Derivatives Flashcards
Why index arbitrage can be difficult to implement?
Selling or purchasing shares in all 500 stocks in the S&P 500 is impractical for two reasons.
- Transaction costs which may outweigh any profits to be made from the arbitrage.
2 . Any lags in the execution of such a strategy can destroy the effectiveness of a plan.
Index arbitrage
Whenever the actual futures price differs from its parity value, there is an opportunity for profit.
If the futures price is too HIGHT:
SHORT the FUTURES contract and BUY the STOCK in the index.
If it is too LOW: BUY FUTURES and SHORT the STOCK.
Explain Hedge Ratio
The hedge ratio is the number of futures contracts necessary to hedge the risk of the unprotected portfolio.
In general, we can think of the hedge ratio as the number of hedging vehicles (e.g., futures contracts) one would purchase to offset the risk of a particular unprotected position.
H = Change in value of unprotected position for a given change in exchange rate / Profit derived from one futures position for the same change in exchange rate (futures price X multiplier)
Delta
Delta (Δ): measures change in option value with respect to changes in the underlying asset price
For every option, we should buy Delta shares to hedge against the price risk of holding one option
Delta call = N(d1)
Delta put = N(d1) - 1
If the futures price is too high…
SHORT the FUTURES contract and BUY the STOCK in the index.
Explain the workings of a credit default swap (CDS). Apply the following terminology in your answer: probability of default, recovery rate, and spread.
A CDS is basically an insurance contract against the default of an entity.
In the case of default (credit event), the CDS seller will pay the difference between the notional value and the recovery rate.
The higher the probability of default by the reference entity, the higher the spread of the CDS contract.
Why does a speculator buy a futures contract and not directly the underlying asset?
- Transaction costs, which are far smaller in futures markets.
- Leverage that it provides. Required margins are considerably less than the value of the contract. Therefore, they allow speculators to achieve much greater leverage than is available from direct trading in a commodity.
You manage a $23 million portfolio, currently all invested in equities, and believe that the market is on the verge of a big but short-lived downturn. You would move your portfolio temporarily into T-bills, but you do not want to incur the transaction costs of liquidating and reestablishing your equity position. Instead, you decide to temporarily hedge your equity holdings with E-mini
S&P 500 index futures contracts.
a. Should you be long or short the contracts? Why?
b. If your equity holdings are invested in a market-index fund, into how many contracts should you enter? The S&P 500 index is now at 2,300 and the contract multiplier is $50.
c. How does your answer to part (b) change if the beta of your portfolio is .6?
a. You should be short the index futures contracts. If the stock value falls, you need futures profits to offset the loss.
b. Each contract is for $50 times the index, currently valued at 1,950. Therefore, each contract controls stock worth: $50 * 2,300 = $115,000
In order to hedge a $23 million portfolio, you need:
$ 23,000,000 / $ 115,000 = 200 contracts
c. Now, your stock swings only 0.6 as much as the market index. Hence, you need 0.6 as many contracts as in (b):
0.6 * 200 = 120 contracts
A manager is holding a $1 million stock portfolio with a beta of 1.25. She would like to hedge the risk of the portfolio using the S&P 500 stock index futures contract. How many dollars’ worth of the index should she sell in the futures market to minimize the volatility of her position?
If the beta of the portfolio were 1.0, she would sell $1 million of the index. Because it is 1.25, she should sell $1,000,000 * 1.12 = $1.25 million of the index.
Yields on short-term bonds tend to be more volatile than yields on long-term bonds. Suppose that you have estimated that the yield on 20-year bonds changes by 10 basis points for every 15-basis-point move in the yield on 5-year bonds. You hold a $1 million portfolio of 5-year
maturity bonds with a modified duration of 4 years and desire to hedge your interest rate exposure with T-bond futures, which currently have modified duration 9 years and sell at F0 = $95. How many futures contracts should you sell?
Suppose the yield on your portfolio increases by 15 basis points. Then the yield on the T-bond contract is likely to increase by 10 basis points.
The loss on your portfolio will be: $1 million x Δy x D* = $1,000,000 x 0.00015 x 4 = $600
The change in the futures price (per $100 par value) will be: $95 x 0.0001 x 9 = $0.0855
This is a change of $85.50 on a $100,000 par value contract.
Therefore you should sell: $600 / $85.50 = 7 contracts
A manager is holding a $1 million bond portfolio with a modified duration of 8 years. She would like to hedge the risk of the portfolio by short-selling Treasury bonds. The modified duration of T-bonds is 10 years. How many dollars’ worth of T-bonds should she sell to minimize the variance of her position?
She must sell: $1million x 8 / 7 = 0.8 million of T-bonds
Firm ABC enters a 5-year swap with firm XYZ to pay LIBOR in return for a fixed 6% rate on notional principal of $10 million. Two years from now, the market rate on 3-year swaps is LIBOR for 5%; at this time, firm XYZ goes bankrupt and defaults on its swap obligation.
a. Why is firm ABC harmed by the default?
b. What is the market value of the loss incurred by ABC
as a result of the default?
c. Suppose instead that ABC had gone bankrupt. How do you think the swap would be treated in the reorganization of the firm?
a. The swap rate moved in favour of firm ABC. ABC should have received 1% more per year than it could receive in the current swap market. Based on a notional principal of $10 million, the loss is 0.01 x $10 million = $100,000 per year.
b. The market value of the fixed annual loss is obtained by discounting at the current 5% rate on three-year swaps. The loss is $100,000 x Annuity factor (5%, 3) = $272,325
(PMT = $100,000; N = 3; I = 5; FV = $0; Solve for PV = $272,325)
c. If ABC had become insolvent, XYZ would not have been harmed. XYZ would be happy to see the swap agreement canceled. However, the swap agreement ought to be treated as an asset of ABC when the firm is reorganized.
A corporation plans to issue $10 million of 10-year bonds in three months. At current yields, the bonds would have a modified duration of 8 years. The T-note futures contract is selling at F0 = 100 and has a modified duration of 6 years. How can the firm use this futures contract to hedge the risk surrounding the yield at which it will be able to sell its bonds? Both the bond and the
contract are at par value.
If yield changes on the bond and the contracts are each 1 basis point, then the bond value will change by $10,000,000 x 0.0001 x 8 = $8,000
The contract will result in a cash flow of $100,000 x 0.0001 x 6 = $60
Therefore, the firm should sell: 8,000 / 60 = 133 contracts
The firm sells the contracts because you need profits on the contract to offset losses as a bond issuer if interest rates increase
The higher the probability of default by the reference entity, the (…) the spread of the CDS contract.
HIGHER
Go through the steps of a protective put strategy. How much should be placed in equity how much in T-bills?
What will happen if your portfolio falls 2%?
- Calculate d1, d2 and find N(d1) , N(d2)
- Calculate the Put price from formula
𝑃 = 𝑋𝑒−𝑟𝑇[1 − 𝑁(𝑑2)] − 𝑆0[1 − 𝑁(𝑑1)] - Calculate the total funds managed: portfolio in mill + put price * 10^6
- place portfolio * N(d1) - 1 % in equity and the rest in t bills
If the portfolio falls 2% the new portfolio value is port. value * (1 - 0.02%)
calculate the new d1 , N(d1) etch….
