**Question 1: Option Pricing and MGRM (35 pts total)**

MGRM created a market by selling long-term forward contracts. Assuming the prices for oil given the first traded day in 2000, MGRM entered into a deal to sell yearly forward contracts for delivery to an unspecified counterparty over the next 10 years (100 contracts for each year). Instead of hedging with the stack-and-roll strategy using futures contracts, MGRM hedged used 1-year ATM call contracts.

a.) Use the risk-free rate at the time and value the options using a 12-step American binomial valuation (since the clients could take delivery of the oil early), find the total cost of the hedging strategy (the sum of all the option contracts) using both a stack-and roll approach and the matched strip approach assuming zero storage cost and a 1% convenience yield? (**15 points**)

b.) Graph the PnL of the hedge performance of the two strategies, including costs, through time. (5 points)

c.) If you used a closed form solution versus the binomial tree approach, how different would the cost be (just do the matched-strip)? Briefly explain why? (5 points)

d.)Now assume that instead of using traditional call options MGRM used a European ATM knock-out down for the stack and roll (don’t do the matched strip), where the knock-out price at initiation is 10% below the spot price. How costly is this strategy (Hint: You can use any technique you like to price the options)? (5 points)

e.)What are the drawbacks of using the knock-out option for MGRM? (5 points)

**Question 2: Risk Management and Barings (30 pts total)**

Assume Nick Lesson, on December 15th, 1994, knowing that he was in trouble, and having perfect foresight, instead of doubling down and selling options, devised a strategy to offset his long exposure of 20,000 futures contracts…He was already down 208mm pounds and wanted to break even by the end of Q1 1995.

a.) Come up with a strategy that would have allowed him to be back to zero by end of Q1. (**15 points**)

-Calculate the cost (or collect) of entering the strategy

-The effect on margin (if any)

-Number of contracts

-Graph the PnL of the Perfect Foresight strategy

b.) Let’s say Nick Leeson had employed a different index arbitrage strategy. Using the historical prices from the Hang Seng and the Nikkei 225, use a one week return convergence rule to try profit from return deviations. Use the Jan 1996 to Dec 2005 period as the estimation window for the daily mean and standard deviation calculations, and use Jan 2007 through Dec 2016 for the investment period analysis (use overlapping windows). Invest when the daily return discrepancies are 1 standard deviation away from the mean.

The investment will be:

When Hang Seng Return-Nikkei 225 Return>1 stdev from mean

Hang Seng =Protective Put

Nikkei =Sell a Put/ Short Index

Hang Seng Return-Nikkei 225 Return<1 stdev from mean

Hang Seng = Sell a Put/ Short Index

Nikkei = Protective Put

Assume each time is an investment opportunity there is an ATM option with 22-days to maturity and use the EWMA volatility. Also assume a RFR of 3% for all options.

Your initial portfolio is $1mm. Each option investment will invest in 1 combined position (size will be 100). The rest of the capital earns interest at the RFR. Calculate your final portfolio value on Dec 30th 2016 and graph the time-series of the portfolio value. (10 points)

c.) How would Nick Leeson have reported the Quarterly PnL of this strategy given the Baring’s setup in Singapore (5 points)?

**Question 3: Volatility, Hedging, and LTCM (35 pts total)**

When the bond-arbitrage strategy dried up, one of the strategies LTCM employed was volatility arbitrage. Let us examine how profitable this is, with a twist. Calculate the GARCH (2,1) volatility of the S&P 500 from 1999 through April 2019. Calculate the daily differences between this series and the VIX index (VIX-GARCH). Now calculate 1 day ahead S&P 500 returns

a.)Form 10-volatility difference baskets (highest to lowest) from the 1999 to 2019 estimation sample, and report what the annualized average return and standard deviation is for each basket. Also report the GARCH parameters. (10 points)

b.)Assume you go 2X long the S&P 500 in 2 highest baskets, short the S&P 500 in the 2 lowest, and are in the S&P 500 the rest. Use the average return and volatility from this strategy, then simulate 5 yearly return paths starting with a price of 100 using Monte-Carlo simulation with daily frequency, and using the Black-Scholes PDE. Report the mean and stdev from the estimation period, the ten-day 99% C-VAR (% not $) based on historical simulation using the 1990-2019 data, and the final 5 prices you get from the MC (10 points)

c.)Does this seem like a good strategy to employ? Explain your answer assuming LTCM leverage (5 points)

d.)Assume that there was an ETF built on this product that is valued at 100 starting 2007. Calculate daily rolling delta and vega hedge ratios assuming you sell an ATM put on the ETF with a strike price of 95, a maturity of three years at initiation and 1% risk free rate. Report the initial put price and the

-delta and vega at t=0

-delta and vega at t=.5

-delta and vega at t=.75

-delta and vega at t=1

-delta and vega at t=1.25

-delta and vega at t=1.5

-delta and vega at t=1.75

-delta and vega at t=2

-delta and vega at t=2.25

-delta and vega at t=2.5

-delta and vega at t=2.75

-delta and vega at t=3

Now explain how leverage could kill a product like this (even though it has historically outperformed the S&P 500) (10 points)