Here is the annual risk-free rate and is the length of a period in years. Thus the end-of-period value of the stock is if is the initial stock price. Let’s see what happens when stock are expected to earn at the risk-free rate. risky assets such as stock are expected to earn at the risk-free rate. We further assume that in a risk-neutral world investors are willing to hold risky assets without a risk premium, i.e. In a risk-neutral world, investors are indifferent between these two investment choices. Normally a risk premium is needed in order to entice a risk-averse investor to hold the second investment. Both investments have the same expected value but the second one is much riskier. Another investment with equally likely payoff of $50 or $0. For example, one investment pays $25 with certainty. Imagine a world where investors are indifferent between a sure thing and a risky investment as long as both investments have the same expected value. Let’s look at the implication of investing in a risk-neutral world. There is no reason to not use risk-neutral pricing. Thus the risk-neutral pricing approach is easy to implement and produces the correct price. Even though using the more standard approach is possible, it is more cumbersome. Our goal in this post is to show that the risk-neutral pricing approach produces the same option price as from using the more standard approach of using a true probability of a stock price up move and using a realistic discount rate. ![]() It is natural to think that discounting the value of an option should be done using the risk-free rate and instead using a rate of return equivalent to the option. Thus an option is riskier than the stock. a call is equivalent to borrowing the amount to partly finance the purchase of shares). In the earlier posts on the binomial pricing model, we see that an option is equivalent to a leverage investment in the stock (e.g. On the other hand, the expected value is counted from one period to the previous period using the risk-free rate. Why is the true probability of stock price movement not used? What is ? Is it really the probability that the stock will go up? There is no reason to believe that is the true probability of an up move in the stock price in one period in the binomial tree. The expected value is calculated using and. Something is peculiar about the expected value calculation and the discounting in formula (1). Using formula (1) in this recursive fashion is called the risk-neutral pricing.įrom a computational standpoint, formula (1) is clear. Using this formula, the price of the option is calculated by working backward from the end of the binomial tree to the front. The formula (1) uses the risk-free rate to discount the expected value back to that given node. The value inside the parentheses in (1) can thus be interpreted as the expected value of the option payoff in the next period that follows a given node. Thus they can be interpreted as probabilities. The values of and sum to 1 and are positive ( discussed in the post #2 on the binomial option pricing model). ![]() The calculation uses the probabilities and : The expected value refers to the result inside the parentheses, which is the expected value of the option value (when stock price goes up) and the option value (when stock price goes down). The formula has the appearance of a discounted expected value. In the post #1 on the binomial option pricing model, the following option pricing formula is derived (formula (4) in that post). In this post, we examine why this is the case. Even though the risk-neutral probabilities are not the true probabilities of the up and down moves of the stock, option pricing using risk-neutral probabilities is the simplest and easiest pricing procedure and more importantly produces the correct option price. Then the price of the option is calculated by working backward from the end of the binomial tree to the front. In risk-neutral pricing, the option value at a given node is a discounted expected payoff to the option calculated using risk-neutral probabilities and the discounting is done using the risk-free interest rate. The idea of risk-neutral pricing is that the binomial option pricing formula can be interpreted as a discounted expected value. Post #6: To revisit the notion of risk-neutral pricing. This is post #6 on the binomial option pricing model.
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