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No-arbitrage pricing is a fundamental concept in finance that assures fairness in the market by preventing any free monetary profit from discrepancies. In the case of the European put option, no-arbitrage arguments ensure that the price of the option we calculate aligns with current market conditions, allowing us to determine a value that doesn’t permit riskless profits through arbitrage. By creating an hypothetical scenario where future payoffs are equated to today's prices using certain probabilities, we align the future expected value with the present one, under risk-neutral circumstances. This process ensures that market efficiency is maintained, as any deviation from this computed price would allow easy profit through arbitrage, thus, correcting the mispricing.

The concept of a risk-neutral measure is pivotal when pricing options in the theoretical landscape. It refers to an imaginary world where investors are indifferent to risk. Here, the expected return on all securities is the risk-free rate, making future cash flows simpler to calculate, as we can maintain a neutral stance on risks associated. This assumption simplifies complex probabilities into easier computations, allowing us to focus on discounted expected payoffs. For the European put option, this means we use probabilities based on future price outcomes that establish equilibrium. Hence, we calculated a 50% probability for up and down price scenarios, leading to a straightforward evaluation of expected payoffs.

Continuous compounding is an interest calculation method where interest is computed and added continuously, resulting in a more frequent accumulation of interest. This method is often used in finance to simplify complex calculations and yield precise results. It is widely seen in option pricing as it allows a smooth adjustment to time-dependent factors. In the case of the European put option, the risk-free rate applied continuously facilitates adjusting the value of expected future payoffs into today’s dollars accurately. The formula used, \[ e^{-rt} \] allows the conversion of the expected payoff from the risk-neutral scenario into its present value, which then reflects the true cost of securing the put option today.

Option payoffs for a European put option are straightforward to calculate once future scenarios are understood. A put option gives the holder the right, but not the obligation, to sell an asset at a predetermined strike price at option expiry. For our example, with a strike price of $80 and possible future stock prices of either $75 or $85, the payoff structure is:

• If the stock price drops to $75, the payoff is the excess of the strike price over the stock price, resulting in a $5 payoff.
• If the stock price rises to $85, the payoff is zero, as selling at a lower strike price doesn't benefit the holder.

These possible payoffs are then weighted by their probabilities to find an expected payoff, considering the no-arbitrage principle to ascertain the net present value located in a risk-neutral framework.