Introduction to Financial Derivatives

Transcription

1 Introuction to Financial Derivatives November 4, 213 Option Analysis an Moeling The Binomial Tree Approach Where we are Last Week: Options (Chapter 9-1, OFOD) This Week: Option Analysis an Moeling: Binomial Tree Approach (Chapter 12, OFOD) Next Week: Weiner Process an the Ito Lemma (Chapter 13, OFOD) Last Day of Classes: December 4 th Final Exam Final: Dec 17 th ; 9:am Noon, Mergenthaler Assignment Assignment For This Week (November 4 th ) Rea: Hull Chapter 12 (Binomial Trees) Problems (Due November 4 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 11 th ): Chapter 12: 1, 5, 6, 11; 2 Chapter 11(7e): 1, 5, 6, 11; For Next Week (November 11 th ) Rea: Hull Chapter 13 (Wiener Process an Ito s Lemma) Problems (Due November 11 th ): Chapter 12: 1, 5, 6, 11; 2 Chapter 11(7e): 1, 5, 6, 11; 2 Problems (Due November 18 th ) Chapter 13: 3, 5, 9, 11; 12 Chapter 12(7e): 3, 5, 9, 11;

2 The Plan for this Week The Concept of Risk-Neutral Analysis Maybe, the most important iea in financial erivative analysis. No, It IS! We introuce this through a moeling approach of Binomial Trees The Essence of the Binomial Moel 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 = $ The Essence of the Binomial Moel The Essence of the Binomial Moel At expiration in 3-months the option is worth either $ or $1 Can we figure-out the value of the option toay? Stock price = $2 Option Price=? Stock Price = $22 Option Price = $1 Stock Price = $18 Option Price = $ Consier the Portfolio: Long shares Short 1 call option What is the value of this PF at expiration? 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 shoul return the risk-free rate

3 The Essence of the Binomial Moel In Summary, with =.25 an a risk-free rate of 12% per annum with continuous compouning 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 An the value of the portfolio toay is 4.5e.12x.25 = The Essence of the Binomial Moel The portfolio that is long.25 shares short 1 option is worth toay Toay s value of the shares in the portfolio 5. (=.25 x 2 ) Therefore, the value of the option is.633 (= ) Generalization of the Binomial Moel Consier a stock at price S, an 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 own Up from S to a new level at T, S u ; u>, an with option value f u or Down from S to a new level at T, S ; >, an with option value f S u S ƒ ƒ u S ƒ 9.11 Generalization of the Binomial Moel Consier the portfolio that is Long shares an Short 1 option S ƒ The portfolio is riskless when S u ƒ u = S ƒ or f ƒ u S u S S u ƒ u S ƒ

4 Generalization of the Binomial Moel Value of the portfolio at time T is S u ƒ u = S ƒ Value of the portfolio toay is S f = (S u ƒ u )e rt = (S ƒ )e rt Hence the value of the option is ƒ = S (S u ƒ u )e rt = S (S ƒ )e rt = S (1 ue rt )+ ƒ u e rt = S (1 e rt )+ ƒ e rt Generalization of the Binomial Moel Substituting for where We obtain Where we efine p as f u f S u S ƒ = [ pƒ u + (1 p)ƒ ]e rt rt e p u For the Generalize Binomial Moel of Option Valuation Risk-neutral Valuation In our main result we tease with nomenclature, p It is natural to want to interpret p an 1 p as the probabilities of an up an own movement with pƒ u + (1 p)ƒ = E[f T ] as the expecte payoff from the option With this interpretation, the option value is the present value of the expecte future payoff S u Remember p has nothing to o with real probability; it is merely a consequence of the risk neutral assumption on the PF resulting from elta heging S ƒ ƒ u S ƒ 9.15 Risk-neutral Valuation When the probability of an up an own movements are p an 1 - p the expecte stock price at time T, E(S T ) is ES T ps u ( 1 p) S ps ( u ) S rt e An substituting for p from our main result p gives E(S T ) = S e rt u 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 expecte return on the unerlying asset is the risk-free rate an iscount at the risk-free rate This is known as Risk-Neutral Valuation We may always make the risk-neutral assumption an etermine correct option values, even when markets are not risk-neutral (as we shall see later); the analytic perfection of elta heging a portfolio

5 Risk-neutral Valuation Irrelevance of Stock s Expecte Return When we are valuing an option in terms of the price of the unerlying asset (but assume a perfect elta hege), the probability of up an own movements of the asset in the real worl are irrelevant This is an example of a more general result stating that the expecte return on the unerlying asset in the real worl is irrelevant to option pricing We come back to this, but first 9.17 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 fin it from 2e.12 x.25 = 22p + 18(1 p ) = 2.61 which gives p =.6523 Alternatively, we can use the formula rt e p u S u = 22 ƒ u = 1 S = 18 ƒ = e Essence Example - Again S ƒ S u = 22 ƒ u = 1 S = 18 ƒ = The value of the option is ƒ = [ pƒ u + (1 p)ƒ ]e rt = e.12x.25 (.6523x x) =.633 as we foun earlier when we took the PV of the riskless PF an solve for the option value What About the Real Worl? When we interprete p (efine earlier) as a probability for an up-own movement in the stock price, we establishe the conition for a risk-neutral worl where investors are inifferent towar risk & all securities return the risk-free rate The real worl is not risk-neutral Investors eman higher return for higher risk Assume in our essence example, investors eman 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 ) = (vs 2.61) An the probability of an up movement, p * =.741 (not.6523 = p as foun before)

6 What About the Real Worl? The real worl is not risk-neutral The expecte payoff from the option in the real worl is 1 x p * + x (1 p * ) =.741 = E[f T ] An we on t know what iscount rate to use to PV this payoff We think the rate shoul 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 iscount rate we require If all we are trying to o is value a erivative-option, we shoul live in the easiest worl we can fin, even if it isn t real; as long as it is vali for what we are trying to o Risk-Neutral = perfect elta heging 9.21 A Two-Step Binomial Example Consier the stock price outcomes Where each time step is 3 months w/ 1% up/own We will value a 6-month Euro-style call option with strike, K=21 an a risk free rate, r=12% 9.22 A Two-Step Binomial Example 6-month Euro-style Call K=21, r=12% (p=.6523, again) 22 B A C F. Move out the tree to expiration an look at the option values, then transverse backwar towar toay Value at noe B = e.12x.25 (.6523 x x ) = Value at noe A = e.12x.25 (.6523 x x ) = D E 9.23 A Two-Step Binomial Example 2-year Euro-style Put K = 52, time step = 1yr, u/ : +2%/-2%, an r = 5% rt.51 e e.8 p.6282 u A 6 B C E D F

7 American-Style Options 2-year American-style Put K = 52, time step = 1yr, u/ : +2%/-2%, an r = 5% D 6 B A E C Since American-style, exercise 12. at any noe when early exercise is optimal F When intrinsic option value is more than PV of pƒ u + (1 p)ƒ At noe C, option is worth 12 (if exercise) vs Delta Delta () is the ratio of the change in the price of a stock option to the change in the price of the unerlying stock the key to risk-neutral analysis Example on last slie, Noes B & C give us a elta of ( )/(6-4)= -.77 Noes D-E: (-4)/(72-48)= Noes E-F: (4-2)/(48-32)= -1. The value of varies from noe to noe, by time an level of unerlying One Last Detail One Last Detail In all of our analysis of options with binomial trees, we ve always been given (assume) the lattice of stock prices Now we look at how to construct that lattice We nee to etermine u & to replicate observations of the stock price in the market, summarize by the volatility measure, σ Volatility, σ, is efine so that σ t is the stanar eviation of the return on the stock price over t Assume the stock price starts at S an over some uniform time increment t the stock price moves either up to S u or own to S The stock price at the en of the first time step is * * * t EST p Su (1 p ) S p S ( u ) S Se where μ is the expecte return on the stock t An the probability, p * * e, can be foun as p u The volatility σ of price is efine so that t = st ev (stock price return; Δt) There are two points in our space, S u an S Making use of the probability result for p * then Variance mean of the squares minus the square of the mean The probability of an up movement (real worl) is p * * Su * S * Su * S p (1 p ) p (1 p ) S S S S

8 One Last Detail So the variance of the stock price return is * 2 * 2 * * 2 2 p u ( 1 p ) [ p u (1 p ) ] t Substituting for p * from above, we get e 2t ( u ) u e t t 2 an one solution for u an is: u e e t t where is the volatility an t is the length of the time step (an when terms in t 2 an higher powers of t are 2 x x ignore) an as t gets small (use e 1 x... ) 2! This is the approach use by Cox, Ross, an Rubinstein Stanar practice for constructing the lattice 9.31 Girsanov s Worl We have seen that a lattice with real-worl probabilities p * (base on e μδt ) or risk-neutral probabilities p (base on e rδt ) give equally vali moels for options analysis Though the risk-neutral moel has preferences ue to the uniformity of using the risk free rate for iscounting 9.32 Girsanov s Worl We have NOW foun how to construct a lattice in the real-worl with u & to replicate observe volatility t e a We have previously shown that if we use p, where a = e rδt u ; that this is the risk-neutral probability of an up move an the expecte stock price at the en of the time step is rt ES ps u (1 p) S ps ( u S S e T u e t ) 9.33 Girsanov s Worl The variance of this stock price return moel is (1 ) [ (1 ) ] [ r t rt pu p pu p e ( u) u e ] t t An when substituting u e t e from before, we fin this too equals σ 2 t (when terms in t 2 an higher powers of t are ignore)! Hence, we see that when moving from the real worl to the risk-neutral worl, the expecte 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 & to replicate observe volatility for the riskneutral approach

9 The General Case with Other Assets Use u & to replicate observe volatility for the asset prices to for the lattice a For up probabilities, use p u Where a = e rδt for a non-ivien paying stock Where a = e (r-q)δt for a ivien yieling stock or for a stock inex where q is the yiel on the inex t ( rq) t ( rq) t Inee, S Se Se psu(1 p) S Se ( rq) t e An p u ( rrf ) t Where a e for a currency where r f is the foreign risk-free rate Where a = 1 for a futures contract