Binomial Moel Hull, Chapter 11 + ections 17.1 an 17.2 Aitional reference: John Cox an Mark Rubinstein, Options Markets, Chapter 5 1. One-Perio Binomial Moel Creating synthetic options (replicating options) by taking positions in the unerlying asset an borrowing Pricing by replication Option s elta Deriving a one-perio binomial option pricing formula an emonstrating that it oes not epen on the real-worl probabilities Risk-neutral probabilities Risk-neutral valuation from the no-arbitrage argument 2. Two-Perio Binomial Moel Extening the one-perio moel Deriving the two-perio binomial option pricing formula 3. Multi-tep Cox-Ross-Rubinstein (CRR) Binomial Moel Binomial moel of asset price ynamics Working backwars through the binomial tree: the backwar inuction algorithm Deriving close-form solutions for European options Expressing option pricing formulas through the complementary binomial istribution function Pricing general European-style contingent claims Pricing American options: ynamic programming approach Dynamic heging: elta heging on a binomial tree (lattice) 4. Extensions an Generalizations of the Basic Binomial Moel; Convergence of Binomial Moel to the Black-choles-Merton Moel 1
Binomial Moel --- Introuctory Example: Current stock price: = $50 Call strike K = $50 r = 25% per annum with annual compouning In the next time perio (one year) the stock can go either up to u = $100 or own to = $25: = $50 u = $100 = $25 Question: Fin the value of the call with K = 50. The Power of No-Arbitrage Arguments: If profitable arbitrage is not possible we can fin the fair (arbitrage-free) value of the call C from the given ata alone without any information about the probabilities of the stock going up or own! trategy: 1. Write 3 calls for C 2. Buy 2 shares for = $50 3. Borrow $40 at 25% to be pai back at the en of the perio (one year) 3 C - 100 + 40 u = 100, -150 + 200 50 = 0 = 25, 0 + 50 50 = 0 The future value of the portfolio is 0 in both cases inepenent of the outcome for the stock =100 or 25, i.e., the portfolio is riskless an, assuming no arbitrate, its present value must also be zero: 3 C - 100 + 40 = 0! C = $20. If the call is not price at $20, arbitrage profits are possible. 2
To price the call we neee the following ata: K,, r, T (time to maturity), range of the stock movement: u, u We i not nee: The probabilities q an (1 - q) that the stock will go up or own. We have price the option relative to the unerlying stock an the risk-free rate. One Perio Binomial Option Pricing: Replicating Portfolio We assume the unerlying asset pays no iviens (we will relax this assumption shortly). We also assume that 0 < < e < u (this is an important assumption) q 1 - q u, C u = max(u - K,0), C = max( - K,0) 0 t T Form the following portfolio (goal replicate the option with positions in stock an borrowing): Borrow B ollars at risk-free rate (equivalently, short sell a zero-coupon risk-free bon with current price B an face value e B pai at the en of the perio) Buy shares of stock Portfolio value at inception: - B. - B q 1 - q u - e B - e B Let us select an B so that the en-of-perio values of our portfolio are equal to the call values for each possible outcome: 3
u - e B = C u - e B = C olve these two equations for an B: Cu C = ( u ) (Delta of the option) B C uc u u = e (Amount to borrow) Replicating portfolio (one perio): Thus we have replicate a call option with a long position in the unerlying stock an borrowing (equivalently, short position in the bon). If the values of two portfolios are equal at the en of the perio in all possible states! they must be equal at the beginning of the perio (the no-arbitrage argument): C = B. One-Perio Binomial Pricing Formula ubstitute the expressions for an B into the pricing formula: C = B Cu C Cu uc = e ( u ) u e u e = e Cu + C u u = e pc + (1 p) C, where [ ] u r t e p = u u e an 1 p = u. Just like pricing forwars, we constructe a synthetic option (a replicating portfolio) using the no-arbitrage argument. The crucial feature of our result is that the probabilities q an 1 q o not appear anywhere in the formula! 4
In this moel, even if ifferent investors have ifferent beliefs about probabilities for the stock to go up or own, they still must agree on the relationship between the option price C, the stock price an the risk-free rate of return r. At the same time, the call price relative to an r ose not epen on investors attitues towar risk. The iscount rate is the risk-free rate r. It shoul be stresse that the pricing formula is a relative pricing relationship that etermines C, given an r. Risk-Neutral Probabilities: The quantities p an (1 p) are calle risk-neutral probabilities r t e p = u u e an 1 p = u As long as 0 < < e < u, they are between 0 an 1. Calculate expecte stock price at the en of the perio using risk-neutral probabilities: e u e pu + (1 p) = u + = e u u Thus, p an (1- p) are inee risk neutral probabilities: expecte rate of return on the stock uner these probabilities is the risk-free rate r: e. In the risk-neutral worl, investors are risk-neutral an o not require any risk premium for holing risky assets. Expecte rate of return on all assets is equal to the risk-free rate. One-Perio Risk-Neutral Valuation Formula C = e - [pc u + (1 - p)c ] One-Perio Binomial Option Pricing: Hege Portfolio (alternative an equivalent erivation) et up a portfolio: 1. hort 1 call 2. Long shares Π 0 = - C 5
At time T: Π 0 u C u, u = u with probability q C, = with probability 1 q For the portfolio to be riskless: Π = Π, u u C u = - C There is no uncertainty about the future value of our portfolio: it is worth the same in both states (up state an own state). olve for : Cu C =. ( u ) Delta is the ratio of the change in the option price to the change in the unerlying price (hege ratio). Portfolio is riskless! must earn interest at the risk-free rate, i.e., Π 0 = e - Π u, C = e - ( u C u ), C = - e - ( u C u ), Cu C C = (1 ue ) e Cu ( u ) + 1 ue ue 1 = Cu e + + C u u = e [ pc + (1 p) C ] u In this erivation we hege out all the risk of the short call position by buying shares of the unerlying ( is our hege ratio). The resulting hege portfolio is risk-free an as such must earn the risk-free rate of return to prevent arbitrage (no-arbitrage argument). This erivation of the option pricing base on the hege portfolio is completely equivalent to the first erivation where we replicate a call with positions in the unerlying an borrowing (replicating portfolio). Example: One-Perio Pricing = $20 K = $21 r =12% 6
t = 0.25 year $22, 22-1 20 20 - C $18, 18-0 22-1 = 18! = 0.25 Π 0 = e - Π u, Π,u = 22 x 0.25 1 = 4.5, Π 0 = e -0.12x0.25 4.5 = 4.367 20 x 0.25 C = 4.367! C = 0.633 Existence an Uniqueness of Risk-Neutral Probabilities, Absence of Arbitrage Opportunities, an Market Completeness The one-perio binomial moel contains one risky asset (a stock ) an one risk-free asset (one-perio money market account or risk-free zero-coupon bon). More general moels are multi-perio (or continuous time) an inclue multiple risky assets, along with the risk-free asset (money market account). The very simple binomial moel alreay contains many of the features of more general moels. Risk-neutral probabilities (risk-neutral probability measure) exist if an only if there are no arbitrage opportunities in the moel (this statement is known as The First Funamental Theorem of Asset Pricing). A contingent claim is sai to be replicable if it can be exactly replicate by an equivalent replicating portfolio of primitive unerlying securities. A moel is sai to be complete if all contingent claims are replicable in this moel. The risk-neutral probability measure is unique if an only if the market is complete (this statement is known as The econ Funamental Theorem of Asset Pricing). The binomial moel with 0 < < e < u amits no arbitrage opportunities, is complete (any contingent claim can be replicate by positions in stock an money market account or bon), an there exists a unique risk-neutral probability measure (risk-neutral probabilities p an 1-p). r t Now suppose we starte with a binomial moel where e < < u. Then the stock return ominates the risk-free return in both states. Then there is an arbitrage opportunity: borrow cash an invest in stock. Initial value of this portfolio is zero, while the final value of this portfolio is positive in both states. The arbitrage portfolio is basically a 7
eterministic money machine. Clearly, this is not a realistic market moel. This moel amits arbitrage, an if you look at our risk-neutral probability formulas, you will see that p < 0 an 1 p > 1! Thus, in this case the risk-neutral probability measure oes not exist. r t Now suppose 0 < < u< e. Then, the risk-free return ominates the return on the stock in both states. Then there is an arbitrage opportunity: sell the stock short an invest cash. Initial value of this portfolio is zero, while the final value of this portfolio is positive in both states. This moel amits arbitrage, an has p > 1 an 1 p < 0. Thus, in this case there risk-neutral probability measure oes not exist. Two-Perio Binomial Moel u 2 C uu = max(u 2 K,0) u C u C u C u = max(u - K,0) t = 0 t 1 T t t 2 C = max( 2 K,0) Two perio option pricing formula Two-perio tree = combination of 3 one-perio sub-trees: C u = e - (pc uu +(1 - p)c u ), C = e - (pc u + (1 - p)c ), C = e - (pc u + (1 - p)c ) = e -r(2 t) [p 2 C uu + 2p(1 - p)c u + (1 - p) 2 C ] Cox-Ross-Rubinstein (CRR) Multi-Perio Binomial Moel T = N t; N time perios (N + 1 noes along the path) N + 1 ifferent terminal prices 2 N possible price paths from (0, ) to (T, T ) i, = u i : price at the noe (i,) i, i = 0,1, N : number of time steps to the noe (i,), = 0,1,, i : number of up moves to the noe (i,) 8
Payoff after N perios: Call option payoff: C N, = max(u N - K,0) (3,3) (2,2) (1,1) (3,2) (0,0) (2,1) (1,0) (3,1) (2,0) (3,0) 0 t 1 t 2 t 3 = T Working Backwars through the Tree: the Backwar Inuction Algorithm tep N C N,, = max(u N K,0), = 0,1,, N tep N -1 C N-1, = e - [pc N,+1 + (1-p)C N, ], = 0,1,, N -1 tep i C i, = e - [pc i+1,+1 + (1-p)C i+1, ], = 0,1,, i tep 0 C = C 0,0 = e - [pc 1,1 + (1-p)C 1,0 ] Close-form solution for European options For European options, the algorithm can be solve in close form: N C = e p p C N rt N (1 ) N, = 0, C N, = max(u N K,0): terminal payoff at (N,), e -rt = e -r(n t) : N-perio iscount factor (T = N t), 9
N N! = : number of paths to a particular outcome (N,) at maturity (binomial!( N )! coefficient), N N pn, = p (1 p) : risk-neutral probability of an outcome with up movements an N own movements (risk-neutral binomial probability). Multi-Perio Risk-Neutral Valuation Formula for European-style Contingent Claims with General Payoffs uppose a contingent claim is efine by its payoff F( T ), where F is a given function of the terminal asset price. Then its present value (price) is given by the iscounte riskneutral expectation of the payoff: rt PV = e p F( ), N, N, where p N, are risk-neutral probabilities. Consistency check: From the Binomial Theorem: N N N rt rt N N e pn, N, = e p (1 p) u 0 0 = = rt N rt rn t = e ( pu+ (1 p) ) = e e = Thus, as expecte, the price of the stock is equal to its current price. American Options (No Diviens) 1. American calls: always worth more alive than ea. It is never optimal to exercise prior to expiration. 2. American puts: at each time step we maximize the value of our position -- optimal exercise strategy maximizes the option value to the option holer. Assuming the rational option holer maximizes his portfolio value, the ynamic programming (Bellman) equation for American puts reas: P = max( K u, e [ pp + (1 p) P ]) i i, i+ 1, + 1 i+ 1, At each noe we perform the test for making early exercise ecision: is the option worth more ea (exercise) or alive (hol)? 10
Example of Delta-Heging on a Binomial Tree: A Riskless Traing trategy to Lock in Profits from Option Mispricing From Cox an Rubinstein, Options Markets, Ch. 5 Three-perio binomial tree (N = 3) for a call option with K = 80 expiring at the en of the thir perio. Initial stock price is = 80 at the noe (0,0), the risk free rate r is such that exp() = 1.1, an the iscount factors for one, two an three perios are: exp(- ) = 0.909, exp(- 2) = 0.826, exp(- 3) = 0.751. Also, suppose that u = 1.5 an = 0.5. The risk-neutral probability of an up move is: 1.1 0.5 p = = 0.6 1.5 0.5 Fair value of the call accoring to our binomial moel is C = $34.065. uppose that at time t = 0 the call is actually trae in the market for C = $36, i.e., it is overprice accoring to our moel. Arbitrage strategy: sell the call for $36 an hege by creating a synthetic call. tep 0 (i = 0). ell the call for 36. Take 34.065 of the procees an invest in a portfolio consisting of = 0.719 shares of stock by borrowing the ifference 0.719 x 80 34.065 = 23.455. Take the remainer 36 34.065 = 1.935 an put it in the bank to grow at the risk free rate. tep 1 (i = 1). uppose the stock goes up to 120 so that the new = 0.848. Buy 0.848 0.719 = 0.129 more shares at 120 per share for a total of 15.480. Borrow to pay for the stock. With the interest rate of 0.1, you alreay owe 23.455 x 1.1 = 25.801. Thus your total current ebt is 25.801 + 15.480 = 41.281. tep 2 (i = 2). uppose the stock now goes own to 60. The new = 0.167. ell 0.848 0.167 = 0.681 shares at 60 per share, taking in 0.681 x 60 = 40.860. Use it to pay back part of your ebt. ince you now owe 1.1 x 41.281 = 45.409, the repayment will reuce this to 45.409 40.860 = 4.549. tep 3 (i = 3). uppose the stock goes own to 30. The call you sol expire worthless. You own 0.167 shares of stock selling at 30 per share, for a total value of 0.167 x 30 = 5. ell the stock an repay the ebt 4.549 x 1.1 = 5 that you now owe. Go back to the bank an withraw your original eposit of 1.935, which has now grown to 1.935 x (1.1) 3 = 2.575. Congratulations! This is your arbitrage profit. 11
tep 3u (i = 3). uppose that instea the stock price goes up to 90. The call you sol is inthe-money at expiration. Buy one share of stock an let the call be exercise, incurring a loss of 90 80 = 10. You also own 0.167 shares of stock currently traing at 90/share, for a total value of 0.167 x 90 = 15. ell the stock an use the procees to repay the ebt an cover the loss of 10 for exercising the call. Go back to the bank an withraw your original eposit of 1.935, which has now grown to 1.935 x (1.1) 3 = 2.575. Congratulations! This is your arbitrage profit. In summary, if our moel of stock price movements is correct an we faithfully aust our portfolio as prescribe by the formula for the option elta, then we can be assure of walking away in the clear at expiration ate, while still keeping the original price ifference plus the interest it has accumulate. In conucting the heging operation, the essential thing was to maintain the proper proportional relationship: for each call we are short, we hol shares of stock an we borrow at the risk free rate the ollar amount B, thus creating a synthetic option. Notation: Next to each noe the following information is given: Noe inexes (i,) Risk neutral probability of getting to the noe at the noe C at the noe C C C C at the noe (i,) is compute accoring to i, = = ( u ) i+ 1, + 1 i+ 1, i+ 1, + 1 i+ 1, i+ 1, + 1 i+ 1, i, 12
(3,3) 0.216 270 190 1.0 (2,2) 0.36 180 107.272 1.0 (1,1) (3,2) 0.6 0.432 120 90 60.463 10 0.848 1.0 (0,0) (2,1) 1 0.48 = 80 60 C = 34.065 5.454 = 0.719 0.167 (1,0) (3,1) 0.4 0.288 40 30 2.974 0 0.136 0 (2,0) 0.16 20 0 0 (3,0) 0.064 10 0 0 13