Question $2$

In your role as a Junior Derivatives Analyst within the Derivatives Trading department at Celestial Investments, you answer to Martin ONeill, CFA. Martin has assigned you the task of creating a concise report comparing various options trading strategies. A critical aspect of Jordan's directive is a thorough examination of the costs and potential payoffs associated with each strategy. For this assignment, you have the discretion to employ both the Black$-$Scholes$-$Merton $($BSM$)$ model and the Binomial option pricing model for valuing European puts and calls. The main focus is on evaluating European calls and puts for shares of BP plc$,$ actively traded on the Eurex or ICE. These calculations should essentially demonstrate derivative pricing using the No Arbitrage Principle. To make progress in this assignment, careful consideration in selecting exercise prices, expirations, the risk$-$free rate, and an appropriate measure of volatility is paramount for pricing these options. BSM and Binomial option pricing model aims to provide a comprehensive exploration of option trading strategies, meeting Martin ONeill' expectations and contributing to the overarching objectives of Celestial Investments. Required:

I$)$ Employ both the Black$-$Scholes$-$Merton $($BSM$)$ model and the two$-$period Binomial option pricing model to estimate the prices of European put and call options associated with BP plc shares. Choose two distinct expiration dates and two exercise prices for each model, then compute the estimated prices for these options. Clearly articulate the fundamental assumptions underlying each model's evaluation, providing reasoned justifications for each input.

II$)$ Illustrate the fairness of the estimated call price in the two$-$period binomial model by employing the hedge portfolio calculation over the two periods and adjusting the hedge ratio accordingly. Note: Choose only one option contract from the Q$($I$).$ III$)$ Conduct a thorough discussion comparing historical volatility to implied volatility within the framework of option pricing. Additionally, calculate the implied volatility specific to BP$\text{'}$s options as outlined in the initial requirement.

IV$)$ Provide an overview of the Strangle $($bottom vertical combination$)$ strategy and illustrate its application using options on BPs shares. Additionally, analyse the performance of the Strangle strategy with various spot prices at expiration for BP$,$ determining the Maximum and Minimum profit and Breakeven price.

V$)$ Evaluate the effect of different holding periods for the Strangle constructed from options on BP$.$