## Binomial two state option pricing model

- Lecture : Option Pricing Models: The Binomial Model
- Two-State Option Pricing Model - University at Albany
- Binomial Option Pricing Tutorial and Spreadsheets

Binomial option pricing model is a risk-neutral model used to value path-dependent options such as American options. Under the binomial model, current value of an option equals the present value of the probability-weighted future payoffs from the options.

## Lecture : Option Pricing Models: The Binomial Model

The payoff pattern of a put option , an option that entitles the holder to sell the underlying at the exercise price is exactly opposite, . its value in case of an up (p + ) or down movement (p - ) in the underlying price S is max(X-uS,5) and max(X-dS,5) respectively. The value of a put option using single-period binomial model can be calculated using the following formula:

### Two-State Option Pricing Model - University at Albany

Binomial model is best represented using binomial trees which are diagrams that show option payoff and value at different nodes in the option’s life. The following binomial tree represents the general one-period call option.

#### Binomial Option Pricing Tutorial and Spreadsheets

Using these final pay-offs, we can find out the call option value at the end of Year 6. In case of an upward movement in Year 6, there is a probability of that the option will be worth $ (corresponding to underlying value of $) and a probability of that it will be 5 (corresponding to underlying value of $) at the end of Year 7. This information can be used to find out the option value at the end of Year 6:

Where r is the risk-free rate , u equals the ratio the underlying price in case of an up move to the current price of the underlying and d equals the ratio of the underlying price in case of a down move to the current price of the underlying.

You have an American call option expiring in 7-years with exercise price of $85 on a stock which currently trades at $89. You expect the stock to increase by a factor of and decrease by a factor of . If the interest rate is 8%, determine the option value.

Similarly, the price of the underlying and associated call option in case of one down and one up movement in either of Year 6 or Year 7 equals $ (=$89 × × ) and $5 respectively. The call option value at end of Year 7 in this case is 5 because the spot price is lower than the exercise price. In case of a down movement in both years, the spot price at end of Year 7 will be reduced to $ and the call option will be worthless.

The price of the underlying and the pay-off of the call option , at the end of Year 7, in case of up movement in both Year 6 and Year 7, equals $ (=$89 × × ) and $ ($ - $85) respectively.