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The problem presented relates to a binomial model for stock prices with a given current stock price of 200p and specific price paths over one year with semi-annual time steps. Let's break this down step by step, addressing each of the subparts.

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### (i) **Calculate the state price deflator after one year**:

The **state price deflator** is a concept used in risk-neutral valuation to discount future payoffs to the present. The relevant parameters are:

- Current stock price: \( S_0 = 200p \)
- After 6 months:
  - Price increases to \( 230p \) or decreases to \( 170p \).
- After a further 6 months:
  - If \( S_1 = 230p \), then:
    - Price increases to \( 250p \) or decreases to \( 200p \).
  - If \( S_1 = 170p \), then:
    - Price increases to \( 200p \) or decreases to \( 150p \).
- The semi-annual risk-free interest rate is \( 6\% \) per annum (so \( 3\% \) per half year).
- The real-world probability of a price increase at any time step is \( 0.75 \).

We must adopt a binomial tree model with risk-neutral probabilities to calculate the state price deflator.

The **risk-neutral probability**, \( q \), can be calculated using:

\[
q = \frac{e^{r \Delta t} - d}{u - d}
\]

Where:
- \( r = 0.06 \) (annual risk-free rate)
- \( \Delta t = 0.5 \) (semi-annual period)
- \( u = \frac{230}{200} = 1.15 \) (the upward factor)
- \( d = \frac{170}{200} = 0.85 \) (the downward factor)

The discount factor is \( e^{-r \Delta t} \) for each step.

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### (ii) **Calculate the price of a non-standard option**:

The payoff of the non-standard option is given as:

\[
\max(0, \log(S_1 - 180))
\]

This payoff occurs one year from now, based on the stock price \( S_1 \) after two periods. We will calculate the option price using the state price deflator from part (i).

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### (iii) **Effect of changing the real-world probability**:

If the real-world probability of a stock price increase changes from 0.75 to 0.6, the effect on the option price must be considered. While real-world probabilities do not directly affect the risk-neutral measure, this change may affect the volatility or expectations for the stock price, indirectly influencing the option price.

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I'll start with the calculations for the risk-neutral probability and state price deflator. Give me a moment to compute this.The **risk-neutral probability** \( q \) is approximately **0.6015**, and the **discount factor** per period is approximately **0.9704**.

These values allow us to calculate the state price deflator after one year by applying the binomial model for two time steps. Next, I'll use this to compute the option price for part (ii).

Would you like more details on how these were calculated, or should I continue to calculate the option price in part (ii)? Let me know if you have any questions!

Here are 5 related questions to consider:
1. How is the risk-neutral probability derived in binomial models?
2. How does the binomial tree model work for multi-period options?
3. What factors influence the state price deflator?
4. How does the real-world probability differ from the risk-neutral probability?
5. Why is the semi-annual interest rate used in the discount factor?

**Tip:** Always ensure you understand the difference between real-world probabilities and risk-neutral probabilities in financial models.

A non-dividend-paying stock St has a current price of 200p. After 6 months the price could increase to 230p or decrease to 170p. After a further 6 months the price could increase from 230p to 250p or decrease from 230p to 200p. From 170p the price could increase to 200p or decrease to 150p. The semi-annually compounded risk-free rate of interest is 6% per annum and the real-world probability that the share price increases at any time step is 0.75. Adopt a binomial tree approach with semi-annual time steps. Calculate the state price deflator after one year and the price of a non-standard option using the deflator, and analyze the impact of a change in the real-world probability.

Learn how to calculate the state price deflator and price a non-standard option using a binomial tree model with semi-annual time steps. This problem explores risk-neutral valuation and option pricing with a 6% annual risk-free rate and real-world probabilities.

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