Explain the no-arbitrage valuation approach to valuing European options in a one-period binomial tree setting. Discuss how the no-arbitrage valuation approach is evolved into the risk-neutral valuation in the one-period setting and how the risk-neutral valuation is extended to a multi-period setting.

How does the synthetic portfolio insurance approach work? What is the theoretical justification? How is the approached evolved to institutional portfolio managers resorting to stock index futures contracts instead of selling the underlying portfolio? Does this approach work? Please carefully explain your answers.

What is the implied volatility and how is it calculated? What is the historical volatility and how is it estimated? When do you use the implied volatility instead of the estimated historical volatility?

What is the reason to create weather derivatives? What are the underlying variables used for temperature-related weather derivatives?