Financial Derivatives

Chapter 2
No-Arbitrage Valuation and Market Foundations

Graduate Program in Finance Management
Financial Derivatives • Graduate

No-Arbitrage Valuation and Market Foundations

Introduction

Derivative pricing begins with a simple but powerful proposition: two portfolios that generate identical future cash flows under every relevant state of the world should have the same current value. If they do not, a trader can buy the cheaper portfolio, sell the more expensive one, and lock in a risk-free profit. This process is arbitrage.

No-arbitrage valuation does not require investors to forecast returns accurately or to agree about the probability of future events. It requires consistency between prices, cash flows, financing arrangements, and trading opportunities. The framework supports pricing of forwards, futures, swaps, and options and provides the conceptual foundation for more advanced models such as binomial trees and Black–Scholes–Merton.

The analysis in this chapter assumes frictionless or approximately frictionless markets unless stated otherwise. In practice, transaction costs, taxes, funding spreads, collateral, short-sale restrictions, credit risk, and liquidity can weaken exact arbitrage relationships. These limitations do not make no-arbitrage theory irrelevant; they determine the size of pricing deviations that can persist.

Learning objectives

By the end of this chapter, you should be able to:

explain arbitrage, replication, the law of one price, and market completeness;
distinguish simple interest, effective compounding, and continuous compounding;
discount and compound cash flows consistently;
construct cash-and-carry and reverse-cash-and-carry strategies;
derive forward prices using cost of carry;
explain the role of short selling and funding in arbitrage trades;
use risk-neutral valuation in a one-period model;
derive and apply put–call parity;
evaluate assumptions, practical limitations, and governance implications of derivative pricing.

1. Core principles of no-arbitrage valuation

Arbitrage

An arbitrage opportunity is a trading strategy with:

zero or negative initial cost;
no possibility of a future loss; and
a strictly positive payoff in at least one possible state.

A stronger textbook definition describes an opportunity with zero initial net investment, nonnegative terminal payoff in every state, and a positive payoff in at least one state. Markets in which such opportunities are absent satisfy the no-arbitrage condition.

A risk-free profit is not the same as a high expected return. A risky investment may have a high expected return but can lose money. Arbitrage is defined by the absence of loss across all relevant outcomes, not by statistical expectations.

Law of one price

The law of one price states that identical cash flows must have identical prices when they are traded in the same market and under the same contractual conditions. If portfolio A and portfolio B have identical cash flows, then:

P_A = P_B

where P_A and P_B are their current prices. If P_A > P_B, buy B and sell A; if P_A < P_B, buy A and sell B. The positions offset in the future, leaving the initial price difference as a locked-in gain.

The law applies to cash-flow equivalence, not merely to similar economic exposure. A corporate bond and a government bond may both pay interest, but differences in default risk, liquidity, tax treatment, and embedded options mean they are not identical claims.

Replication

A replicating portfolio is a combination of traded assets whose future cash flows exactly match those of a target derivative. If a portfolio replicates a derivative, the derivative and the portfolio must have the same value under no arbitrage:

V₀^derivative = V₀^{replicating ext{ portfolio}}

Replication is central because it converts valuation into construction. Rather than estimating a derivative’s value directly, the analyst identifies traded instruments that reproduce its cash flows.

Dominance

If portfolio A produces at least as much as portfolio B in every state and strictly more in at least one state, while costing no more, A dominates B. A rational no-arbitrage market cannot assign a higher price to B. Dominance is particularly useful for establishing option bounds.

2. Time value of money and discounting

A future cash flow is generally worth less than the same amount received today because today’s funds can be invested. Let r be an annual interest rate and T the time in years.

Simple interest

Under simple interest, interest is not reinvested:

FV = PV(1 + rT)

Here, PV is present value, FV is future value, and r is the annual simple rate. The corresponding discounting formula is:

PV = FV ÷ (1 + rT)

Simple interest is appropriate mainly for short periods or instruments whose market convention specifies it. The product rT must be dimensionless: an annual rate must be multiplied by years.

Discrete compounding

If interest is compounded m times per year, the future value is:

FV = PV(1 + r/m)^mT

and:

PV = FV ÷ (1 + r/m)^mT

The rate r is the nominal annual rate convertible m times per year. The effective annual rate is:

EAR = (1 + r/m)^m − 1

For example, a 6% nominal rate compounded quarterly has an effective annual rate of (1 + 0.06/4)⁴ − 1 = 0.06136, or approximately 6.136%.

Continuous compounding

With continuous compounding:

FV = PV e^rT

and:

PV = FV e^−rT

where e is the base of natural logarithms. The continuously compounded rate corresponding to an effective annual rate EAR is:

r_c = ln(1 + EAR)

Continuous compounding is a convenient mathematical convention in derivative pricing. It does not mean that a bank literally credits interest at every instant; it is a limiting representation of increasingly frequent compounding.

Worked example: discounting

An institution will receive $1,000,000 in two years. The continuously compounded annual risk-free rate is 5%. Its present value is:

PV = 1,000,000e^−0.05×2 = 904,837.42

Thus, the cash flow is worth approximately $904,837 today, assuming the rate is appropriate, default-free for valuation purposes, and continuously compounded.

3. Short selling, borrowing, and lending

Arbitrage strategies frequently require selling an asset that the trader does not own. In a short sale, the trader borrows the asset, sells it, and later repurchases and returns it. The trader must compensate the lender for distributions such as dividends and may face margin requirements, recall risk, and borrowing fees.

Borrowing and lending are also essential. A cash-and-carry trade typically borrows money to buy an asset and carries it until delivery. A reverse-cash-and-carry trade sells or shorts the asset and lends the proceeds. These strategies are feasible only if financing and short selling are available at relevant rates.

In professional markets, there may be several interest rates rather than one universal risk-free rate: secured funding, unsecured borrowing, repo rates, collateral rates, and institution-specific funding rates. Consequently, an apparent theoretical arbitrage may be unprofitable after funding costs and margin requirements.

4. Forward prices and the cost of carry

A forward contract is an agreement made today to exchange an asset at a specified delivery price on a future date. Consider a non-income-producing asset with spot price S₀, continuously compounded annual interest rate r, and maturity T. Under no arbitrage, the forward price is:

F₀ = S₀e^rT

The strategy is a cash-and-carry trade:

borrow S₀ today;
buy one unit of the asset;
hold it until time T;
deliver it into the forward contract;
repay S₀e^rT.

The delivery price must equal the repayment amount. If the quoted forward price is higher than theoretical value, buy the asset and sell the forward. If it is lower, execute the reverse-cash-and-carry strategy, subject to the ability to short the asset.

Assets with known income

Suppose the asset provides a known cash income with present value I during the life of the forward. Buying the asset today gives ownership of that income, whereas a forward buyer does not receive it before delivery. The forward price becomes:

F₀ = (S₀ − I)e^rT

If the asset pays a known proportional yield q, continuously compounded, the formula is:

F₀ = S₀e^(r−q)T

For a foreign currency, q is often replaced by the foreign risk-free interest rate r_f:

F₀ = S₀e^(r−r_f)T

Here, S₀ is domestic-currency units per unit of foreign currency, and rates must use consistent compounding conventions.

Storage costs and convenience yield

For commodities, storage costs increase the cost of carrying the physical asset. If storage costs have present value U, a simplified formula is:

F₀ = (S₀ + U)e^rT

If storage costs are represented as a proportional continuous yield u, and convenience yield as y, then a common conceptual expression is:

F₀ = S₀e^(r+u−y)T

Convenience yield is the non-cash benefit of physically holding a commodity, such as assurance of production continuity. Unlike a contractual dividend, it is generally difficult to observe directly. Cost-of-carry formulas must therefore be interpreted carefully for commodities.

5. Risk-neutral valuation

Risk-neutral valuation is a pricing technique, not a claim that investors are actually risk neutral. In a no-arbitrage complete market, derivative prices can be calculated as discounted expected payoffs under a probability measure chosen so that traded assets earn the risk-free rate in expectation.

One-period binomial model

Assume an asset currently costs S₀. At time T, it will be worth either S_u in the up state or S_d in the down state. Let the continuously compounded risk-free discount factor be e^−rT. A derivative pays V_u or V_d in the two states.

A replicating portfolio contains ∆ units of the asset and B invested in a risk-free account. Its state payoffs satisfy:

∆S_u + Be^rT = V_u

∆S_d + Be^rT = V_d

Subtracting the equations gives:

∆ = (V_u − V_d) ÷ (S_u − S_d)

The amount invested in the risk-free asset is:

B = e^−rT(V_u − ∆S_u)

The derivative value is:

V₀ = ∆S₀ + B

Define the risk-neutral probability:

p^* = (e^rTS₀ − S_d) ÷ (S_u − S_d)

Provided S_d < e^rTS₀ < S_u, 0 < p^* < 1. The derivative can then be valued as:

V₀ = e^−rT[p^*V_u + (1−p^*)V_d]

The condition that the forward value of the current asset lies between the down and up values prevents an immediate arbitrage. If it fails, the model’s prices are internally inconsistent.

Worked example: a one-period call

Suppose S₀ = $100, S_u = $120, S_d = $90, the continuously compounded annual risk-free rate is 5%, maturity is one year, and a call has strike K = $100. Its payoffs are V_u = $20 and V_d = $0.

The risk-neutral probability is:

p^* = (e^0.05×100 − 90) ÷ (120 − 90) = (105.127 − 90) ÷ 30 = 0.5042

Therefore:

C₀ = e^−0.05(0.5042×20 + 0.4958×0) = $9.61 approximately

This probability is a pricing device. It need not equal the analyst’s estimate of the actual probability of an up move.

6. Put–call parity

European put–call parity follows from replicating cash flows. Consider a European call and put with the same underlying asset, strike K, and maturity T. Assume no dividends initially. Compare:

Portfolio A: one call plus cash equal to the present value of the strike;
Portfolio B: one put plus one unit of the underlying asset.

At maturity, if the asset price S_T exceeds K, both portfolios deliver one asset. If S_T is below K, both deliver K in cash. Thus, their current values must be equal:

C₀ + Ke^−rT = P₀ + S₀

Equivalently:

C₀ − P₀ = S₀ − Ke^−rT

where C₀ and P₀ are European call and put prices.

If the underlying pays known income with present value I, the parity relation becomes:

C₀ + Ke^−rT = P₀ + S₀ − I

If it pays a continuous proportional yield q, the asset value in the parity relationship is adjusted to S₀e^−qT:

C₀ + Ke^−rT = P₀ + S₀e^−qT

Put–call parity applies to European options with matching strike, maturity, underlying, and settlement terms. It does not directly apply to American options because early exercise creates different rights.

Example: detecting mispricing

Suppose S₀ = $50, K = $48, r = 4%, T = 0.5, no dividends, and observed prices are C₀ = $5.00 and P₀ = $2.20. Parity implies:

C₀ − P₀ = 5.00 − 2.20 = $2.80

The theoretical difference is:

S₀ − Ke^−rT = 50 − 48e^−0.04×0.5 = 50 − 47.05 = $2.95 approximately

The $0.15 discrepancy might suggest an arbitrage, but only after allowing for bid–ask spreads, commissions, financing, taxes, dividends, exercise and settlement conventions, and execution risk. A professional trader would construct the offsetting positions and test whether the net profit remains positive after all costs.

7. Option bounds and consistency checks

No-arbitrage also provides bounds. For a non-dividend-paying underlying, a European call must satisfy:

max(0, S₀ − Ke^−rT) ≤ C₀ ≤ S₀

The lower bound follows from call–bond dominance. A call cannot be worth less than the present value of its guaranteed intrinsic-value benchmark. The upper bound follows because a call gives no more than the right to acquire an asset that currently costs S₀.

A European put must satisfy:

max(0, Ke^−rT − S₀) ≤ P₀ ≤ Ke^−rT

These are necessary conditions, not complete pricing models. A price within the bounds may still be unreasonable because of volatility, liquidity, credit, or model considerations.

8. Market foundations and practical limitations

Completeness and uniqueness

In a complete market, every relevant contingent claim can be replicated using traded assets. If no arbitrage holds, the replicating price is unique. In an incomplete market, some claims cannot be perfectly replicated. No-arbitrage may then determine an interval of acceptable prices rather than one unique price; additional assumptions, preferences, or risk-management criteria are required.

Transaction costs and funding

The textbook equations generally assume zero transaction costs, unrestricted borrowing and lending at the same rate, unlimited liquidity, no taxes, continuous trading when required, and reliable settlement. Real markets violate these assumptions. Pricing should therefore use executable bid and ask prices, realistic funding curves, collateral agreements, margin rules, and balance-sheet constraints.

Model risk

A model can be internally consistent but economically inappropriate. Errors may arise from incorrect dividend forecasts, volatility assumptions, interest-rate curves, settlement calendars, or treatment of early exercise. Independent validation, sensitivity analysis, back-testing where appropriate, and clear documentation are essential controls.

Governance and ethical considerations

Derivative valuation affects financial statements, collateral calls, regulatory capital, executive incentives, and client fairness. Institutions should maintain segregation between trading, valuation control, risk management, and model validation. Valuation adjustments should not be concealed to improve reported performance. Ethical practice requires transparent disclosure of assumptions, uncertainty, liquidity limitations, and material model risks.

9. Applied case: an index forward discrepancy

The following is an illustrative case, not a report of a specific institution.

Background

A portfolio manager observes an equity index at 4,000. The continuously compounded annual financing rate is 5%, the maturity is three months, and the present value of dividends expected before maturity is 28 index points. A three-month forward is quoted at 4,040.

Problem

Determine whether the quote is consistent with a simple no-arbitrage cost-of-carry relationship.

Analysis

The theoretical forward price is:

F₀ = (S₀ − I)e^rT = (4,000 − 28)e^0.05×0.25

Since e^0.0125 ≈ 1.01258:

F₀ ≈ 3,972 × 1.01258 = 4,021.99

The quoted forward is about 18.01 points above the theoretical value. A simplified cash-and-carry strategy would buy the index exposure, finance it, receive the dividends, and sell the forward. However, the manager must check whether the index can be purchased precisely, whether dividends are certain, whether financing is available at the assumed rate, and whether trading costs and taxes exceed the discrepancy.

Approach and outcome

The trading desk estimates implementation costs of 7 points, funding spread costs of 6 points, and dividend uncertainty valued at 4 points. Total expected implementation cost is 17 points, leaving only approximately 1 point of theoretical excess. The trade is therefore not economically attractive after costs, despite appearing to offer an arbitrage before implementation adjustments.

Learning points

The case demonstrates that theoretical arbitrage is a benchmark, not an automatic profit. Accurate valuation requires cash-flow matching, executable prices, realistic financing, dividend treatment, and operational analysis. It also illustrates why market prices can deviate modestly from frictionless formulas without creating exploitable opportunities.

10. Best practices for professional application

Identify every contractual cash flow, including dividends, coupons, storage costs, taxes, and settlement payments.
Use consistent units, day-count conventions, compounding frequencies, and maturity dates.
Distinguish a theoretical mid-market value from an executable bid or ask price.
Verify that borrowing, lending, short selling, and collateral are actually available.
Build replication or arbitrage trades explicitly, state by state where relevant.
Include transaction costs, funding spreads, margin, liquidity, and credit adjustments.
Test formulas through limiting cases—for example, zero maturity, zero income, or zero interest rate.
Document assumptions and subject material models to independent validation.
Use sensitivity analysis for rates, dividends, volatility, funding, and settlement timing.
Treat risk-neutral probabilities as pricing constructs rather than forecasts of physical probabilities.

Chapter Summary

No-arbitrage valuation rests on the law of one price, replication, dominance, and the absence of risk-free profits. Time value of money converts between present and future values through simple, discrete, or continuous compounding. Forward prices follow from the cost of carrying an asset, adjusted for income, storage costs, and convenience yield. Short selling, borrowing, lending, and settlement access determine whether theoretical arbitrage can be implemented.

In a complete one-period market, a derivative can be replicated and valued either by solving for the replicating portfolio or by discounting its payoff under a risk-neutral probability. Put–call parity is a direct replication result and provides a powerful consistency check for European options. Bounds and parity relationships are necessary conditions, not substitutes for judgment about liquidity, funding, model risk, and market frictions. Professional valuation therefore combines mathematical consistency with disciplined governance and implementation analysis.

Key Terms

Arbitrage: A strategy with no possible loss and a positive gain in at least one state, requiring no net initial investment in the strict definition.

Law of one price: Identical cash flows should have identical prices.

Replication: Constructing a portfolio whose future cash flows match a target claim.

Dominance: A relationship in which one portfolio is never worse and is better in at least one state.

Present value: The current value of a future cash flow after appropriate discounting.

Future value: The value of a current amount after compounding.

Simple interest: Interest calculated without reinvesting prior interest.

Discrete compounding: Interest added at specified intervals during a period.

Continuous compounding: A limiting compounding convention represented by the exponential function.

Forward contract: An agreement to exchange an asset at a future date for a specified delivery price.

Spot price: The current price for immediate delivery of an asset.

Cost of carry: Net financing and holding costs associated with carrying an asset to a future date.

Cash-and-carry: Buying and carrying an asset while selling a forward against it.

Reverse cash-and-carry: Shorting or selling an asset, investing proceeds, and buying a forward.

Risk-neutral probability: A pricing probability under which discounted expected payoffs equal current values.

Complete market: A market in which every relevant claim can be perfectly replicated.

Put–call parity: The no-arbitrage relationship among European calls, puts, the underlying, and the present value of the strike.

Model risk: The possibility that a valuation model or its assumptions are unsuitable or incorrect.

Convenience yield: The non-cash benefit of physically holding a commodity.

Review Questions

5 Short Answer Questions

What conditions define an arbitrage opportunity?
Why does replication support derivative valuation?
Distinguish nominal and effective interest rates.
What is the purpose of a risk-neutral probability?
State the basic put–call parity relationship for a non-dividend-paying asset.

5 Long Answer Questions

Explain the law of one price and discuss its practical limitations in real markets.
Derive the forward-price formula for a non-income-producing asset using a cash-and-carry strategy.
Explain how short selling, borrowing, and lending support reverse-cash-and-carry arbitrage.
Derive the one-period binomial risk-neutral valuation formula from a replicating portfolio.
Discuss why model governance and funding assumptions are important in derivative valuation.

3 Analytical / Application Questions

An asset is priced at $80, the continuously compounded annual rate is 6%, and maturity is nine months. Calculate the theoretical forward price when the asset pays no income. Explain the arbitrage direction if the quoted forward is $86.
A European call and put have the same strike and maturity. Explain how you would test their prices for put–call parity when the underlying pays a known dividend.
A quoted forward appears mispriced by 1% relative to a frictionless cost-of-carry formula. Develop a decision framework to determine whether the discrepancy is a genuine arbitrage after funding, liquidity, margin, taxes, and execution costs.

Multiple Choice Questions with Answers

Question 1

Which condition is most directly associated with a strict arbitrage opportunity?

A. Positive expected return with some probability of loss

B. No possible loss and a positive payoff in at least one state

C. A return greater than the market average

D. A payoff that is positive only in the most likely state

Correct Answer: B. No possible loss and a positive payoff in at least one state

Question 2

A $1,000 cash flow is due in two years and the continuously compounded annual rate is 5%. What is its approximate present value?

A. $900.00

B. $904.84

C. $950.00

D. $1,102.50

Correct Answer: B. $904.84

Question 3

For a non-income-producing asset under continuous compounding, the theoretical forward price is:

A. S₀e^(rT)

B. S₀e^(−rT)

C. (S₀−I)e^(−rT)

D. S₀(1+rT) only

Correct Answer: A. S₀e^(rT)

Question 4

In a one-period binomial model, the risk-neutral probability is used primarily to:

A. Estimate the statistically expected probability of an up move

B. Guarantee that the asset price rises

C. Discount expected derivative payoffs at the risk-free rate

D. Measure the investor’s risk aversion

Correct Answer: C. Discount expected derivative payoffs at the risk-free rate

Question 5

Which portfolio replicates a European call plus the present value of its strike under put–call parity?

A. A put plus one unit of the underlying

B. A put minus one unit of the underlying

C. A call plus one unit of the underlying

D. A bond plus a short put

Correct Answer: A. A put plus one unit of the underlying

Question 6

If an asset pays a known income with present value I, the simple continuous-compounding forward-price formula is:

A. (S₀ + I)e^(rT)

B. (S₀ − I)e^(rT)

C. S₀e^((r+I)T)

D. (S₀ − I)e^(−rT)

Correct Answer: B. (S₀ − I)e^(rT)

Question 7

A market is complete when:

A. All assets have the same expected return

B. Every relevant contingent claim can be perfectly replicated

C. Short selling is prohibited

D. All derivatives are exchange traded

Correct Answer: B. Every relevant contingent claim can be perfectly replicated

Question 8

For a non-dividend-paying underlying, which is a valid lower bound for a European call?

A. S₀ + Ke^(−rT)

B. max(0, S₀ − Ke^(−rT))

C. max(0, Ke^(−rT) − S₀)

D. Ke^(rT) − S₀

Correct Answer: B. max(0, S₀ − Ke^(−rT))

Question 9

Why might a theoretical forward-price discrepancy fail to produce an actual arbitrage?

A. Arbitrage never uses financing

B. The law of one price applies only to stocks

C. Transaction costs, funding spreads, and liquidity constraints may absorb the discrepancy

D. Forward contracts have no maturity date

Correct Answer: C. Transaction costs, funding spreads, and liquidity constraints may absorb the discrepancy

Question 10

In a one-period replication, the hedge ratio Δ is calculated as:

A. (Vᵤ + Vd) ÷ (Sᵤ + Sd)

B. (Vᵤ − Vd) ÷ (Sᵤ − Sd)

C. (Sᵤ − Sd) ÷ (Vᵤ − Vd)

D. Vᵤ ÷ Sᵤ

Correct Answer: B. (Vᵤ − Vd) ÷ (Sᵤ − Sd)

Financial Derivatives
Chapter 2 of 15

Chapter 2 No-Arbitrage Valuation and Market Foundations