Arbitrage & No-Arbitrage Pricing

Medium3–4 hrsArbitrageNo-ArbitrageReplicationLaw of One Price

Overview

Arbitrage is the foundation of all derivative pricing — if two things have the same payoff, they must have the same price. The cash-and-carry argument shows why: any deviation from fair forward price creates a riskless profit opportunity that will be instantly exploited. In interviews, the question is almost always: 'Are these prices consistent?' Your answer: construct the arbitrage if they're not.

How to Recognize

→Two assets with identical payoffs trading at different prices
→'Construct a riskless profit'
→Forward price, futures basis, ETF premium/discount
→Currency triangular arbitrage

Step-by-Step Approach

1.Identify: do two portfolios have IDENTICAL payoffs in all scenarios?
2.If yes and prices differ: buy cheap, sell expensive → riskless profit
3.Cash-and-carry arbitrage: buy spot + carry costs, sell forward
4.Forward price: F = S·e^(rT) (no dividends, continuous compounding)

Key Formulas

F = S \cdot e^{rT}

(fair forward price, no dividends)

F = S \cdot e^{(r-q)T}

(with continuous dividend yield

q

)

F = (S - PV(\text{dividends})) \cdot e^{rT}

(discrete dividends)

04Cost of carry:

F = S + \text{storage} - \text{convenience yield}

Worked Examples

0Quick Example

100, r=5%, T=1yr. Fair forward = 100·e^0.05 ≈

105.13. If F=

107: sell forward, buy spot, finance at 5%, deliver at expiry. Profit =

107 −

105.13 =

1.87.

Problem

Gold spot = $1800/oz. 1-year forward =$ 1950. Risk-free rate = 5%, storage cost = 1%/year. Is this a fair price?

Solution

Fair forward with storage: F = S·e^((r+storage)T) = 1800·e^(0.06·1) = 1800·1.0618 = $1911.

Market forward = $1950 >$ 1911. Forward is overpriced.

Arbitrage: borrow $1800 at 5%, buy gold spot ($ 1800), pay storage ($18/yr ≈ prepaid).

Simultaneously: sell 1-year forward at $1950.

At expiry: deliver gold, receive $1950. Repay loan:$ 1800·e^0.05 = $1892, pay storage$ 18 = $1910 total.

Net profit = $1950 −$ 1910 = $40 per ounce. Risk-free.

Answer

Fair forward = $1911. Market forward =$ 1950. Arbitrage profit = $39/oz. Buy spot + carry, sell forward.

Common Mistakes

!
Ignoring carrying costs (storage, financing, convenience yield). Fair forward ≠ spot price.
!
Confusing futures and forward prices. Futures are marked-to-market daily (introduces interest on daily settlement), making them slightly different from forwards.
!
Not specifying which leg is buy vs. sell in the arbitrage — always be explicit.

Practice Problems

Click "Show Answer" to reveal

S= $50, r=10%, T=0.5yr, no dividends. What is the fair forward price? If the market quotes F=$ 53, what is the arbitrage?

Hint: F = S·e^(rT). F_fair = 50·e^0.05.

A stock pays a $2 dividend in 3 months. S=$ 100, r=5%, T=1yr. What is the fair 1-year forward?

Hint: F = (S − PV(div)) × e^(rT). PV(div) = 2·e^(−0.05·0.25).

Back to Study Plan

Powers, Roots & Logarithms Volatility — Realized, Implied, & VIX