## Valuing options using binomial model

- CHAPTER 5 OPTION PRICING THEORY AND MODELS
- Binomial Option Pricing Model Definition
- Binomial Option Pricing Model | Formula & Example
- Understanding the Binomial Option Pricing Model
- Valuing an Option Embedded Bond using Binomial Interest

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.

## CHAPTER 5 OPTION PRICING THEORY AND MODELS

Under this scenario, we will use the Geometric Brownian Motion of the stock price which implies that the stock follows a random walk. Random walk Random Walk Theory The Random Walk Theory or the Random Walk Hypothesis is a mathematical model of the stock market. Proponents of the theory believe that the prices of means that the future stock prices cannot be predicted by the historical trends because the price changes are independent of each other.

### Binomial Option Pricing Model Definition

The Black-Scholes model is mainly used to calculate the theoretical value of European-style options and it cannot be applied to the American-style options due to their feature to be exercised before the maturity date.

#### Binomial Option Pricing Model | Formula & Example

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.

##### Understanding the Binomial Option Pricing Model

We utilized the lattice model previously to price convertible bonds. In this post, we’re going to use it to value an American equity option. We use the same input parameters as in the previous example. Using our Excel workbook, we obtain a price of $, which is smaller than the price determined by the analytical approximation (Barone-Andesi-Whaley) approach.

###### Valuing an Option Embedded Bond using Binomial Interest

The binomial pricing model traces the evolution of the option’s key underlying variables in discrete-time. This is done by means of a binomial lattice (tree), for a number of time steps between the valuation and expiration dates. Each node in the lattice represents a possible price of the underlying at a given point in time.

Under the binomial model, we consider that the price of the underlying asset will either go up or down in the period. Given the possible prices of the underlying asset and the strike price of an option, we can calculate the payoff of the option under these scenarios, then discount these payoffs and find the value of that option as of today.

The simplest method to price the options is to use a binomial option pricing model. This model uses the assumption of perfectly efficient markets. Under this assumption, the model can price the option at each point of a specified time frame.

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: