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Futures Markets (Part 2)

Futures Markets (Part 2). BKM Chapter 18, continued. How are futures/forward prices determined?. In the pricing discussion, I ignore daily resettlement and margin considerations. S t Spot price of underlying at time t F t Futures price at time t T Settlement date.

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Futures Markets (Part 2)

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  1. Futures Markets (Part 2) • BKM Chapter 18, continued

  2. How are futures/forward prices determined? • In the pricing discussion, I ignore daily resettlement and margin considerations. • St Spot price of underlying at time tFt Futures price at time tT Settlement date. • Futures- (or Forward-) spot convergence. • As t gets very close to T, the futures and spot transactions become essentially equivalent. • No matter how far apart St and Ft diverge prior to settlement, at settlement ST=FT. (“Convergence”)

  3. Futures- (or Forward-) spot parity • If I know that I will need gold in 1 year, I can • Enter into a forward contract. Lock in a price F0 today. Pay and take delivery in 1 year. Or … • Borrow the spot price S0 at interest rate r. Buy the gold in the spot market today. Store the gold until I need it in one year. In one year, to get the gold, I’ll pay: S0(1+r) + Storage. • Market prices should be set so that I’m indifferent: • If this relationship is not satisfied, arbitrage is possible.

  4. Example • Spot price of gold (per ounce), S0 = 400Borrowing and lending rate, r = 5%Annual storage cost 4/ounce • By forward-spot parity, the forward price for one-year delivery should be F0 = S0(1+r) + Storage = 400(1.05) + 4 = 424.

  5. Suppose that F0=430. Arbitrage… • Today: • Short gold forward at F0(I agree to deliver gold in 1 yr at 430/oz.) • Borrow 400 • Buy gold in spot market • In one year: Repay loan –420 Pay storage –4 Deliver gold (and receive pmt) +430 Net +6 • If everyone does this, F0 will  and S0 (tending to restore the parity relationship).

  6. Arbitrages and how to build them… • An arbitrage is a risk-free transaction that returns a positive profit. • Construction • Do actual market prices violate the parity relation? • If “yes”, identify what is relatively overvalued and what is relatively undervalued. • Buy (or go long) what is undervalued. • Sell (or sell short) what is overvalued. • If you need any money today, borrow it. • If you have surplus funds today, invest them. • Compute your net profit at maturity.Verify that this profit is equal to the mispricing (up to rounding error)

  7. Suppose that F0=410. • Someone who currently owns gold can • Buy gold forward • Sell gold spot • Invest the sale proceeds • In one year:Investment matures +420Receive gold and make pmt –410Savings on storage cost +4Net +14 • If everyone does this, F0, S0 (tending to restore parity)

  8. Other complications: maturity T1 yr. • F0=S0(1+c)Twhere c is the cost of carry expressed as “% per year” • If there is no storage cost, then c=r andF0 = S0(1+r)T

  9. Stock index futures: cost of carry • As an alternative to going long the futures contract, I can borrow money; buy the stock; hold until maturity. • If I do this, I will pay (at maturity)S0(1+r) – dividends I receive on the stock • F0 = S0(1+r–d) where d is the dividend yield • F0=S0(1+r)–D where D is the dividend yield in index points.

  10. Example of Index Arbitrage • S0 = 650, rf= 5%, d = 3%, T=9 months • Parity: F0 = 650(1 + 0.05 – 0.03)3/4 = 659.7 • If F0=665, then future is overvalued relative to the spot. • Arbitrage:Borrow $650 +650Buy index – 650Short the future 0 • “Unwind” at maturity:Collect divs [(3/4) x 3%(650)] 14.6Sell stock (index) + STSettle future – (ST –665)Repay loan [1+ (3/4)x5%]x650 – 674.4Net 5.2

  11. Example (continued) • If F0 = 655, the future is undervalued relative to spot. • Arbitrage:Long futures 0Short stock +650Invest short-sale proceeds – 650 • At maturity:Settle futures (ST –655)Pay dividends [3/4 x 3% of 650)] –14.6Repurchase stock –STInvestment matures 674.4Total +4.8 • Due to the difficulties in selling short, these strategies are usually practiced by institutions that already own the stock.

  12. The preceding day’s close was 1,222=S0

  13. Arbitrage • In principle, if F0 > S0+4, then the spot (S&P 500 index) is undervalued relative to the futures contract. • To set up the arbitrage: • Short the futures contract • Buy the index (the component stocks) • These must be done simultaneously. • We must make allowances for transaction costs (commissions, bid-ask spread, etc.)

  14. Stock Index Futures and Portfolio Management: Synthetic stock positions. • As an alternative to buying the S&P index, buy the future and invest cash in risk-free security. • Example: An S&P index mutual fund can manage inflows and outflows from a synthetic stock position. • Inflow: buy S&P index future, invest in T-bills. • Outflow: sell S&P index future, divest T-bills

  15. Stock Index Futures and Portfolio Management: Synthetic risk-free positions. • If you already own the stock, you can “shift into the risk-free security” by shorting the index futures.Any gain/loss on the stock is offset by index futures. • These strategies may be used to • minimize transaction costs(particularly in dynamic hedging strategies) • profit from small misvaluations (“augmented index funds”)

  16. Stock index futures and speculation • If you think that small firms are likely to outperform large firms . . .long $1 in the Value Line index futureshort $1 in the S&P index future • If you think that Japanese firms are likely to outperform U.S. firms . . .long $1 in the Nikkei 225 index futureshort $1 in the S&P index future

  17. Stock index futures and selective speculations • To bet that IBM is going to outperform the market:long $1 in IBM stockshort $1 S&P index futures(Your return is rIBM – rM) • To bet that IBM is going to underperform the market:short $1 in IBM stocklong $1 S&P index futures

  18. Interest rate futures contracts • CME T-bill contract • The underlying is 13-week T-bills with a total par value of $1 Million. Settled in kind. • The CBT 30-year T-bond contract • The underlying is 30-year US T-bonds with a total par value of $100,000. Settled in kind. • Other • US Agency bonds • Federal funds • Eurodollars • Municipal bonds

  19. Using the T-bond future: Hedging by an underwriter • Underwriter trying to sell an inventory of corporate bonds can hedge against movements in the U.S. bond rate by going short the T-bond contract. • Model: yCORP = yGOVT + credit spread with credit spread constant. • If yGOVT then yCORP • The prices of both bonds fall. • We’ll have a loss on the corporate bonds we have in inventory. • We’ll have an offsetting gain on our short T-bond contract.

  20. Using the T-bond future: Hedging by borrowers and lenders • A pension fund that is due to receive an inflow of cash can lock in a long-term bond rate by going long the futures contract, and taking delivery of the bonds. • A corporate treasurer who wishes to lock in a borrowing rate can go short the T-bond future.(If y’s go up, there will be a loss associated with increased borrowing expense and a gain on the T-bond contract.)

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