1/1 Options, Derivatives, Risk Management (Welch, Chapter 27) Ivo Welch UCLA Anderson School, Corporate Finance, Winter 2014 January 13, 2015 Did you bring your calculator? Did you read these notes and the chapter ahead of time?
2/1 Maintained Assumptions We assume perfect markets, so we assume 1. No differences in opinion. 2. No taxes. 3. No transaction costs. 4. No big sellers/buyers we have infinitely many clones that can buy or sell. We assume uncertainty and risk-aversion, as in the portfolio optimization and CAPM chapters (Chapters 8 and 9).
3/1 Definition of Derivative A financial instrument whose payoff depends on some other asset. Often a (side-)bet between investors. Thus, for every long there is a short. (Opposite=Stock is in positive net supply.) A priori, both should be better off contracts are voluntary. But after the fact, one will lose, one will win. Examples: Life Insurance. Car Insurance. A bet paying off $50 if the price of oil on December 31, 2020 will be > $100. The price of oil on December 31, 2020, minus $100. The price of oil, squared, on December 31, 2020, minus $100, squared. The most prominent financial derivatives: Forward and Future an agreement to buy or sell something for an agreed upon time in the future. Often written so that no money exchanges hands today. Future = settled every day to make price zero (reduces run-away default). Option an agreement that one party has the right but not the obligation to buy or sell something to the other party for an agreed-upon time period. Many others, such as swaps.
4/1 Option The right but not obligation to transact at a predetermined strike price K at a predetermined time (or time range) in the future. The final time is usually abbreviated by capital T. Call option: The right to buy 1 share. Its value at expiration is C(K, T) = max(0, S T K) Put option: The right to sell 1 share. Its value at expiration is P(K, T) = max(0, K S T ) Jargon: Out-of-the-money = if it expired now, you would get nothing. In the money = if it expired now, you would get something. At the money = right at the cusp. Above-the-money = S > K. American option = the option holder can exercise anytime before the final moment. European option = exercise only at the final instant. Exchange-traded options have compensating rules for cases in which shares split, but not for dividend payments, even liquidating ones. PS: (Usually) sold in lots of 100 only, called a contract.
5/1 Call Option Example: IBM on May 31, 2002 The expiration date T, July 20, 2002, was 0.1333 years away. The prevailing interest rates were 1.77% over 1 month, and 1.95% over 6 months. Underlying Strike Call Put Base Asset S t Expiration T Price K Price C t Price P t IBM $80.50 July 20, 2002 $85 $1.900 $6.200 C Jul 20 = max(0, S Jul 20 $85) C @May 31 = $1.90 P Jul 20 = max(0, $85 S Jul 20 ) P @May 31 = $6.20 Different Strike Prices IBM $80.50 July 20, 2002 $75 $7.400 $1.725 IBM $80.50 July 20, 2002 $80 $4.150 $3.400 IBM $80.50 July 20, 2002 $90 $0.725 $10.100 Different Expiration Dates IBM $80.50 Oct. 19, 2002 $85 $4.550 $8.700 IBM $80.50 Jan. 18, 2003 $85 $6.550 $10.200
Why would someone want to purchase a call option? 6/1
Why would someone want to sell a call option? 7/1
Does the seller of an option need to own the underlying (IBM) stock? 8/1
Why would someone want to purchase a put option? 9/1
Why would someone want to sell a put option? 10/1
As a function of the stock price at expiration T, what is the payoff table and payoff diagram of a call option with strike price K=$90? (Long and short. Final payoff only. Ignore upfront cost.) 11/1
As a function of the stock price at expiration T, what is the payoff table and payoff diagram of a put option with strike price K=$90? (Long and short. Final payoff only. Ignore upfront cost.) 12/1
Is a put option the exact flip side of a call option? That is, is it the same to buy 1 call option or to sell 1 put option? 13/1
Spread (Long and Short of Same Time): As a function of the stock price at expiration T, what is the payoff table and payoff diagram of a position with one put option long with strike price K=$90 and one put option short with strike K = $70? 14/1
15/1 Other Common Positions Spread: (Long Call, Short Call). Or (Long Put, Short Put). Combination: (Long Call, Short Put). (Short Call, Long Put). Straddle: (Long Call, Long Put). (Short Call, Short Put). Most popular position. Speculates on...? Your brokerage has special buttons to purchase many such special positions. (By smartly combining puts and calls, you can even construct binomial options, e.g., which pay off $1 when the price is between $50 and $51, and $0 otherwise.)
16/1 No-Arbitrage Relationships (All Options:) C 0 S 0. C 0 0. P 0 0. C 0 (K Low ) C 0 (K High ) P 0 (K Low ) P 0 (K High ) (American Options w/o Dividends, but usually w/ Divs, too:) C 0 max(0, S 0 K). P 0 max(0, K S 0 ). P 0 (T later) P 0 (T earlier). C 0 (T later) C 0 (T earlier). (If I use subscript 0, it means any time before expiration.)
Say S = $80. C(K = $100) = $30. P(K = $100) = $50. To expiration, r = 10%. European. How do you get rich? 17/1
18/1 Put-Call Parity For European call and put options without dividends C 0 (K) = P 0 (K) + S 0 PV 0 (K)
If there were no European put options, but you could buy/sell European call options, could you manufacture a European put option? 19/1
20/1 Assume zero dividends. 1. What is the value of immediate exercise for an American call option? 2. Is an American call option worth more exercised or unexercised? 3. What is the value of an American call option compared to a European call option?
21/1 Assume zero dividends. 1. What is the value of immediate exercise for an American put option? 2. Is an American put option worth more exercised or unexercised? 3. What is the value of an American put option compared to a European put option?
22/1 The American Put Why is the Put s American feature worthwhile, while the Call s American feature is not? Take an expiration date of +1 million years. If the stock price is $100 and the call s strike price is $50 (ITM), does time work for you or against you? If the stock price is $50 and the put s strike price is $100 (ITM), does time work for you or against you?
23/1 What is the Price of the Call? Our ultimate goal is still to price the following call: Stock Price Now S 0 $80.50 Agreed-Upon Strike Price K $85.00 Time Remaining to Maturity t 0.1333 years Interest Rate on Risk-Free Bonds R F 1.77% per year Volatility (Standard Deviation) σ 30% per year of the Underlying Stock If you know the put option, you can tell me the price of the call. But, without the put, you are stuck. The solution: binomial pricing.
24/1 Let s say you buy (hold) δ = 0.5 shares at a stock price of $120 and borrow $50 (flow b = ( $50)) in bonds. What is the net cost of your position today? If the price of shares will be S = $150 and the interest rate is 1% from now to next period, then what will your position be worth?
25/1 Binomial Stock Price Movements: up = 5%. down = 4%. rr= 0.1%. = u = 1.05. d = 0.96. r = 1.001. Instant 0 = Now Instant 1 Instant 2 S0 = $50.00. B = $100 ST1 = u S0 = $52.50 B = $100.1 ST1 = d S0 = $48.00 B = $100.1 ST2 = u 2 S0 = $55.125 B = $100.2 ST2 = u d S0 = $50.40 B = $100.2 ST2 = d 2 S0 = $46.08 B = $100.2
What is the expected rate of return on the stock? 26/1
27/1 Binomial Pricing Price a call with a strike price $50, expiring at Instant 2. Instant 0 = Now Instant 1 Instant 2 S0 = $50.00 C 0 =? C u T1 =? ST1 = u S0 = $52.50 ST1 = d S0 = $48.00 C d T1 =? C uu T2 = $5.125 C du T2 = $0.40 ST2 = u 2 S0 = $55.125 ST2 = u d S0 = $50.40 ST2 = d 2 S0 = $46.08 C dd T2 = $0.00 You can now forget about the contract we have used all its information (t and K). We could actually price anything that is a payoff that depends only on S T2.
28/1 Binomial Pricing: Price? Instant 0 = Now Instant 1 Instant 2 = Expiration Ignore Ignore Ignore Ignore C du T2 = $0.40 ST2 = u d S0 = $50.40 ST2 = d 2 S0 = $46.08 C dd T2 = $0.00
How many δ shares and how many bonds b do you have to issue (i.e., borrow) today in order to receive next instant $0.40 if S = $50.40 and $0.00 if S = $46.08? What are the equations? 29/1
What is the expected rate of return on the stock? Do we need it? 30/1
31/1 How much do have to lay out to hold 0.0926 shares if you borrow $4.262 and shares cost $48? Instant 0 = Now Instant 1 Instant 2 = Expiration Ignore Ignore Ignore ST1 = d S0 = $48.00 C d T1 =? (you are here)
32/1 Binomial Pricing: Working Backwards Instant 0 = Now Instant 1 Instant 2 = Expiration Ignore Ignore Ignore ST1 = d S0 = $48.00 C d T1 =? C du T2 = $0.40 ST2 = u d S0 = $50.40 (you are here) ST2 = d 2 S0 = $46.08 C dd T2 = $0.00 A portfolio consisting of 0.0926 shares and $4.262 borrowed from the bank costs $0.182 if S = $48. It is worth $0.40 if the stock goes to $50.40 and $0.00 if the stock goes to $46.08.
33/1 Can you do the upper branch? Instant 0 = Now Instant 1 Instant 2 Ignore C u T1 =? ST1 = u S0 = $52.50 Ignore C uu T2 = $5.125 C du T2 = $0.40 ST2 = u 2 S0 = $55.125 ST2 = u d S0 = $50.40 Ignore
34/1 Can you do the left twig? Instant 0 = Now Instant 1 Instant 2 Ignore S0 = $50.00 C 0 =? ST1 = u S0 = $52.50 C u T1 = $2.550 ST1 = d S0 = $48.00 C d T1 = $0.182 Ignore Ignore
35/1 The Solved Tree (and Binomial Price) Instant 0 = Now Instant 1 Instant 2 S0 = $50.00 C0 = $1.26, δ 0.5 ST1 = u S0 = $52.50 C u T1 = $2.550, δ = 1 ST1 = d S0 = $48.00 C d T1 = $0.182, δ 0.1 C uu T2 = $5.125 C du T2 = $0.40 ST2 = u 2 S0 = $55.125 ST2 = u d S0 = $50.40 ST2 = d 2 S0 = $46.08 C dd T2 = $0.00
How often do you have to readjust your mimicking stock+bond portfolio to have the same payoffs as the option? 36/1
Intuition You buy a fraction of the stock and borrow some money to create a portfolio that will respond just like a real option to a tiny change in the underlying stock market basis, at least over the next instant of time. By the law of one price, the two portfolios should cost the same amount of money. To replicate the call, you need to buy the stock and borrow some money. So, what matters for your own replication ability is the interest that you have to pay for borrowing money from now to the expiration of the put. In an imperfect market, this may not be the same as your lending interest rate. Also, in an imperfect market, you may be able to lend out your stock (to hedge funds who want to short), which can earn you extra fees. Net in net, B-S type arbitrage is something that experts in the control of transaction costs should do, not you. (Warning: day-end prices can be misleading. the NYSE closing prices occur before the CBOE closing prices.) 37/1
Do Stock Prices (and Returns) Seem Binomial? 1 Binomial Outcome 5 Levels 0.5 0.30 0.4 2 Levels 0.25 5 Levels Probability 0.3 0.2 0.1 Probability 0.20 0.15 0.10 0.05 0.0 50 100 150 200 250 0.00 50 100 150 200 250 Stock Price (in $) Stock Price (in $) 50 Levels 500 Levels 0.12 Probability 0.10 0.08 0.06 0.04 0.02 50 Levels Probability 0.03 0.02 0.01 500 Levels 0.00 50 100 150 200 250 0.00 50 100 150 200 250 Stock Price (in $) Stock Price (in $) Over an infinite number of periods, this would look log-normal. 38/1
39/1 What is the Price of the Call? Get back to at our original question. How do we price: Stock Price Now S 0 $80.50 Agreed-Upon Strike Price K $85.00 Time Remaining to Maturity t 0.1333 years Interest Rate on Risk-Free Bonds R F 1.77% per year Volatility (Standard Deviation) σ 30% per year of the Underlying Stock Let s say you have an infinite number of levels, not just 3 levels. Does this not make the problem even harder?
40/1 Nice Black-Scholes Formula The five inputs are as follows: C 0 (S 0, K, t, R F, σ) = S 0 (d 1 ) PV(K) (d 2 ) d 1 = log N S0 /PV(K) (σ t) + 1 t) (σ 2 d 2 = d 1 σ t S 0 is the current stock price. t is the time left to maturity. K is the strike price. PV(K) is the present value of K. Thus, it depends on the risk-free rate, R F. σ is the standard deviation of the underlying stock s continuously compounded rate of return (i.e., of log(1 + r t )). It is often casually called just the stock volatility. is the cumulative normal distribution (Excel normsdist()) The time units must be the same on PV(K), sigma, and t. Often all annualized. log N is the natural logarithm, not the base-10 logarithm. Calculators often use ln for this.
41/1 Insights S 0 ( ) PV(K) ( ) still looks like buy some stock and borrow (short some bonds). Hedge ratio : Instead of δ, we now have (d 1 ). (d [1,2] ) is like a probability: a number between 0 and 1. σ t is the standard deviation of log-returns to expiration. σ only enters this way. PV(K) is the discounted strike price. K enters only in this form. t enters only to scale K and σ. S/PV(K) is how much your call is in the money. The log thereof is positive iff S 0 > PV(K). If you are out of the money, and volatility to expiration is 0, d 1 =, so (d 1 ) = 0. If you are far in the money, and volatility to expiration is 0, d 1 = +, so (d 1 ) = 1.
42/1 Steps Calculate PV(K). Calculate σ t Calculate d 1 Calculate d 2 Calculate the rest. Let s do the full calculation once by hand.
43/1 S 0 = $80.50. K = $85. t = 0.133y. R F = 1.78%/y. σ = 0.3/y. What is the PV(K)?
44/1 S 0 = $80.50. PV(K) = $84.80. t = 0.133y. σ = 0.3/y. What is d 1?
What is d 2? 45/1
What is ( 0.4204)? What is ( 0.5300) 46/1
47/1 The Cumulative Normal Distribution 1.0 0.3 0.8 0.6 n(z) 0.2 N(z) 0.4 0.1 Area =15.87% 0.2 N( 1)=15.87% 0.0 3 2 1 0 1 2 3 3 2 1 0 1 2 3 z z The right-side figure plots normsdist() in Excel.
48/1 Cumulative Normal Distribution z (z) z (z) z (z) z (z) z (z) z (z) 4.0 0.00003 3.5 0.00023 3.0 0.0013 2.0 0.0228 1.0 0.1587 0.0 0.5000 1.0 0.8413 2.0 0.9772 2.9 0.0019 1.9 0.0287 0.9 0.1841 0.1 0.5398 1.1 0.8643 2.1 0.9821 2.8 0.0026 1.8 0.0359 0.8 0.2119 0.2 0.5793 1.2 0.8849 2.2 0.9861 2.7 0.0035 1.7 0.0446 0.7 0.2420 0.3 0.6179 1.3 0.9032 2.3 0.9893 2.6 0.0047 1.6 0.0548 0.6 0.2743 0.4 0.6554 1.4 0.9192 2.4 0.9918 2.5 0.0062 1.5 0.0668 0.5 0.3085 0.5 0.6915 1.5 0.9332 2.5 0.9938 2.4 0.0082 1.4 0.0808 0.4 0.3446 0.6 0.7257 1.6 0.9452 2.6 0.9953 2.3 0.0107 1.3 0.0968 0.3 0.3821 0.7 0.7580 1.7 0.9554 2.7 0.9965 2.2 0.0139 1.2 0.1151 0.2 0.4207 0.8 0.7881 1.8 0.9641 2.8 0.9974 2.1 0.0179 1.1 0.1357 0.1 0.4602 0.9 0.8159 1.9 0.9713 2.9 0.9981 3.5 0.99977 4.0 0.99997
What is IBM s Black-Scholes European Call Option Value? 49/1
50/1 Quasi-Binomial BS Pricing Trust me IBM = $80.50. Call Strike = $85. r= 1.77%/1 month, 1.95%/3 months. Time to Expiration = 0.1333 years. If IBM goes to $80.51, the call increases in value by 0.3371 cents. If IBM goes to $80.49, the call decreases in value by 0.3371 cents. Thus, if you buy 33.71% $80.50 = $27.14 of IBM stock, then a 1 cent change in IBM stock means your portfolio changes by 0.3371 cents. The $27.14 position and the option react the same way. You also need to finance this purchase, though, borrowing $25.28. Your portfolio net cost is $1.86. Using such mimicking portfolios and the law of one price, you can work back an infinite tree from the final instant before expiration to determine the share price today. (Or just use B-S.) The call price is thus determined by arbitrage. It is called dynamic arbitrage, because every instant, you may have to change your hedge portfolio a little.
If there were no options on a traded base asset, could you manufacture the payoffs of an option? 51/1
If IBM s European Call Option costs $1.86, what should be IBM s American Call Option value? 52/1
What is IBM s European Put Option value? 53/1
What is IBM s American Put Option value? 54/1
How do you get the Black-Scholes inputs? 55/1
56/1 Remaining Problem How do you get the volatility??
57/1 Method 1: Historical Volatility Use Historical Volatility. Volatilities are easier to forecast than means. Why? Often modeled with complex GARCH etc. models: volatility is both strongly auto-regressive and mean-reverting.
58/1 Method 2: Implied Volatility S = $80.50, K = $85, t = 0.133, r = 1.78% 12 Call Option Value (in $) 10 8 6 4 2 sigma=30.38% P=$1.90 0 20 40 60 80 100 120 Sigma (in %) There is no closed-form solution. You must plot the B-S formula for all possible sigmas, and find the one that matches the actual price. (When not otherwise qualified, implied-vol refers to the B-S model.)
59/1 Table With Implied Vols Underlying Expira- Strike Option Option Implied Option Option Implied Base Asset tion T Price K Type Price Volatility Type Price Volatility IBM $80.50 July 20, 2002 $85 Call $1.900 30.38% Put $6.200 29.82% Different Strike Prices IBM $80.50 July 20, 2002 $75 Call $7.400 34.89% Put $1.725 34.51% IBM $80.50 July 20, 2002 $80 Call $4.150 32.58% Put $3.400 31.67% IBM $80.50 July 20, 2002 $90 Call $0.725 29.24% Put $10.100 29.18% Different Expiration Dates IBM $80.50 Oct. 19, 2002 $85 Call $4.550 31.32% Put $8.700 31.61% IBM $80.50 Jan. 18, 2003 $85 Call $6.550 31.71% Put $10.200 31.40% If the B-S model held, all implied vols should be identical. Note: the implied vol is called delta. there are also other greeks : Vega: price sensitivity to changes in volatility. Theta: price sensitivity to passage of time. Rho: price sensitivity to changes in interest rate. Lambda, Omega: gearing (leverage). delta times S/V. Many other second derivatives.
60/1 Where BS works and where it fails. Below-of-the-money options have higher prices than BS suggests. This is probably due to sudden risk of catastrophic drops. It is also partly due to market imperfections if you want to sell $1 million of puts that are far below the money, your counterparty will worry that you know something that (s)he does not. Above-of-the-money options have modestly higher prices than BS suggests. This is called the option smirk when the strike price is on the X axis and the imp vol is on the Y axis. There are more complex models than B-S, e.g., Merton jump models. There are also models taking into account dividends and models that work with futures instead of stocks.
61/1 Vol of Vol? There is even an implied volatility index, the VIX. It is the implied volatility of various S&P500 options. You can buy and sell options on the VIX itself, too! You are then basically speculating whether the implied volatility will go up or down. You can speculate that the vol of the vol is lower than other people think! These are very popular (incl as hedges against increases in risk).
Comparative Statics: How does BS change with its parameters? 62/1
63/1 Volatility Estimation Volatility estimation is a big deal. There is a whole army of people on Wall Street (not just, but mostly quants) engaged in the business of forecasting it. If you can estimate volatility better than others, you can sell expensive options and buy cheap options! For example, if the market prices options at an implied volatility of 30% and you think it is 20%, then sell puts and calls!
Why is the expected rate of return on the stock not in the BS price? 64/1
65/1 With the Price at Time 0, you can do other things See next two figs.
66/1 BS Values Prior To Expiration 60 Current Call Option Value (in $) 50 40 30 20 10 5 Years Remaining 1 Year Remaining 1 Day Remaining 0 0 20 40 60 80 100 120 140 Current Stock Value (in $) Note that the y-axis here is the value at any point in time, not just the value at the instant of expiration.
67/1 BS Riskier Purchases Call option rate of return ( in %) 300 200 100 0-100 Call(K=$100) Call(K=$90) Call(K=$70) Call(K=$0) (= buy the stock) 0 50 100 150 Final stock value (in $) Note that the y-axis here is rates of return, not payoffs. The point is: calls with higher strike prices (and puts with lower strike prices) are riskier gambles. They don t pay off anything more often, but when they do, it could be much more.
Have you seen options before this chapter? 68/1
69/1 Corporate Risk Management and Hedging Uses (synthetic) securities to offset risk. E.g., a gold mine may sell calls on gold. or futures on gold. In a perfect market, investors can hedge and unhedge at will. So, hedging is irrelevant. Eliminating unnecessary risk (that is not the strength of the company) may reduce the probability of bankruptcy, and in a non-m&m world, may thus enhance value. BUT: Is it clear whether it is good or bad for Southwest Airlines if jet fuel increases in price? Should SWA really hedge? Without good self-discipline and controls, hedging can quickly deteriorate into gambling. Every few years, some 30-year old trader gets caught having gambled away billions of dollars. (Most of the time, they get caught only having gambled away a few million dollars, and are let go quietly.) Most large publicly-traded firms, not in the business of speculation, should avoid risk-management, except in the most obvious of cases and with excellent controls.