Risk Neutral Probability Calculator

Risk Neutral Probability Formula:

\[ \pi = \frac{((1 + \frac{R_f}{100}) \times P_0) - S_d}{S_u - S_d} \]

Risk Free Rate (Rf):

Initial Stock Price (P0):

Stock Down Price (Sd):

Stock Price Up (Su):

Risk Neutral Probability (π):

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1. What is Risk Neutral Probability?

Risk Neutral Probability is the probability measure where investors are indifferent to risk, ensuring that the expected return on an asset equals the risk-free rate. It is a fundamental concept in option pricing and financial derivatives valuation.

2. How Does the Calculator Work?

The calculator uses the Risk Neutral Probability formula:

\[ \pi = \frac{((1 + \frac{R_f}{100}) \times P_0) - S_d}{S_u - S_d} \]

Where:

$ \pi $ — Risk Neutral Probability
$ R_f $ — Risk Free Rate (%)
$ P_0 $ — Initial Stock Price ($)
$ S_d $ — Stock Down Price ($)
$ S_u $ — Stock Up Price ($)

Explanation: This formula calculates the probability that makes the expected return from the stock equal to the risk-free rate, assuming a binomial model where the stock can only move up or down.

3. Importance of Risk Neutral Probability

Details: Risk neutral probabilities are crucial in option pricing models like the binomial options pricing model. They allow pricing derivatives without considering investors' risk preferences, simplifying the valuation process.

4. Using the Calculator

Tips: Enter risk-free rate as a percentage, and all price values in dollars. Ensure that the up and down prices are different from each other for valid calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is the difference between real and risk-neutral probabilities?
A: Real probabilities reflect actual market expectations, while risk-neutral probabilities are adjusted probabilities that make expected returns equal to the risk-free rate, used for derivative pricing.

Q2: When should I use risk-neutral probabilities?
A: Risk-neutral probabilities are primarily used in option pricing and financial derivatives valuation, particularly in binomial and trinomial pricing models.

Q3: What assumptions does this model make?
A: The model assumes a binomial world where the stock price can only move to one of two possible prices in the next period, and that markets are efficient and complete.

Q4: Can risk-neutral probability be greater than 1 or less than 0?
A: For the model to be arbitrage-free, the calculated risk-neutral probability should be between 0 and 1. Values outside this range indicate arbitrage opportunities.

Q5: How does risk-free rate affect the probability?
A: Higher risk-free rates generally increase the risk-neutral probability of the stock moving up, as the expected return needs to compensate for the higher risk-free alternative.