1. FEC FINANCIAL ENGINEERING CLUB

2. AN INTRO TO OPTIONS

3. AGENDA  More on Spread strategies  Greeks  Delta  Theta  Gamma  Vega

4. SPREAD STRATEGIES

5. 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?

6. 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:

7. 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?

8. 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?

9. 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 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75

10. STRANGLE UnderlyingLong CallLong PutLong Strangle 0-33431 5-32926 10-32421 15-31916 20-31411 25-396 30-341 35-3-4 40-3-4 4521 5076 551211 601716 652221 702726 753231

11. 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)

12. MORE SPREAD STRATEGIES UnderlyingLong Call (K=35)Position 1 000 500 1000 1500 2000 2500 3000 3500 40510 451020 501530 552040 602550 653060 703570 754080 UnderlyingLong Call (K = 30)Long Call (K = 40)Position 2 0000 5000 10000 15000 20000 25000 30000 35505 40100 4515520 50201030 55251540 60302050 65352560 70403070 75453580

13. MORE SPREAD STRATEGIES

14. 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:

15. GENERAL APPROACHES TO SPREADS 38 50 51 52 37

16. REPLICATION WITH CALLS  Evaluate positions from left to right 38 50 51 52 37 1) 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)

17. REPLICATION WITH PUTS  Evaluate positions from right to left 38 50 51 52 37 5) 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 52 +3 Put(52) +2 Put(51) -5 Put(50) -10 Put(38) +10 Put(37)

18. GREEKS

19.  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.

20. 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 of 60 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

21. DELTA-HEDGING

22. DELTA-HEDGING 1

23. Buy 39 Shares

24. DELTA-HEDGING 2

25. Sell 1220 Shares

26. DELTA-HEDGING 2 Sell 860 Shares

27. MORE ON DELTA

29. GAMMA  The sensitivity of delta as the underlying price changes.  Effectively the second derivative of an option’s price with respect to time.

30. GAMMA

32. WHAT DOES THIS MEAN FOR THE OPTIONS TRADER?  As underlying increases,  Call options become more in the money  Delta approaches 1  Observation: The more in the money a call option is, the more it “behaves” like the underlying (delta is closer to 1, other greeks less relevant)

33. WHAT DOES THIS MEAN FOR THE OPTIONS TRADER? NEED PICTURE OF DELTA VS UNDERLYING FOR PUT

34. VEGA, RHO, THETA

35. BLACK-SCHOLES EQUAITON

36. GREEKS  We can take partial derivatives of the above formula to get the following greeks

37. VOLATILITY

38. IMPLIED VOLATILITY  What is the level of volatility that sets our Black-Scholes equation of one option equal to the market value of that options?  Effectively reverse-engineer volatility from options prices  Implied volatility is an annualized metric  Ex) Consider our old option example: A call options with 0.3 years until maturity and strike is $40. The risk-free interest rate is 0 and the expected standard deviation of returns over the next 0.3 years is 0.2. The underlying is at $41.  The market values this option at $2.50  What is the implied volatility of this? Vol =.2216 

39. IMPLIED VOLATILITY  How did we get this number?  Numerically, via a root-finding method such as Newton-Raphson or Bisection Method (See lecture 1)  Skew and smiles  Black-Scholes assumes that volatility is constant across strikes  Data does not reflect this. Why?  Lognormal assumptions about tail returns. Many believe tail events are more likely than indicated by Black-Scholes  Options traders bid up the prices of these tail options to match the likelihood of them being in the money.

40. NEXT LECTURE  Continuous models for option valuation  Implied volatility  Stochastic calculus  Black-Scholes-Merton  Calculation of greeks

41. THANK YOU!  Facebook: http://www.facebook.com/UIUCFEChttp://www.facebook.com/UIUCFEC  LinkedIn: http://www.linkedin.com/financialengineeringclubhttp://www.linkedin.com/financialengineeringclub  Email: uiuc.fec@gmail.comuiuc.fec@gmail.com Internal Vice President Matthew Reardon mreardon5@gmail.com President Greg Pastorek gfpastorek@gmail.com