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The concept of risk-neutral probability is crucial when valuing options in financial markets. It allows us to compute expected outcomes in a theoretical, risk-neutral world. To determine this probability, we use the formula: \[p = \frac{e^{rT} - d}{u - d}\] Here's the breakdown: - We compute the growth factor, using the risk-free interest rate over the period. For example, with a 10% annual rate over 2 months, \(e^{rT} = e^{0.10 \times \frac{2}{12}}\) gives us about 1.0167. - The up-factor \(u\) is the ratio of the possible increased stock price to the current price, like \(\frac{53}{50} = 1.06\). - The down-factor \(d\) is similarly computed as \(\frac{48}{50} = 0.96\). By plugging these into the risk-neutral formula, we find \(p = 0.567\). This probability doesn't imply real-world behavior but is used to simplify calculation by assuming investors are indifferent to risk.

The payoff calculation is critical to understand because it tells us what an option might be worth under different scenarios. For a call option, the payoff is the potential profit you gain if the option ends 'in the money'. This is determined by the formula: \[ ext{Payoff} = \max(S_T - K, 0)\] In this exercise, if the stock price rises to 53, the payoff is \(\max(53 - 49, 0) = 4\). Conversely, if the price drops to 48, the payoff becomes zero because the strike price exceeds the market price, making exercise unprofitable. The total expected payoff combines these scenarios, weighted with risk-neutral probability. Understanding these payoffs is essential for determining option valuation.

Discounting the expected payoff back to the present value is a fundamental aspect of pricing options, reflecting the time value of money. We calculate the present value to understand what future gains are worth today. The formula is: \[ ext{Present Value} = \frac{ ext{Expected Payoff}}{e^{rT}}\] We use the expected payoff previously calculated, which here was 2.268. This is adjusted by the factor \(e^{0.10 \times \frac{2}{12}}\), leading to discounting the future payoff by about 1.0167. The result, approximately 2.230, is thus the present value of the expected future cash flows from the option. Accurate discounting is key in financial calculations because it prevents overvaluation of future cash flows.

The principle of no-arbitrage is a cornerstone of modern financial theory. It mandates that there be no way to make a riskless profit instantly and requires that the price of a derivative, like our call option, must be set where arbitrage opportunities do not exist. Using the present value of the expected payoff (2.230), the no-arbitrage price means that if the call option is valued at this amount, investors can't exploit price differences for guaranteed gains without risk. The calculated present value in our example thus ensures that the market price aligns with theoretical predictions, allowing neither undervaluation nor overvaluation under current conditions. It's this alignment that helps traders make informed decisions and keeps the markets fair.