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### Problem Interpretation

1. We are asked to derive an expression for the price of a European put option at time \( T - t \) with strike price \( K \) and expiry date \( T \), similar to the Cox-Ross-Rubinstein (CRR) formula for a call option.
2. We also need to confirm that the put-call parity holds for these CRR prices at time \( T - t \), given by the formula:
   \[
   P + S_{T-t} = C + K(1 + r)^{-t}
   \]
   where:
   - \( P \) is the price of the put option,
   - \( C \) is the price of the call option,
   - \( S_{T-t} \) is the stock price at time \( T - t \),
   - \( r \) is the risk-free interest rate.

### Solution Outline

1. **CRR Model for a European Call and Put Option:**
   In the CRR binomial model, the price of a European option at time \( T - t \) can be represented as a risk-neutral expectation. For a put option with strike price \( K \) expiring at \( T \), the formula would be analogous to the call option formula but adjusted for the put's payoff.

2. **Put Option Price in CRR Model:**
   The put option price \( P \) at time \( T - t \) can be given by:
   \[
   P = E\left[ \max(K - S_{T}, 0) \right] (1 + r)^{-t}
   \]
   where \( S_{T} \) is the stock price at expiry \( T \).

3. **Put-Call Parity Verification:**
   To verify the put-call parity, we plug in the CRR expressions for \( P \) and \( C \) to show that:
   \[
   P + S_{T-t} = C + K(1 + r)^{-t}
   \]
   holds true in the CRR model setup.

Let me know if you'd like detailed calculations for each step, including how we derive the put option price in the CRR model or the verification of the put-call parity condition.

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### Follow-Up Questions for Exploration

1. What assumptions are required for the Cox-Ross-Rubinstein model to be valid?
2. How does the binomial approach change if we consider American options instead of European options?
3. How does the risk-neutral probability affect the pricing of options in the CRR model?
4. Can we use the Black-Scholes formula to approximate the CRR model prices as the number of steps goes to infinity?
5. How would you adapt the CRR model for an option on an asset with dividends?

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### Tip
In the CRR model, ensure that you correctly calculate the up and down factors and risk-neutral probabilities for accurate option pricing.

Find an expression for the price at time T - t of a European put option with strike price K and expiry date T, analogous to the Cox-Ross-Rubinstein formula for the price of a call option. Confirm that the CRR prices at time T - t for a call and put with the same strike price K and expiry date T satisfy the put-call parity formula P + S_{T-t} = C + K(1 + r)^{-t}.

Explore how to find the price of a European put option at time T - t using the Cox-Ross-Rubinstein model. This problem involves calculating the put option price and verifying the put-call parity formula in a binomial model setup. Suitable for finance and math students studying options pricing and binomial models.

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