16.1 Pricing, Discounting & Arbitrage

Pricing, Discounting & Arbitrage

Valuation and Risk Models’ fixed-income block starts where every certain cash flow is priced: discount factors. Once you can turn a schedule of sure dollars into a present value, the law of one price and cash-and-carry arbitrage follow immediately—and so do the market conventions that turn a theoretical PV into the clean and dirty prices you see on a screen.

Discount Factors as the Primitive

A discount factor DF(t) is the price today of $1 delivered for certain at time t (measured in years from the settlement date, or in periods if you work on a discrete grid). If a Treasury strip pays $1 at maturity T with no credit risk and no optionality, its price is exactly DF(T).

For a package of certain cash flows C_{t1}, C_{t2}, …, C_{tn},

PV = Σ_i C_{ti} × DF(ti)

That linearity is the entire pricing engine for default-free, option-free instruments. You do not need a yield yet; yields are summaries of the same discount curve. Risk managers care about DF(t) because every DV01, carry, and arb check ultimately moves through these factors.

Worked discounting

Suppose annual discount factors DF(1) = 0.9709, DF(2) = 0.9426, DF(3) = 0.9151. A 3-year 4% annual coupon bond with $100 face pays 4, 4, and 104. Its no-arbitrage price is

PV = 4×0.9709 + 4×0.9426 + 104×0.9151 = 3.8836 + 3.7704 + 95.1704 = 102.8244

If the bond is offered at 103.10 with the same certain cash flows, it is rich to the curve; if offered at 102.50, it is cheap.

Law of One Price

The law of one price (LOOP) says that two portfolios that deliver identical certain cash flows in every state (here: on every date, with certainty) must trade at the same price. Otherwise you sell the expensive package, buy the cheap package, pocket the difference today, and face zero net future liability.

In fixed income with a complete strip curve, LOOP is operational: any coupon bond’s cash flows can be matched by a basket of zeros. Therefore

P_coupon = Σ Coupon_or_principal_at_ti × P_zero(ti)

must hold (up to transaction costs and financing frictions). LOOP is not a soft “should”; for certain cash flows it is an arbitrage constraint.

Arbitrage for Certain Cash Flows

Arbitrage here means a self-financing strategy with non-positive initial outlay, non-negative future payoffs in all scenarios, and a strictly positive payoff in some scenario (or a positive cash inflow today with zero future net cash flows). For default-free Treasury cash flows, the scenarios collapse to calendar dates.

Classic cash-and-carry:

1. Observe P_market(bond) versus P_replicating = Σ C_t DF(t).
2. If P_market > P_replicating: short the bond, buy the replicating zeros (or synthetic zeros via strips / repo financing of coupons). Initial cash inflow is positive; future cash flows cancel.
3. If P_market < P_replicating: buy the bond, short the zeros.

Worked arb

Using the DF table above, the 4% bond’s fair value is 102.824. Suppose it trades at 103.50. Short $100 face of the coupon bond (receive 103.50), buy 4 face of the 1y zero, 4 face of the 2y zero, and 104 face of the 3y zero. Cost of zeros = 102.824. Net cash today = 103.50 − 102.824 = +0.676 per 100 face. At each coupon date the zeros cover the short bond’s obligations exactly. That locked-in 0.676 is the arb profit (before frictions).

Frictions that kill textbook arb: bid–ask on strips, repo specialness, inability to short cheaply, taxes, and the fact that many “bonds” have embedded options so cash flows are not certain.

Coupon Bonds Versus STRIPS (P vs C)

STRIPS (Separate Trading of Registered Interest and Principal of Securities) split a coupon Treasury into individual zero-coupon pieces. Market slang:

• P-STRIP: the principal (corpus) payment at final maturity.
• C-STRIP: each coupon payment stripped into its own zero.

Economically, a coupon bond = basket of C-STRIPS + one P-STRIP with matching dates and amounts. Reconstitution stitches strips back into the coupon bond; stripping does the reverse. No-arbitrage requires

Dirty price(coupon bond) ≈ Σ prices of matching C- and P-STRIPS

Persistent gaps create strip/reconstitution flows until LOOP is restored within costs.

Exam cue: if a question contrasts P and C strips, remember they are both zeros—they differ only in which original cash flow they came from, not in pricing math.

Replicating Portfolios

Replication means holding instruments whose net cash flows match a target. To replicate a coupon bond with zeros: hold face equal to each cash flow at each date. To replicate a zero with coupon bonds is harder (you may need a portfolio of bonds and financing), but with a full strip curve the zero is already the primitive.

Replication is also how you prove a price. You do not argue “fair value feels like 102”; you exhibit a hedge portfolio that offsets every cash flow and show the hedge costs 102.824. That is the risk manager’s language for LOOP.

Clean Price, Dirty Price, and Accrued Interest

Bonds accrue coupon continuously between payment dates. The dirty price (invoice / full / settlement price) is what the buyer actually pays:

Dirty = Clean + Accrued interest

The clean price is the quoted price with accrued stripped out so the quote does not jump by a full coupon on the ex-dividend / payment date in a mechanically confusing way. Accrued interest (AI) is the fraction of the next coupon earned by the seller since the last coupon date:

AI = Coupon × (days accrued / days in coupon period)

under the applicable day-count convention.

Worked clean / dirty

Semi-annual bond, 6% coupon on $100 face → $3 per coupon period. Suppose 73 days have accrued in a 182-day period under the market’s convention. AI = 3 × (73/182) ≈ 1.203. If the clean quote is 101.250, dirty = 101.250 + 1.203 = 102.453. Discount-factor PV equals the dirty price (present value of remaining cash flows), not the clean quote.

Day-Count Conventions

Day-count rules define the fraction of the year (or coupon period) used for accrued interest and sometimes for interest on money-market instruments.

A 1-day difference in accrued can move AI by Coupon/period_days. For risk P&L, always mark dirty and know which convention the ISIN uses—mixing Actual/Actual accrued with a 30/360 assumption is a classic ops error.

Putting it together for the exam

1. Build or read DF(t) from strips or bootstrapped instruments.
2. Price certain cash flows as Σ C_t DF(t) → dirty fair value.
3. Compare to market dirty price; gaps invite replication arb.
4. Convert dirty ↔ clean with accrued under the right day-count.
5. Treat STRIPS as the visible zero curve that enforces LOOP on coupon Treasuries.

Master that chain and every later VRM topic—spots, forwards, YTM, DV01—sits on top of the same discount factors.