BU7508: Explain why the futures price 𝐹𝑡 of stock with price 𝑆𝑡 satisfies: Derivatives Assignment, TCD, Ireland

Question 1
a) Explain why the futures price 𝐹𝑡 of stock with price 𝑆𝑡 satisfies 𝐹𝑡 = 𝑆𝑡𝑒 𝑟(𝑇−𝑡) where r is the risk-free rate, and T is the maturing date. Can you use the same arguments to price all other types of futures contracts? Furthermore, if the stock price 𝑆𝑡 follows a Geometric Brownian Motion, what is the process followed by the futures price 𝐹𝑡 Interpret your result.

b) Demonstrate the manner in which the Black-Scholes model is adapted to accommodate European futures options, i.e. the Black’s model for futures options.

c) Explain the exponentially weighted moving average (EWMA) model for estimating volatility from historical data. Explain how to apply the GARCH methodology to derive and forecast an asset’s volatility. Demonstrate how to apply the two methods to estimate the volatility of a chosen stock and critically discuss your results.

d) Discuss an exotic option that was not covered in our class. Give the definition, valuation method, pricing formula, and properties that are different from plain vanilla options.

Question 2 
a) With reference to the Black Scholes model, explain the concept of risk-neutral valuation. Outline the Monte Carlo valuation procedures. Use the Monte Carlo method to price an option of your own choice, compare the obtained price with the market price, and discuss your results.

b) Define Value at Risk. Critically discuss the advantages and disadvantages of Value at Risk as a measure of risk. You must choose TWO companies listed on ISEQ, New York Stock Exchange, NASDAQ, London Stock Exchange, or the Exchange of the country where you come from, say Stock A and B. Download a minimum of 61 days of price data of the two companies A and Bending February 2021. (1) What is the 99%, 5-day VaR for a 1 million dollar investment in stock A? (2) What is the 99%, 5- day VaR for a 1 million dollar investment in stock B? (3) What is the 99%, 5-day VaR for a 1 million dollar investment in stock A and 1 million dollar investment in stock B? (4) What is the benefit of diversification for the 99% VaR?

c) On 20 April 2020, the price of one American oil futures contract plunged to be negative for the first time in history. Explain the reasons behind this and critically discuss its implications on risk management and investment.

d) Within the Black-Scholes framework, is the following a possible value for a derivative security D(t) on the underlying asset S(t)? For all t≥0,𝐷(𝑡) = 𝑆(𝑡)−2𝑟/𝜎2

Question 3 
a) Outline the mean-variance approach to hedge ratio construction. What are the speculative demand and the hedging demand? Interpret them economically. Compare and contrast it with the minimum variance hedge ratio, and critically discuss its advantages and disadvantages, and its
possible extension.

b) What are volatility smiles, describe the key features of volatility smiles, and
why do you think they exist?

c) What is the implied volatility? Critically review the recent literature on the estimation of implied volatility. Explain how the Bisection OR the Newton-Raphson procedure is utilized to calculate the implied volatility of options priced according to the Black-Scholes model. Use market data from an option to calculate its implied volatility using the Bisection OR the Newton-Raphson method.

d) What is credit risk? Explain the risk-neutral and real-world default probabilities and the difference between them. Which should be used for (i) valuation and (ii) scenario analysis? How are recovery rates usually defined and how is the recovery rate used to approximately calculate default probability?