2. Lattice Methods. Outline. A Simple Binomial Model. 1. No-Arbitrage Evaluation 2. Its relationship to risk-neutral valuation.

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1 . Lattice Methos. One-step binomial tree moel (Hull, Chap., page 4) Math69 S8, HM Zhu Outline. No-Arbitrage Evaluation. Its relationship to risk-neutral valuation. A Simple Binomial Moel A stock price is currently $ After 3 months it will be either $ or $8 A 3-month European call option on the stock with K=$ Stock Price = $ Option Price = $ Stock price = $ Option Price=? 3 months Stock Price = $8 Option Price = $ 3

2 No-Arbitrage Evaluation Consier the risklessportfolio: long shares short call option Portfolio Value 8 Portfolio is riskless when = 8 i.e. =.5 4 Valuing the Portfolio an the Option (Risk-Free Rate r = is %). What is the value of the portfolio at the expiry of the option?. What is the value of the portfolio toay? 3. What is the value of the option toay? 5 General One-Step Binomial Tree (No-arbitrage evaluation) A erivative lasts for time T an is epenent on a stock S u (u > ) v u S v ( < ) S v 6

3 General One-Step Binomial Tree (No-arbitrage evaluation) Consier the portfolio: long shares, short erivative S v S u v u S v The portfolio is riskless when S u v u = S v or = v u v S u S 7 General One-Step Binomial Tree (No-arbitrage evaluation) Value of the portfolio at time T: S u v u Value of the portfolio toay: S v =(S u v u )e rt Hence v = S (S u v u )e rt 8 Another approach: risk-neutral evaluation Substituting for, we obtain Where rt [ pv + ( p) v ] e v = u rt e p = u S v p ( p ) S u v u S v The variables p an ( p ) can be interprete as the riskneutral probabilities of up an own movements The value of a erivative can be consiere as its expecte payoff in a risk-neutral worl iscounte at the risk-free rate 9 3

4 Comments an example Note: When we are valuing an option in terms of the unerlying stock the expecte return on the stock is irrelevant Use risk-neutral evaluation to calculate option price in the original example (r = %, T = 3/): p S u = v u = S v ( p ) S = 8 v = Original example revisite (risk-neutral evaluation) rt..5 e e.9 p = = =.653 u..9 S v S u = v u = S = 8 v = Therefore, the value of the option is v = e -. 3 / =. 633 [ ] Risk-neutral evaluation In a risk-neutral worl, all investors are inifferent to risk, i.e., investors require no compensation for risk an the expecte returns of all securities is the riskfree interest rate The option price resulte from risk-neutral evaluation is correct not only in the risk-neutral worl, but also in other worl as well Further reaing: Cox, Ross, an Rubinstein, Option pricing: a simplifie approach, J. of Financial Economics 7:9-64 (979) 4

5 . Lattice Methos. Binomial Methos (Hull, Sec.7, Sec. 7.) Math69 S8, HM Zhu Outline. Extension to N-step binomial metho. Determine the parameters 3. Binomial methos in option pricing 4 Binomial Trees Binomial trees are frequently use to approximate the movements in the price of a stock or other asset In each small interval of time ( t) the stock price is assume to move up by a proportional amount u or to move own by a proportional amount p us S p S Stock movements in time t 5 5

6 Assumption To moel a continuous walk by a iscrete walk in the following fashion : S p -p t t us S p p -p -p u S u S S u > < p : probability time T=M t M 3 6 Assumption (cont) The parameters u,, an p in such a way that mean an variance of the iscrete ranom walk coincie with those of the continuous ranom walk 7 Assumption : Risk-Neutral Valuation We may assume investors are risk-neutral (even though most are not) In terms of option pricing (or other erivatives), we can assume: The expecte return from all trae securities is the risk-free interest rate Future cash flows can be value by iscounting their expecte values at the risk-free interest rate The tree is esigne to represent the behavior of a stock price in risk-neutral worl 8 6

7 Binomial Metho: Assumption Uner the assumption of risk-neutral worl, the value V m of the put option at m t = ( ) m+ the expecte value of option at V m+ t iscounte by the risk-free interest rate r m r t m+ V = E e V 9 Tree parameters p, u, an for nonivien paying stock The parameters u,, an p in such a way that mean an variance of the iscrete ranom walk coincie with those of the continuous ranom walk in risk-neutral worl, i.e.,: m+ m m+ m E c S S E b S S = an m+ m m+ m var c S S var b S S = The parameters p, u, an must give correct values for the mean an variance of stock price changes uring t. Tree parameters u,, an p m+ m m+ m E c S S = E b S S ( ) ( ) e S = pu+ p S r t m m r t e = pu+ p () 7

8 Tree parameters u,, an p m+ m m+ m var c S S = var b S S m r t σ t ( S ) e ( e ) pu ( p) r t m e ( S ) = + e = pu + p r + t σ t ( ) ( ) In orer to etermine these three unknowns uniquely, we nee to a another conition (somewhat arbitrary). Note: ST follows a lognormal istribution. It can be shown that ) The expecte value of ST : µ ( T t) E( ST ) = Se ) The variance of ST : var ( ST ) = S e σ e µ ( ) ( ) T t T t One further conition: u = One of the popular choices is u =. It generates a symmetric tree w.r.t. the initial value S : σ σ σ + t r t σ + t r t r t e u= e, = e, p =. u [reference: Cox, Ross an Rubinstein (979)] Often, ignoring the terms in t an higher power of t gives r t σ t σ t e u= e, = e, p =. u Note: If t is too large, p or ( - p) <. The binomial metho fails. 3 Symmetric Tree: u = 4 8

9 Another further conition: p = One of the popular choices is p =. It generates an asymmetric tree, oriente in the irection of the rift: ( σ ) ( σ ) r t t r t t u= e + e, = e e, p= [ Kwok, 998; Wilmott et al, 995] Note: If t is too large, <. The binomial metho fails. 5 Asymmetric Tree: p = 6. Lattice Methos.3 General Binomial Tree Moels (Hull, Sec. 7.,, page 39) Math69 S8, HM Zhu 9

10 The Iea of Binomial Methos. Buil a tree of possible values of asset prices an their probabilities, given an initial asset price. Once we know the value of the option at the final noes, work backwar through the tree using risk-neutral valuation to calculate the value of the option at each noe, testing for early exercise when appropriate Avantages: easily eal with possibility of early exercise an with ivien payments 8 Binomial Metho for valuing nonivien-paying options Step. Buil up a tree of possible asset prices for all time points an their probabilities, given the initial price S S = u S for n =,,,m m n m n n m S : the n-th possible value of S at m-th time-step n 9 Buil A Complete Tree of Asset Prices S u S u S u 3 S u 4 S u S u S S Tree structure use the relationship u = / S S S S S S 3 S 4 3

11 Step : Valuing an European option: Step. Use this tree to calculate the possible value of option at expiry. Then work back own the tree to caculate the price of the option.. Value the option at expiry, i.e, time-step M t M ( n ) M Put: V = max K-S, n =,, M n Call:.. Fin the expecte value of the option at a time step prior to expiry ( ) V = E e V = e pv + p V an so on, back to time-step. m r t m+ r t m+ m+ n n n+ n 3 Example : European -year Put Option S = 5; K = 5; r =5%; σ = 3%; T = years; t = year The parameters imply 3. u = e = ; = =. 748 ; a = e =. 53 ; p = = p = Example : European -year Put Option D 5? A B C E F year year 33

12 Example : European Put Option S = 5; K = 5; r =%; σ = 4%; T = 5 months =.467; t = month =.833 The parameters imply u σ t = e =. 4 ; σ t = e =. 899 ; a r t = e = 84. ; p = a =. 576 u - p = Example : European Put Option Example : European Put Option

13 Step : Valuing an American put option k At time step k an at asset price S, there are two possibilities:. Exercise the option which yiel a profit m m ( n ) = ( - n ) payoff S max K S,. Retain the option, the value of the option is then m r t m+ r t m+ m+ Vn E e V = e pvn+ + p Vn Therefore, the value of the option is the maximum of two possibilities,i.e., 37 i = ( ) + + ( ( ) ( ) + ) V = max payoff S,e pv + p V m m r t m m n n n n Example : American -year Put Option S = 5; K = 5; r =5%, σ=3%, T = years, t = year 5? A B? 37.4 C? E D F year year 38 Example : American put option of the same stock: S = 5; K = 5; r =%; σ = 4%; T = 5 months; t = month

14 Step : Valuing an American call option k At time step k an at asset price S, there are two possibilities :. Exercise the option which yiel a profit payoff =? m ( S ) =?. Retain the option, the value of the option is then V m n Therefore, the value of the option is the maximum of two possibilities, i. e., m V =? n n i 4 Exercise : American call option of the same stock: S = 5; K = 5; r =%; σ = 4%; T = 5 months; t = month Convergence of the Price of the American Put on Non-Divien-Paying Option Example 4 4

15 . Lattice Methos.4 Dealing with Options on Divien-paying Stocks (Hull, Sec. 7.3, page 4) Math69 S8, HM Zhu With Continuous Divien Yiels Assume there is a constant ivien yiel D pai on the unerlying. Then the expecte return of the unerlying is at the rate ( r D ). To accommoate the constant ivien yiel in the tree moel,. Replace r by r D in the parameters u,, an p in the tree construction of stock prices. For example, in the case u = /, it becomes: σ u= e, = e, ( r D) t e p =. u t σ t For the case when p =, it becomes: ( ) ( + ) ( ) ( ) σ t r D t r D t σ t u= e e, = e e. 44 With Continuous Divien Yiels. Use this tree to calculate the possible value of option at expiry. Then work back own the tree to caculate the present value of the option at previous time points is obtaine using V m? n = E e t m+ V n 45 5

16 With Continuous Divien Yiels. Use this tree to calculate the possible value of option at expiry. Then work back own the tree to caculate the present value of the option at previous time points is obtaine using V m n = E e r t m+ V n 46 With Dollar Divien A better proceure: Draw the tree for the stock price less the present value of the iviens Create a new tree for the stock price by aing the present value of the iviens at each noe This ensures that the tree recombines an makes assumptions similar to those when the Black-Scholes moel is use 47 With Dollar Divien Assume that there is one ex-ivien ate, τ, uring the life of the option. * -- Construct a tree for the uncertain component,.., r( τ i t ) * S De, if i t < τ S = S, if i t > τ * * Using the volatility σ of S, a tree can be constructe in usual * way to moel S. -- To moel S, we simply a back the present value of future ivien, ie.., * n i n r( τ i t ) i Su + De, if i t < τ Sn =, n=,,, i * n i n Su, if i t> τ S ie 48 6

17 Example : American put option on a stock: S = 5; K = 5; D = $.6 r =%; σ = 4%; T = 5 months; τ = 3.5 months t = month Example : Tree moel for S* $ Example : American put option on a stock: S = 5; K = 5; D = $.6 r =%; σ = 4%; T = 5 months; τ = 3.5 months t = month 7

18 . Lattice Methos.5 Further Comments Math69 S8, HM Zhu Delta ( ) An important parameter in the pricing an heging of options It is the ratio of the change in the price of a stock option to the change in the price of the unerlying stock It is # of the units of the stock we shoul hol for each option shorte to create a riskless hege. Such a construction of riskless heging is calle elta heging. of a call option is positive whereas of a put option is negative 53 Valuing Delta s.57 Value at t: = = Values at t: 3. If upwar movement over the first time step, = = Otherwise, D = = B A E.83 8 C. F

19 Delta The value of varies over time an from noe to noe To maintain a riskless hege using an option an the unerlying stock, we nee to ajust our holings in the stock perioically 55 Trees for Options on Inices, Currencies an Futures Contracts (Hull, Sec..9, Sec. 7.) As with Black-Scholes: For options on stock inices, replace the continuous ivien yiel D with the ivien yiel on the inex For options on a foreign currency, D equals the foreign risk-free rate r f For options on futures contracts D = r 56 Time Depenent Interest Rate an Divien Yiel (page 49) Making interest rate r or ivien yiel D a function of time oes not affect the geometry of the tree. The probabilities on the tree become functions of time Discounting factor becomes a function of time as well 57 9

20 Time Depenent Volatility (page 49) Changing σ at each time step oes affect the geometry of the tree. (The probabilities on the tree become functions of time) Or we can make σ a function of time by making the lengths of the time steps inversely proportional to the variance rate. 58 Trinomial Tree (see Technical Note 9, u = e σ 3 t = / u Su t σ p + u = r σ 6 pm = 3 t σ p = r σ + 6 S p u p m p S S 59 Explicit FDM =Trinomial Tree V i +, j + V i, j V i +, j V i +, j Explicit Metho

21 Further Reaing. D. Leisen an M. Reimer. Applie Mathematical Finance, 996 (evelope a general convergence rate theory). L. Clewlow an C. Stricklan. Implementing Derivatives Moels. Wiley, Chichester, West Sussex, Englan, 998 (relationship between finite ifferences an trinomial trees) 3. D J Higham. Nine Ways to Implement the Binomial Metho for Option Valuation in Matlab. SIAM review, 44:66-677, (issues of implementing binomial trees) 4. G. Levy. Computational Finance. Numerical Methos for Pricing Financial Instruments. Elsevier Butterworth-Heinemann, oxfor, 4 (implie lattices an efficient implementations) 6 Aaptive Mesh Moel This is a way of grafting a high resolution tree on to a low resolution tree We nee high resolution in the region of the tree close to the strike price an option maturity Numerically efficient over a binomial or trinomial tree Figlewski an Gao, The aaptive mesh moel: a new approach to efficient option pricing, J. of Financial Ecomonics, 53:33-35 (999) 6