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Chapter 8

Chapter 8. Arbitrage. Suppose that a particular stock is selling for $53 on the New York Stock Exchange and simultaneously selling for $50 on the Pacific Coast stock exchange.

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Chapter 8

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  1. Chapter 8 Arbitrage

  2. Suppose that a particular stock is selling for $53 on the New York Stock Exchange and simultaneously selling for $50 on the Pacific Coast stock exchange. • On arbitrageur can simultaneously buy on the Pacific Coast exchange for $50 and sell on the New York stock exchange for $53.

  3. NYSE PAC Sell Buy +$53 -$50 = $3. • The arbitrageur makes an instant, risk-free profit of three dollars. The ability to repeatedly carry out this transaction will force the prices to be the same in equilibrium.

  4. Assumptions for Arbitrage • No transactions costs. • No default. • No collateral. • The ability to shortsell securities and use the proceeds from the shortsale. This is called unrestricted shortselling.

  5. Short Position Is Established Sale of certificate Shortseller Buyer $ IOU Certificate Lender of certificates Short Position Is Closed Purchase of certificate Shortseller buys Seller $ ReturnIOU Certificate Lender of certificates

  6. Shortseller must buy back at some future. • Profit: Shortsale price > Purchase price. • Loss: Shortsale price < Purchase price. Potential shortsale losses have no upper bound, implying shortselling is very risky.

  7. For stocks, shortsellers must pay dividends to lender of certificates. $ After-tax value of dividends Time Ex-dividend point Not an issue for bonds because of daily accrued interest.

  8. Shortselling a Bond Equals Borrowing Points in Time 0 1 2 Cash flows +$82.64 0 -$100

  9. Hypothetical Strips Prices Points in time 0 1 2 $70 $100 – $80 $100

  10. Arbitrage Cash Flows Action Points in Time 0 1 2 Buy one-period strip -$70 +$100 0 Shortsell two-period strip +$80 0 -$100 Net cash flows +$10 +$100 -$100 Cumulative net cash flows +$10 +$110 +$10

  11. In a multi-period context, a sufficient condition for arbitrage is for the cumulative cash flows to never be negative and have the possibility of being positive at a future point in time.

  12. Points in time 0 1 2 $88 $100 – $80 $100

  13. A Non-arbitrage Position Action Points in Time 0 1 2 Shortsell one-period strip +$88 -$100 0 Buy two-period strip -$80 0 +$100 Net cash flows +$8 -$100 +$100 Cumulative net cash flows +$8 -$92 +$8

  14. Arbitrage and Bond Coupons

  15. Two-period Bonds Points in Time 0 1 2 Bond G -$100 +$6 +$106 Bond H -$100 +$8 +$108

  16. Arbitrage for Two-period Bonds Action Points in Time 0 1 2 Shortsell Bond G +$100 -$6 -$106 Buy Bond H -$100 +$8 +$108 Net cash flows 0 +$2.00 +$2.00 Cumulative net cash flows 0 +$2.00 +$4.00

  17. Two-period Bonds: No Arbitrage Profit Action Points in Time 0 1 2 Shortsell Bond G +$100 -$6 -$106 Buy Bond H -$103.60 +$8 +$108 Net cash flows -$3.60 +$2 +$2 Cumulative net cash flows -$3.60 -$1.60 +$.40

  18. Cash Flows Points in Time 0 1 2 Bond G 100 6 106 Bond H 106 8 108

  19. Arbitrage Points in Time 0 1 2 Buy G -100 +6 +106 Short H +106 -8 -108 Net +6 -2 -2 Cumulative Net +6 +4 +2

  20. Price Arbitrage 104 100 S  P = c[PVA] + PAR[PV]  Arbitrage Coupon 0 6 8

  21. Suppose Points in Time 0 1 2 Bond G $100 $6 $106 Bond H $102 $8 $108

  22. There is an arbitrage profit as follows Points in Time 0 1 2 Short 1.02 units Bond G +$102 -$6.12 -$108.12 Buy Bond H -$102 +$8 +$108 Net 0 +$1.88 -$0.12 Cumulative Net 0 +$1.88 +$1.76

  23. The forward interest rate is negative

  24. 0 1 2 PG 6 106 PH 8 108 116 Total Future Inflows 112

  25. To avoid arbitrage (a negative forward rate)

  26. Bonds of Different Maturities Total Future Inflows 0 1 2 3 4 5 Bond G 100 6 6 6 106 124 Bond H 100 4 4 4 4 104 120

  27. Arbitrage 0 1 2 3 4 5 Buy Bond G -100 +6 +6 +6 +106 Short Bond H +100 -4 -4 -4 -4 -104 Net 0 +2 +2 +102 -104 +2 Cum Net +2 +4 +6 +108 +4 0

  28. General Case Total Future Inflows 0 1 2 3 4 5 CG+PAR PG CG CG CG 4CG+PAR PH CH CH CH CH CH+PAR 5CH+PAR Arbitrage unless

  29. 0 1 2 94.34 100 85.73 100 100 6 106 Replicating Portfolio (94.34)(.06) + (1.06)(85.73) = $96.53

  30. Arbitrage between Coupon-bearing Bonds and Strips Action Points in Time 0 1 2 Two-period bond $100 $6 $106 One-period strip $94.34(6%) $100(6%)  Two-period strip $85.73(106%) $100(106%) (94.34)(.06) + (1.06)(85.73) = $96.53

  31. Arbitrage between Coupon-bearing Bonds and Strips Action Points in Time 0 1 2 Short two-period bond +$100 -$6 -$106 Buy 6% of a one-period strip -$5.66 +$6  Buy 106% of a two-period strip -$90.87  +$106 Net cash flows +$3.47 0 0 Cumulative net cash flows +$3.47 +$3.47 +$3.47

  32. Cash Flows in Equilibrium When Price of Two-period Strip is $89 Action Points in Time 0 1 2 Short two-period bond +$100 -$6 -$106 Buy 6% of a one-period strip -$5.66 +$6  Buy 106% of a two-period strip -$94.34  +$106 Net cash flows 0 0 0 Cumulative net cash flows 0 0 0

  33. U.S. Treasury Strips 0 1 2 3 . . . n C C C C + PAR S3 Sn + Sp,n S1 S2 • Individual bonds can be stripped and reconstituted. • Principal (Par) strips from the specific bond must be used to reconstitute the bond. • Coupon strips from any bond can be used to reconstitute the coupons.

  34. 0 1 2 3 4 5 C5+PAR5 5-Year Bond C1 C2 C3 C4 • To reconstitute a 4-year bond, coupon strips from either bond can be used for the coupons at times 1, 2, 3, 4. • Only Par4 can be used to reconstitute the 4-year par value. C4+PAR4 4-Year Bond C1 C2 C3

  35. Principal Strips vs. Coupon Strips • In reconstituting a bond, the principal strip coming from the original bond must be used. • Principal strips with the appropriate maturing may be used to reconstitute any bond.

  36. In practice prices of principal and coupon strips with the same maturity may be different Par SHORT Sp   Sc LONG Time Maturity

  37. Arbitrage positions between principal and coupon strips in practice require collateral. • The differences in price must be big enough to justify investing this collateral. • Price differences may get larger over time and more collateral may be required.

  38. Creating Forward Contracts from Spot Securities

  39. Long Forward Position Points in Time 0 1 2 Long forward 0 -Forward +Par

  40. S2 S1 Numerical Example 0 1 2 Spot 85.73 = S2 100 Strips 96.15 = S1 100 Long Forward 0 -F +100 85.73 96.15 = = 0.8916.

  41. A Numerical Example ofCreating a Long Forward Position Action (at time 0) Points in Time 0 1 2 Long two-period strip -$85.73  +$100 Short 0.8573/0.9615 one-period bonds +$85.73 -$89.16  Net = Long forward 0 -$89.16 +$100

  42. A Numerical Example of Creating a Short Forward (Borrowing) Position Action (at time 0) Points in Time 0 1 2 Short 1 two-period strip +$85.73  -$100 Long 0.8573/0.9615 one-period bonds -$85.73 +$89.16  Net = Short forward 0 +$89.16 -$100

  43. Creating a Long Forward Position Action (at time 0) Points in Time 0 1 2 Long 1 two-period strip -S2 +$100 Short S2/S1 one-period bonds +S1/(S2/S1) -1(S2/S1)  Net = Long forward 0 -(S2/S1) +$100

  44. Arbitrage and Forward Interest Rates Suppose that R0,1 = 4%, R0,2 = 8%, implying that a forward loan can be created with an interest rate of 12.15%. F = = 89.16.

  45. Suppose the actual forward rate is 15%, while the rate implied by strips is 12.15%. Action (at time 0) Points in Time 0 1 2 Lend forward at 15% 0 -$100/1.15 +$100 = -$86.96 Short 1 two-period strip +$85.73  -$100 Long 0.8573/0.9615 -$85.73 +$89.16  one-period strips Net 0 +$2.20 0

  46. Suppose the actual forward rate is 15%, while the rate implied by strips is 12.15%. Action (at time 0) Points in Time 0 1 2 Lend forward at 15% 0 -$100/1.15 +$100 = -$86.96 Borrow forward at 5% +$89.16 -$100 ________________________________________________ Net 0 +$2.20 0

  47. Suppose the actual forward rate is 5% and the implied forward rate is 12.15%. Action (at time 0) Points in Time 0 1 2 Borrow forward at 5% 0 +$100/1.05 -$100 = +$95.24 Long 1 two-period strip -$85.73  +$100 Short 0.8573/0.9615 +$85.73 -$89.16  one-period strips Net 0 +$6.08 0

  48. Suppose the actual forward rate is 5% and the implied forward rate is 12.15%. Action (at time 0) Points in Time 0 1 2 Borrow forward at 5% 0 +$100/1.05 -$100 = +$95.24 Lend forward at 12.15% -$89.16 +$100 ________________________________________________ Net 0 +$6.08 0

  49. Price P2High P3  P2 P2Low P1  Coupon C1 C2 C3 Arbitrage if P2High > P2 rr if P2Low < P2

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