Fi8000 Valuation of Financial Assets Spring Semester 00 Dr. Isabel katch Assistant rofessor of Finance Valuation of Options Arbitrage Restrictions on the Values of Options Quantitative ricing Models Binomial model A A formula in the simple case An algorithm in the general case Black-Scholes model (a formula) Binomial Option ricing Model Assumptions: A single period wo dates: time t0 time t (expiration) he future (time ) stock price has only two possible values he price can go up or down he perfect market assumptions No transactions costs, borrowing lending at the risk free interest rate, no taxes Binomial Option ricing Model Example he stock price Assume S $50, u 0% d (-3%) S S u S (+u) S d S (+d) S$50 S u $55 S d $48.5 Binomial Option ricing Model Example he call option price Assume X $50, year (expiration) Binomial Option ricing Model Example he bond price Assume r 6% C u Max{Su-X,0} C u $5 Max{55-50,0} (+r) $.06 C C $ C d Max{Sd-X,0} C d $0 Max{48.5-50,0} (+r) $.06
Replicating ortfolio At time t0, we can create a portfolio of N shares of the stock an investment of B dollars in the risk-free bond. he payoff of the portfolio will replicate the t payoffs of the call option: N $55 + B $ $.06 $5 N $48.5 + B $.06 $0 Obviously, this portfolio should also have the same price as the call option at t0: N $50 + B $ C We get N0.769, B( (-35.959) the call option price is C$3.656. Replicating ortfolio he weights of Bonds Stocks in the replicating portfolio are: Z B (-$35.959) / $3.656-0.78 Z S (0.769 * $50) / $3.656.78 Say you invest $00. he two equivalent investment t strategies are:. Buy call options for $00 (i.e., buy $00 / $3.656 30.6 call options). Sell bonds for 0.78 * $00 $,078 Buy stocks for.78 * $00 $,78 Binomial Option ricing Model Example Continued he put option price Assume X $50, year (expiration) u Max{X-Su,0} d Max{X-Sd,0} u $0 Max{50-55,0} d $.5 Max{50-48.5,0} Replicating ortfolio for the ut Option At time t0, we can create a portfolio of N shares of the stock an investment of B dollars in the risk-free bond. he payoff of the portfolio will replicate the t payoffs of the put option: N $55 + B $ $.06 $0 N $48.5 + B $.06 $.5 Obviously, this portfolio should also have the same price as the put option at t0: N $50 + B $ We get N( (-0.308), B.9739 the put option price is $0.4354. Replicating ortfolio A Different Replication he weights of Bonds Stocks in the replicating portfolio are: Z B ($.9739) / $0.4354 7.5 Z S (-0.308 * $50) / $0.4354-6.5 Say you invest in one put options contract (i.e. 00 options). he two equivalent investment strategies are:. Buy one put options contract for $0.4354*00 $43.54. Buy bonds for 7.5 * $43.54 $,97.35 Sell stocks for 6.5 * $43.54 $,53.8 he price of $ in the up state: $ q u $0 he price of $ in the down state: $0 q d $
Replicating ortfolios Using the State rices We can replicate the t payoffs of the stock the bond using the state prices: q u $55 + q d $48.5 $50 q u $.06 u + q d $.06 $ Obviously, once we solve for the two state prices we can price any other asset in that economy. In particular we can price the call option: q u $5 + q d $0 C We get q u 0.653, q d 0.903 the call option price is C$3.656. Binomial Option ricing Model Example he put option price Assume X $50, year (expiration) u Max{X-Su,0} d Max{X-Sd,0} u $0 Max{50-55,0} d $.5 Max{50-48.5,0} Replicating ortfolios Using the State rices We can replicate the t payoffs of the stock the bond using the state prices: q u $55 + q d $48.5 $50 q u $.06 $06+q q d $.06 $ But the assets are exactly the same so are the state prices. he put option price is: q u $0 + q d $.5 We get q u 0.653, q d 0.903 the put option price is $0.4354. wo eriod Example Assume that the current stock price is $50, it can either go up 0% or down 3% in each period. he one period risk-free interest rate is 6%. What is the price of a European call option on that stock, with an exercise price of $50 expiration in two periods? he Stock rice S $50, u 0% d (-3%) S uu $60.5 he Bond rice r 6% (for each period) $.36 S u $55 $.06 S$50 S ud S du $53.35 $ $.36 S d $48.5 S dd $47.05 $.06 $.36 3
he Call Option rice State rices in the wo eriod ree X $50 periods C C u C d C uu Max{60.5-50,0}$0.5 C ud Max{53.35-50,0}$3.35 C dd Max{47.05-50,0}$0 We can replicate the t payoffs of the stock the bond using the state prices: q u $55 + q d $48.5 $50 q u $.06 + q d $.06 $ Note that if u, d r are the same, our solution for the state prices will not change (regardless of the price levels of the stock the bond): q u S (+u) + q d S (+d) S q u (+r) t + q d (+r) t (+r) (t (t-) herefore, we can use the same state-prices in every part of the tree. he Call Option rice q u 0.653 q d 0.903 C uu $0.5 wo eriod Example What is the price of a European put option on that stock, with an exercise price of $50 expiration in two periods? C u C C d C u 0.653*$0.5 + 0.903*$3.35 $7.83 C d 0.653*$3.35 + 0.903*$0.00 $.9 C 0.653*$7.83 + 0.903*$.9 $5.75 C ud $3.35 C dd $0 What is the price of an American call option on that stock, with an exercise price of $50 expiration in two periods? What is the price of an American put option on that stock, with an exercise price of $50 expiration in two periods? wo eriod Example European put option - use the tree or the put-call parity he European ut Option rice q u 0.653 q d 0.903 uu $0 What t is the price of an American call option - if there are no dividends u d ud $0 dd $.955 American put option use the tree u 0.653*$0 + 0.903*$0 $0 d 0.653*$0 + 0.903*$.955 $0.858 EU 0.653*$0 + 0.903*$0.858 $0.5 4
he European ut Option rice Another way to calculate the European put option price is to use the put-call-parity restriction: C + V ( X ) S + C + V ( X ) S 5.75 + 50 ( +.06) 50 $0.5 he American ut Option rice q u 0.653 q d 0.903 Am u d uu $0 ud $0 dd $.955 American ut Option Note that at time t the option buyer will decide whether to exercise the option or keep it till expiration. If the payoff from immediate exercise is higher than the value of keeping the option for one more period ( European ), then the optimal strategy is to exercise: If Max{ X-S u,0 } > u ( European ) > Exercise he American ut Option rice q u 0.653 q d 0.903 u d uu $0 ud $0 dd $.955 u Max{ 0.653*0 + 0.903*0, 50-55 } $0 d Max{ 0.653*0 + 0.903*.955, 50-48.5 } 50-48.5 $.5 Am Max{ 0.653*0 + 0.903*.5, 50-50 } $0.4354 > $0.490 Eu Determinants of the Values of Call ut Options Variable S stock price X exercise price σ stock price volatility time to expiration r risk-free interest rate Div dividend payouts C Call Value Decrease Decrease ut Value Decrease Decrease Black-Scholes Model Developed around 970 Closed form, analytical pricing model An equation Can be calculated easily quickly (using a computer or even a calculator) he limit of the binomial model if we are making the number of periods infinitely large every period very small continuous time Crucial assumptions he risk free interest rate the stock price volatility are constant over the life of the option. 5
Black-Scholes Model he N(0,) Distribution C S N( d ) Xe N( d ) r S ln + r+ σ X d, σ d d σ C call premium S stock price X exercise price time to expiration r the interest rate σ std of stock returns ln(z) natural log of z e -r exp{ exp{-r} (.783) -r N(z) stard normal cumulative probability pdf(z) N(z) μ z 0 z Black-Scholes example Black-Scholes Example r C S N( d C? ) Xe N( d) S 05years 0.5 ln + r+ σ X r 0.05 (5% annual rate d, σ d d σ σ 0.30 (or 30%) d ( 0.836) N( d ) 0.47 d ( 0.3336) N( d ) 0.3693 C $.056 C? 05years 0.5 r 0.05 (5% annual rate σ 0.30 (or 30%) Black-Scholes Model Continuous time therefore continuous compounding N(d) loosely speaking, N(d) is the risk adjusted probability that the call option will expire in the money (check the pricing for the extreme cases: 0 ) ln(s/x) approximately, a percentage measure of option moneyness Black-Scholes Model [ ] [ ] r Xe N( d ut premium ) S N( d) S stock price X exercise price time to expiration S ln + r + σ r the interest rate X d σ std of stock returns, σ ln(z) natural log of z e -r exp{ exp{-r} (.783) -r N(z) stard normal d d σ cumulative probability 6
Black-Scholes Example he ut Call arity d ( 0.836) N( d ) 0.47 d ( 0.3336) N( d ) 0.3693 $3.935? 05years 0.5 r 0.05 (5% annual rate σ 0.30 (or 30%) he continuous time version (continuous compounding): C + V ( X ) S+ r C+ Xe S + 0.05 0.5 $.056 $50? $47.5 $3.935 + e + Stock Return Volatility One approach: Calculate an estimate of the volatility using the historical stock returns plug it in the option formula to get pricing n Est( σ ) return average return n ( ) t t S return ln t t S t Stock Return Volatility Another approach: Calculate the stock return volatility implied by the option price observed in the market (a trial error algorithm) C S N( d ) Xe N( d ) r S ln + r + σ X d d d σ σ Option rice Volatility Let σ < σ be two possible, yet different return volatilities; C, C be the appropriate call option prices;, be the appropriate put option prices. We assume that the options are European, on the same stock S that pays no dividends, with the same expiration date. Note that our estimate of the stock return volatility changes. he two different prices are of the same option, can not exist at the same time! hen, C C Implied Volatility - example r C S N( d C $.5 ) Xe N( d) S 05years 0.5 ln + r+ σ X r 0.05 (5% annual rate d, σ d d σ σ? 7
Implied Volatility - example d ( 0.836) N( d ) 0.47 d ( 0.3336) N( d ) 0.3693 C $.056 < $.5 σ < 0.3 or σ > 0.3? C? 05years 0.5 r 0.05 (5% annual rate σ 0.30 (or 30%) Implied Volatility - example d ( 0.0940) N( d ) 0.466 d ( 0.940) N( d ) 0.3844 C $.99 > $.5 C? 05years 0.5 r 0.05 (5% annual rate σ 0.40 (or 40%) Implied Volatility - example Application: ortfolio Insurance d ( 0.34) N( d ) 0.4466 d ( 0.309) N( d ) 0.3786 C $.505 C? 05years 0.5 r 0.05 (5% annual rate σ 0.35 (or 35%) Options can be used to guarantee minimum returns from an investment in stocks. urchasing gp portfolio insurance (protective put strategy): Long one stock; Buy a put option on one stock; If no put option exists, use a stock a bond to replicate the put option payoffs. ortfolio Insurance Example You decide to invest in one share of General ills (G) stock, which is currently traded for $56. he stock pays no dividends. You worry that the stock s price may decline decide to purchase a European put option on Gs stock. he put allows you to sell the stock at the end of one year for $50. If the std of the stock price is σ0.3 (30%) rf0.08 (8%, what is the price of the put option? What is the CF from your strategy at time t0? What is the CF at time t as a function of 0<S <00? ortfolio Insurance Example What if there is no put option on the stock that you wish to insure? - Use the B&S formula to replicate the protective put strategy. What is your insurance strategy? What is the CF from your strategy at time t0? Suppose that one week later, the price of the stock increased to $60, what is the value of the stocks bonds in your portfolio? How should you rebalance the portfolio to keep the insurance? 8
ortfolio Insurance Example he B&S formula for the put option: - -Xe -r [-N(d )]+S[-N(d )] herefore the insurance strategy (Original portfolio + synthetic put) is: Long one share of stock Long X [-N(d )] bonds Short [-N(d )] stocks ortfolio Insurance Example he total time t0 CF of the protective put (insured portfolio) is: CF 0 -S 0-0 -S 0 -Xe -r [-N(d )]+S 0 [-N(d )] -S 0 [N(d )] -Xe -r [-N(d )] And the proportion invested in the stock is: w stock -S 0 [N(d )] /{-S 0 [N(d )] -Xe Xe -r [ [-N(d )]} ortfolio Insurance Example he proportion invested in the stock is: w stock S 0 [N(d )] /{S 0 [N(d )] +Xe -r [-N(d )]} Or, if we remember the original (protective put) strategy: w stock S 0 [N(d )] /{S 0 + 0 } Finally, the proportion invested in the bond is: w bond -w stock ortfolio Insurance Example Say you invest $,000 in the portfolio today (t0) ime t 0: w stock 56 0.7865/(56+.38) 75.45% Stock value 0.7545*$,000 $754.5 Bond value 0.455 *$,000 $45.5 End of week (we assumed that the stock price increased to $60): Stock value ($60/$56) $754.5 $808.39 Bond value $45.5 e 0.08 (/5) $45.88 ortfolio value $808.39+$45.88 $,054.7 ortfolio Insurance Example Now you have a $,054.7 portfolio ime t (beginning of week ): w stock stock 60 0.8476/(60+.63) 8.53% Stock value 0.853*$,054.7 $870.06 Bond value 0.747 *$,054.7 $84. I.e. you should rebalance your portfolio (increase the proportion of stocks to 8.53% decrease the proportion of bonds to 7.47%). Why should we rebalance the portfolio? Should we rebalance the portfolio if we use the protective put strategy with a real put option? ractice roblems BKM Ch. 7th Ed. : 7-0, 7,8 8th Ed. : -4, 4, 7,8 ractice set: 36-4. 9