Problem 2: Hedging with options

Today is the 13th of December. You plan on setting up a manufacturing firm in three months time. You need a large one-off investment in gold to set up your firm. After finalizing your business plan you realise that you may not be able to start your business if the price of gold increases too much beyond the current price. You would like to hedge this risk using a call option contract. Please refer to the data in the sheet titled ‘Problem 2’ in the .xls file to answer parts of this question that require calculation. There are two return series ‘r1’ and ‘r2’, as well as the price series associated with these returns, ‘S1’ and ‘S2’ respectively.

Question 2.1

a) Value the option using the Black-Scholes model you learned in class. You can assume that the risk free interest rate is 1% annualised, the expiration is in 93 days, the daily volatility is given in cell I7 in the excel file in the sheet ‘Problem 2’, and the option is at-the-money with a current gold price of 100.

b) What challenges could you face trying to use this hedging method?

Question 2.2

Instead of purchasing the option you can engage in synthetic replication, i.e. create a position in the underlying asset that mimics the payoff of the option. The data series ‘r1’ and ‘S1’ represents possible future outcomes of the returns and prices of underlying asset.

a) Show the outcome of synthetically creating the option with different rebalancing intervals (e.g. 10 days and 1 day) and comment on your results.

b) Explain any difference between the price of the synthetic hedge portfolio and the price of the option calculated in 2.1.

c) Give some reasons why you may prefer to buy an option rather than trying to synthetically replicate it.

Question 2.3

You decided to buy call options from a bank. It turned out that the future realisation of the process was not driven by the process assumed in the Black-Scholes model (the process in ‘r1’ and ‘S1’). Due to political events the true process is given in the return series ‘r2’ and associated price series ‘S2’. When you answer this question please ensure that the value in cell I10 is greater than 0 (you can refresh the sheet by pressing the F9 key).

a) Please repeat the synthetic replication of the option from 2.2 and comment on the results.

b) What assumption of the Black-Scholes model is violated?

c) Explain carefully why this is an important violation and how it impacts your hedging approach.

d) How would the bank act differently in the case of the price process being like ‘r2’ compared to ‘r1’.

e) Describe how the bank would implement a gamma hedge in this situation. Would it help with their hedging? Why/Why not?

Question 2.4

You would now like to understand the differences between the two series by doing some empirical analysis.

a) Graph the distribution of returns from ‘r1’ and ‘r2’, as well as a time series graph of some realised price paths from the processes.

b) In 2.3 (b) you explained how an assumption of the Black-Scholes model is violated. Can you give a quantitative measure of the violation?

c) How do you expect a graph of option implied volatility on the underlying asset as a function of strike price to be shaped in a market where returns follow ‘r1’ compared to a market where returns follow the process given in ‘r2’. Explain your answer.

Question 2.5

a) If the market was valuing options under the assumption of S1, but you had information that the true process in the future would be S2, what would be your trading strategy if you wanted to profit from the information?

b) Show some analysis of the expected profit from your strategy in a)

c) What would be the effect of many people learning about this discrepancy?

d) What about the opposite situation, where the rest of the market believes that prices follow the process in S2 but the true process is that in S1, what would be your strategy? Comment on your ability to profit from the strategy.

Question 2.6

a) What is the formula for the price of an option that pays $1 if the stock price is greater than some fixed value K at time T and zero otherwise? (note: you can use the Black-Scholes assumptions)

b) What is the price of the option in part a) you can assume that the risk free interest rate is 1% annualised, the expiration is in 93 days, the daily volatility is given in cell I7 in the excel file in the sheet ‘Problem 2’, K=100, and the current gold price is 100.

c) Do you expect the price to be higher or lower if you assume the underlying process has characteristics like series ‘S2’ instead of the Black-Scholes assumption (like the series S1)?