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Option Pricing the short story

9-2. The problem. Recall that call value at expiration, CT, isCT = max[0, ST - X]implies that C is a function of S and X.The problem:What is C0?From above,C0 = Cte-rtC0 = max[0, Ste-rt - Xe-rt]C0 = max[0, S0 - Xe-rt]implies that C is a function of S, X, r, and tBut at this point we have not captured the probability that S will be in the moneythis probability will depend upon volatility (?).

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Option Pricing the short story

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    1. 9-1 Option Pricing (the short story)

    2. 9-2 The problem Recall that call value at expiration, CT, is CT = max[0, ST - X] implies that C is a function of S and X. The problem: What is C0? From above, C0 = Cte-rt C0 = max[0, Ste-rt - Xe-rt] C0 = max[0, S0 - Xe-rt] implies that C is a function of S, X, r, and t But at this point we have not captured the probability that S will be in the money this probability will depend upon volatility (?)

    3. 9-3 Black-Scholes (BS) Assumptions Assumptions about stock return distribution Stock price changes are described by a stochastic differential equation S(t) is the stock price, dS(t) is the instantaneous change in the stock price, alpha is the continuously compounded expected return, sigma is the continuously compounded standard deviation and Z(t) is a normally distributed random variable that follow a process called Brownian motion. Brownian motion is a stoachastic process that is a random walk occurring in continuous time, with movements that are continuous rather than discrete. Continuously compounded returns on the stock are normally distributed and independent over time No ‘jumps’ in the stock price The volatility of continuously compounded returns is known and constant Future dividends are known as continuous dividend yield

    4. 9-4 Black-Scholes (BS) Assumptions (cont’d) Assumptions about the economic environment The risk-free rate is known and constant There are no transaction costs or taxes It is possible to short-sell costlessly and to borrow at the risk-free rate

    5. 9-5 Black-Scholes method Consider an European call (or put) option written on a stock, and assume that the stock pays dividend at the continuous rate d

    6. 9-6 Black-Scholes Formula (cont’d) Call Option price Put Option price where and

    7. 9-7 Option price sensitivity to parameters

    8. 9-8 Structured Notes Debt obligations ‘structured’ to achieve a particular risk profile or payoff structure for example, equity-linked CD’s

    9. 9-9 Start with a problem ExxonMobil is an oil producer, they have a natural long position in oil They need to hedge against a drop in oil prices Basic hedges: Short oil futures/forwards (sell forward) Swap oil (sell oil through swap) Buy put options on oil (sell through options)

    10. 9-10 Basic problems with these hedges Is timing good? Is it a direct hedge? Can the firm fund interim losses?

    11. 9-11 Alternative solutions Engineer a product that also sells oil in the future. Widely used are structured notes for example, a “commodity linked bond” Get cash from investors and pay interest and principal in oil rather than in cash Selling gold forward through a debt instrument

    12. 9-12 Pricing and Designing Structured Notes A structured note has interest or maturity payments that are not fixed in dollars, but are contingent in some way Structured notes can make payments based on stock prices, interest rates, commodities, or currencies Structured notes can have options embedded in them

    13. 9-13 Structured Note Basics

    14. 9-14 Profit loss on this bond

    15. 9-15 Suppose the following… If the oil company issued the bond, they raise $19.245 on a per barrel basis Oil company might want to raise more per bond The bond buyers might want to receive cash during the life of the bond.

    16. 9-16 New problem… What quarterly cash coupon must the firm pay to make the bond worth $24.00? The promised barrel has a PV of $19.245 The PV of 4 coupon payments must have a PV of $4.755 Coupon*(.9852+.9701+.9546+.9388)=4.755 Coupon*(3.8487)=4.755 Coupon = $1.235 each quarter

    17. 9-17 Embed an option in the bond: Suppose the corporation wants to “buy a put” and offers the following: If oil < $18, payment = $24 – ($18-ST), otherwise payment = $24 Same as 24 – max(0,18-ST) There are no coupon payments What is the price of this bond? Assume volatility of oil is 35% 1-year interest rate: 0.9388 = 1e-r(1), therefore r=.0631 Since F=20.5 = 20.90e(.0631-?)1, therefore ?=0.0824

    18. 9-18 What does the bond buyer get? Bond price = PV($24) - PV(put) First, value the put option component (S=20.9, K=18, r=.0631, ?=.0824, ?=.20, t=1) Bond Price = (24)e-.0631 -1.5 = $21.03

    19. 9-19 Bond buyer’s payoff

    20. 9-20 Bond issuer’s payoff

    21. 9-21 Another example – embed call option in coupon paying commodity-linked bond Suppose now that the bond promises a $1.235 quarterly coupon, 1 barrel of oil at the end of the year, and 3*max(0,ST-$20.5), where $20.5 is the 1 year forward price of oil. What is the value of this bond if the volatility of oil is 18%?

    22. 9-22 Valuing the structured note Bond price = PV{quarterly coupon stream + S1 + 3*max(0,ST-$20.5)}, PV 4 coupon payments = 4.755 PV of S1 = 19.245 Need to value a call option where the underlying asset pays a lease rate 1-year interest rate: 0.9388 = 1e-r(1), therefore r=.0631 Since F=20.5 = 20.90e(.0631-?)1, therefore ?=0.0824 Option price = Bond value = 4.755+19.245+3*1.38 = $28.14

    23. 9-23 Payoff to the bondholder

    24. 16-24 More Corporate Applications Acquisitions Alternative Debt Structures Compensation Real Options (not covered)

    25. 16-25 Introduction There are four major corporate contexts in which options regularly appear, explicitly or implicitly Acquisitions Capital structure (equity, debt, and warrants) Compensation Real options

    26. 16-26 Acquisitions

    27. 16-27 The Use of Collars in Acquisitions When firm A buys firm B, it can pay cash to B’s shareholders or it can pay shares, exchanging A shares for B share Cash offers have relatively less risk than a share offer. Why? Once Company B accepts a share offer, the acquisition will take time to complete Which company bears the risk of a change in the stock price of company A? What is the magnitude of this exposure?

    28. 16-28 The Use of Collars in Acquisitions (cont’d) There are 4 common ways to structure an offer Fixed stock offer A offers to pay B a fixed number of A shares per B share. Floating stock offer A offers to pay B however many shares have a given dollar value, based on A’s share price just before the merger is completed Fixed collar offer There is one range for A’s share price within which the offer is a fixed stock offer Floating collar offer There is one range for A’s share price within which the offer is a floating stock offer

    29. 16-29 WorldCom/MCI acquisition On October 1, 1997, WorldCom Inc. CEO (Bernard Ebbers) sent the following note to the CEO of MCI (Bert Roberts), and it was also released through the typical wires: “I am writing to inform you that this morning WorldCom is publicly announcing that it will be commencing an offer to acquire all the outstanding shares of MCI for $41.50 of WorldCom common stock per MCI share. The actual number of shares of WorldCom to be exchanged for each MCI share in the exchange offer will be determined by the stock price of WorldCom on the closing day of the offer, but will be no less than 1.0375 shares (if WorldCom’s average stock price exceeds $40) or no more than 1.2206 shares (if WorldCom’s average stock price is less than $34).

    30. 16-30 Profile of the offer The payoff is contingent upon price of WCOM:

    31. 16-31 Structuring the offer The offer can be decomposed into three products A risk-free bond paying 41.5 1.2206 short put options with K=34 1.0375 long call options with K=40

    32. 16-32 GTE enters the mix Two weeks later, on October 15th, GTE bid for MCI, offering $40 in cash. On that day, WCOM stock closed at $35.4375. Suppose that, if approved on October 15, the WorldCom offer would have taken exactly four months to close (due to anti-trust regulation), while th eGTE offer could have been closed immediately. Finally, assume that on October 15, 1997, the continuously compounded risk free rate of return was 5.5%, the volatility of WCOM was 0.38, and WCOM was not expected to pay any dividends over the life of the offer.

    33. 16-33 Picture of the competitive offers

    34. 16-34 Which offer is greater? Need to compare PV of WCOM offer to $40 PV call = 1.695 PV put = 2.088 PV bond = 41.5e(-.055*.3333)=40.746 Sum = 40.746-1.2206*2.088+1.0375*1.695=$39.95 < $40

    35. 16-35 Apache / Amoco acquisition In 1991 Amoco decided to sell marginal oil and gas properties Created a new organization, MW Petroleum Corporation 9,500 oil wells 300 producing fields Hired Morgan Stanley to market the unit to potential buyers

    36. 16-36 Apache Corporation An independent oil and gas company was an aggressive acquirer of oil and gas fields Apache’s strategy “…is a bit like a pig following a cow through the cornfield. The scpas are pretty good for someone with our particular mission.” They viewed MW Petroleum as good scraps

    37. 16-37 It all hinges on the price Amoco argues that oil prices will be great in the future, therefore they ask for a price that assumes high oil prices Apache (of course) believed oil prices would be stable or lower (following Gulf War I) – resulting in a lower bid Bid-ask spread was about 10% Too high to “split the difference”

    38. 16-38 Good news The deal hinged upon a disagreement about a commodity price in the future Amoco doesn’t want to have sold too low if oil prices turn out high Apache doesn’t want to have paid too much if oil prices turn out low Therefore the deal is structured so that both sides shared the risk of future price movements

    39. 16-39 Proposed deal structure Amoco could write Apache a “capped price support guarantee”. If oil prices fall too low over the next 2 years Amoco would make payments of K-ST where $K is the price support level. Simultaneously, Apache would pay Amoco if oil prices exceeded a designated “price sharing” level over the next 5-8 years If oil prices rise over K, Apache pays Amoco ST-K.

    40. 16-40 The commodity collar Apache/Amoco collared the offer on the price of oil Apache gives up some upside in return for downside protection Apache wrote a call and bought a put Note, each party gets the price it had forecast if their forecast was accurate A ‘win-win’ deal structure

    41. 16-41 Tufano’s comments on the case “It did not take financial skills to recognize that the risks of this deal could be shared” “Financial analysts, however, could value the collar by using actual data and financial models. Moreover, their pricing exercise was not merely theoretical. After the deal was closed, both sides were approached to sell off their positions and thus had the choice of monetizing the options and closing their risk exposures.”

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