Calculating Two-Period Binomial Stock Price Probabilities

Summary

The document checks two probability calculations in a two-period binomial stock-price model. With the stated up and down factors and physical probability of an upward move, the questioner identifies the paths that satisfy each event: an upward move in both periods for the terminal price to exceed its initial level, and an up move followed by a down move for the price to finish below the first-period price while that first-period price exceeds the initial price.

The accepted response confirms both path-probability calculations and clarifies that the comma in the second event is intended as “and.” The example illustrates how to translate a compound price condition into qualifying paths, then multiply the probabilities of successive moves. It is a narrow worked confirmation rather than a general treatment: it does not discuss risk-neutral probabilities, discounting, or alternative tree structures.

Key ideas

• Identify which paths satisfy each event before calculating its probability.
• For a specified path in the model, multiply the probabilities of its sequential moves.
• The second event combines both price inequalities, so it requires an up move followed by a down move.
• The calculations use the stated market probability rather than deriving a pricing probability.

Tags

# Two-period binomial model probability question


# Two-period binomial model probability question












I have started to work with given two period binomial model S(0)=100 u=1.25 d=0.8 r=0.05 and the market probability of stock going up each period is p=0.55.

I am trying to calculate two probabilities;

- P(S(2)>S(0)) -> For this one I thought only Suu (156.25)>100 so I assumed probability being 0.55x0.55.

- P(S(2) < S(1),S(1) > S(0)) -> For this I did not know how begin with but I said looks like only up and down gives this so I claimed 0.55x0.45.

So I wanted to ask is this the right way or am I missing something. Any explanation would be appreciated.

## Answer by KaiSqDist (score 0, accepted)

https://quant.stackexchange.com/a/78334

Answering both questions:

- Your answer is correct.

- Is the comma (in the probability function) an "And" or a "Or", what is the mathematical notation used? From what you answered, it seems like an "And", and in which case your answer seems correct as well.

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