Financial Engineering and Risk ManagementFutures

Martin Haugh Garud IyengarColumbia University

Industrial Engineering and Operations Research

Problems with forward contractsNot organized through an exchange.Consequently, no price transparency!Double-coincidence-of-wants: need someone to take the opposite side!Default risk of the counterparty.

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Futures contractSolves the problem of a multitude of prices for the same maturity bymarking-to-market

disbursing profits/losses at the end of each day

Now contracts can be organized through an exchange.Can be written on any underlying security with a settlement price

CommoditiesBroad based indicies, e.g. S & P 500, Russel 2000, etc.Volatility of the market, e.g. VIX futures

http://www.cmegroup.com/market-data/delayed-quotes/commodities.html

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Mechanics of a futures contractIndividuals open a margin account with a brokerEnter into N futures contracts with price F0Deposit initial margin into the account 5 10% of contract valueAll profit/loss settled using margin accountMargin call if balance is low

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Pros/cons of futuresPros:

High leverage: high profitVery liquidCan be written on a wide variety of underlying assets

Cons:High leverage: high riskFutures prices are approximately linear function of the underlying onlylinear payoffs can be hedgedMay not be flexible enough; back to Forwards!

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Pricing futuresNeed martingale pricing formalismDeterministic interest rates: forward price = futures priceAt maturity futures price FT = price of underlying ST

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Hedging using Futures: Long hedgeToday is Sept. 1st. A baker needs 500, 000 bushels of wheat on December 1st.So, the baker faces the risk of an uncertain price on Dec. 1st.

Hedging strategy: buy 100 futures contracts maturing on Dec. 1st each for5000 bushels

Cash flow on Dec. 1stFutures position at maturity: FT F0 = ST F0Buy in the spot market: STEffective cash flow: ST F0 ST = F0

Price fixed at F0!

Did this cost anything? Cash flows associated with margin calls.

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Perfect hedges are not always possibleWhy?

The date T may not be a futures expiration date.PT may not correspond to an integer number of futures contractsA futures contract on the underlying may not be availableThe futures contract might not be liquidThe payoff PT may be nonlinear in the underlying

Basis = Spot price of underlying - futures pricePerfect hedge: basis = 0 at time TBasis risk: basis 6= 0 at time TBasis risk arises because the futures contract is on a related but differentasset, or expires at a different time.

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Hedging problem with basis riskToday is Sept. 1st. A taco company needs 500, 000 bushels of kidney beans onDecember 1st. So, the taco company faces the risk of an uncertain price.

Problem: No kidney bean futures available. Basis risk inevitable.

Hedge: Go long y soybean futures each for 5000 bushels of soybeans

Cash Flow in 90 daysFutures position at maturity: (FT F0)yBuy kidney beans in the spot market: PTEffective cash flow: CT = y(FT F0) + PT

PT 6= yFT for any y: Perfect hedge impossible!

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Minimum variance hedgingVariance of the cash flow

var(CT) = var(PT) + var(y(FT F0)

)+ 2cov

(y(FT F0),PT

)= var(PT) + y2var(FT) + 2ycov(FT ,PT)

Set the derivative with respect to y to zero:

dvar(CT(y))dy = 2yvar(FT) + 2cov(FT ,PT) = 0

Optimal number of Futures contracts:

y = cov(FT ,PT)var(FT)

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Mod 2: Futures and Options