Bond Price Consistency and Arbitrage Across Maturities

Summary

The document poses a no-arbitrage relationship for risk-free zero-coupon bonds when interest rates are deterministic. The cost today of investing through two successive periods should equal the cost today of buying a bond that pays at the end of the same total horizon. Otherwise, investors could construct two portfolios with identical future cash flows but different initial costs.

For the stated case where the product of the one-period bond prices exceeds the two-period bond price, the two-step investment is more expensive. An arbitrage argument is to buy the cheaper two-period bond and short the more expensive strategy of buying the first-period bond now and reinvesting its maturity proceeds in the next-period bond. The initial price difference is received up front, while the future obligations and receipts offset under the assumptions. The document asks for this derivation but supplies no answer; the argument relies on deterministic rates, risk-free borrowing and lending, and the ability to trade both positions without frictions.

Key ideas

• No-arbitrage requires equal prices for portfolios with identical risk-free cash flows.
• A bond held through successive periods can be compared with a bond spanning the combined horizon.
• If the sequential investment costs more, buying the cheaper longer bond and shorting the sequential strategy creates an arbitrage under the assumptions.
• The argument depends on deterministic rates and frictionless access to borrowing, lending, and bond trading.

Tags

# Arbitrage argument with bonds


# Arbitrage argument with bonds












Let $B(t,T)$ denote the cost at time t of a risk-free 1 euro bond, at time T. Assume that the interest rate is a deterministic function. Show that the absence of arbitrage requires that:

$ B(0,1) B(1,2) = B(0,2)$

For instance, could you give a detailed explanation of what to do if $B(0,1)B(1,2) > B(0,2)$ ?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.