A market for options exists because options have a unique, calculable price. For this price to exist, the market must have a mechanism for ignoring the different risk tolerances of the different players in the market. Risk neutral probabilities is a tool for doing this and hence is fundamental to option pricing. While most option texts describe the calculation of risk neutral probabilities, they tend to gloss over their importance. Failing to incorporate risk neutral probabilities can lead to incorrect conclusions.
Risk Neutral Probabilities
In order to calculate the expected value of some outcome, you need two quantities--the value of an outcome and the probability of the outcome. Consider the outcome of flipping a fair coin. If you guessed correctly, you would get USD2; otherwise, you would get USD0. The expected payoff for this bet is USD1. (USD2*0.5 + USD0*0.5 = USD1). You can think of expected payoff as a weighted average, where the probabilities are used as weights.
Suppose you offered this bet to 100 people and each one gave you USD1. However, 80 of the people picked heads. If the coin came up heads, you would loose USD60. (USD100 (income) - USD2*80 (payout) = -USD60); if the coin came up tails, you would gain USD60. (USD100 (income) - USD2*20 (payout) = USD60). Over several games, you would break even. But on any one particular game there is a chance you could lose money.
Since you don't have any appetite for losing money, you decide to change the payouts. Rather than paying USD2 for guessing correctly, you change the payouts in the following fashion. If people pick heads and heads show up, you pay USD1.25. If they pick tails and tails show up, you pay USD5.00. In either case, if they picked incorrectly, you don't pay anything. Now, if the coin comes up heads, you break even. (USD100 - USD1.25*80 = USD0); if it comes up tails, you still break even. (USD100 - USD5*20 = USD0). Thus, regardless of the outcome, you have eliminated your chances of losing money.
Let us look at this from the viewpoint of somebody who picked heads. If you told them it was a fair coin, their expected payoff would be USD0.625. (USD1.25*0.5 + USD0*0.5 = USD0.625). They would never enter a game for USD1, if the expected payoff were USD0.625. So rather than using a regular coin, you find a coin that has an 80% chance of coming up heads. Then the expected payoff for selecting heads is USD1. (USD1.25*0.8 + USD0*0.2 = USD1). From the viewpoint of somebody who picked tails, the expected payoff is also USD1. (USD0*0.8 + USD5*0.2 = USD1). Note that we are not changing the actual probabilities of the coin coming up heads or tails; we are only changing the weights associated with it. This new set of weights is called "risk neutral probabilities."
There is another reason we need to determine the risk neutral probabilities. With the original payoffs, you had a chance of losing USD60 that you wanted to avoid. But, there might have been somebody else, with a different appetite for risk, willing to lose USD60 for the chance of gaining USD60. This person would be willing to take the bet. However, if we shift to risk neutral probabilities, then everybody will have look at the game in the same manner. This is important in option pricing. Without risk neutral probabilities, each person will value an option differently. In order to trade options effectively, we need to be able to determine the price of an option that does not depend to anybody's risk tolerance; we determine this price using risk neutral probabilities.
Application To Options
Now we will use risk neutral probabilities to determine the price options. We want to determine the price of a European call option today, V0, which expires in one year and has a strike price of K. Suppose the underlying stock is valued today at S0. A year from now, the price of this stock, S1, will only have two values, S+ or S-. For convenience, we use "u" and "d" to represent the ratio by which the price changes; that is, S1 = S+ = uS0 or S1 = S- = dS0.
The value of the option a year from now is V1 = Max {S1 - K, 0}. If the stock goes up, the value of the option is V+ = Max {S+ - K, 0} = Max {uS0 - K, 0}; if the value of the stock goes down, the value of the option is V- = Max {S- - K, 0} = Max {dS0 - K, 0}.
Suppose we sell such a call. We will receive V0 an amount which is yet to be determined. We take some of this money and buy (also an amount not yet determined) shares of the stock at S0. The remainder, V0 S0 , we invest at a risk free rate. At the end of the year, the value of the option has to equal the value of this portfolio
Since the stock can have two states,
Subtracting one equation from the other, we get a value for
Substituting back into the formula, we get the value of the call today
We now define two new values,
We will call these our risk neutral probabilities or risk neutral weights. Then, we get the value of the option today.
V0 is the present value of E[V1]. E[V1] is the expected value of the option at expiration using the risk neutral weights. Since we used risk neutral probabilities, everyone will have the same expected value for E[V1] and thus everyone will have the same value for V0. Notice that once again the risk neutral probabilities do not depend on the actual odds of the stock going up or down.
Mis-Application To Insurance
Because of the variability in insurance company liabilities, insurance regulators have to worry about the cost of an insurance company going out of business. For simplicity, assume an insurance company invests in a risk-less asset. Then the cost of bankruptcy is solely a function of the liabilities. If the liabilities are greater than then assets, the company's deficit is the difference between the liability amount and the assets available. (See graph.) Using a variety of simulation techniques, we can generate the probability distribution of liabilities. The expectation of the costs exceeding the assets is called the expected policyholder deficit, or EPD.
The graph at left resembles the payoff of a long call option at expiration. Thus, people have suggested that EPD is equivalent to E[V1] in the example above and the present value of the EPD is "completely equivalent to a financial option." However, the probability used in this calculation is based on the simulation of the liabilities and represents the actual probability of a liability value occurring. As we have show above, option prices are calculated using risk neutral probabilities and not actual probabilities. We can fairly say that EPD represents the average cost to the guarantor of the insurance company bankruptcy. However, it does not represent the value a guarantor should place on the guaranty.
Most financial texts use expectation calculations without emphasizing the use of risk neutral probabilities in this calculation. The result is the incorrect use of these formulae.
Expected Policyholder Deficits
This week's Learning Curve was written Raghu Ramachandran, senior portfolio strategist at Brown Brothers Harriman in New York.
