55.444 Introduction to Financial Derivatives November 5, 212 Option Analysis and Modeling The Binomial Tree Approach Where we are Last Week: Options (Chapter 9-1, OFOD) This Week: Option Analysis and Modeling: Binomial Tree Approach (Chapter 12, OFOD) Next Week: Weiner Process and the Ito Lemma (Chapter 13, OFOD) No Class: Wednesday, November 21, 212 Last Day of Class: Wednesday, Dec 5 th, 212 Final Exam: Thursday, Dec 2 th ; 9:am Noon 9.1 9.2 Assignment Assignment For This Week (November 5 th ) Read: Hull Chapter 12 (Binomial Trees) Problems (Due November 7 th ): Chapter 1: 7, 14, 15, 18, 19; 23 Chapter 9(7e): 7, 14, 15, 18, 19; 23 Look at DerivaGem problems 1.21 & 1.26 (7e) 9.21 & 9.26 Problems (Due November 12 th ): Chapter 12: 1, 5, 6, 11; 2 Chapter 11(7e): 1, 5, 6, 11; 2 9.3 For Next Week (November 12 th ) Read: Hull Chapter 13 (Wiener Process and Ito s Lemma) Problems (Due November 12 th ): Chapter 12: 1, 5, 6, 11; 2 Chapter 11(7e): 1, 5, 6, 11; 2 Problems (Due November 19 th ) Chapter 13: 3, 5, 9, 11; 12 Chapter 12(7e): 3, 5, 9, 11; 12 9.4 1
Plan for This Week Options Carry Over from Last Week Upper and Lower Bounds for Prices Put-Call Parity Details: Early Exercise, Dividends and American vs European style Plan for This Week Options Binomial Tree Approach to Option Modeling Generalized Binomial Model & Risk-Neutral Valuation When Valuation is not Risk-Neutral Volatility of Asset Prices General Applicability of the Binomial Model What s Ahead: Options (9-1); Binomial Trees (12); Wiener Process & Ito Lemma (13); Black-Scholes-Merton Model (14); BSM for Options on Indexes, Currencies & Futures (16-17); The Greeks (18) 1.5 1.6 Lower Bound for European Call Option Prices (No Dividends) Consider the two portfolios Portfolio A: One European Call option plus cash = Ke rt Portfolio B: One Share For PF A at expiration, T, is worth max (S T, K) If the cash is invested, it will grow to K at T If S T > K, then the option is exercised; PF A is worth S T If S T < K, then the option is worthless; PF A is worth K PF B at expiration is worth S T At expiration, PF A PF B; No Arbitrage => c + Ke -rt S or c S Ke rt today, so cmax S Ke rt, 9.7 Calls: An Arbitrage Opportunity? Suppose that c = 3 S = 2 T = 1 r = 1% K = 18 D = What is the arbitrage opportunity? rt.1 S Ke 2 18e 3.713 3 c Which violates the lower bound which says c S Ke -rt Hence, sell the stock, buy the call, CF = 2 3 = 17 At expiration, have 17e.1 = 18.79 If stock is >18, exercise call; deliver into short, pocket.79 If stock is <18, forget call; buy stock, deliver into short, pocket >.79 1.8 2
Lower Bound for European Put Option Prices (No Dividends) Consider the two portfolios Portfolio C: One European Put option plus One Share Portfolio D: Cash = Ke rt For PF C at expiration, T, is worth max (S T, K) If S T < K, then the option is exercised; PF C is worth K If S T > K, then the option is worthless; PF C is worth S T PF D at expiration is worth K At expiration, PF C PF D; No Arbitrage => p + S Ke rt or p Ke -rt S today, so rt pmax Ke S, 9.9 Put: An Arbitrage Opportunity? Suppose that p = 1 S = 38 T =.25 r = 1% K = 4 D = What is the arbitrage opportunity? rt.25 Ke S 4e 38 39.138 1.1 1 p Which violates the lower bound which says p Ke -rt S Hence, buy the stock, buy the put, borrow 39 At expiration, owe 39e.25 = 39.98 If stock is < 4, exercise put; deliver stock for 4, pay loan, pocket.2 If stock is > 4, forget put; sell stock, pay loan, pocket >.2 1.1 Put-Call Parity European Options (No Dividends) Put-Call Parity European Options (No Dividends) Consider the two portfolios Portfolio A: One European Call option plus cash = Ke rt Portfolio C: One European Put option plus One Share PF A at expiration, T If the cash is invested, it will grow to K at T If S T > K, then the option is exercised; PF A is worth S T We pay for the share with the cash and have a share worth S T If S T < K, then the option is worthless; PF A is worth K Hence PF A is worth max (S T, K) 9.11 PF A is worth max (S T, K) at expiration PF C at expiration, T If S T < K, then the option is exercised; PF C is worth K We deliver the share and receive K If S T > K, then the option is worthless; PF C is worth S T Hence PF C is worth max (S T, K) At expiration, PF A has value equal to PF C In the absence of arbitrage, this must also be true today Hence c + Ke -rt p + S This is known as Put-Call Parity 9.12 3
Put-Call Parity: An Arbitrage Opportunity? Suppose that c = 3 S = 31 T =.25 r = 1% K = 3 D = What are the arbitrage opportunities when: p=2.25? Then PF A: And PF C: So PF C is overpriced to PF A. Sell PF C, buy PF A. p=1.? Then PF A: And PF C: rt.25 cke 3 3e 32.26 ps 2.25 31 33.25 rt.25 cke 33e 32.26 ps 131 32. So PF A is overpriced to PF C. Sell PF A, buy PF C. 9.13 Early Exercise Usually there is only a very small chance that an American Call option will ever be exercised early on a non-dividend paying stock A Call owner that thinks a stock is overpriced and otherwise has no long term desire to hold the stock could sell (not exercise) to capture option premium; shorting the stock is another alternative, but still no exercise Couldn t the price always go higher?! As a general rule, never exercise an American call early on a nondividend paying stock Indeed, rt We know that C c S Ke And given that r>, C>S K If it were optimal to exercise early, C= S K, so it is never optimal to exercise early In the absence of external knowledge, never exercise the American Call early on a non-dividend paying stock 1.14 Put-Call Parity (American-Style & No Dividend) Put-Call Parity (American-Style & No Dividend) Equality in Put-Call Parity only holds for European Options Consider the case of American Options on a non-dividend paying stock, P p and C = c : So P c + Ke -rt -S from P-C Parity & relation of A & E options And P C + Ke -rt -S or C - P S -Ke -rt Consider the two portfolios (options have identical strike and expiration) Portfolio A: One European Call option plus cash = K Portfolio E: One American Put option plus One Share The cash in PF A is invested at risk free rate If Put is NOT exercised early, PF E is worth max (S T, K) at T 1.15 At expiration PF A is worth max (S T K, ) + Ke rt = max (S T, K) K + Ke rt PF A is therefore worth more than PF E at expiration Suppose that the put in PF E is exercised early, at α This means that PF E is worth K at time α Even if the Call option were worthless, at t = α, PF A is worth Ke rα Therefore, PF A is always worth at least as much as PF E Hence c + K P + S And since C = c : C + K P + S or C P S K Combining with first result: S K C P S Ke -rt 1.16 4
Qualitative Reasons For Not Exercising an American Call Early No income is sacrificed in a non-dividend paying stock Payment of the strike price is delayed Holding the call provides insurance against stock price falling below strike price this comes at a cost, however Remember, insurance cost is commensurate with how much coverage you are buying Should an American Put Be Exercised Early? Like the Call, a Put can be viewed as insurance. But the price of a stock can never go below zero It may be optimal for the investor to forego the purchased insurance in order to realize the strike immediately As with Calls, we know p Ke rt S But even with r>, it may be optimal to exercise if the stock price is rt rt sufficiently low (e.g., if K S Ke or S K(1 e ) ) in this case, P K S (equality may hold, e.g., when S = ) Because there are circumstances when it is desirable to exercise an American Put early, it follows that it is always worth more than the corresponding European Put, so P>p. rt For example, when S =, K P p Ke 1.17 1.18 The Impact of Dividends on Lower Bounds to Option Prices The extension to the bounds for European Options in the case of dividends is simply to adjust the cash positions in PF A or PF D to include the PV of the dividend, D For the Call rt c S D Ke For the Put p D Ke rt S 1.19 Extension of Put-Call Parity for Dividend Paying Stock European options; D >, Include PV of Dividend with Cash in PF A c + D + Ke -rt = p + S American options; D >, P > p and C c We have C P < S Ke -rt D > reduces C & increases P so this relation still holds To refine further, consider the American P-C Parity PFs with dividends (Options have identical strikes and expiration) Portfolio A: One European Call option plus cash = D + K Portfolio E: One American Put option plus One Share Cash is invested at the risk-free rate 9.2 5
Extension of Put-Call Parity for Dividend Paying Stock If the put is not exercised early, PF E is worth max (S T, K) + De rt at T PF A is worth, at expiration, max (S T K, ) + (D + K)e rt = max (S T, K) + De rt + Ke rt K Portfolio A is therefore worth more than PF E at expiration Suppose the put in PF E is exercised early, at α This means that PF E is worth De rα + K at time α Even if the Call were worthless, at α PF A is worth (D+K)e rα It follows that PF A is worth more than PF E in all cases, α> The End for Inequalities Next? Equalities for Option Value and Option Models The Concept of Risk-Neutral Analysis Maybe, the most important idea in financial derivative analysis. No, It IS! Hence, c + D + K > P + S Because C c; C + D + K > P + S or C P > S D K Hence S D K<C P<S Ke -rt 9.21 9.22 The Essence of the Binomial Model The Essence of the Binomial Model Example: Euro-style 3-month Call on Stock at $21 Stock price is currently $2 In 3-months we know it will be either $22 or $18 Stock Price = $22 Stock price = $2 Stock Price = $18 At expiration in 3-months the option is worth either $ or $1 Can we figure-out the value of the option today? Stock price = $2 Option Price=? Stock Price = $22 Option Price = $1 Stock Price = $18 Option Price = $ 9.23 9.24 6
The Essence of the Binomial Model Consider the Portfolio: Long shares Short 1 call option What is the value of this PF at expiration? 22 1 18 The PF is riskless when 22 1 = 18 or =.25 That is, irrespective of the future, the PF has a single value at expiration; therefore the PF should return the risk-free rate 9.25 The Essence of the Binomial Model In Summary, with =.25 and a risk-free rate of 12% per annum with continuous compounding The riskless portfolio is: Long.25 shares Short 1 call option The value of the portfolio in 3 months is 22 x.25 1 = 4.5 or 18 x.25 = 4.5 And the value of the portfolio today is 4.5e.12x.25 = 4.367 9.26 The Essence of the Binomial Model The portfolio that is long.25 shares short 1 option is worth 4.367 today Today s value of the shares in the portfolio 5. (=.25 x 2 ) Therefore, the value of the option is.633 (= 5. 4.367 ) 9.27 Generalization of the Binomial Model Consider a stock at price S, and an option on the stock, with expiration T, who s current price is f During the option s life, the stock can move up or down Up from S to a new level at T, S u ; u>, and with option value f u or Down from S to a new level at T, S d ; d>, and with option value f d S u S ƒ ƒ u S d ƒ d 9.28 7
Generalization of the Binomial Model Consider the portfolio that is Long shares and Short 1 option S ƒ The portfolio is riskless when S u ƒ u = S d ƒ d or ƒ u S u S S u ƒ u S d ƒ d f d d Generalization of the Binomial Model Value of the portfolio at time T is S u ƒ u = S d ƒ d Value of the portfolio today is S f = (S u ƒ u )e rt = (S d ƒ d )e rt Hence the value of the option is ƒ = S (S u ƒ u )e rt = S (S d ƒ d )e rt = S (1 ue rt )+ ƒ u e rt = S (1 de rt )+ ƒ d e rt 9.29 9.3 Generalization of the Binomial Model Substituting for where We obtain Where we define p as f u f d S u S d ƒ = [ pƒ u + (1 p)ƒ d ]e rt rt e d p u d For the Generalized Binomial Model of Option Valuation 9.31 Risk-neutral Valuation In our main result we tease with nomenclature, p It is natural to want to interpret p and 1 p as the probabilities of an up and down movement with pƒ u + (1 p)ƒ d = E[f T ] as the expected payoff from the option With this interpretation, the option value is the present value of the expected future payoff S u Remember p has nothing to do with real probability; it is merely a consequence of the risk neutral assumption on the PF resulting from delta hedging S ƒ ƒ u S d ƒ d 9.32 8
Risk-neutral Valuation When the probability of an up and down movements are p and 1 - p the expected stock price at time T, E(S T ) is ES T ps u ( 1 p) S d ps ( u d) S d rt e d And substituting for p from our main result p gives E(S T ) = S e rt u d This shows that the stock price earns the risk-free rate Binomial trees illustrate the general result that to value an option, assume that the expected return on the underlying asset is the risk-free rate and discount at the risk-free rate This is known as Risk-Neutral Valuation We may always make the risk-neutral assumption and determine correct option values, even when markets are not risk-neutral (as we shall see later); the analytic perfection of delta hedging a portfolio 9.33 Risk-neutral Valuation Irrelevance of Stock s Expected Return When we are valuing an option in terms of the price of the underlying asset (but assume a perfect delta hedge), the probability of up and down movements of the asset in the real world are irrelevant This is an example of a more general result stating that the expected return on the underlying asset in the real world is irrelevant to option pricing We come back to this, but first 9.34 Essence Example - Again S ƒ Since p is the probability that gives a return on the stock equal to the risk-free rate. We can find it from 2e.12 x.25 = 22p + 18(1 p ) = 2.61 which gives p =.6523 Alternatively, we can use the formula rt e d p u d S u = 22 ƒ u = 1 S d = 18 ƒ d =.12.25 e.9.6523 1.1.9 9.35 Essence Example - Again S ƒ S u = 22 ƒ u = 1 S d = 18 ƒ d = The value of the option is ƒ = [ pƒ u + (1 p)ƒ d ]e rt = e.12x.25 (.6523x1 +.3477x) =.633 as we found earlier when we took the PV of the riskless PF and solved for the option value 9.36 9
What About the Real World? When we interpreted p (defined earlier) as a probability for an up-down movement in the stock price, we established the condition for a risk-neutral world where investors are indifferent toward risk & all securities return the risk-free rate The real world is not risk-neutral Investors demand higher return for higher risk Assume in our essence example, investors demand 16% return on the stock where otherwise, the risk-free rate r=12% Then 2e.16 x.25 = 22p * + 18(1 p * ) = E(S T ) = 2.816 (vs 2.61) And the probability of an up movement, p * =.741 (not.6523 = p as found before) 9.37 What About the Real World? The real world is not risk-neutral The expected payoff from the option in the real world is 1 x p * + x (1 p * ) =.741 = E[f T ] And we don t know what discount rate to use to PV this payoff We think the rate should be higher than the stock s return as a call option is thought to be more risky than the stock In fact since we know the correct value for the call option.633 =.74e -Rx.25, R=42.58% gives the discount rate we require If all we are trying to do is value a derivative-option, we should live in the easiest world we can find, even if it isn t real; as long as it is valid for what we are trying to do Risk-Neutral = perfect delta hedging 9.38 A Two-Step Binomial Example Consider the stock price outcomes 2 22 18 24.2 19.8 16.2 Where each time step is 3 months w/ 1% up/down We will value a 6-month Euro-style call option with strike, K=21 and a risk free rate, r=12% 9.39 A Two-Step Binomial Example 6-month Euro-style Call K=21, r=12% (p=.6523, again) 22 B 2 2.257 A 1.2823 18 24.2 3.2 19.8. C. 16.2 F. Move out the tree to expiration and look at the option values, then transverse backward toward today Value at node B = e.12x.25 (.6523 x 3.2 +.3477 x ) = 2.257 Value at node A = e.12x.25 (.6523 x 2.257 +.3477 x ) = 1.2823 D E 9.4 1
A Two-Step Binomial Example 2-year Euro-style Put K = 52, time step = 1yr, u/d : +2%/-2%, and r = 5% rt.51 e d e.8 p.6282 ud 1.2.8 5 4.1923 A 6 B 1.4147 4 C 9.4636 E D F 72 48 4 32 2 American-Style Options 2-year American-style Put K = 52, time step = 1yr, u/d : +2%/-2%, and r = 5% D 6 B 5 1.4147 A E 5.894 4 C Since American-style, exercise 12. at any node when early exercise is optimal F 72 48 4 32 2 When intrinsic option value is more than PV of pƒ u + (1 p)ƒ d At node C, option is worth 12 (if exercised) vs. 9.4636 9.41 9.42 The End for Today Questions? Delta Delta () is the ratio of the change in the price of a stock option to the change in the price of the underlying stock the key to risk-neutral analysis Example on last slide, Nodes B & C give us a delta of (1.4147-12)/(6-4)= -.77 Nodes D-E: (-4)/(72-48)= -.16667 Nodes E-F: (4-2)/(48-32)= -1. The value of varies from node to node, by time and level of underlying 9.43 9.44 11
One Last Detail One Last Detail In all of our analysis of options with binomial trees, we ve always been given (assumed) the lattice of stock prices Now we look at how to construct that lattice We need to determine u & d to replicate observations of the stock price in the market, summarized by the volatility measure, σ Volatility, σ, is defined so that σ t is the standard deviation of the return on the stock price over t Assume the stock price starts at S and over some uniform time increment t the stock price moves either up to S u or down to S d The stock price at the end of the first time step is * * * t EST p Su (1 p ) Sd p S ( u d) Sd Se where μ is the expected return on the stock t And the probability, p * * e d, can be found as p u d The volatility σ of price is defined so that t = std dev (stock price return; Δt) There are two points in our space, S u and S d Making use of the probability result for p * then Variance mean of the squares minus the square of the mean 2 2 2 The probability of an up movement (real world) is p * * Su * Sd * Su * Sd p (1 p ) p (1 p ) S S S S 9.45 9.46 One Last Detail So the variance of the stock price return is * 2 * 2 * * 2 2 p u ( 1 p ) d [ p u (1 p ) d] t Substituting for p * from above, we get e 2t ( u d) ud e t t 2 and one solution for u and d is: u e d e t t where is the volatility and t is the length of the time step (and when terms in t 2 and higher powers of t are 2 x x ignored) and as t gets small (use e 1 x... ) 2! This is the approach used by Cox, Ross, and Rubinstein Standard practice for constructing the lattice 9.47 Girsanov s World We have seen that a lattice with real-world probabilities p * (based on e μδt ) or risk-neutral probabilities p (based on e rδt ) give equally valid models for options analysis Though the risk-neutral model has preferences due to the uniformity of using the risk free rate for discounting 9.48 12
Girsanov s World We have NOW found how to construct a lattice in the real-world with u & d to replicate observed volatility t u e t d e a d We have previously shown that if we use p, where a = e rδt u d ; that this is the risk-neutral probability of an up move and the expected stock price at the end of the time step is rt ES ps u (1 p) S d ps ( u d S d S e T ) 9.49 Girsanov s World The variance of this stock price return model is 2 2 2 2 2 (1 ) [ (1 ) ] [ r t rt pu p d pu p d e ( ud) ud e ] t t And when substituting u e t d e from before, we find this too equals σ 2 t (when terms in t 2 and higher powers of t are ignored)! Hence, we see that when moving from the real world to the risk-neutral world, the expected return on the stock changes, from μ to r, but the stock price volatility remains the same (at least in the limit as the time step becomes arbitrarily small) Girsanov s Theorem Use u & d to replicate observed volatility for the riskneutral approach 9.5 The General Case with Other Assets The End for Today Use u & d to replicate observed volatility for the asset prices to for the lattice a d For up probabilities, use p u d Where a = e rδt for a non-dividend paying stock Where a = e (r-q)δt for a dividend yielding stock or for a stock index where q is the yield on the index t ( rq) t ( rq) t Indeed, S Se Se psu(1 p) Sd Se ( rq) t e d And p u d ( rrf ) t Where a e for a currency where r f is the foreign risk-free rate Where a = 1 for a futures contract 9.51 Questions? 7.52 13
Schedule Lecture Encounters Monday & Wednesday, 3-4:15pm, Gilman 132 Section: Friday 3-3:5pm, Hodson 213 Section: Thursday 3-3:5, Barton 117 (3) Final Exam Thursday, December 2 th ; 9:am Noon Gilman 132 Schedule Lecture Encounters Monday & Wednesday, 3-4:15pm, Gilman 132 Section: Friday 3-3:5pm, Hodson 213 Section: Thursday 3-3:5, Gilman 77 Final Exam Thursday, December 2 th ; 9:am Noon Gilman 132 1.53 1.54 Principals David R Audley, Ph.D.; Sr. Lecturer in AMS david.audley@jhu.edu Office: WH 212A; 41-516-7136 Office Hours: 4:3 5:3 Monday Teaching Assistant(s) Yu Du, (ydu1@jhu.edu) Tao Wang (twang55@jhu.edu) Tian Xia (summerzju@gmail.com) Office Hours: TBA 1.55 Resources Textbook John C Hull: Options, Futures, and Other Derivatives, Prentice-Hall 212 (8e) Recommended: Student Solutions Manual On Reserve in Library Text Resources http://www.rotman.utoronto.ca/~hull/ofod/errata8e/index.html http://www.rotman.utoronto.ca/~hull/technicalnotes/index.html 1.56 14
Resources Supplemental Material As directed AMS Website http://jesse.ams.jhu.edu/~daudley/444 Additional Subject Material Class Resources & Lecture Slides Industry & Street Research (Optional) Consult at your leisure/risk Interest can generate Special Topics sessions Measures of Performance Mid Term Exam (~1/3 of grade) Final Exam (~1/3 of grade) Home work as assigned and designated and Quizzes (~1/3 of grade) Blackboard 1.57 1.58 Protocol Attendance Lecture Mandatory (default) for MSE Fin Math majors Quizzes Section Strongly Advised/Recommended Assignments Due as Scheduled (for full credit) Must be handed in to avoid incomplete Exceptions must be requested in advance 1.59 15