I get the c dollars from Mr. Trustme, but my liability is to Mr. Buyer, and not to Mr. Trustme. Now if Mr. Buyer wants to exercise his option at expiry T, he will come to ask me for the payoff because I have borrowed the option from him. With all said and done, it is clear that when I short sell an option to Mr. Trustme with the option from Mr. Buyer for c dollars, it is equivalent to I sell the option (or a certificate) to Mr. Buyer for c dollars. 4 Why Would Anyone Buy Options? Options have two primary uses: (i) speculation, and (ii) hedging. For speculators, options are cheap ways to rip off large profits or if they are unlucky, large losses. For instance, an investor who believes that a particular stock, XYZ share say, is going to rise can purchase some shares in that company. If he is correct, he makes money, if he

18 MAT4210 Notes by R. Chan is wrong, he loses money. The investor is speculating. Refer to Example 1 again. We have noted, if the share price rises from $250 to $270 he makes a profit of $20 or 8%. If it falls to $230, he makes a loss of $20 or 8%. Alternatively, suppose that the investor thinks that the share price is going to rise in about 4 months time and that he buys a call with exercise price $250 and expiry date in 4 months time, say. We have seen that, if such an option costs $10, then the profit or loss is magnified to 100%. In short, options can be a cheap way of exposing one s investment plan to a large amount of risk. If, on the other hand, an investor thinks that XYZ shares are going to fall he can, conversely, sell shares or buy puts. If he speculates by short selling then he will profit from a fall in XYZ shares. The same argument concerning the exaggerated movement of the option prices applies to puts as well as calls, and if he wants to speculate, he may decide to buy puts instead of selling the asset. However, as in Example 3, suppose that the investor already owns XYZ shares as a long-term investment. In this case he might wish to insure against a temporary fall in the share price, while being reluctant to liquidate his XYZ holdings only to buy them back again later, possibly at a higher price if his view of the share price is wrong, and certainly having incurred some transaction costs on the two deals. Hedging is the opposite to speculation. It is the avoidance of risk. Risk is commonly described as being of two types: specific and non-specific. The latter is also called market or systematic risk. Specific risk is the component of risk associated with a single asset (or a sector of the market, e.g., electronic), whereas non-specific risk is associated with factors affecting the whole market. An unstable management would affect an individual company but not the market; the company would show signs of specific risk, a highly volatile share price perhaps. On the other hand, the possibility of a change in interest rates would be a non-specific risk, as such a change would affect the market as a whole. A popular definition of the risk of an investment is the variance of the return. A bank account which has a guaranteed return has no variance and is thus termed risk-less or risk-free. On the other hand, a highly volatile stock with a very uncertain return and thus a large variance is a risky asset. This is perhaps the simplest and commonest definition of risk, but it does not take into account the distribution of the return, but rather only one of its properties, the variance. There are, generally speaking, three ways of transferring risk to others: hedging, insuring, and diversifying. Insuring and diversifying are easy to understand. We have illustrated hedging by Examples 1.2 and 1.5. In the examples, the call option on British pounds is kind of like an insurance. You pay a premium and then yours HK$15,600 is guarded against the fluctuation of the exchange rate. In a more general setting, hedging is used to guarantee that the lost in one financial product is compensated by the gain of another financial product. As an example, observe that the value of a put option p(t) rises when the underlying asset price S(t) falls. What happens to the value of a portfolio (i.e. an investment plan) containing both assets and puts? The answer depends on the ratio of assets and options in the portfolio. A portfolio that contains only assets falls when the asset price falls, while one that is all put options rises. Somewhere in between these two extremes is a ratio at which a small unpredictable movement in the asset does not result in any unpredictable movement in the value of the portfolio. This ratio is instantaneously risk-free. The reduction of risk by taking advantage of such correlations between the asset and option price

Options 19 movements is called hedging. This idea is central to the theory and practice of option pricing. To illustrate, let us consider a portfolio: Buy x units of XYZ shares at S(t) and buy y units of put on the XYZ shares at p(t). The portfolio value, denoted by Π(t), is equal to Π(t) = x S(t) + y p(t). When S(t) increases, p(t) decreases; when S(t) decreases, p(t) increases. A natural question one would like to ask is for which (x, y) the risk of the above portfolio is minimized. The activity of finding such (x, y) is called hedging. 5 Why Would Anyone Write an Option? The discussion so far has been from the point of view of the holder of an option. Let us now consider the position of the other party to the contract, the writer. While the holder of a call option has the possibility of an arbitrarily large payoff, with the loss limited to the initial premium, the writer has the possibility of an arbitrarily large loss, with the profit limited to the initial premium. Similarly, but to a lesser extent, writing a put option exposes the writer to large potential losses for a profit limited to the initial premium. One would therefore ask Why would anyone write an option? The first likely answer is that the writer of an option expects to make a profit by taking a view on the market. Writers of a calls are, in effect, taking a short position in the underlying: they expect its value to fall. It is usually argued that such people must be present in the market, for if everyone expected the value of a particular asset to rise its market price would be higher than, in fact, it is. Similarly, there must also be people who believe that the value of the underlying will rise or the price would be lower than, in fact, it is. An extension of this argument is that writers of the options are using them as insurance against adverse movements in the underlying, in the same way as holders do. Although this motivation is plausible, it is not the whole story. Market makers, such as banks or securities firms, have to make a living, and in doing so they cannot necessarily afford to bear the risk of taking exposed positions. Instead, their profit comes from selling at slightly above the true value and buying at slightly below; the less risk associated with this policy, the better. If a market maker can sell an option for more than it is worth and then hedge away all the risk for the rest of the option s life, he has locked in a guaranteed, risk-free profit. This idea of reducing or eliminating risk brings us back to the subject of hedging. We explain it by another example. Example 5. Suppose a market maker has $100 to invest for the next three months. He decides to use the money to buy a stock at S(t) = 100. Then he will lose money if the stock price falls, see Figure 2 (left). To protected against the possible lost, he can buy a put as in Example 3. But it is more fun to write a call option say with exercise price $105 and expiry date three months from now. The option is out-of-money now, but it still has the time value. Say the fair price for the option is $3 and the market maker can sell it above the fair price at $4. Then the payoff at expiry for the market maker is given in Figure 2 (right). Notice that by writing a call, the market maker 1. is insured from the downside with $4 to spare; i.e., the breakeven point is shifted from $100 down to $96, and 2. earns more than he should if S(T ) $109.

20 MAT4210 Notes by R. Chan Of course, the market maker has forfeited part of the upside when S(T ) > $109, he would earn less than he should have if he had not written the call. 10 8 6 4 Long Stock 10 8 6 4 Payoff 2 0 2 4 6 8 Payoff 2 0 2 4 6 8 Long Stock and Short Call 10 90 95 100 105 110 Stock Price 10 90 95 100 105 110 Stock Price Figure 2. Payoff functions for different portfoilos. What if S(T ) < $96? The market maker will lose money eventually. One way to avoid that is to sell the stock whenever S(t) drops below $96 and buy it back when it gets back to $96. If there is no transaction cost and there is no sudden jump in the stock price, then the market maker is completely protected against the downside. 6 Portfolios A portfolio is a combination of buying and selling certain financial products. That is, a portfolio is an investment plan. The value of a portfolio, denoted by Π(t), is the sum of the values of all financial products within the portfolio. This will also be the amount of money a third party paid to the holder of the portfolio so that the holder can transfer the ownership of the portfolio to the third party. For instance, if my portfolio consists of holding 1 unit of share with the price S(t) at time t and I wish to get rid of the portfolio, I can find someone in the market to pay me S(t). After that he becomes the holder of the portfolio and I have got rid of the portfolio. Here, the value of the portfolio is just S(t). More precisely, if Π(t) > 0, then you need to pay Π(t) to own it (consider the portfolio as a handbag of price Π(t), and you want to own it). If Π(t) < 0, then you should pay someone Π(t) to get rid of it (consider the portfolio as a trash-bag that you want to throw away then you should pay the garbage collector to take it away). Example 6. Let us look at the following simple portfolios: (i) Consider a portfolio consisting of just 1 share at price S(t). The value of this portfolio is Π(t) = S(t), which is positive. That means you need money to set up (i.e. own) such a portfolio, and earn money when selling it. (ii) You have just borrowed E amount of money from the bank, then Π(t) = E. The portfolio in this case is the mortgage payment agreement. Its value is negative as you earn money when setting it up, but you need to pay money to disown it. The value of the portfolio will become more negative as time goes by because of the mortgage interest. The negative in value means that you owe some obligation by having or setting up this portfolio.

Options 21 (iii) You have just short sold 1 share at price S(t). If you hold this portfolio and you want to transfer the ownership of it to a third party. Then you have to pay the third party S(t). Namely, the third party has to pay me S(t) in order to transfer the ownership. So in this case, the value for this portfolio is Π(t) = S(t). Again the negative portfolio price means that the holder owes some obligation. If I am getting this portfolio, I am getting your obligation. Hence I should get money from you when I get this portfolio. (iv) You have just bought 1 share at price S(t) and 1 call option c(t). The value for this portfolio at t is Π(t) = S(t)+c(t). The value of this portfolio is positive. That means you need money to set this up. (v) As in Example 5, you have just bought 1 share at price S(t) and sold (or written) 1 call option c(t). The value for this portfolio at t is Π(t) = S(t) c(t). One important point to remember is if Π(t) > 0, like Example 7 (i), then there is a positive cash flow to the holder if he wants to sell it, and there is a negative cash flow to the buyer or anyone wanting to set up such a portfolio, i.e. one has to pay to own or set up the portfolio. On the other hand, if Π(t) < 0, like Example 7 (ii), then there is a negative cash flow for the holder if he wants to sell it, i.e. he has to pay money to the buyer in order to sell it. However, there will be a positive cash flow to the buyer of such portfolio, or more precisely, one earns money in order to buy or set up such a portfolio. Example 7. Consider this more useful example. Suppose that a local XYZ company is negotiating a joint venture project with PCCW in January, of which the outcome and the details of the contract will be made known to investors in late September. It is predicted that the XYZ share price will either rise or fall by more than 10% after the announcement is made. Say, today s XYZ share price is $10, and European calls and puts are available with expiry dates in April, July, October, December and with strike prices of $10 plus or minus $0.5, respectively. Under such a circumstance, what portfolio of options will do well? Consider buying a European call c with strike price at $9.50, maturity in October; and a European put p with strike at $10.50 at the same expiry date. Such a portfolio (with a call and a put on the same expiry date) is called a strangle. The value of this portfolio is Π(t) = c(s(t), 9.50, T t, r) + p(s(t), 10.50, T t, r). At T (in October), the terminal payoff is given by Π(T ) = max (S(T ) 9.50, 0) + max (10.50 S(T ), 0) 10.50 S(T ), if S(T ) 9.50, = 1, if 9.50 < S(T ) 10.50 S(T ) 9.50, if S(T ) > 10.50. It is plotted in Figure 3, where again we do not consider the premium of buying this portfolio upfront, which should be something more than $1 if the interest rate is zero (why?). We see that this portfolio speculates a big swing in the stock price far above or below its current price of $10. What if we buy one call with exercise price at $10.5 and one put at $9.5 instead? Can you draw the payoff diagram as in 3? What can you say about the premium in this case? We will study strangles and other combinations of options in Chapter 5.

22 MAT4210 Notes by R. Chan 3 2.5 Payoff 2 1.5 1 0.5 0 8 8.5 9 9.5 10 10.5 11 11.5 12 S Figure 3. Value of the portfolio at expiry. 7 No Arbitrage Arguments Knowing how to construct a correct portfolio is essential when one uses no arbitrage arguments to derive a desired financial proposition. Let us illustrate that with a simple example. Proposition 1. The current stock price S(t) is always greater than or equal to the call option price c(t), i.e. S(t) c(t). Proof. Suppose c(s(t), E, τ, r) > S(t) where τ = T t. We form a portfolio as follows: buy 1 unit of the share at S(t) and sell 1 unit of the European call at c(t). (Remember the slogan: buy low, sell high and it will be easier for you to construct the portfolio.) Note that, when you sell 1 unit of a call, you become a writer of this call option. Observe that the value of this portfolio at time t is given to be Π(t) = S(t) c(s(t), E, τ, r) < 0. That is, if you accept to set up this portfolio at t, you have a positive cash flow. At T, the expiry date, the payoff of this portfolio is equal to the asset price at T less the price of the call at T, i.e. the terminal payoff is Π(T ) = S(T ) c(s(t ), E, 0, r) = S(T ) max (S(T ) E, 0) = S(T ) + min (E S(T ), 0) = min (S(T ) + E S(T ), S(T ) + 0) = min (E, S(T )) 0. The payoff at T of this portfolio is nonnegative, while its value at t is negative. Since the profit of a portfolio is equal to the terminal payoff less the initial premium, we have the profit Π(T ) Π(t) > 0 always. One can also check that it is precisely Case 1(b) of arbitrage in Section 1.9. Thus, there would be an arbitrage opportunity, a contradiction.

Options 23 Consider a more complicated example. Proposition 2. If 0 E 1 E 2, then 0 c(s(t), E 1, τ, r) c(s(t), E 2, τ, r) E 2 E 1 (4) and 0 p(s(t), E 2, τ, r) p(s(t), E 1, τ, r) E 2 E 1. (5) Proof. If c(s(t), E 1, τ, r) < c(s(t), E 2, τ, r), then we buy one c(s(t), E 1, τ, r) and sell one c(s(t), E 2, τ, r). (Remember buy low, sell high again.) At t, the value of this portfolio is Π(t) = c(s(t), E 1, τ, r) c(s(t), E 2, τ, r) < 0. At T, Π(T ) = max (S(T ) E 1, 0) max (S(T ) E 2, 0) 0, if S(T ) E 1, = S(T ) E 1, if E 1 < S(T ) E 2, E 2 E 1, if E 2 S(T ). In any case, Π(T ) 0. Hence the profit Π(T ) Π(t) > 0 always. This shows an arbitrage opportunity exists. A contradiction! Thus, the inequality c(s(t), E 2, τ, r) c(s(t), E 1, τ, r) is established. Suppose now that c(s(t), E 1, τ, r) c(s(t), E 2, τ, r) > E 2 E 1. Form the following portfolio: buy one c(s(t), E 2, τ, r), sell one c(s(t), E 1, τ, r) and keep E 2 E 1 cash. Note that, as it were assumed that c(s(t), E 1, τ, r) c(s(t), E 2, τ, r) > E 2 E 1, after having bought one c(s(t), E 2, τ, r) and sold one c(s(t), E 1, τ, r), we would have cash more than E 2 E 1 at hand, so we can afford to keep E 2 E 1 cash in the portfolio. The value of this portfolio at t is At T, the payoff is equal to Π(t) = E 2 E 1 + c(s(t), E 2, τ, r) c(s(t), E 1, τ, r) < 0. Π(T ) = (E 2 E 1 ) + max (S(T ) E 2, 0) max (S(T ) E 1, 0) (E 2 E 1 ), if S(T ) E 1, = (E 2 E 1 ) (S(T ) E 1 ), if E 1 < S(T ) E 2, (E 2 E 1 ) (E 2 E 1 ), if E 2 < S(T ). It is easy to see that Π(T ) 0 always. Therefore, the profit of this portfolio, Π(T ) Π(t), is always > 0, which means an arbitrage opportunity exists. A contradiction! Inequalities (5) can be derived in a similar fashion, and will be left as an exercise. Similarly, you can construct portfolios to verify the following proposition. It is a corollary of a famous equality Put-Call Parity which will be studied in Chapter 4. Its proof is simple and is also left as an exercise. Proposition 3. For all t T, we have c(t) S(t) E.

24 MAT4210 Notes by R. Chan By now we have seen a few examples in which the so-called arbitrage arguments are used. When using the no-arbitrage principle to prove a financial statement, one usually forms a portfolio by using the buy low, sell high principle and use it to show that an arbitrage opportunity exists if the statement is assumed to be wrong. Note that the portfolio to be formed in the argument is not necessarily unique. Namely, we can sometimes form several different portfolios to show that each of them provides an arbitrage opportunity if the statement is assumed to be wrong. But one of such portfolios is enough to prove the statement. 8 American Options European options can only be exercised at expiry. Besides European options, many options nowadays are what is called American. An American option is one that may be exercised at any time prior to expiry by its holder. Mathematically speaking, American options are more interesting since they can be interpreted as free boundary problems. A question that the holder of an American option must determine is when it is best time to exercise the option. We will see that the best time to exercise is not subjective, but that it can be determined in a natural and systematic way. The value of an American option depends also on factors such as S(t), the asset price at time t; E, the strike price; τ = T t, the length of the period of the current time to the time of the contract expiration; r, the interest rate; volatility; etc. Throughout this course, c and p will denote the value (or the price) of the European call option and put option, respectively while C and P denote that of the American call option and put option, respectively. We have c(t) = c(s(t), E, τ, r); p(t) = p(s(t), E, τ, r); C(t) = C(S(t), E, τ, r); P (t) = P (S(t), E, τ, r). As American options can be exercised at any time prior to T, when we, the holder of an American call, would like to exercise it, we must determine if it is optimal to do so. Suppose that, at time t (t < T ), we decide to exercise it, then we must have C(S(t), E, τ, r) = S(t) E > 0. Otherwise, when exercising it, we would get S(t) E dollars which is non-positive. That means either we get 0 dollars, or we have to pay S(t) E to the option writer. In either case, we lose money in the deal. That contradicts the idea of exercising it in an optimal way. On the other hand, if we do not exercise it and hold the option till the contract expiration, then the same consideration given to European calls can be applied to this case and we get C(T ) = C(S(T ), E, 0, r) = max (S(T ) E, 0) = (S(T ) E) + 0. More precisely, we state the following proposition.

Options 25 Proposition 4. At expiry, the values of options are given by c(t ) = c(s(t ), E, 0, r) = max (S(T ) E, 0) 0; p(t ) = p(s(t ), E, 0, r) = max (E S(T ), 0) 0; C(T ) = C(S(T ), E, 0, r) = max (S(T ) E, 0) 0; P (T ) = P (S(T ), E, 0, r) = max (E S(T ), 0) 0. Since the American option gives its holder greater rights than the European option, via the right of early exercise, potentially it has a higher value. Indeed, the following arbitrage argument shows how this can happen. Proposition 5. For all τ = T t 0, C(S(t), E, τ, r) c(s(t), E, τ, r). Proof. Suppose C(t) < c(t). Then form the following portfolio: buy 1 unit of this American call and sell (i.e. write) 1 unit of this European call. Since we are the holder of the American call, we have the right to decide when to exercise it (earlier or at T ). Let us choose to hold it till T. Then the terminal payoff is equal to Π(T ) = C(T ) c(t ) = max (S(T ) E, 0) max (S(T ) E, 0) = 0. How much would we gain in this investment? The gain is given by Π(T ) Π(t) = 0 (C(t) c(t)) > 0. This shows there would be an arbitrage opportunity, which is a contradiction. European and American call and put options form a small section of the available derivative products. They are called vanilla options because they are ubiquitous and simple. Nowadays the trade in the vanilla options is so great that it can, in some markets, have a value in excess of that of the trade in the underlying assets. There are however other more complicated options, including the so-called exotic or pathdependent options. These options have values which depend on the history of an asset price, not just on its value on exercise. An example is an option to purchase an asset for the arithmetic average value of that asset over the month before expiry. An investor might want such an option in order to hedge sales of a commodity, say, which occur continually throughout this month. There are also options which depend on the geometric average of the asset price, the maximum or the minimum of the asset price. For instances, Digital option: an option that at maturity pays a fixed amount if in-the-money, zero otherwise. A digital option is also called a binary option. Barrier options: the option which can either come into existence or become worthless if the underlying asset reaches some prescribed value before expiry. Asian options: the price S(T ) depends on some form of average. If S(T ) is replaced by the average, it is called Asian price option. If E is replaced by the average, it is called Asian strike option. Lookback options: the price depends on the asset price maximum or minimum. Example 8. Warrants in the HK Stock Exchange are like European options. They can only be exercised at the expiry date. Of course, one can sell them before expiry if the price of the warrant is already high enough. One question is how one computes the

26 MAT4210 Notes by R. Chan stock price at expiry. The use of the closing price of the stock on the expiry date may subject the stock price and hence the warrant price to big fluctuation. Thus HKEx uses the average of the closing prices in the final 5 days as the price of the stock at T when computing the payoff of the warrant. In this sense, warrants are like Asian price options. Example 9. One very hot derivative product in the HK Stock Exchange is the callable bull/bear contracts (CBBC). They are indeed barrier options. Consider a Bull CBBC with exercise price $80 and barrier (known as the call price in the market) $83. On the date of issue, let the underlying stock price be $100. Then the price of the Bull CBBC on the issue date will be around $100 $80=$20. If some time between the issue date and the expiry date, the stock price equals call price, then the contract is called (i.e. terminated). The residual value of the Bull CBBC at the call is $83 $80=$3. If the Bull CBBC has never been called, and if at expiry, the 5-day average closing price is $120, then the value of the Bull CBBC at expiry is $120 $80=$40. Options like this are called down-and-out options. There are down-and-in, up-and-out, and up-and-in options. Most of the options we discussed are what we call stock options because the underlying assets are shares of stocks. There are in fact foreign currency options, index options, futures options, etc. We have considered one such example (call option on British pounds) in Example 1.5, where the underlying is the exchange rate. There are many different index options traded now in the markets. The two most popular that are traded in the United States are those on the S&P 100 and S&P 500 traded on the Chicago Board Options Exchange (CBOE). For instance, consider an index option on S&P 100 to buy or sell 100 times the index at the specified strike price. Say, the value of the index is 992 at expiry, and this call option with the strike price 980 is exercised. The writer of this contract pays the holder (992 980) $100 = $1,200. The cash payment is based on the index value at the end of the day on which exercise instructions are issued. In Hong Kong, we have the Hang Seng Index (HSI) which comprises of 33 constituent stocks. For HSI futures, contract multiplier is HK$50 per index point.