In options pricing, risk-neutral probability is a concept that refers to the probability of an event occurring based on the assumption that the market is risk-neutral. A risk-neutral market is one in which the expected return on all assets is the risk-free rate. In such a market, the expected value of an option at expiration is the present value of the option’s payoff discounted at the risk-free rate.
To calculate the risk-neutral probability of an event occurring, we use the risk-neutral valuation method. The steps involved in this method are as follows:
1. Determine the expected cash flows from the option at expiration under different market scenarios.
2. Discount the expected cash flows at the risk-free rate to obtain the option’s present value under each scenario.
3. Calculate the probability of each scenario occurring under the risk-neutral assumption such that the expected value of the option is equal to its present value.
For example, suppose you are considering a call option on a stock with a current price of $100 and a strike price of $110. Assume that the risk-free rate is 5%, and there are two possible outcomes at expiration: either the stock price will increase to $120 or decrease to $90. You can use the risk-neutral probability to calculate the option’s fair price.
Using the risk-neutral valuation method, you can calculate the expected value of the call option at expiration under the two scenarios as follows:
- If the stock price increases to $120, the option is "in the money," and its payoff is $10 ($120 - $110). The option’s expected value is therefore $10. Since the risk-free rate is 5%, the option’s present value under this scenario is $9.52 [= $10 / (1 + 5%)].
- If the stock price decreases to $90, the option is "out of the money," and its payoff is $0. The option’s expected value is therefore $0, and its present value under this scenario is also $0.
Next, you need to calculate the risk-neutral probabilities of the two scenarios such that the expected value of the option is equal to its fair price. Let P be the probability of the stock price increasing to $120, and (1-P) be the probability of the stock price decreasing to $90. Then, the fair price of the call option is:
$9.52P + $0(1-P) = $9.52P
Since the expected value of the call option is $9.52, we can set this equal to its fair price and solve for P:
$9.52 = $9.52P P = 1
Therefore, the risk-neutral probability of the stock price increasing to $120 is 1, and the probability of the stock price decreasing to $90 is 0. This implies that the market is assuming a risk-neutral stance, and the fair price of the call option is $9.52.
In summary, risk-neutral probability is a key concept in option pricing as it allows one to calculate the fair price of an option based on the assumption of a risk-neutral market. This approach has become widely used in quantitative finance because it provides a powerful tool for valuing complex financial instruments.