Multi-Period Binomial Option Pricing - Outline

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June Chapman
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1 Multi-Period Binomial Option - Outline 1 Multi-Period Binomial Basics Multi-Period Binomial Option European Options American Options 1 / 12

2 Multi-Period Binomials To allow for more possible stock prices, we can combine individual binomial steps to create multi-period trees S C 0 us C u ds C d u 2 S C uu uds dus C ud C du d 2 S C dd Note that when u and d are constant, the tree will recombine. I.e., uds dus Multi-Period Binomial Basics 2 / 12

3 Binomial Models in the Limit Let T be the option s expiration and n be the number of binomial steps between time 0 and time T. (Note that h T {n.) Then as n Ñ 8: 1 Forward, lognormal and CRR trees will all result in the same option price 2 S T will be distributed lognormal Multi-Period Binomial Basics 3 / 12

4 Solving Multi-Period Binomials To solve multi-stage binomial option problems: 1 Start by computing the option payoffs at expiration at the far right of the tree 2 Work backwards through the tree (i.e., right to left) solving each individual binomial step in the tree for the binomial option price Note that in recombining trees, p will remain constant throughout the tree; whereas, and B will not Thus, the risk-neutral pricing method is generally preferred for multi-period problems Multi-Period Binomial Option 4 / 12

5 Multi-Period Binomial Example Example Given S 0 50, δ 0.1, σ 0.2, h 6 months, and r 2%, use a forward tree to price an at-the-money European call option expiring in 1 year. 1 Solve for u and d u e pr δqh σ?h , d e p q ? Complete stock tree us S 50 ds u 2 S uds d 2 S Multi-Period Binomial Option 5 / 12

6 Multi-Period Binomial Example Example (continued) 3 Calculate the call payoffs at expiration 50 C C u C d C uu maxp0, q C ud maxp0, q C dd maxp0, q 0 4 Calculate p p epr δqh d u d ep.02.1qp.5q Multi-Period Binomial Option 6 / 12

7 Multi-Period Binomial Example Example (continued) 5 Solve for C u and C d C u e rh rp C uu p1 p qc ud s e.02p.5q rp0.4647qp11.244q p1.4647qp0qs C d e rh rp C ud e.02p.5q r0 0s 0 p1 p qc dd s 6 Solve for C 0 C 0 e rh rp C u p1 p qc d s e.02p.5q rp0.4647qp5.1731q p1.4647qp0qs 2.38 Multi-Period Binomial Option 7 / 12

8 Binomial Tree Probabilities If you label the end nodes from i 0 to n, the number of paths n to reach the ith node in an n-period binomial tree is i E.g., consider a tree with n 4 periods i 0 i 1 i 2 i 3 i 4 1 path 4 paths 6 paths 4 paths 1 path Let p be the risk-neutral probability of an up move, then the risk-neutral probability of reaching node i is: pp q n i n p1 p q i i European Options 8 / 12

9 Multi-period Binomial of European Options Since early exercise is not possible, we can price a European option by discounting the expected risk-neutral payoff at time T back to time 0 in a single step: ņ C e rt pp q n i n p1 p q i max 0, u n i d i S 0 K i i0 Applying to the previous example: C e pp.02p1q q 2 p1qp11.244q p p1 p qp2qp0q p1 p q 2 p1qp0q 2.38 European Options 9 / 12

10 Multi-Period Binomial of American Options The above procedure will not work for American options (except an American call on a non-dividend paying stock) To price American binomial options: 1 At every intermediate node, starting at the right, decide whether early exercise is optimal I.e., check if payoff from immediate exercise calculated value at that node for corresponding European option 2 If early exercise is optimal, then the payoff from early exercise becomes the new value at that node 3 Continue working backwards through the tree using this procedure American Options 10 / 12

11 American Options Example Example What would be the price of the call in the previous example if it was an American option? 50 C ˆ Value at node u for European call is ˆ Value at node u from immediate exercise is: Ñ exercise early ˆ Call is out of the money at node d, so early exercise not optimal American Options 11 / 12

12 American Options Example Example (continued) ˆ Replace the value of C u with the payoff from early exercise ˆ Solve for C 0 : 50 C C 0 e.02p.5q r.4647p5.3371q 0s ˆ Option is not in the money at node 0, so early exercise is not optimal. The price of our American call is $ American Options 12 / 12

Put-Call Parity. Put-Call Parity. P = S + V p V c. P = S + max{e S, 0} max{s E, 0} P = S + E S = E P = S S + E = E P = E. S + V p V c = (1/(1+r) t )E

Put-Call Parity. Put-Call Parity. P = S + V p V c. P = S + max{e S, 0} max{s E, 0} P = S + E S = E P = S S + E = E P = E. S + V p V c = (1/(1+r) t )E Put-Call Parity l The prices of puts and calls are related l Consider the following portfolio l Hold one unit of the underlying asset l Hold one put option l Sell one call option l The value of the portfolio