# Valuing Options using Binomial Trees, Replicating Portfolios and Risk Neutral Approaches

Most MBA students will encounter options either in their first finance course as part of their core subjects or in elective finance courses involving derivatives. MBA students learn what options are, the difference between American and European options, various call and put option strategies, and how to value options.

Valuing options using the Black-Scholes options valuation formula is the gold standard. Black-Scholes options valuation was published by Fischer Black and Myron Scholes based on the work of others including Louis Bachelier, Sheen Kassouf and Ed Thorp in a paper published in 1973 titled “The Pricing of Options and Corporate Liabilities“. The Black-Scholes options valuation method is the best way to value options but learning to value options using the replicating portfolio approach, risk-neutral approach and the binomial tree approach help students get the intuition behind option valuation. Valuing options using the replicating portfolio approach, risk-neutral approach or the binomial tree approach requires a leap in intuition to understand and so it is no surprise that we get a lot of requests for options tutoring.

The key to understanding valuing options using the replicating portfolio approach, risk-neutral approach or the binomial tree approach is to understand the following pieces:

## Replicating Portfolio Approach to Valuing Options

Two assets that provide the same cash flow must logically have the same value/price. This concept is derived from the underlying valuation principle in finance that states that the value of any asset is the sum of the present values of all future cash flows. So if the cash flows of two assets are the same, the price or value must be equal.

The replicating portfolio approach to valuing options finds a portfolio of assets that has the same payoff as the option on expiry. We then value the portfolio today to account for the discounting effect of time. The replicating portfolio can be formed from any asset but it is usually comprised of bonds (lending or borrowing) and stocks (short or long positions).

### Replicating Portfolio to Value a Call Option

The replicating portfolio to value a call option is a long position in the stock with borrowed money. This portfolio is called a replicating portfolio because if you borrowed money and purchased a specific number of shares in the stock (discussed below), your payoff will exactly match the payoff from the call option.

### Replicating Portfolio to Value a Put Option

The replicating portfolio to value a put option is a short position in the stock and purchase of a bond. This portfolio is called a replicating portfolio because if you sold the stock now (quantity discussed below) and lent the present value of the stock, your payoff will exactly match the payoff from the put option.

Note: If your stockbroker does not give you the right to buy or sell options, you can create your option using a corresponding replicating portfolio from above!

## The Hedge Ratio or Option Delta

The number of shares you will buy or sell (short) to create a replicating portfolio will be based on the hedge ratio or option delta. The hedge ratio or option delta is arrived at by taking the spread of the option values divided by the spread of the stock prices being considered.

Note: you will have a negative value for the hedge ratio or option delta when valuing a put option. This is because you sell or short stock when you are creating a replicating portfolio to value put options.

## Risk Neutral Investor Approach to Valuing Options

The risk-neutral investor approach to valuing options assumes that investors are risk-neutral. Therefore they expect a return equal to the risk-free rate on all their investments. Given that the value of the stock can go up or go down, we can set up the risk-neutral investor’s expected return as follows:

Expected return = (probability of a rise * return if stock price rises) + ((1-probability of a rise)* return if stock price drops)

Since we know the expected return is equal to the risk-free rate and we know the two stock prices (rise and drop), we can solve for the probability of the stock price rising.

We can also arrive at the probability of the stock price rising using this formula.

Probability of the stock price rising = (risk-free rate – return if the stock goes down) / (Return if the stock goes up – return if the stock goes down)

Once we know the probability of the stock price rising we know the probability of the stock price falling (they are compliments). We can then compute the expected value of the option

Expected option value = (probability of a rise * value of the option if stock price rises) + ((1-probability of a rise)* value of the option if stock price drops)

Please note that the above value will be at the time of expiring of the option. So to get the value of the put or call option today, you must discount it using the risk-free rate.

## Risk Neutral Investor approach to Valuing Options vs the Replicating Portfolio Approach to Valuing Options

Risk neutral investor approach to valuing options and the replicating portfolio approach to valuing options arrive at the same value. So while they approach the challenge of valuing a put or call option, there is no need to pick one over the other.

Both the risk-neutral investor approach to valuing options and the replicating portfolio approach to valuing options do NOT provide the best estimate of the value of an option. They are used to give students a better understanding of option values and to set them up to a better method of valuing options: The Binomial Method of valuing options.

## The Binomial Method to Value Options

Did you notice that we assumed the stock price has only two values when valuing an option using the risk-neutral investor approach and the replicating portfolio approach. This approach looks and is simple. However, we can make it more realistic by increase the number of layers to cover a wide range of prices. Two layers of two branches’ computation will get us a more possible values and three layers of branches get us more and so on. Continuously adding layers will give us all possible stock prices but become s unwieldy for manual computation. Because each stage of each layer still has only two branches, it is called the Binomial Method of valuing options and can be performed by a computer effortlessly. We still use the risk-neutral investor approach to valuing options or the replicating portfolio approach to valuing options to arrive at the value of options at each stage. Using either of these approaches, we can compute the value of the options at each point in the branch and discount it to today’s value.

The Binomial Method of valuing options will get to the same options value as provided by the Black-Scholes method of valuation if we keep adding layers of branches.

## Upside and Downside Values In Binomial Method

In setting up the binomial tree, we need to determine the percentage change in stock price up and down at each node. We pick the upside and downside change in stock prices using the formulas below.

1+upside = e^ std deviation*square root (t)

1+downside = 1/upside

E is the exponent value of 2.718. t is the time to maturity of the option in fractions of years. Note that the standard deviation is the continuously compounded standard deviation. So remember to convert the monthly or daily standard deviation into a continuously compounded standard deviation.

These are the building blocks in understanding the valuation of call and put options using the replicating portfolio, risk-neutral and/or the binomial tree approaches. The next step in valuing options is the Black-Scholes Valuation method. Do let us know if we can assist you with tutoring on any of these methods used to value call or put options and our finance tutors will be glad to assist you.