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Binomial Pricing Model: Tackling Exotic Options

Binomial Pricing Model: Tackling Exotic Options. By: Joe Ejzak and James Belasco. p. S u ƒ u. S ƒ. e rt – d u – d. (1 – p ). S d ƒ d. P =. Binomial Pricing: Recap. ƒ = S D – ( S u D – ƒ u ) e – rT D= ( ƒ u - ƒ d )/ ( S u - S d ) ƒ = [ p ƒ u + (1 – p )ƒ d ] e – rT.

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Binomial Pricing Model: Tackling Exotic Options

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  1. Binomial Pricing Model:Tackling Exotic Options By: Joe Ejzak and James Belasco

  2. p Su ƒu S ƒ ert – d u – d (1– p ) Sd ƒd P = Binomial Pricing: Recap • ƒ = SD – (SuD – ƒu)e–rT • D= (ƒu- ƒd)/(Su- Sd) • ƒ = [ p ƒu + (1 – p )ƒd ]e–rT SuD – ƒu S ƒ SdD – ƒd

  3. American Options • Options can be exercised early • Optimal Time to exercise: • Compare value of selling the option to the value of exercising the option then take the maximum of the two • Directly adaptable to binomial pricing model

  4. Non-Vanilla Exotic Options • Path-Dependent • Asian option • Payoff determined by average underlying price during some preset time period • Lookback Option • Owner has the right to buy (sell) underlying at lowest (highest) price over a preceding interval • Barrier Option • Option to exercise depends on the underlying reaching a certain barrier level

  5. Asian Options • The payoff at expiry is determined by the average price of the underlying over a previously defined period of time • The Asian options have a lower volatility than vanilla options • Useful for assets that a company must buy consistently to hedge against price volatility (airlines and oil prices)

  6. Binomial Pricing: Asian Options • Pricing is path dependent • Value of the option is dependent on the average price of the path taken • Multiple paths are possible • The approximate average is calculated at each node • The payoff is determined via linear interpolation • The price is then discounted to the current time

  7. Barrier Options • Become activated or void only if underlying reaches a certain level (barrier) • A “Knock-In” option starts worthless and becomes active when a certain barrier is breached • A “Knock-Out” option starts active and becomes void if a barrier is breached • Allows the same insurance as an option but is cheaper than normal put and call options

  8. Lookback Options • Path dependent option where the payoff is dependent on the maximum or minimum price of the underlying over a specified period • The deal is valued by considering all possible values of a stock at each node • The max and min of the path function must be calculated at each node • Max and min are used to price the option at each level

  9. 70.70 70.70 0.00 62.99 62.99 56.12 56.12 3.36 62.99 56.12 6.87 0.00 56.12 50.00 50.00 5.47 4.68 56.12 50.00 44.55 44.55 6.12 2.66 56.12 50.00 50.00 11.57 5.45 36.69 6.38 50.00 35.36 10.31 50.00 14.64 Lookback Option Pricing Example S0 = 50, s = 40%, r = 10%, dt = 1 month, A [1]

  10. Introducing Dividends • The iterative path calculation can be adapted for many different derivatives including those with dividends • A dividend payment can be added at any step • The discrete time step allow for the introduction of dividend values at specific points

  11. References [1] : Chung, Dr. San-Lin. Binomial Models. Dept. of Finance, National Taiwan University [2] : Global Derivatives v3.0 : global-derivatives.com [3] : Benninga, Simon and Zvi Wiener. “Binomial Option Pricing, the Black-Scholes Option Pricing Formula, and Exotic Options.” Mathematica in Education and Research. Vol. 6 No. 4 (1997) [4]: Dai, Tian-Shyr, Guan-Shieng Huang, and Yuh-Dauh Lyuu. “Extremely Accurate and Efficient Tree Algorithms for Asian Options with Range Bounds.” [5]:Cheng, Kevin. “An Overview of Barrier Options.” http://www.global-derivatives.com

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