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# Risk Neutral Probabilities within the Binomial Option Pricing Model

Option pricing models often rely on the concept of risk neutral probabilities. However, few investors readily grasp the concept as presented in academic literature and texts. Simply stated a risk neutral probability is the the probability that exists when one is indifferent with respect to risk or more specifically when one is indifferent with respect to the ultimate outcome of a future event. If a portfolio is hedged such that the ending value of the portfolio is known at the beginning of a period of time then the investor holding that portfolio is indifferent with respect to the final values of the individual securities in that portfolio given that the value of the portfolio as a whole is not in question. If the portfolio's beginning and ending values are known with certainty then the the portfolio is riskless and the rate of return for the portfolio is also known with certainty. Assuming further that there are no arbitrage opportunities, this risk free portfolio must earn the same risk free rate as other risk free investments. Furthermore, knowing the ending value of the portfolio and knowing that it is constructed to be riskless allows one to find the value of the portfolio at the beginning of the time period by discounting the ending value at the risk free rate of interest.

Cox, Ross and Rubinstein used this logic to present their binomial option pricing model. The binomial option pricing model assumes just two possible states at the end of a given time period. Either the underlying security will rise or it will fall. Clearly, one is uncertain as to which state will occur. However, it is possible to construct a portfolio composed of long stock and short call(s) such that the ending value o f the portfolio is certain. By purchasing the right number of shares in conjunction with a short position in the call option the ending value of the portfolio becomes certain. Once the ending value of the portfolio is certain the portfolio is riskless and because there are no arbitrage opportunities the portfolio cannot earn more or less than the risk free rate. Therefore, the ending value of the portfolio may be discounted at the risk free rate to determine the beginning value of the portfolio. Recall that the portfolio consists of only two securities, underlying stock and the short call option. Since the value of the portfolio at the beginning of the period is now known and the price of the stock is readily observable, subtracting the value of the long stock from the value of the portfolio at the the beginning of the period yields the value of the short call option.

Notice that the probability of the stock going up or down are conspicuously absent from the model. The model does not require any estimate of the probability that the stock will go up or down. The model is risk neutral. At first glance this is counter intuitive. An explanation is clearly in order. Once the portfolio is perfectly hedged by purchasing just the right amount of stock to ensure the same ending value for the portfolio regardless of the ending value of the stock, all of the risk to the portfolio has been eliminated and the actual probability of the stock rising or falling becomes irrelevant. The final value of the hedged portfolio is certain. The actual ending value of the stock does not matter. Furthermore, because the actual ending value of the stock does not matter, the probability with which any ending value occurs does not matter either. Again, in all cases the value of the portfolio as a whole is certain.

Recall that the portfolio was constructed so that the value of the portfolio was the same regardless of the value of the stock. Since there are only two states in this model, this is the same as saying the value of the portfolio in the case where the stock rises is the same as the value of the portfolio when the stock falls.

Stated mathematically,

(S*u*delta – call up) = (S*d*delta – call down)

Where S is the current price of the stock, u is the factor that is applied to the stock to obtain its value in the up state and d is the factor that is applied to the stock to give the stock its value in the down state. Delta is the “right number” of shares to purchase to ensure that the portfolio reaches the same ending value whether the stock rises or falls.

The equation can be rearranged to solve for delta

delta = (call up – call down)/(S*u*delta - S*d*delta)

Since the portfolio's value is the same whether the stock goes up or down, the ending value of the portfolio is S*u*delta – call up or S*d*delta – call down since they are equal. Discounting the ending value of the portfolio to the beginning time period is accomplished by applying a present value discount factor of e-r*t.

Where e is the e constant, r is the risk free rate of interest and t is the period of time in years. For simplicity, t is assumed to be one year.

Value of the portfolio at the beginning of the time period = (S*u*delta – call up)e-r*t

Subtracting the present value of the long stock in the portfolio from the present value of the entire portfolio yields the value of the short call option. (Represented by the negative sign in front of the call.)

-call = (S*u*delta – call up)e-r*t -(S*delta)

The call has been valued and the probabilities of the stock reaching the up state or the down state are nowhere to be found. Substituting [(call up – call down)/(S*u - S*d)]for delta and performing some algebraic rearranging the equation becomes:

call = [(er*t -d/u-d) call up + (1-(er*t t-d/u-d))* call down]* e-r*t

Let p = (er*t-d)/(u-d)

the equation then becomes

call = [(p)*call up + (1-(p))* call down]* e-r*t

This equation is intuitively pleasing. The value of the call is the sum of the call's value in the up state times the probability of reaching that state, and the value of the call in the down state times the probability of it reaching the down state, with the quantity discounted back to the present value using the discount factor e-r*t.

Using this same logic the expected value of the stock at the end of a given time period can be thought of as the sum of the probability that the stock will reach the up state times its value in the up state and the probability that the stock will reach the down state time the probability that it will reach the down state. Notice that these are values that are obtained in the future so they are not discounted back to the beginning of the period.

Mathematically stated,

expected value of a stock = (p)*S*u + (1-p)*S*d

Substituting (er*t-d)/(u-d) for p, the equation becomes:

expected value of a stock = S*er*t

and the following equation must hold

S*er*t = (p)*S*u + (1-p)*S*d

Given S, u, and d the equation can be solved for p

p = (er*t-d)/(u-d)

The risk neutral probability that the stock will end in the up state is p.

As an illustration, once again consider a portfolio with only one stock and a world with only two possible outcomes. One year from now a stock will either be up or it will be down. The probability that it will go up is p and the probability that it will go down is (1-p). The stock is currently trading at 50, u is 1.25 and d is .75. If the stock goes up it will be worth 62.5 if it goes down it will be worth 37.50. There are also call options on the stock. If the portfolio only consists of stock it is risky and one would be very concerned with the actual probabilities of an up move or a down move occurring. However, if the portfolio is hedged properly it is riskless and must return the risk free rate.

Next, create a portfolio of stock and short call(s) such that portfolio is worth 11.25 whether the stock is worth 62.50 or whether it is worth 37.50. In either case, our portfolio has exactly the same value. This is accomplished by buying 0.30 shares of stock and selling 1 call with a 55 strike price. If the stock goes up the portfolio is worth (.30*62.5 – 7.5) if the stock goes down the portfolio is worth (.3*37.5). The 0.30 is the delta obtained from the formula described earlier. Assume the risk free rate of interest is 6% and t is equal to one year. The ending value of the portfolio, 11.25, is discounted back to the beginning of the period using the present value discount factor of e-.06*1 to arrive at a portfolio value of 10.5949. Next we can subtract the value of the stock 15 (0.30 shares trading at 50) to arrive at the value of our short call which is 4.405. The call has been priced using a simple two state binomial pricing model.

Now, to price the call using the risk neutral probabilities. Plugging the values in the preceding paragraph into the equation for p; p equals 0.6237.

Recall this equation:

call = [(p)*call up + (1-(p))* call down]* e-r*t

The value of the call is equal to the sum of the probability that it reaches the up state times its value in the up state and probability that it reaches the down state times its value in the down state with the entire quantity discounted back to the beginning of the period.

Using the numbers in the example, the call is equal to [(0.6237)*7.50 + (0.3763)*0]* e-r*t or 4.405 the same value obtained when pricing the call with the binomial model under the no arbitrage assumption.

In conclusion, it is important to realize that risk neutral probabilities are the probabilities derived under the no arbitrage assumption. Risk neutral probabilities are not the actual probabilities of an event occurring. In the same way that, the odds offered on a thoroughbred do not reflect the actual probability of the horse actually winning the race. They merely reflect the odds that the the bookmaker must offer in order to insure a risk free profit on his portfolio of bets offered on individual horses. Likewise risk neutral probabilities are not the actual probability of an event occurring they are merely the odds implied by with respect to a risk free portfolio.

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