Vitalik Buterin, the co-founder and inventor of the decentralized open-source blockchain Etherium, 3 currently has a portfolio of cryptocurrency worth USD800 million (including about 350,000 Ether, or ETH, the cryptocurrency generated by the Ethereum protocol). The portfolio also has an income yield of 0.5% per annum with simple compounding. Vitalik has seen the value of his portfolio significantly decline over the first half of 2022 , due in large part to the latest 'crypto winter'. 4 He expects the bear market to end before the year is out, but he is worried that there will be further falls before then. As such, he has approached your institution for advice on how best to protect the value of his existing crypto portfolio over the next 6 months. He also does not want to miss out on any potential upside if the crypto market starts to rise sooner than he expects. You know just the derivative for the job and suggest that a position in Ether (ETH) options will provide the desired protection. Since Vitalik's portfolio is highly correlated with Ether prices, options on Ether can be used as a portfolio hedge (in the same way that stock index options can be used to hedge an equity portfolio). To this end, you estimate from historical price data that the 'Ether beta'5 of Vitalik's portfolio is 1.4. You also note that Ether does not pay any income (and hence has a zero income yield) and that the risk-free interest rate in the US is currently 3.1% per annum with simple compounding for all maturities. The current spot price of Ether is USD1,485 and the following table provides the market prices (in USD) for various European call and put options written on Ether with different strikes and (a) Describe the options portfolio insurance strategy that would insure against Vitalik's cryptocurrency portfolio falling below USDS00 million over the next six months. 6 Explain why this strategy fulfils his request and why hedging with Ether futures does not suffice. Also, highlight some of the potential downsides of the strategy if implemented in practice. (b) Calculate the gains/losses on the strategy if the price of Ether in six months is either (i) $500, or (ii) $3,000, and prepare a short summary for your Vitalik to discuss the outcome of the insurance strategy in these two scenarios. Note that here you should include the cost of the options purchase (inferred from the table above) in your calculation of the P\&L in each scenario. Ignoring the time value of money, what value of Ether in six months would the portfolio, including the option costs, breakeven (i.e., be worth USD800 million)? Vitalik is somewhat interested in the proposed strategy, but he is surprised by how expensive the options are. You state that this is due to the very high volatility in ETH and the wider crypto market. To explain things further you decide to investigate the implied volatility of ETH using the option prices given above. (c) Use Excel's GoalSeek (or otherwise) to estimate the implied volatility of the spot price of ETH, based on the market prices of the ETH options above. Specifically, complete the following table with the estimated implied volatilities (to 3 significant figures): You should also perform the following tasks and answer the following questions: i. Plot the implied volatility from each option type (put/call) as a function of the strike. ii. Is the implied volatility of one option class higher than the other? If so, explain why. iii. Does the implied volatility depend on the moneyness of the option? Is so, explain. iv. What do your results tell you about the observed market prices and their consistency with the assumptions underlying the Black-Scholes model? 6 Note that the USDSOO million value should not include the cost of the options used for insurance and each option contract is written on exactly one unit of ETH. Vitalik Buterin, the co-founder and inventor of the decentralized open-source blockchain Etherium, 3 currently has a portfolio of cryptocurrency worth USD800 million (including about 350,000 Ether, or ETH, the cryptocurrency generated by the Ethereum protocol). The portfolio also has an income yield of 0.5% per annum with simple compounding. Vitalik has seen the value of his portfolio significantly decline over the first half of 2022 , due in large part to the latest 'crypto winter'. 4 He expects the bear market to end before the year is out, but he is worried that there will be further falls before then. As such, he has approached your institution for advice on how best to protect the value of his existing crypto portfolio over the next 6 months. He also does not want to miss out on any potential upside if the crypto market starts to rise sooner than he expects. You know just the derivative for the job and suggest that a position in Ether (ETH) options will provide the desired protection. Since Vitalik's portfolio is highly correlated with Ether prices, options on Ether can be used as a portfolio hedge (in the same way that stock index options can be used to hedge an equity portfolio). To this end, you estimate from historical price data that the 'Ether beta'5 of Vitalik's portfolio is 1.4. You also note that Ether does not pay any income (and hence has a zero income yield) and that the risk-free interest rate in the US is currently 3.1% per annum with simple compounding for all maturities. The current spot price of Ether is USD1,485 and the following table provides the market prices (in USD) for various European call and put options written on Ether with different strikes and (a) Describe the options portfolio insurance strategy that would insure against Vitalik's cryptocurrency portfolio falling below USDS00 million over the next six months. 6 Explain why this strategy fulfils his request and why hedging with Ether futures does not suffice. Also, highlight some of the potential downsides of the strategy if implemented in practice. (b) Calculate the gains/losses on the strategy if the price of Ether in six months is either (i) $500, or (ii) $3,000, and prepare a short summary for your Vitalik to discuss the outcome of the insurance strategy in these two scenarios. Note that here you should include the cost of the options purchase (inferred from the table above) in your calculation of the P\&L in each scenario. Ignoring the time value of money, what value of Ether in six months would the portfolio, including the option costs, breakeven (i.e., be worth USD800 million)? Vitalik is somewhat interested in the proposed strategy, but he is surprised by how expensive the options are. You state that this is due to the very high volatility in ETH and the wider crypto market. To explain things further you decide to investigate the implied volatility of ETH using the option prices given above. (c) Use Excel's GoalSeek (or otherwise) to estimate the implied volatility of the spot price of ETH, based on the market prices of the ETH options above. Specifically, complete the following table with the estimated implied volatilities (to 3 significant figures): You should also perform the following tasks and answer the following questions: i. Plot the implied volatility from each option type (put/call) as a function of the strike. ii. Is the implied volatility of one option class higher than the other? If so, explain why. iii. Does the implied volatility depend on the moneyness of the option? Is so, explain. iv. What do your results tell you about the observed market prices and their consistency with the assumptions underlying the Black-Scholes model? 6 Note that the USDSOO million value should not include the cost of the options used for insurance and each option contract is written on exactly one unit of ETH