Fin 500Q – Quantitative Risk Management Homework #5
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Fin 500Q – Quantitative Risk Management
Homework #5 Solutions
1. Suppose a hedger is short one unit of the stock A with price hA = 100, and plans to close this short position in one month. She attempts to hedge the risk in this stock by trading another stock B with price hB = 200. Assume that the standard deviations of monthly stock returns are 7A = 0.3 and
7B = 0.56, and the correlation between the returns is 0AB = -0.7.
(a) Find the optimal hedge ratio that she should use.
Answer: Suppose the hedger invests in stock B with a hedge ratio of o. Then her optimal hedge ratio is o = 0 . 7A /7B , which equals (-0.7) . 0.3/0.56 = -0.375.
(b) Find the optimal number of units of stocks B she should buy or sell.
Answer: Since she currently has a short position in stock A and o is negative, she should sell stock B for hedging purposes. The hedge ratio tells us the dollar investment in stock B for each dollar in asset A. The exposure to stock B should therefore be -hA . (-o) = -37.5 dollars. To find the optimal number of units zB of stock B, we compute zB = o .hA /hB = -0.375 . 100/200 = -0.1875. She should sell 0.1875 units of stock B .
(c) What is the variance of her position in one month? Answer: The variance of her position is
u P(2) = (-1)2 . hA(2) . 7A(2) + zB(2) . hB(2) . 7B(2) + 2 . 0 . (-1) . zB . 7A . hA . 7B . hB ,
which is 459.
2. Suppose that a firm will sell one unit of output in three months, accepting payment in the cryptocur- rency Ethereum (ETH) at a price of 100 ETH. To hedge risk in the ETH price in dollars, the firm can trade a three-month futures contract on the Bitcoin (BTC) price in dollars. Each contract is on
5 Bitcoins. At this horizon, the volatility of changes in the ETH price is 1 000 and the volatility of changes in the BTC price is 12 500. The correlation between the price changes is 0 = 0.9.
(a) What is the optimal hedge ratio that the firm should use?
Answer: We compute the hedge ratio that gives the number of BTC futures contracts per unit of ETH exposure. Since the futures contract is on 5 Bitcoins, the optimal hedge ratio is
o = 0.9 . = 0.0144.
(b) Find the optimal number of futures contracts that the firm should buy or sell.
Answer: Since the firm is long in ETH, it should sell futures contracts for hedging purposes.
The optimal number of futures contracts that the firm should sell is
100 . o = 1.44.
2023-05-02