No-Arbitrage and One-Period Pricing
The single principle underneath all of derivative pricing, made concrete in a one-period market.
Leads to: Replication (14.2) and the fundamental theorems (14.3) generalize this one-period picture.
Learning Objectives
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- Define an arbitrage opportunity precisely and explain why prices must exclude it.
- Model a one-period market with a bond and a stock and identify its payoff vectors.
- Derive the no-arbitrage bounds \(d\lt R\lt u\) and interpret their failure as free money.
- Compute the price of a claim by the law of one price in a complete one-period market.
- Solve a one-period replication problem for the hedge \((\Delta,B)\) and the implied price.
Key Vocabulary
- Arbitrage
- A self-financing strategy with zero cost today and, tomorrow, a nonnegative payoff that is strictly positive with positive probability - risk-free profit.
- One-period market
- Two dates \(t=0,1\) and finitely many states; assets are a risk-free bond and a risky stock with known state payoffs.
- Gross return \(R\)
- One plus the risk-free rate over the period; \(\$1\) in the bond becomes \(\$R\) tomorrow.
- Law of one price
- Two portfolios with identical payoffs in every state must have identical prices today, or an arbitrage exists.
- Replicating portfolio
- Holdings \((\Delta,B)\) in stock and bond whose date-1 payoff equals the claim’s in every state.
- State prices
- Nonnegative numbers \(\psi_s\) pricing an \(\$1\)-in-state-\(s\) security; a claim’s price is \(\sum_s\psi_s\,\text{payoff}_s\).
Intuition & Motivation
The one-period market
Two dates, \(t=0\) (now) and \(t=1\) (period end), and two states of the world, up (\(\omega_u\)) and down (\(\omega_d\)). Two traded assets:
- A bond: costs \(\$1\) now, pays \(\$R\) in both states, where \(R=1+r\gt 0\) is the gross risk-free return.
- A stock: costs \(S_0\) now, pays \(S_0u\) up and \(S_0d\) down, with \(0\lt d\lt u\).
A portfolio holds \(\Delta\) shares of stock and \(B\) dollars in the bond; its cost is \(\Delta S_0+B\) and its date-1 payoff is \(\Delta S_0u+BR\) (up) or \(\Delta S_0d+BR\) (down).
What arbitrage is
The pricing axiom is that observed prices admit no arbitrage. This is not an empirical law of markets so much as a consistency condition: any violation is competed away almost instantly by traders, so any sensible pricing model must be arbitrage-free.
The no-arbitrage bound \(d\lt R\lt u\)
Why: if \(R\le d\) the stock dominates the bond (it returns at least \(R\) in every state and more in the up state), so borrow at \(R\), buy stock, and profit risk-free. If \(R\ge u\), short the stock and lend at \(R\). Only \(d\lt R\lt u\) prevents both. Solving \(R=qu+(1-q)d\) gives the risk-neutral probability
Replication prices the claim
A claim (e.g. a call) pays \(C_u\) up and \(C_d\) down. Find \((\Delta,B)\) matching both states:
Subtracting gives the delta (the hedge ratio), then back-substitute for \(B\):
By the law of one price the claim must cost exactly the replicating portfolio’s price \(V_0=\Delta S_0+B\). Remarkably this equals a discounted risk-neutral expectation, \(V_0=\tfrac1R\big(qC_u+(1-q)C_d\big)\) - the seed of everything in 14.2.
Interactive: solve the replication linear system
- Discounting the claim with the real-world probability \(p\) of an up move; the price uses \(q\), which depends only on \(u,d,R\), not on beliefs.
- Forgetting that \(d\lt R\lt u\) is required; if it fails, a static stock–bond arbitrage exists and no consistent price can be assigned.
- Sign errors on the bond position \(B\); a negative \(B\) means borrowing - part of the hedge, not an inconsistency.
- Believing the option’s price reflects its ‘expected payoff’ under real probabilities; it reflects the cost of the replicating hedge.
- First compute \(\Delta\) (the hedge), then \(B\), then the price - this order works in every one-period problem.
- Cross-check any one-period price two ways: replication cost and \(\tfrac1R(qC_u+(1-q)C_d)\). They must match exactly.
- State prices \(\psi_u=q/R,\ \psi_d=(1-q)/R\) price any claim by \(\psi_uC_u+\psi_dC_d\); a compact bookkeeping device.
Knowledge Check
Practical Exercise
A one-period market has \(S_0=50,\ u=1.3,\ d=0.8,\ R=1.10\). A put has strike \(K=55\). (a) Find the put payoffs \((P_u,P_d)\). (b) Compute \(q\), the hedge \((\Delta,B)\), and the put price. (c) Verify no-arbitrage holds.
(a) \(S_0u=65,\ S_0d=40\), so \(P_u=(55-65)^+=0,\ P_d=(55-40)^+=15\).
(b) \(q=(R-d)/(u-d)=(1.1-0.8)/0.5=0.6\). Delta: \(\Delta=(P_u-P_d)/(S_0(u-d))=(0-15)/(50\cdot0.5)=-0.6\) (short 0.6 shares - a put hedge). Bond: from \(\Delta S_0d+BR=P_d\), \(-0.6\cdot40+1.1B=15\Rightarrow B=(15+24)/1.1=35.4545\). Price: \(V_0=\Delta S_0+B=-30+35.4545=5.4545\). Check: \(\tfrac1{1.1}(0.6\cdot0+0.4\cdot15)=6/1.1=5.4545\). Agreed.
(c) \(d=0.8\lt R=1.1\lt u=1.3\), so \(q\in(0,1)\) and the market is arbitrage-free.
Lesson Summary
Retrieval Practice
Close the lesson and answer from memory before checking. This is deliberate, effortful recall - the single highest-yield study action.
A: Arbitrage-free iff \(d\lt R\lt u\); then \(q=(R-d)/(u-d)\in(0,1)\) is the unique probability making the stock’s expected gross return equal \(R\).
A: Replicate it with \((\Delta,B)\) and charge the portfolio cost, equal to \(\tfrac1R(qC_u+(1-q)C_d)\). The physical \(p\) does not enter the price.
Completion Checklist
- I can explain the core ideas in my own words
- I worked the derivations/examples by hand
- I completed the interactive workbench(es)
- I passed the knowledge check