Unformatted text preview: TBS 907 AUTUMN 2005 LECTURE 9 1 Options and Option Markets An option is the right to force a transaction to occur at some future time on terms and conditions decided now. Contract which gives the purchaser the right, but not the obligation, to assume a long (buy) or short (sell) position in the relevant underlying financial instrument at a pre-determined exercise (strike) price, at a time in the future.
2 Options and Option Markets (cont.) In return for this right the purchaser pays a premium to the seller/writer of the option. Unlike FRAs and futures contracts, options allow the benefits of favourable price movements and provide protection against unfavorable price movements.
3 Option Terminology Call option Purchaser has the right but not the obligation to buy an asset at the specified exercise price. Purchaser's risk is limited to the premium paid. Writer's profit is limited to the premium paid and has unlimited upside risk should prices rise.
4 Option Terminology (cont.) Put option Gives the purchaser the right, but not the obligation, to sell a specified asset at a specified exercise price. The purchaser's risk is limited to the premium paid. The writer of the put option has unlimited risk should prices fall, and profit is limited to the premium 5 received. Option Terminology (cont.) Exercise price (strike price) The price at which a particular option can be exercised. American-type options Buyer may exercise option at any time up to (and including) the expiry date. European-type options Buyer may only exercise option on the expiry date (and not before). 6 Option Terminology (cont.) In the money Call option whose exercise price is below, or a put option whose exercise price is above, the current price of the asset on which the option is written. Out-of-the-money Term used to refer to a call option with an exercise price above, or to a put option with an exercise price below, the current price of the asset 7 on which the option is written. How Options are Created and Traded Options can be created by the company whose shares underlie the option contract. May be issued to raise capital for the company. May be issued to employees of the company to form part of a compensation package. Options can also be created by parties who may have no association with the company. For example, two share market observers may enter into a private option contract on the shares of BHP. 8 Payoff Structures for Calls Set of future cash flows, example is for a call with an exercise price of $34.00
Table 19.1 IF THE SHARE PRICE ($) ON THE CALL'S EXPIRY DATE IS 32.00 32.50 33.00 33.50 34.00 34.50 35.00 35.50 36.00 36.50 THEN THE PAYOFF (CASH FLOW) ($) TO THE CALL HOLDER IS 0 0 0 0 0 0.50 1.00 1.50 2.00 2.50
9 Payoff Structures for Calls (bought)
Payoff 1.00 0.50 33.00 33.50 34.00 34.50 35.00 35.50
10 Share price on call's expiry date ($) Figure 19.1 Payoff Structures for Puts
Put with an exercise value of $30.50
IF THE SHARE PRICE ($) ON THE PUT'S EXPIRY DATE IS 28.00 28.50 29.00 29.50 30.00 30.50 31.00 31.50 32.00 THEN THE PAYOFF (CASH FLOW) ($) TO THE PUT HOLDER IS 2.50 2.00 1.50 1.00 0.50 0 0 0 0 Table 19.2
11 Payoff Structures for Puts (Bought)
Payoff ($) 30.5 20 10.5 0 10 20 30.5 40 Share price on put's expiry date ($) 50 Figure 19.2
12 Option Pricing At expiry a call is worth, and should therefore be priced at: MAX [ 0, P * C] At expiry a put is worth, and should therefore be priced at: MAX [ 0, C P * ] where P * = date C the share price on the call's expiry = the exercise price of the option
13 Option Pricing Option price depends on: underlying physical price strike price the risk-free interest rate time to maturity volatility of underlying asset expected dividends
14 Factors Affecting Call Options The current share price The higher the current share price, the greater is the probability that the share price will increase above the exercise price and, therefore, the higher the call price becomes, other things being equal. The exercise price The higher the exercise price, the lower is the probability that the share price will increase above the exercise price and, therefore, the lower the call price becomes, other things being equal. 15 Factors Affecting Call Options (cont.) The term to expiry A longer-term option dominates a shortterm option, because there is more time for the share price to increase above the exercise price. Therefore, the longer the term to expiry, the greater the call price, other things being equal. 16 Factors Affecting Call Options (cont.) The volatility of the share Higher share price volatility increases the chance of both large increases and large decreases in the share price. However, the asymmetric features of options mean that the holder of a call gains more from the increased chance of a large increase in the share price than is lost from the increased chance of a large decrease. Higher volatility increases the price of a call, other things being equal.
17 Factors Affecting Call Options (cont.) The risk-free interest rate The buyer of a call option can defer paying for the shares. Because interest rates are positive, money has a time value, so the right to defer payment is valuable. The higher the interest rate, the more valuable is this right. Therefore, the higher the risk-free interest rate, the higher the price of a call, other things being 18 equal. Factors Affecting Call Options (cont.) Expected dividends If a company pays a dividend to its ordinary shareholders, the share price will fall on the ex-dividend date. Therefore, a call on a share that will go ex-dividend before the expiry of the call, is worth less than if the share either never pays dividends or, if it does pay dividends, will not reach the next exdividend date until after the call has expired. 19 Factors Affecting Call Options (cont.) Other things being equal, call prices should be higher (lower): the higher (lower) the current share price the lower (higher) the exercise price the longer (shorter) the term to expiry the more (less) volatile the underlying share the higher (lower) the risk-free interest rate the lower (higher) the expected dividend to be paid following an ex-dividend date that occurs during the term of the call 20 PutCall Parity For European options on shares that do not pay dividends, there is an equilibrium relationship between the prices of puts and calls that are written on the same underlying share, are traded simultaneously, and have the same exercise price and term to expiry. 21 PutCall Parity (cont.) C x = w -P + 1 +r ' where x = the price of the European put w = the price of the corresponding European call P = the share price C = the exercise price r = the risk-free interest rate for borrowing or lending for a period equal to the term of the put and the call
22 PutCall Parity (cont.) While there is no simple equation linking the values of American puts and calls, the following upper and lower bounds have been established: C W - P + X W - P + C 1 +r "
X = the price of the American put W = the price of the corresponding American call P = the share price C = the exercise price r = the risk-free interest rate for where borrowing 23 The Minimum Value of Calls The minimum value of a European call on a nondividend paying share: C Min w = Max 0, P 1 + r In the absence of dividends, American call options should not be exercised before expiry. Supposing that P > C and the holder of the option decided to dispose of the call, the payoff from selling the call [P C / (1 + r )] will always be greater than the payoff from exercising the call [P C ].
24 [ The Minimum Value of Puts The minimum value of a European put is: C Min x = Max 0, P 1+r [ Unlike calls, it can be rational to exercise an American put before expiry. In some circumstances, the benefit of receiving an early cash flow from early exercise will outweigh the cost of forfeiting some of the option's time value.
25 Black-Scholes Model of Call Option Pricing Black and Scholes presented a model that determines the price of a call as a function of five variables: the current price of the underlying share the exercise price of the call the call's term to expiry the volatility of the share (as measured by the variance of the distribution of returns on the share) the risk-free interest rate 26 Black-Scholes Model of Call Option Pricing (cont.) Assumptions: Constant risk-free interest rate at which investors can borrow and lend unlimited amounts. Share returns follow random walk in continuous time with a variance proportional to the square of the share price. The variance rate is a known constant. No transaction costs or taxes. Short selling is allowed, with no restrictions or penalties. There are no dividends or rights issues. The call is of the European type.
27 Option Value
Black-Scholes Option Pricing Model OC = [ N (d1 ) P ] - [ N (d 2 ) PV ( EX )] 28 Black-Scholes Option Pricing Model
OC = [ N ( d1 ) P ] - [ N ( d 2 ) PV ( EX )]
OC- Call Option Price
P - Stock Price N(d1) - Cumulative normal density function of (d1) PV(EX) - Present Value of Strike or Exercise price N(d2) - Cumulative normal density function of (d2) r - discount rate (90 day comm paper rate or risk free rate) t - time to maturity of option (as % of year) v - volatility - annualized standard deviation of daily returns Black-Scholes Option Pricing Model
OC = [ N ( d1 ) P ] - [ N ( d 2 ) PV ( EX )] PV ( EX ) = EX (e
-rt -rt ) 1 = rt = continuous compounding discount factor e Black-Scholes Option Pricing Model d1 = ln( P EX ) + (r + v t v2 2 )t N(d1)= 32 34 36 38 40 Cumulative Normal Density Function d1 = ln( P EX ) + (r + v t v2 2 )t d 2 = d1 - v t Call Option
Example What is the price of a call option given the following? P = 36 r = 10% v = .40 EX = 40 t = 90 days / 365
P ln( EX ) + ( r + v2 )t d1 = v t
2 d1 = -.3070 N ( d1 ) = 1 - .6206 = .3794 Call Option
Example What is the price of a call option given the following? P = 36 r = 10% v = .40 EX = 40 t = 90 days / 365 d 2 = d1 - v t d 2 = -.5056 N ( d 2 ) = 1 -.6935 = .3065 Call Option
Example What is the price of a call option given the following? P = 36 r = 10% v = .40 EX = 40 t = 90 days / 365 OC = [ N ( d1 ) P ] - N ( d 2 ) ( EX )e -rt OC OC = $1.70 [ = [.3794 36] - [.3065 ( 40)e -(.10 )(.2466 ) Black Scholes Comparisons
Establishment Industries INPUTS Stock price(P) Exercise price (EX) Interest rate, percent Maturity in years (t) Annual standard deviation, percent () Are these rates compounded annually (A) or continuously ? equivalent continously compounded rate, percent INTERMEDIATE CALCULATIONS: PV(EX) d1=log[P/PV(EX)]/....... d2=d1-..... N(d1)= delta N(d2) OPTION VALUES: Call value = N(d1) * P - N(d2)* PV(EX) Put value = Call value + PV(EX) - S 22 25 4 5 24 a 3.92 Digital Organics 22 25 4 5 36 a 3.92 20.5482 0.3955 -0.1411 0.6538 0.4439 20.5482 0.4873 -0.3177 0.687 0.3754 5.26 3.81 7.4 5.95 36 Contingent Claims (cont.) Rights Issue A shareholder is given the right to purchase new shares in the company at an issue price set by the company. The rights must be sold or taken up by a specified date. Simply a call option issued by the company.
37 Contingent Claims (cont.) Convertible Bonds A type of debt security that, in addition to paying interest, gives the investor the right to convert the security into shares of the company. Equivalent to ordinary debt plus a call option on the shares of the company. 38 Contingent Claims (cont.) Project Evaluation and `Real' Options The NPV approach is based on an analogy between a proposed investment project and a bond. However, sometimes the bond analogy is not the best one available. For example, once a project has commenced, the bond analogy assumes its cash flows cannot be changed. However, in practice the investing company may intervene in the project after it has started. In short, there may be `options' such as abandonment available. 39 Contingent Claims (cont.) Project Evaluation and `Real' Options Some real options include: The option to defer, abandon, re-open, to study or to expand. These sorts of options are written into contracts on a regular basis, which suggests they are valuable. The point is that having such options cannot reduce the value of a project or asset: at worst they have zero value. 40 Example Mark I Microcomputer ($ millions)
Year 1984 159 0 100 59 Microcomputer Forecasts
1982 1983 100 0 50 60 1985 295 0 100 195 1986 185 0 -125 310 1987 0 0 -125 125 After-tax operating cash flow (1) Capital investment (2) Increase in working capital (3) Net cash flow (1)-(2)-(3) 450 0 -450 NPV at 20% = - $46.45, or about -$46 million 41 Example Mark II Microcomputer
900 = 676 3 1.1 Microcomputer Forecasts PV (exercise price) = OC = [ N ( d1 ) P ] - [ N ( d 2 ) PV ( EX )]
d1 = log[P / PV ( EX )] / t + t / 2 = log[.691 / .606] + .606 / 2 = -.3072 d 2 = d1 - t = -.3072 - .606 = -.9134 N ( d1 ) = .3793 N ( d 2 ) = .1805 Call Value = [.3793 467} - [.1805 676] = $55.12million
42 Example Mark II Microcomputer ($ millions) Forecasted cash flows from 1982
1982 After-tax operating cash flow Increase in working capital Net cash flow Present Value @ 20% Investment, PV @ 10% Forecasted NPV in 1985 .......... Year 1985 1986 220 100 120 1987 318 200 118 Microcomputer Forecasts 1988 590 200 390 1989 370 -250 620 1990 0 -250 250 467 676 807 900 -93 NPV(1982) =PV(inflows) -PV(investment) = 467 676 = - $209 million 43 Example Mark II Microcomputer (1985) Microcomputer Forecasts
Distribution of possible Present Values Probability Present value in 1985 Expected value ($807) Required investment ($900)
44 Option to Wait
Option Price Stock Price Option to Wait
Intrinsic Value + Time Premium = Option Value Time Premium = Vale of being able to wait
Option Price Stock Price Option to Wait
More time = More value Option Price Stock Price Option to Wait
Example Development option
Cash flow Office Bldg Office Bldg NPV>0 240 Wait 100 NPV<0 1 Hotel NPV>0 240 Cash flow from hotel 48 Option to Abandon
Example - Abandon Mrs. Mulla gives you a non-retractable offer to buy your company for $150 mil at anytime within the next year. Given the following decision tree of possible outcomes, what is the value of the offer (i.e. the put option) and what is the most Mrs. Mulla could charge for the option? Use a discount rate of 10% Example - Abandon Option to Abandon Mrs. Mulla gives you a non-retractable offer to buy your company for $150 mil at anytime within the next year. Given the following decision tree of possible outcomes, what is the value of the offer (i.e. the put option) and what is the most Mrs. Mulla could charge for the option? Year 0 Year 1 Year 2 120 (.6) 100 (.6) 90 (.4) NPV = 145 70 (.6) 50 (.4) 40 (.4) Example - Abandon Option to Abandon Mrs. Mulla gives you a non-retractable offer to buy your company for $150 mil at anytime within the next year. Given the following decision tree of possible outcomes, what is the value of the offer (i.e. the put option) and what is the most Mrs. Mulla could charge for the option? Year 0 Year 1 Year 2 120 (.6) 100 (.6) 90 (.4) NPV = 162 Option Value =
150 (.4) 162 - 145 = $17 mil Option to Abandon
Example Ms. East - Revenues
3.73 3.05 2.50 2.50 2.05 1.68 52 Option to Abandon
Example Ms. East Cash Flows
3.03 2.35 1.80 1.80 1.35 .98 53 Option to Abandon
Example Ms. East Value
PV = 2.5 = $2.29million 1.09 3.05 p + 2.05(1 - p ) Expected return = = .06 2.29 Prob of up change = .382 Prob of down change = .618 APV = -1.108 + 3.803 = +$2.695million 54 Tanker Example
Value of Tanker
Value in operation Cost of reactivating Mothballing costs Value if mothballed Tanker Rates 55 ...
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