Financial Derivatives

1. Financial Derivatives Daniel Thaler December 1, 2009

2. What are financial derivatives? • They are financial instruments whose value is derived from some other asset, index, event, value, or condition. • Those from which it is derived is known as an underlying asset.

3. Conceptual Example • It’s Super Bowl XLII between the Giants and the Patriots and the Patriots are a 4-1 favorite. Your friend places a $1,000 bet on the Giants to win the game. How much would you pay your friend to have the option to purchase his bet? (Question 1)

4. Scenarios • Would you pay him more than $1,000 for this option at the start of the game? • No, you could make the bet yourself • The Giants are winning 3-0 at the end of the first quarter how would the price of the option change? • The price of the option would increase • The Patriots are winning 14-10 with 2:42 left in the game how would the price of the option change? • The price of the option would decrease • The Giants score a late touchdown to make it 17-14 with 0:35 left in the game would you pay him more than $1,000 for this option? • Yes, considering if the Giants win the payout is $4,000.

5. Background information • Rather than trade or exchange the underlying asset itself, derivative traders enter into an agreement to exchange cash or assets over time based on the underlying asset. • Derivates are often highly levered, so a small change in the underlying asset can cause a large change in the value of the derivative.

6. More background • Derivatives can be used by investors to speculate and to make a profit if the value of the underlying moves the way they expect • Traders can use derivatives to hedge or mitigate risk in the underlying, by entering into a derivative contract whose value moves in the opposite direction to their underlying position and cancels part or all of it out.

7. Back to the Super Bowl • You are a Patriots fan and bet $4,000 on them to win the game (remember the odds are 1-4 so the payout is $1,000). It’s the end of the 3rd Quarter and the Patriots are only up 4pts. You want to hedge your risk so you find someone to sell you a $500 option on a $1,000 bet that the Giants win. Answer Question 2.

8. Hedging Solutions How much will you win/lose if the Patriots win/lose? • Pats win, you win $1,000 - $500 = $500 • Pats lose, you lose $4,000 – $4,000 - $500 = -$500 • How much would have had to wager without options if you wanted to win $500 • $2,000

9. Categories • The type of the underlying • Stocks, Bonds, Commodity • The market which they trade • Over-the-counter (OTC), Exchange-traded derivates (ETD) • The relationship between the underlying and the derivative • Options, Futures/Forwards, Swaps

10. Options • Contracts that give the owner the right, but not the obligation to buy or sell an asset • The strike price is the price at which the transaction would take place • The option must also have a maturity date • 1 Options contract usually represents the right to buy 100 shares of the underlying security

11. Types of Options Trades • Long Call • Long Put • Short Call (“Write a Call”) • Short Put (“Write a Put”)

12. Long Call • Buy the right to purchase the stock at the strike price. • Will only exercise if the stocks price is higher than the strike price plus the price paid for the option • Believe the price will INCREASE • For the same amount of money you can obtain a larger amount of options than shares

13. Long Call

14. Long Put • Buy the right to sell the stock at the strike price • Will exercise only if the stock price plus the premium is below the strike • Believe the stock price will DECREASE

15. Long Put

16. Write a Call • Selling a call option to a buyer and had the obligation to fulfill the contract at a strike price • Will profit only if the stock price remains below that of the strike price plus the premium • Potential loss is unlimited • Believe the stock price will DECREASE

17. Write a Call

18. Write a put • An obligation to buy the stock from the put buyer at the strike price • Will profit if stock price plus the premium is above the strike price. • Loss is capped at the full value of the stock • Believe the stock price will INCREASE

19. Write a put

20. Identifying options • Google’s stock price is $570 and Bill has bought 3 option contracts for $15($5 per contract) with a Strike Price of $580. • If the Stock price goes above $600 Bill will exercise the option, What has he bought? • He has a call option • How much will he make? • ($600-$580)*300 – $15 = $5,985

21. Identifying options • Google’s stock price is $570 and Bill has written 3 option contracts for $9,000 ($3,000 per contract) with a Strike Price of $600. • If the Stock price goes down to $560 Bill will have to exercise the option, What has he written? • He has written a put • How much will he lose? • ($560-$600)*300 + $9,000 = -$3,000

22. How Risky are Options? • They can expire worthless and they increase leverage • Example: Stock A is selling at 100 and its options are selling at $2.50 with a strike price of $120 • You want to invest $1,000 • So you can buy 10 shares of stock or…. • 4 options contracts • In a week the price of the stock is now at 110 so your profit with just the stocks is 10*10 = 100 but lets say the value of the option went up to $4.50(very reasonable) your profit is $2 * 400 = 800

23. Option Strategies • Combine any of the four basic options trades (possibly with different exercise prices) • Can also use the two basic kinds of stock trades (long and short) • Used to engineer a particular risk profile to movements in the underlying security.

24. Option Strategies • Bull Call Spread • Combines a short call and a call

25. Option Strategies • Long Strangle • Combines a call and a put

26. Other Option Strategies • Bullish Strategies Bearish Strategies Neutral Strategies • Bull Put Spread Bear Put Spread Short Straddle • Bull Call Spread Bear Call Spread Long Straddle • Covered Straddle Put Time Spread Short Combo • Call Time Spread Ratio Put Spread Guts • Ratio Call Spread Condor Strangle • Long Butterfly

27. Excel

28. How are options priced? • Want to find a way to quantify the expected payoffs that would occur if the stock price goes up or goes down. • Also, it must incorporate the length for which the option is available

29. How are options priced? • Binomial Options Pricing Model • Uses a “discrete-time” model of the varying price over time of the underlying financial instrument • Black-Scholes Model • A continuous extension of the binomial model

30. Binomial Model • Provides a general numerical model • Process is iterative • Each node represents a possible price at a particular point in time

31. Binomial Model • Steps: • Price tree generation • Calculation of option value at each final node • Calculation of option value at each earlier node • The value at the first node is the price of the option

32. Price Tree Generation • It assumed that at point in the tree the underlying instrument will move either up or down. • Let S = Current Price • Let Su = S * u = Price when stock moves up (u >1) • Let Sd = S * d = Price when stock moves down (0<d<1)

33. Price Tree Generation • To determine d and u we will use the volatility of the underlying asset which is σ • u = e ^ σrad(t) • d = e ^ - σrad(t) = 1/u • t is the time between periods measured in years • This ensure that the tree is recombining which accelerates the computation of the option price

34. Price Tree Generation • Answer Question 5 part (a)

35. Value at Final Nodes • The final node is expiration, there if it is profitable to exercise the option you will if not you will let it expire. For a call option For a put option Max [ (K – S), 0 ] Max [ (S – K), 0 ] K is the strike price, S is price of underlying asset

36. Value at Final Nodes • Answer Question 5 part (b)

37. Value of Option at earlier nodes • Use the value of the option at an intermediate node using the value of the option at the following nodes. • First: need to assign a probability to the price will increase by u or decrease by d (We will use 50/50 chance to keep it simple)

38. Value of Option at earlier nodes • The value of an option at an earlier node is then equal to the following: Max [ (S – Strike), p × Vu+ (1-p) × Vd] × e(- r × t)

39. Value of Option at earlier nodes • Answer question 5 part (c), calculate the value of the option

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41. Black-Scholes Model • A continuous continuation of the binomial model • The binomial model assumes that movements in the price follow a binomial distribution; for many trials, this binomial distribution approaches the normal distribution assumed by Black-Scholes.

42. Black-Scholes Model • Developed by Fischer Black and Myron Scholes in a 1973 paper. • They received the 1997 Nobel Prize in economics for this and related work.

43. Black-Scholes Model • It assumes the underlying asset follows a geometric Brownian Motion and using partial differential equations to get the Black-Scholes PDE:

44. Black-Scholes Model • The value of a call option is found by solving the PDE and the result is N(•) is the standard normal distribution T - t is the time to maturity S is the price of the underlying asset K is the strike price r is the risk free rate σ is the standard deviation

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46. Futures/Forwards • Futures contract is a standardized contract to buy or sell a specified commodity at a certain date in the future for a certain price. • Forwards are similar to futures except that they are traded OTC and as such are more customizable

47. Futures • A futures contract gives the holder the obligation to make or take delivery under the terms of the contract • Differs from an options contract in that both parties must fulfill the contract at the settlement date.

48. Who Buys Them? • Speculators who seek to make a profit by predicting market moves. • Producers and consumers purchase futures contracts to guarantee a certain price.

49. Types of Futures • Crude Oil • Corn • Soybean • Sugar • Wool • Cotton • Coffee • Cocoa • Wheat • Lumber • Orange Juice • Silver • Gold • Copper

50. Trading Places