Finance-Theory · Unit 7 · Video 2 · Interactive Practice

The Law of One Price: Same Cash Flows, Same Price

IKey Formulas

FormulaNameWhat it needs
asset=(CF1,CF2,…,CFT)\text{asset} = (\text{CF}_1, \text{CF}_2, \dots, \text{CF}_T)What an asset isNothing more than amounts of money paid on particular dates
CFt(1)=CFt(2) for every t  ⟹  P1=P2\text{CF}^{(1)}_t = \text{CF}^{(2)}_t \text{ for every } t \;\Longrightarrow\; P_1 = P_2Law of one priceThe same amounts on the same dates in every outcome, known to be identical, and one person who prefers more money to less. No equilibrium: supply need not equal demand
P1<P2:today P2−P1>0,each date t: +CFt−CFt=0P_1 \lt P_2:\quad \text{today } P_2 - P_1 \gt 0, \quad \text{each date } t\text{: } +\text{CF}_t - \text{CF}_t = 0Arbitrage when the law failsBuy the cheaper asset and sell the dearer one; this takes free buying and selling, including selling what you do not own
P0=C P0,1+C P0,2+⋯+(C+F) P0,TP_0 = C\,P_{0,1} + C\,P_{0,2} + \cdots + (C+F)\,P_{0,T}Coupon bond = package of STRIPSThe law applied: P0,tP_{0,t} is the price of one dollar due at date tt, and the bond pays exactly what the package pays

Key Insight: The law assumes no equilibrium and no pricing formula. If identical cash flows ever sold at two prices, a single person who prefers more money to less would buy the cheap one and sell the dear one, and that trading pushes the prices together.

IIThe Free Lunch, Date by Date

Buying the cheaper of two identical assets and selling the dearer pays the price gap today and nets zero on every later date.

IIIOne Arbitrageur Closes the Gap

Each round of buying the cheap asset and selling the dear one earns the gap and narrows it.

IVIdentical Is the Precondition

Once the two cash flow streams differ, the same price gap no longer guarantees a free lunch.

💡 The cash flows must also be known to be identical. A wrapped package that auctions for about the price of an unwrapped iPod tells you what the bidders believe is inside: prices carry what the entire market knows.

VQuiz Questions

Problem 1 · Buy Cheap, Sell Dear

Given: Assets A and B promise identical cash flows on identical dates in every outcome. A sells for $96.40 and B for $97.10. Which trade is the arbitrage, and what does it pay per unit?

✅ Correct! You receive 97.10−96.40=0.7097.10 - 96.40 = 0.70 dollars today, and on every later date what A pays you is exactly what you owe on B.
❌ Wrong direction. Buying B costs 97.10 and selling A brings in 96.40, so today you are 0.70 dollars out of pocket. Buy the cheap one.
❌ Not quite. The gap is collected once, today. Afterwards you receive CFt\text{CF}_t on A and owe the same CFt\text{CF}_t on B, so each later date nets +CFt−CFt=0+\text{CF}_t - \text{CF}_t = 0.
❌ Not quite. Whatever the outcome, A and B pay the same amount on the same date. If the cash flows are low, you receive less on A and owe exactly that much less on B: no risk remains.
Show solution

A is cheaper, so buy A and sell B. Today:

PB−PA=97.10−96.40=0.70>0P_B - P_A = 97.10 - 96.40 = 0.70 > 0

On every later date tt, in every outcome, you receive CFt\text{CF}_t from A and owe CFt\text{CF}_t on B:

+CFt−CFt=0+\text{CF}_t - \text{CF}_t = 0

So you keep $0.70 per unit with no risk and no net obligation after today: a free lunch.

Problem 2 · What the Law Assumes

Given: A classmate says the law of one price is an equilibrium result that holds only when supply equals demand. What does the arbitrage argument actually assume about investors?

✅ Correct! One person who prefers more to less keeps buying the cheap asset and selling the dear one for as long as the prices differ.
❌ Not quite. Supply may very well not equal demand, and the law still holds; the argument never mentions equilibrium.
❌ Too strong. Not even people in general are needed: a single person in the economy who prefers more money to less is enough to exploit the gap.
❌ Not quite. The argument uses no pricing formula for any instrument. It compares two prices for the same cash flows; it never says what either price should be.
Show solution

Suppose identical cash flows sell at P1<P2P_1 \lt P_2. Buying asset 1 and selling asset 2 gives P2−P1>0P_2 - P_1 \gt 0 today and +CFt−CFt=0+\text{CF}_t - \text{CF}_t = 0 on every later date. Anyone who prefers more money to less takes that trade, and keeps taking it while the prices differ.

Nothing in that argument needs supply to equal demand, every investor to behave the same way, or a formula for the price. It needs one person who prefers more to less (plus the ability to buy and sell freely, including selling what you do not own).

Problem 3 · Scaling the Trade

Given: Two assets promise identical cash flows. Asset 1 trades at $48.25 and asset 2 at $48.60. You buy 2,000 units of asset 1 and sell 2,000 units of asset 2. At date 2 each unit of either asset pays $3.10. Find your net cash today and at date 2.

Net cash today?

Net cash at date 2?

✅ Correct! 2,000×0.35=7002{,}000 \times 0.35 = 700 dollars today, and 2,000×3.10−2,000×3.10=02{,}000 \times 3.10 - 2{,}000 \times 3.10 = 0 at date 2.
❌ Check today's cash. You sell the dear asset and buy the cheap one: you receive 2,000×48.602{,}000 \times 48.60 and pay 2,000×48.252{,}000 \times 48.25.
❌ Per unit only. 48.60−48.25=0.3548.60 - 48.25 = 0.35 is the gain on one unit; the position is 2,000 units.
❌ One leg only. 2,000×48.60=97,2002{,}000 \times 48.60 = 97{,}200 is what the sale brings in, but the purchase of asset 1 costs 2,000×48.25=96,5002{,}000 \times 48.25 = 96{,}500.
❌ Check date 2. Count both legs: what asset 1 pays you and what you owe on asset 2.
❌ One leg only. 2,000×3.10=6,2002{,}000 \times 3.10 = 6{,}200 comes in on asset 1 and the same 6,200 goes out on asset 2; they cancel.
❌ Not quite. The price gap is collected once, when you trade. Later dates only carry the cash flows, and those cancel.
Show solution

Today: sell 2,000 units of asset 2 and buy 2,000 units of asset 1:

2,000×48.60−2,000×48.25=97,200−96,500=7002{,}000 \times 48.60 - 2{,}000 \times 48.25 = 97{,}200 - 96{,}500 = 700

so $700 is received.

Date 2: asset 1 pays you 2,000×3.10=6,2002{,}000 \times 3.10 = 6{,}200 and you owe the same on asset 2:

+6,200−6,200=0+6{,}200 - 6{,}200 = 0

The same holds on every other date, so $700 is the whole profit, with no net obligation after today.

Problem 4 · A Coupon Bond Against Its STRIPS

Given: A 2-year bond pays a $4 coupon at date 1 and $104 (the last coupon plus $100 face) at date 2. STRIPS cost $0.96 per $1 due at date 1 and $0.91 per $1 due at date 2. The bond trades at $99.10. Which trade is the arbitrage, and what does it pay today?

✅ Correct! The package costs $98.48 and pays exactly what the bond pays, so selling the bond at $99.10 and buying the package locks in $0.62.
❌ Missing a cash flow. At date 2 the bond pays the last coupon too: 104, not 100. With only 100 STRIPS at date 2 you would still owe 4 on that date, so the position is not riskless.
❌ Wrong direction. The bond (99.10) is dearer than the package (98.48). Buying the bond and selling the package costs you 0.62 dollars today.
❌ Not quite. The law compares cash flows, not labels. The package pays 4 at date 1 and 104 at date 2, exactly what the bond pays, so it must have the same price.
Show solution

The package that copies the bond's cash flows costs

4×0.96+104×0.91=3.84+94.64=98.484 \times 0.96 + 104 \times 0.91 = 3.84 + 94.64 = 98.48

The bond sells for 99.10, more than the identical package. Sell the bond and buy the package:

99.10−98.48=0.62>099.10 - 98.48 = 0.62 > 0

At date 1 the STRIPS pay you 4 and you owe the bond's coupon of 4; at date 2 they pay 104 and you owe 104. Every later date nets to zero, so $0.62 per bond is a free lunch. If you do not own the bond, selling it means selling what you do not own, the assumption the next video tests.

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