OPTIONS Professor Anant K. Sundaram THUNERBIR Spring 2003 Options: efinitions Contingent claim; derivative Right, not obligation when bought (but, not when sold) More general than might first appear Calls, puts American, European Can buy, sell (write)» Notation:» S, T, S T,»,» C, C T,» P, P T,» r, e rt efinitions (Cont) Price of Call at Maturity and Payoff Price of call at maturity: C T = Max{0, S T } Payoff S T Price of Put at Maturity and Payoff Payoff from Holding Stock and Riskfree Bond Price of put at maturity: P T = Max{0, S T } Payoff Payoff Payoff S T Buying a Stock S T Buying a Riskfree Bond S T 1
Put-Call Parity Basic idea in options theory. Establishes a formal link between the four fundamental types of securities, stocks, riskfree bonds, calls and puts. Says (in FV terms): S T + P T = C T + Put-Call Parity (Cont) Holding a portfolio that is long one share and one put option on that stock (with exercise price ) gives us a payoff that is identical to that of a portfolio long one call option (at the same exercise price and maturity as the put) and $ worth of riskfree bonds. In PV terms: S + P = C + PV() Buying Stock Buying Call Put-Call Parity: Graphical escription + + Buying Put Buying Riskfree Bond = Portfolio of Stock+Put = Portfolio of Call+Riskfree bond Put-Call parity: Example Say, = $105 on European calls and puts of the same maturity for the same underlying asset. S T P T C T S T + P T C T + 101 4 0 105 101+4 = 0+105 102 3 0 105 102+3 = 0+105 103 2 0 105 103+2 = 0+105 104 1 0 105 104+1 = 0+105 105 0 0 105 105+0 = 0+105 106 0 1 105 106+0 = 1+105 107 0 2 105 107+0 = 2+105 108 0 3 105 108+0 = 3+105 109 0 4 105 109+0 = 4+105 Binomial Model Example = $50; T = 1 year; r = 5% S u = $55 S = $50 C =? S d = $45 C u =? C d =? Value of call if stock price goes up: C u = Max{0, S T } = Max{0, 5} = $5 Value of call if stock price goes down: C d = Max{0, S T } = Max{0, 5} = $0 2
We can create a synthetic portfolio of the underlying stock and a certain amount of riskfree bonds (whose values we already know), to mimic the payoff characteristics of the call option that we are trying to value. Once we create such a portfolio and figure out its value, we would have indirectly priced the option (by using the no-arbitrage logic). General principles in creating the synthetic portfolio:» Such a synthetic portfolio will consist of the underlying stock and the riskfree bond: long the stock, and short the bond.» There will be a specific ratio in which this portfolio value will be related to the option value, called the hedge ratio. Create the synthetic portfolio in this example:» Buy one share of stock ($50) and borrow $45/(1+r f ) = $45/(1.05) = $42.86.» You owe $45 to the lending institution one year from now.» If the stock goes up, this portfolio is worth:$55 $45 = $10» If the stock goes down, this portfolio is worth:$45 $45 = $0 Payoff comparison between synthetic portfolio and the option we are trying to value, given the same underlying states of nature, S u and S d : Portfolio Call Option S u $10 $5 S d $ 0 $0 The portfolio payoff is twice the payoff to the option fi The portfolio is worth twice as much as the option fi The option is worth half as much as the portfolio. Portfolio is worth: $50 $42.86 = $7.14 Therefore, the call option is worth $7.14/2 = $3.57 3
What is the put worth? Apply put-call parity: P = C + PV() S = $3.57 + $(50/1.05) $50 = $1.19 Multiperiod Binomial Option Valuation (Optional) Similar to single period valuation, except that we want to work backwards from the last period. For instance, if there are N-periods, start with Period N option values, then Period N 1, then N 2, and so forth. Note that in all periods but N, the value of the underlying security is the option value as derived from the subsequent period Equity in a levered firm is like a call option on the value of the firm s assets with an exercise price equal to the face value of debt... Value of equity + Value of debt Owning riskless debt is like owning risky debt and buying a put option with an exercise price equal to the face value of debt... Value of riskless debt = V V Value of risky debt Value of put option Value of equity plus debt = V = Firm value = Face value of debt + V Other corporate finance insights from options... An equity warrant is like a long-dated call option (after adjusting for dilution effects). Convertible debt is just a combination of straight debt plus a warrant. Buying insurance is like buying a put option. Other corporate finance insights from options... A loan guarantee is like a put option on the face value of the loan. A forward contract is the equivalent of buying a call and selling a put for an exercise price equal to the forward price, and for the maturity of the forward contract. 4
Assume that markets are perfect and that trading takes place continuously; Further assume that, over time, the movement in asset returns can be modeled as a diffusion process (this is a random walk in continuous time): ds/s = µdt + sdz This means the following: at any point in time, the expected instantaneous percentage change in asset prices is a function of two variables: (i) an expected rate of return µ in a small interval of time, dt, and (ii) an expected standard deviation in the rate of return, s, that gets moved around by a Weiner process, dz. B-S-M observed that we can create a hedge portfolio with the underlying asset and call options which will mimic the payoff from having risk-free debt. Since we know the rate of return on risk free debt, we can calculate its price; we can also observe the price of the underlying asset; thus we can derive the value of the only unknown, the call option. Once we know the value of the call, we can derive the value of the put using put-call parity. The mathematics produces the following formulae for pricing (European) call and put options: C = SN(d 1 ) e rt N(d 2 ) P = e rt N( d 2 ) SN( d 1 ), N = Cumulative normal probability that s variable takes on value d 1 ln(s/) + [r+s 2 /2]T d 1 = s T d 2 = d 1 s T Thus, need to know five variables: S,, T, r, and s, to value European options. N(d 1 ) is the option delta i.e., the amount of shares to buy in order to create the replicating portfolio for pricing one option. The amount to borrow in order to create this portfolio is e rt N(d 2 ). 5
B-S-M for a ividend-paying When the underlying asset pays dividends, one additional variable is introduced into the option pricing model: the (expected) annualized dividend yield. Call this annualized dividend yield y. B-S-M for a ividend-paying ividends lower the value of the underlying asset (since the ex-dividend price is lower by the extent of the pershare dividend) and therefore lower call option values (and increase put option values). B-S-M for a ividend-paying Early exercise may be optimal in the case of dividend-paying stocks if the dividends are high enough. The reason is that, by exercising prior to the dividend and obtaining the underlying asset, the option holder gets the dividends in addition to the underlying asset. B-S-M for a ividend-paying Likelihood of early exercise will also be affected by the time to expiration. B-S-M for a ividend-paying Stock (Cont) With dividends, the B-S-M model gets modified slightly to the following: C = Se yt N(d 1 ) e rt N(d 2 ) P = e rt N( d 2 ) Se yt N( d 1 ), where y is the annualized dividend yield B-S-M for a ividend-paying and, ln(s/) + [r y+s 2 /2]T d 1 = s T d 2 = d 1 s T Thus, we need to know six variables: S,, T, r, y, and s, to value American options on dividend-paying stocks. 6
Calculating Annualized Volatility Can be calculated using daily stock price data (But: How far back to go? How many data points?) Step 1: Calculate time series of returns using the formula ln(p t+1 /P t ) where ln is the natural logarithm, P t+1 is the asset price on day t+1 and P t the price on day t. Step 2: Calculate the s of the time series of returns. Calculating Annualized Volatility (Cont) Step 3: Multiply this daily s by the square root of 250 (If using weekly data, multiply by the square root of 50), to get the annualized s. Note: We can also calculate implied volatilities based on observed option price data. Volatility Calculation: Example Suppose the following are daily stock prices: ay 1 $25 ay 2 $26 ay 3 $23 ay 4 $24 ay 5 $25 : : : : ay 59 $25 ay 60 $22 Volatility Calculation: Example Step 1: Calculate daily returns: ay 1 $25 ln(26/25) = 3.92% ay 2 $26 ln(23/26) = 12.26% ay 3 $23 ln(24/23) = 4.26% ay 4 $24 ln(25/24) = 4.08% ay 5 $25 : : : : : : : ay 59 $25 : ay 60 $22 ln(22/25) = 12.78% aily returns Volatility Calculation: Example Step 2: Calculate the standard deviation of the daily returns ( STEV in Excel); suppose this worked out to 2.3%. Step 3: Since we are using daily data, multiply this number by the square root of 250: (0.023)*( 250) =.3637 The volatility of this stock (annualized) for B-S-M input purposes is 36.37% B-S-M Example: American Option Consider the following data for IBM on October 9, 2002:» Stock price, S = $57.05» Exercise price, = $60.00» Time to maturity, T = 1 month (26 days)» Riskfree rate (3 month T-bill yield), r F = 1.60%» ividend yield = 1.10%» Volatility, s = 42.75% 7
B-S-M Example: American Option Using the NUMA options calculator:» Price of the call = $1.45» Price of the put = $4.38 The actual price of the call and put were $1.45 and $4.30 respectively. Pretty decent estimates! How Changes in Underlying Variables Affect Call & Put Values Variable Impact on Call Impact on Put Explanation Increase in S Call: Further in-the-money Put: Further out-of-the-money Increase in Increase in s Increase in T Increase in r f Call and Put: Reduces the PV Increase in y of the underlying asset Call: Less to gain on exercise Put: More to gain on exercise Call: on t care about downside Put: on t care about upside Call and Put: Increased prob. of finishing in-the-money Call and Put: Reduces the PV of exercise price 8