Fin 533 module5 binomial step by step (corrected)
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1) The document describes how to use a single-period and two-period binomial option pricing model to value European call and put options. 2) It provides a step-by-step example of using a single-period model to value a European call and put. The steps include creating an asset price lattice, determining terminal option values, calculating probabilities, and determining the option and hedge ratios. 3) It then demonstrates a two-period binomial model example to value a European put option, extending the process over two periods rather than one. 4) Finally, it notes that an American option would be valued differently than a European option using binomial modeling, as the early exercise feature has additional value.
Fin 533 module5 binomial step by step (corrected)
- 1. Binomial Option Pricing Model Step by Step Dr. Nivine Richie
- 2. How to use a single-period binomial model to price a European call and a put First, the call…
- 3. Before we start… Recall, the value of the call option at expiration is the greater of zero or the value of the underlying minus the exercise price (X). c = Max(0,S – X) The value of the put option at expiration is the greater of zero or the value of the exercise price minus the underlying. p = Max(0,X – S)
- 4. Example Assume an initial stock price of GBP 60 on the underlying asset, and a risk-free rate of 5%. Assume the asset price can move up 15% or down 10%, so the up jump and down jump factors are u = 1.15 and d = 0.90. Price a European call and put option with a strike price of GBP 60 using a single period binomial model. Calculate the hedge ratio and interpret it.
- 5. Step 1. Create the lattice of values for the underlying asset showing the asset’s possible terminal values t=0 t=1 S+ = 60 (1.15) = 69.00 S0 = 60 S- = 60 (0.90) = 54.00
- 6. Step 2. Determine the terminal values of the call option t=0 t=1 S+ = 60 (1.15) = 69 C+ = max(0, 69 – 60) = 9 S0 = 60 S- = 60 (0.90) = 54 C- = max(0, 54 – 60) = 0
- 7. Step 3. Calculate the risk-neutral probabilities π = 1 + r − d u − d = 1 + 0.05 − 0.90 1.15 − 0.90 = 0.15 0.25 = 0.60 1 − π = 0.40
- 8. Step 4. Calculate C0 c = πc+ + 1 − π c− 1 + r = 0.6 9 + 0.4 0 1.05 = GBP 5.14 Note that this 5.14 call value is the present value of the call’s expected value based on the risk-neutral probabilities
- 9. t=0 t=1 S+ = 60 (1.15) = 69 C+ = max(0, 69 – 60) = 9 S0 = 60 C0 = 5.14 S- = 60 (0.90) = 54 C- = max(0, 54 – 60) = 0 Step 4. Calculate C0
- 10. Step 5. Calculate the hedge ratio h = c+ − c− S+ − S− h = 9 − 0 69 − 54 h = 0.60 Interpreting the hedge ratio (h): • One call can be replicated with 0.6 shares of the asset purchased with borrowed money C0 = S0 – PV(X) • A riskless portfolio can be created by going long one call and going short 0.60 shares of the asset PV(X) = C0 – hS0
- 11. How to use a single-period binomial model to price a European call and a put …and now for the put
- 12. Step 1. Same as for a call—create the lattice of values for the underlying asset showing the asset’s possible terminal values t=0 t=1 S+ = 60 (1.15) = 69.00 S0 = 60 S- = 60 (0.90) = 54.00
- 13. Step 2. Determine the terminal values of the put option t=0 t=1 S+ = 60 (1.15) = 69 P+ = max(0, 60 – 69) = 0 S0 = 60 S- = 60 (0.90) = 54 P- = max(0, 60 – 54) = 6
- 14. Step 3. Same as for call—calculate the risk- neutral probabilities π = 1 + r − d u − d = 1 + 0.05 − 0.90 1.15 − 0.90 = 0.15 0.25 = 0.60 1 − π = 0.40
- 15. Step 4. Calculate P0 p = π𝑝+ + 1 − π p− 1 + r = 0.6 0 + 0.4 6 1.05 = GBP 2.29 Note that this 2.29 put value is the present value of the put’s expected value based on the risk-neutral probabilities
- 16. Step 4. Calculate P0 t=0 t=1 S+ = 60 (1.15) = 69 P+ = max(0, 60 – 69) = 0 S0 = 60 P0 = 2.29 S- = 60 (0.90) = 54 P- = max(0, 60 – 54) = 6
- 17. Step 5. Calculate the hedge ratio h = p+ − p− S+ − S− h = 0 − 9 69 − 54 h = −0.60 Interpreting the hedge ratio (h): • One call can be replicated with 0.6 shares of the asset purchased with borrowed money P0 =hS0 + PV(X) • A riskless portfolio can be created by going long one call and going short 0.60 shares of the asset PV(X) = C0 – hS0
- 18. Now we complicate things NOTE that a 1-period model is only useful for understanding the model. A more realistic model would be to assume the life of the contract is divided into multiple jumps, rather than a single jump. So let’s see a two-period binomial model
- 19. Example Consider the following data and determine the values of a European put option using the two-period binomial option- pricing model: • S0 = $56 • X = $60 • u = 1.3 • d = 0.625 • T = 2 • r = 2.5%
- 20. Step 1. Create the lattice of underlying asset values t=0 t=1 S++ = 56 (1.3)2 = 94.64 S+ = 56 (1.3) = 72.80 S0 = 56 S+- = 56 (1.3)(0.625)= 45.50 S- = 56 (0.625) = 35 S- = 56 (0.625)2 = 21.875
- 21. Step 2. Determine option terminal values t=0 t=1 S++ = 56 (1.3)2 = 94.64 P++ = max(0, 60 – 94.64) = 0 S+ = 56 (1.3) = 72.80 S0 = 56 S+- = 56 (1.3)(0.625)= 45.50 P+- = max(0, 60 – 45.50) = 14.50 S- = 56 (0.625) = 35 S- = 56 (0.625)2 = 21.875 P-- = max(0, 60 – 21.875) = 38.125
- 22. Step 3. Calculate the risk-neutral probabilities π = 1 + r − d u − d = 1 + 0.025 − 0.625 1.3 − 0.625 = 0.59 1 − π = 0.41
- 23. Step 4. Calculate P+, P-, P0 p+ = πp++ + 1 − π p+− 1 + r = 0.59 0 + 0.41 014.50 1.025 = 5.763 p− = πp±− + 1 − π p−− 1 + r = 0.59 14.50 + 0.41 38.125 1.025 =25.54 𝑝0 = πp+ + 1 − π p− 1 + r = 0.59 9 + 0.41 0 1.025 = 12.687
- 24. t=0 t=1 t=2 S++ = 56 (1.3)2 = 94.64 P++ = max(0, 60 – 94.64) = 0 S+ = 56 (1.3) = 72.8P P+ = 5.763 S0 = 56 P0 = 12.687 S+- = 56 (1.3)(0.625)= 45.50 P+- = max(0, 60 – 45.50) = 14.50 S- = 56 (0.625) = 35 P- = 23.54 S- = 56 (0.625)2 = 21.875 P-- = max(0, 60 – 21.875) = 38.125 Step 4. Calculate P+, P-, P0
- 25. Step 5. Calculate the hedge ratios at all nodes prior to expiration h = p−+ − p−− S+ − S− h = 14.50 − 38.125 45.50 − 21.875 ℎ = −1 h = p+ − p− S+ − S− h = 5.763 − 23.537 72.80 − 25.537 ℎ = −0.47 h = p++ − p+− S+ − S− h = 0 − 14.50 94.64 − 45.50 ℎ = −0.2951
- 26. Example Consider the following data and determine the values of an AMERICAN put option using the two-period binomial option-pricing model: • S0 = $56 • X = $60 • u = 1.3 • d = 0.625 • T = 2 • r = 2.5%
- 27. t=0 t=1 t=2 S++ = 56 (1.3)2 = 94.64 P++ = max(0, 60 – 94.64) = 0 S+ = 56 (1.3) = 72.8P P+ = 5.763 S0 = 56 P0 =12.687 P0 = 13.269 S+- = 56 (1.3)(0.625)= 45.50 P+- = max(0, 60 – 45.50) = 14.50 S- = 56 (0.625) = 35 P- = 23.54 P- = 25 S- = 56 (0.625)2 = 21.875 P-- = max(0, 60 – 21.875) = 38.125 Re-evaluate each node; if early exercise is better than model value then replace model value with exercise value This shows that an American put will not necessarily have the same value as a European put because the ability to exercise early is valuable when the option is deep in the money
- 28. We can do a 3-period binomial model, and with a spreadsheet, we can create models with 100+ nodes. Note: as we carve the term of an option into smaller and smaller segments of time (and increase the number of nodes), the value of the option using the binomial model will approach the value using the Black-Scholes-Merton model