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Derivatives Question and Answers

Autor:   •  November 22, 2017  •  6,041 Words (25 Pages)  •  763 Views

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- Consider a second 3-month European put option on the same stock, with strike price equal to $56. Compute the option’s corresponding option premium, the delta, and the gamma. How can you use the put option, the call option from part (a) of this question, together with the stock, to gamma and delta neutralize the portfolio? Please indicate clearly whether you go long or short in the stock/options. What is the value of your total portfolio? Hint: You can normalize your portfolio to have 1 share of the stock, and solve for the positions in the call and put options. Note that non-integer units of the option are allowed to solve this exercise.

- You have delta-gamma hedged your portfolio (in part (d) of this question), and you areaway on sick leave. After two months, you come back to your trading desk, and see that the stock price has moved up to $58 (while all other risk factors remain unchanged). Recalculate the value of your delta-gamma hedged portfolio from (d), assuming that you have not rebalanced your portfolio holdings or your hedge. Quantify the change in portfolio value and explain the reasoning behind it.

- What is the approximated price change of the call option over one week when volatility increases by 2 percentage points, and the stock increases by $1.5?

Responses

- Hedging with Futures

(a) The markets for oil, natural gas, and many other commodities are characterized by high levels of volatility. Prices and inventory levels fluctuate considerably from week to week, in part predictably (e.g., due to seasonal shifts in demand) and in part unpredictably. Furthermore, levels of volatility themselves vary over time. In markets for storable commodities, such as oil, inventories play a crucial role in price formation. As in manufacturing industries, inventories are used to reduce the costs of changing production in response to demand fluctuations (whether predictable or not), and to reduce marketing costs by helping to ensure timely deliveries and by avoiding running out of stock. Producers must determine their production levels jointly with their expected inventory draw-downs or buildups. These decisions are typically made in light of two prices: the spot price for sale of the commodity itself, and the cost of storage. At any instant of time, the supply of storage is simply the total quantity of inventories held by producers, consumers, or third parties. In equilibrium, this quantity must equal the quantity demanded, which, similar to any good, is a function of the price. The price of storage is the “payment” by inventory holders for the privilege of holding a unit of inventory. As with any good or service sold in a competitive market, if the price lies on the demand curve, it is equal to the marginal value of the good or service, i.e., the utility from consuming a marginal unit.

In the case of commodity storage, this marginal value is the value of the flow of services accruing from holding the marginal unit of inventory; it is what we usually refer to as marginal convenience yield. Expected depreciation (and thus the price of storage) will be particularly high following periods of supply disruptions or unusual (but temporary increases in demand. We can measure expected depreciation, for example using futures prices. The marginal value of storage is likely to be small when the total stock of inventories is large (because one more unit of inventory will be of little extra benefit), but it can rise sharply when the stock becomes very small. In normal times, for an extractive resource commodity like crude oil, we would expect the futures market to exhibit weak or strong backwardation most of the time, since owning in-ground reserves is equivalent to owning a call option with an exercise price equal to the extraction cost, and with a payoff equal to the spot price of the commodity. If there were no backwardation, producers would have no incentive to exercise this option, and there would be no production. If spot price volatility is high, the option to extract and sell the commodity becomes even more valuable, so that production is likely to require strong backwardation in the futures market.

The convenience yield reflects the market’s expectations concerning the future availability of the commodity. As we are expecting a low chance of shortage in the upcoming years due to the increased global supply of oil in the market, the convenience yield is expected to be low. By looking at the data and assuming a low convenience yield, we observe prices for near maturities to be lower than the distant maturities in most cases. This will lead to an overall contango as the storage costs exceed the convenience yield. The contango behaviour in the crude oil market is consistent among most observed data points in 2015 with the exception of January, February, March, October, and November, when there exists a weak short-term backwardation when the spot price is slightly higher than its nearest futures contract maturity only. (b)

Ft = Ste(r+u−y)T

[pic 1]87%

- The Futures price is too low. It should be:

[pic 2]

In this case, you can arbitrage the market by shorting the oil spot. By doing so you will lock in the profit of 46.2457 - 45.81 = $0.4375 per barrel.

- When expecting a shortage in supply of oil, market prices will go up because of the expected decrease in supply. The exposure to oil price risk can be hedged by taking a long position in the futures contract. Note that this strategy assumes that this information is no common knowledge in the financial markets, otherwise this input would already be reflected in the market’s expectation. You need to buy 3,500,000/(54.38*1000) = 64 contracts.

- Both the initial and maintenance margins are set at $4,250. The total required initialmargin would be 64*4,250= $272,000. The maintenance margin is set to the same amount.

- Duration Hedging with Futures

- CPPIB is currently underfunded given that the present value of assets is below thepresent value of liabilities. The funding ratio is given by the ratio of assets to liabilities, which is equal to 1,000/1,050 = 95.24%. Thus, CPPIB is underfunded by approximately 5%.

- The total duration of the investment portfolio will be given by the weighted averageduration of all positions in the portfolio, i.e. Dp = we × E + wFI × FI = 0.6 × 600 + 0.4 × 400 = 2. Thus, the investment portfolio has a duration of two years.

- If equity markets drop independently by 10%, this implies an investment loss

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