Real-World Probabilities and Risk-Neutral Pricing in the Binomial Model

Summary

The document raises questions about the multiperiod binomial model introduced in a finance text. It asks whether each step uses the same real-world probability of an upward move, whether that assumption is necessary, and whether probabilities may differ across steps. The central pricing issue is the distinction between the probabilities used to describe observed outcomes and risk-neutral probabilities used to price contingent claims.

The text does not provide answers or derive pricing results. It serves as a prompt to examine assumptions about the model’s stepwise structure and the role of real-world probabilities. In a complete treatment, conclusions would depend on the model’s specified stock dynamics and no-arbitrage setup; the question alone does not establish that probabilities are identical or explain the conditions under which they can vary.

Key ideas

• The document asks whether upward-move probabilities are identical across time steps in a multiperiod binomial model.
• It distinguishes real-world probabilities from risk-neutral probabilities used in pricing.
• It asks whether stepwise probabilities can vary but does not answer the question.
• No derivation or pricing example is provided.

Tags

# Shreve multiperiod binomial model


# Shreve multiperiod binomial model












In Section 1.2 in Shreve's Stochastic Calculus for Finance I, he introduces the Multiperiod Binomial Model. There is something about it that I don't quite understand.

He assumes that coins are tossed and depending on heads/tails we go up or down, every step. However, it is unclear to me how this exactly works:

(1) Are all coins, at different time steps, identically distributed (i.e., each having same prob. for heads), or what is the assumption about that?

(2) If they are all identically distributed, I wonder why we even have to assume this (why this assumption is necessary). Namely, it seems to me that the (real-life) probabilities do not matter, since we're going to use the risk-neutral probabilities anyway for pricing.

(3) So: Can we assume in this model that the coin tosses at different times may have different distributions?

If the real-life probabilities are irrelevant, then I find it a bit confusing that he mentions them explicitly. It seems to me that he assumes that the "up" and "down" (real-life) probabilities at different time steps are the same (and I don't understand why such assumption would be necessary).

Thanks in advance.

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