A dealer can strip a Treasury note into its individual cash flows and sell them separately, or buy the strips back and reconstitute the note. If the two prices ever diverge, the dealer collects the difference for free. That trade is the entire logic behind arbitrage-free bond valuation.
Arbitrage-free valuation produces security values consistent with no arbitrage opportunity. An arbitrage opportunity is a transaction requiring no net cash outlay that generates a riskless profit. In well-functioning markets, prices adjust until such opportunities disappear. That is the principle of no arbitrage.
The foundation is the law of one price: two assets that are perfect substitutes must trade at the same price absent transaction costs. If they do not, you buy the cheap one, sell the rich one, and pocket the spread. Violations come in two flavors.
- Value additivity: the whole must equal the sum of the parts. If a package paying 105 in one year costs 97 while 105 units of a one-year zero costing 0.952381...
- Dominance: a risk-free future payoff must carry a positive price today, and two risk-free assets must be discounted at the same rate.
Common mistakes
- Discounting every cash flow at the yield-to-maturity. On the upward-sloping curve in Example 1 that produces 102.7751 instead of 102.8102. The yield overdiscounts the early coupons; with a downward-sloping curve it underdiscounts them.
- Using the next period's rate to discount. At a node, discount with the forward rate at that node, not the rate at the node you are moving toward. Discounting 102.0819 at 4.646% when the node rate is 2.000% is the classic setup error.
- Weighting the branches with anything other than 0.5. The lognormal binomial model assigns equal probability to up and down moves. Candidates sometimes import risk-neutral probabilities from an equity tree; here they are simply one-half.
Bottom line
- No arbitrage: a bond equals a portfolio of zeros, each discounted at its own benchmark spot rate
- Two arbitrage types: value additivity (whole equals sum of parts) and dominance (risk-free payoff needs a positive price)
- Lognormal tree: adjacent same-period rates differ by the multiplier e raised to 2 sigma; the standard deviation of the one-year rate is i-zero times sigma
- Calibration: iterate trial rates until the tree reprices each benchmark bond at par; at zero volatility the tree collapses to the implied forward curve
Exam shortcut
Read the stem for one signal: does the bond have an embedded option? No option means spot-rate discounting is legal and fastest, and any tree answer must match it. Option or prepayment means lattice or Monte Carlo. When a node calculation appears, write the template before touching the calculator: coupon, plus half of up-value plus half of down-value, divided by one plus THIS node's rate.
The full lesson (about 3,182 words, 21 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.
Learning objectives
- arbitrage free valuation
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