FEC FINANCIAL ENGINEERING CLUB

AN INTRO TO OPTIONS

AGENDA  What are options?  Bounds on prices  Spread strategies  Greeks

OPTIONS CONTRACTS  An option contract is a right to buy (call option) or sell (put option) an underlying security at a pre-specified date in the future and at a pre-specified price.  Date is called the maturity or expiration date  Pre-specified price is called the strike price  Ex) AAPL is currently at (about) $ You want to buy a call option with a strike of $ whose expiration is March 21, This means you can ‘exercise’ your option to buy AAPL at $ at the maturity date (European-style option) or before (American-style option)

OPTION VALUE  What is the value of such an option?  Depends on many things—most importantly, the underlying (AAPL) price.  Suppose this call option expired today and AAPL was at $ How much would you be willing to pay for it?  Right to buy AAPL (worth $509.00) for $ Call Intrinsic = (S-K) + = max{S-K,0}, S is the price of the underlying today K is the strike price

INTRINSIC VALUE

Intrinsic Value of a Call Option (Green)Intrinsic Value of a Put Option (Green)

TIME VALUE  However, if there is time left until expiration, the stock price (at time t) S t, could change and thus the value of the option would change.  This component of price that changes over time is known as time value.  Time value is the value associated with the likelihood that the option will become in the money (valuable) by a favorable move in the underlying price  Some determinants of time value:  How volatile is the underlying stock  What is your borrowing rate, are there any dividends from the underlying stock  How much time is left until maturity

BUYING VS SELLING OPTIONS  If you buy an option you have the option to exercise it: Long Call option: You may pay K to receive the stock. Long Put option: You may sell the stock for K.

BUYING VS SELLING OPTIONS  When you sell an option, you give the buyer the right to exercise the option: Short Call option: Buyer may buy the stock from you for K. Short Put option: Buyer may sell the stock to you for K. When will the buyer buy the stock from you? In the same situations you would exercise the call option—when S > K. They would not exercise when S < K.

BUYING VS SELLING OPTIONS  Same logic applies for short puts.  When you sell (called writing) an option and it is exercised by the buyer, it is said to be assigned against you.

OPTION CONTRACT STYLES  European—Option may be exercised at maturity only.  American—Option may be exercised at any time preceding maturity.  Others—Asian, Bermudan, Barrier options.  For this lecture, we will discuss the simplest and most common cases—European and American options.

COMMON SENSE BOUNDS

PUT-CALL PARITY  What happens if I sell a put at strike price K, buy an identical call, and lend PV(K)?  Has the same payoff as a long position in the underlying!  Therefore costs of our combined position must equal that of the underlying.

PUT-CALL PARITY

Cost to be long underlying Cost to be long call option Cost to be short put Cash outflow from lending PV(K) This, and other bounds may be used in the Black-Scholes PDE (next lecture)

SPREAD STRATEGIES

SPREADS  Spread strategies are multi-legged option positions  What is a leg?  A position using one type of options contract  Example: What if we buy a call option and buy an identical put option (same strike, time until maturity, etc)?  One leg is the call option  One leg is the put option  What does our position look like?

SPREADS When S > K, what happens? We exercise our long call option(s) When S < K, what happens? We exercise our short put options(s) Profit/Loss Diagram is:

LONG STRADDLE  This is known as a long Straddle position.  One of the simpler spread positions  When would one want to trade a straddle? A ) When volatility is high ? B) When volatility is low? C) When we are certain the underlying will increase?

LONG STRADDLE  This is known as a long Straddle position.  One of the simpler spread positions  When would one want to trade a straddle? A ) When volatility is high ? B) When volatility is low? C) When we are certain the underlying will increase?

MORE SPREAD STRATEGIES  Underlying is 37  Strategy: Long call (Strike = 40); Long a put (Strike = 35). The call is worth $3. The put is worth $1. UnderlyingLong CallLong PutLong Strangle

STRANGLE UnderlyingLong CallLong PutLong Strangle

MORE SPREAD STRATEGIES  From a payoff standpoint (ignore costs), would you prefer to be long  Position 1: two call options (K = 35), or  Position 2: one call option (K = 30) and another call option (K = 40)

MORE SPREAD STRATEGIES UnderlyingLong Call (K=35)Position UnderlyingLong Call (K = 30)Long Call (K = 40)Position

MORE SPREAD STRATEGIES

GENERAL APPROACH TO SPREADS  Options can replicate any risk profile at maturity with exclusively puts or calls.  That is, you can construct a position like this:

GENERAL APPROACHES TO SPREADS

REPLICATION WITH CALLS  Evaluate positions from left to right ) Slope must be 10— buy 10 Calls at 37 2) Slope from 38 to 50 must be 0— sell 10 Calls at 38 to get flat 3) Slope from 50 to 51 must be -5—sell 5 Calls at 50 4) Slope from 51 to 52 must be -3—buy 2 Calls at 51 5) Slope after 52 is 0—buy 3 Calls at 52 to get flat +10 Calls(37) -10 Calls(38) -5 Calls(50) +2 Calls(51) +3 Calls(52)

REPLICATION WITH PUTS  Evaluate positions from right to left ) Slope must be 0—buy 10 Puts at 37 to get flat 4) Slope from 38 to 37 must be 10—sell 10 Puts at 38 3) Slope from 50 to 38 must be 0—sell 5 Puts at 50 to get flat 2) Slope from 51 to 50 must be -5—buy 2 Puts at 51 1) Slope from 52 to 51 is -3— buy 3 Puts at Put(52) +2 Put(51) -5 Put(50) -10 Put(38) +10 Put(37)

GREEKS

 Recall that there are five drivers of an option’s price:  Price of the underlying  Volatility of the returns on the underlying  Interest rates  Strike price  Time until maturity  What is the risk of an option? How does the price of an option change as the underlying factors change?  These are the greeks.

DELTA  The sensitivity of an option with respect to a change in the underlying’s price.  Ex) Suppose that the underlying is at 60. A call option with strike has a delta of.5 (usually quoted as 50).  What happens if underlying moves to 65?  Option price increases by.5*(65-50) =.5*15 = 7.50

DELTA-HEDGING

Sell 1220 Shares

DELTA-HEDGING Sell 860 Shares

NEXT LECTURE  Continuous models for option valuation  Stochastic calculus  Black-Scholes-Merton  More greeks

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