Week 4

1. Week 4 Options: Basic Concepts

2. Definitions (1/2) • Although, many different types of options, some quite exotic, have been introduced into the market, we shall only deal with the simplest “plain-vanilla” options like calls and puts. • Call: A call gives you the option to buy an underlying asset at a fixed price (called the strike or exercise price) before or on a certain date (called the maturity, or the expiration date of the option). • Put: A put gives you the right to sell the underlying asset.

3. Definitions (2/2) • A European option is one that can be exercised only on the maturity date. • An American option is one that can be exercised any time before the exercise date. • (The American option price is equal to or greater than the European option price. Why?) • As before, we will denote S as the spot exchange rate, and F as the forward.

4. Example • Example: A call on the BP at a strike of $1.60 expiring, say, on 12/15/2005. • This call gives you the option to buy 1 BP for $1.60 on 12/15/2005. You will exercise the call if the BP is stronger than the strike ($/BP > 1.6) on that date. Let us denote ST as the exchange rate on 12/15/05, X as the strike, and T as the exercise date. Then, you will exercise the option if ST >X. For example, if the exchange rate at maturity is 1.70$/BP, you will exercise the call, and your payoff will be (ST -X)=1.70-1.60=$0.10.

5. Payoff of the Call and Put When Exercised • Denote CT and PT we can write the payoff of the call and put on the exercise date as follows: • Payoff on call at T: CT = 0, if ST<X, and CT =ST -X, if ST>X. Thus, CT = Max(ST-X,0). • Payoff on put at T: PT = 0, if X<ST, and PT =X-ST , if X>ST. • Thus, PT = Max(X-ST,0)

6. Payoff Diagrams • The payoff diagram represents the payoff of the option on the expiration date (T) as a function of the price of the underlying security. To trade options, it is essential to understand payoff diagrams. • Examples: How would you draw the payoff of the following options, or portfolio’s of options: • 1. Long/Short call of strike X. • 2. Long/Short Put with strike X. • 3. Long C and Long P, both with same strike X • 4. Long C and short P, both with same strike X.

7. Long Call with Strike=1.60 Payoff ST-X=0.2 0 ST X=1.6 ST=1.8

8. Short Call with Strike=1.60 Payoff 0 ST -(ST-X) = -0.4 X=1.6 ST=2.0

9. Long Put with Strike=1.60 Payoff (X-ST) = 0.2 0 ST X=1.6 ST=1.4

10. Short Put with Strike=1.60 Payoff 0 -(X-ST) = -0.6 ST X=1.6 ST=1.0

11. Other Examples • The basic payoff’s of a long/short call, and a long/short put can be combined into much more complicated payoff structures. • Examples: (Please draw the payoffs) • 1. Straddle: A straddle is an option position of long 1 C + long 1 P, where the call and put have the same strikes. This position allows the trader to take a view on volatility. • 2. Bull Spread: Long 1 call of strike 1.60, short 1 call of strike 1.80. This allows the trader to take a bullish position but with a capped upside. The capped upside makes the position cheaper to implement.

12. Some Terminology • Intrinsic value = value of the option if it is immediately exercised. • For a call, the intrinsic value is St-X. • For a put, the intrinsic value is X- St. • An at-the-money option is one with intrinsic value equal to zero. • An in-the-money option is one whose intrinsic value is positive. • An out-the-money option is one whose intrinsic value is negative.

13. Issues • Here are some of the questions, we are interested in answering: • 1. What is the relation between the call, put, forwards and the spot? • 2. How does one price an option? What factors help determine the option’s price?

14. Basics of Arbitrage Pricing • We will price an option by arbitrage. • Basic Principle: Two securities that have the same payoff at a future date must have the same price today. In other words, the price of an asset should not allow for arbitrage. • The way we will use this principle is as follows: Suppose we can replicate the payoff on an option by trading a portfolio of securities whose prices we know. Then we can argue that if we can create the same payoff using two different securities, then those securities should be priced the same.

15. An Application of Arbitrage Pricing: Put Call Parity (1/5) • Let us figure out today’s (t=0) price of a portfolio of 1 long call and 1 short put ( C - P) with the same strike (X) and maturity (T). • We have to figure out how we can replicate the payoff of this portfolio using existing securities. So, the first question to ask is what is the payoff of this portfolio of C-P? • Payoff of C-P : • If ST > X, then the call is exercised and the put is not, so the combined payoff is (ST-X) = ST - X. • If ST<X, then the put is exercised and the call isn’t, so the combined payoff is -(X-ST) = ST - X.

16. Put Call Parity (2/5) • Thus, for any spot rate at t=T, the payoff on the portfolio of (C - P) is ST-X. • Suppose the underlying asset is 1 British Pound (BP). This means that buying a call and selling a put is the same as receiving one BP at maturity of value ST for a price of X. • Can we replicate this payoff using the underlying securities? That is, can we replicate this payoff by trading the foreign exchange and borrowing/lending at the domestic and foreign interest rates?

17. Put-Call Parity (3/5) • Here is one strategy: buy the present value of 1BP at t=0 and hold it until t=T, and borrow the present value of X dollars. If the option has a maturity of T days, then the value of this portfolio is: PV(S) - PV(X) = S/(1 + r*T/360) - X/(1+rT/360) • What would be the total payoff on this portfolio of PV(S) - PV(X)? • The payoff on this portfolio is exactly ST-X, i.e, the same as that on (C-P). • Note that when you take the present value of the foreign currency, you need to discount at the foreign interest rate.

18. Put-Call Parity (4/5) • Thus, to prevent arbitrage, it must be that the following is true: for currency options • C-P= S/(1 + r*T/360) - X/(1+rT/360) • Although we haven’t managed to determine the price of the C and P individually, we have at least managed to figure out a relationship between C, P and S. This is called the Put-Call Parity.

19. Put Call Parity (5/5) • We can also write the relation for currency options in terms of the forward price F=S(1+rT/360)/(1+r*T/360). Substituting for S in the put call parity we get, • C-P = F/(1+rT/360) - X/(1+rT/360) = (F-X)/(1+rT/360) • Thus, we can either express the put call parity in terms of the spot rate or the forward rate.

20. Summary of Put-Call Parity • These are the important points to note from Put-Call Parity: • 1. There is a precise relation between the prices call and the put of the same strike, given by C-P=S/(1+r*n/360) - X/(1+rn/360). (If the observed option prices do not follow this relation, then there exists an arbitrage.) • It easier in practice to construct an arbitrage using the futures. C-P=(F-X)/(1+rn/360) • Thus, given the price of the call, one can deduce the price of the put (or vice versa).

21. 3. If S=X, then the prices of the at-the-money call and put will be equal to each other only if r=r*. The call will be more expensive than the put if r>r* (and P>C, if r*>r).

22. Bounds on European Call and Put Prices • Because the call and put price can never be negative (C>0, P>0), we can use the put call parity to give a lower bound to the option prices. • C= P + S/(1 + r*T/360) - X/(1+rT/360) • Therefore: C>S/(1 + r*T/360) - X/(1+rT/360) • P=C+ X/(1+rT/360) - S/(1 + r*T/360) • Therefore: P> X/(1+rT/360) - S/(1 + r*T/360) • If this bound is violated, there is an arbitrage opportunity.

23. Bounds on American Call and Put Prices • It is easier to derive the bounds on American calls and puts. • Because an American call or put can be exercised at any time, it has to be priced higher than its intrinsic value. • C >= S-X • P >= X-S