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European options are unique financial derivatives where the option holder can only exercise the option at its expiration date. This differs from American options, which can be exercised at any time before or on the expiration date.

The valuation process for European options involves determining the option's fair price at the present moment. To do this, we look at the final payoff at the end of the option term:

• If the option is a put option, as in our exercise, the payoff at expiration is calculated using: \( \max(K - S_T, 0) \), where \( K \) is the strike price and \( S_T \) is the stock price at expiration.

This value is then discounted back to the present using risk-neutral probabilities. For European options, this calculation provides a straightforward method for determining the current value of the option.

American options offer increased flexibility over their European counterparts because they can be exercised at any point before expiration. This means the valuation process must account for the possibility of early exercise.

In the binomial tree model, while computing the American options, we start from the final payoffs and move backward, just like with European options. However, at each node, we compare the payoff of exercising immediately to the expected payoff of holding the option further, utilizing:

• The intrinsic value is determined by the formula \( K - S \), where exercising the option immediately is beneficial.

This decision-making process at each step leads to the potential early exercise, which can result in a different valuation than what would be seen in European options. It is this complexity that often makes American options more valuable.

Option valuation is the process of determining the fair value of an option. The binomial tree model provides a systematic way to achieve this by simulating potential future outcomes of the stock price.

In the three-step binomial tree, the stock price is calculated at each node, moving upward or downward according to the up factor \( u \) and the down factor \( d \). This reflects possible movements in the market, allowing valuation of both European and American options:

• The model incorporates interest rates, dividends, and volatility to determine these movements.

Using the coherent structure of a binomial tree, we can systematically calculate the option payoff at final maturity and use backward induction to arrive at the present value of the option.

Risk-neutral probabilities are crucial in option pricing because they allow us to discount expected future payoffs to their present values in a way that is consistent with the current market environment.

In the binomial tree model, these probabilities are calculated based on current risk-free interest rates and dividend yields. Specifically, the formula is:

• \( p = \frac{e^{(r-q)\Delta t} - d}{u - d} \)

This ensures that the expected value of the stock price in a risk-neutral world is equivalent to earning the risk-free rate. This assumption simplifies the calculation without needing to account for investors' risk preferences, allowing for a universal method of valuation for options.