Derivative basics and pricing - No arbitrage relationships a...

Learning Outcomes

This article explains how arbitrage-free valuation underpins the pricing of basic derivative contracts for the CFA Level 1 exam, including:

• understanding what is meant by arbitrage, the no‑arbitrage condition, and the law of one price in efficient markets;
• relating the no‑arbitrage principle to the determination of fair values for forwards, futures, and simple options;
• interpreting and applying the concept of a replicating portfolio constructed from the reference asset and borrowing or lending;
• deriving and using the standard pricing formula for a forward contract on a non‑dividend‑paying asset, including continuous compounding of the risk‑free rate;
• determining whether a quoted forward or option price is consistent with its replicating portfolio and the law of one price;
• constructing clear long–short trading strategies that exploit mispriced derivatives to lock in risk‑free profits when arbitrage exists;
• assessing how borrowing costs, dividend payments, and other contract features may alter arbitrage relationships and exam calculations;
• evaluating short calculation-based and conceptual items that test recognition of arbitrage opportunities, interpretation of payoff diagrams, and correct application of replication logic in typical CFA Level 1 question formats.

CFA Level 1 Syllabus

For the CFA Level 1 exam, you are required to understand the principles behind arbitrage-free pricing of derivatives, with a focus on the following syllabus points:

• Define arbitrage and explain its role in financial markets
• Describe the no-arbitrage condition, and explain why derivative prices must preclude riskless profit
• Analyze and identify arbitrage opportunities in derivative pricing
• Compute fair values for basic derivative contracts using replication and the law of one price
• Distinguish between the price and value of a forward contract over its life
• Apply put–call parity and synthetic replication to check option prices for consistency

Test Your Knowledge

Attempt these questions before reading this article. If you find some difficult or cannot remember the answers, remember to look more closely at that area during your revision.

1. What best describes an "arbitrage opportunity" in derivative markets?
   1. A trade that has zero probability of loss but requires a large initial investment
   2. A strategy that earns a risk-free profit with no net investment and no risk
   3. Any strategy that earns a profit above the risk-free rate
   4. A trade that exploits high volatility in option prices
2. A 1‑year forward on a non‑dividend stock has a quoted forward price that allows a trader to earn a risk-free profit with no net investment. What does this imply about the market?
   1. The market is perfectly efficient
   2. The market violates the no‑arbitrage condition
   3. The risk-free rate must be negative
   4. The reference stock must be mispriced
3. Consider a forward on a non‑dividend‑paying stock when interest rates are positive. Which statement is most accurate?
   1. The forward price must always equal the spot price
   2. The forward price must always be less than the spot price
   3. The forward price should exceed the spot price in an arbitrage‑free market
   4. The relationship between spot and forward prices is unrelated to interest rates
4. Which statement best describes the "law of one price" in the context of derivatives?
   1. All derivatives on the same reference asset must trade at the same price
   2. Identical future cash flow streams must have the same price in efficient markets
   3. Derivatives with different maturities must have the same price
   4. The spot and forward prices must always be equal

Introduction

Pricing derivatives—contracts whose value depends on an associated asset—is based on the no-arbitrage principle. This ensures that no market participant can earn a guaranteed, risk-free profit by exploiting price inconsistencies. For CFA Level 1, you must understand how the concept of replication, together with arbitrage arguments, determines fair values for forward, futures, and option contracts.

Key Term: derivative
A financial contract whose value is derived from the price or level of a reference asset, rate, or index.

Key Term: reference asset
The asset, rate, or index on which a derivative contract is based (e.g., a stock, bond, equity index, currency, or interest rate).

Key Term: spot price
The current price at which the reference asset can be bought or sold for immediate delivery.

Key Term: arbitrage
The act of simultaneously buying and selling assets or contracts to exploit price differences and lock in a risk-free profit without net investment.

Key Term: no-arbitrage condition
The fundamental requirement that the prices of assets or derivatives must not permit the existence of arbitrage opportunities in efficient markets.

Key Term: law of one price
The rule stating that identical assets (or sets of future cash flows) must sell for the same price in efficient markets to rule out arbitrage.

Key Term: replication
The process of creating a synthetic payoff for a target contract using traded securities or other derivatives to match its risk and return profile.

Key Term: long position
A position that benefits when the price of the reference asset or derivative rises (e.g., owning the asset or being the buyer in a forward contract).

Key Term: short position
A position that benefits when the price of the reference asset or derivative falls (e.g., selling the asset short or being the seller in a forward contract).

The central idea is that any derivative can be viewed as a package of more basic cash flows. If you can build an identical package using traded instruments (such as the reference asset and borrowing/lending), then the derivative’s fair price must equal the cost of that replicating portfolio. If it does not, arbitrageurs can set up long–short strategies to lock in risk‑free profits, and their trades will quickly push prices back toward the no‑arbitrage level.

This logic is most transparent for forward contracts, then extends naturally to options via replicating portfolios and put–call parity. Typical exam items will ask you to:

• compute a no‑arbitrage forward price;
• compare it with an observed market price;
• decide which side of the derivative to take (long or short); and
• specify the offsetting position in the reference asset and borrowing or lending.

Understanding this sequence is far more important than memorizing formulas. Being able to articulate the economic intuition—what you are effectively doing when you enter into a derivative—is key for conceptual questions.

Before turning to specific contracts, it is useful to clarify a few additional concepts.

Key Term: payoff
The cash flow received (or paid) at expiration or settlement of a derivative, as a function of the reference asset’s price.

Key Term: arbitrage-free price
A theoretical price for an asset or derivative that is consistent with the no-arbitrage condition; if the market price equals this value (before transaction costs), no riskless profit can be locked in.

Derivative Pricing and Replication

The price of most derivatives is determined by constructing a replicating portfolio—using traded assets and riskless borrowing or lending—that mimics the contract's payoff. If the price of the derivative differs from the cost of the replicating portfolio, arbitrage is possible and market forces will act to eliminate the discrepancy.

Replication depends heavily on two ideas:

• The law of one price: two portfolios that generate exactly the same future cash flows in all states of the world must have the same current value.
• The time value of money: cash flows at different dates must be adjusted (via discounting or compounding) using the risk-free rate when comparing them.

The simplest derivatives to analyze in this framework are forward and futures contracts.

Key Term: forward contract
A customized agreement to buy or sell an asset at a fixed price at a specified future date.

Key Term: forward price
The agreed‑upon price in a forward contract at which the reference asset will be exchanged at maturity.

Key Term: forward value
The current economic value (gain or loss) of an existing forward contract to one party, given current market conditions. Under no‑arbitrage, this value is zero at initiation but generally becomes non‑zero as time passes.

Key Term: futures contract
A standardized, exchange‑traded forward contract with daily settlement through a clearinghouse.

A long forward position commits you to buy the asset at the forward price on the settlement date; the short commits you to sell. At maturity, the long’s payoff per unit is:

and the short’s payoff is:

where:

• $S_T$ is the spot price at maturity, and
• $F_{0,T}$ is the delivery price agreed at time 0 for delivery at time $T$.

Economically, entering a long forward today is very similar to:

• agreeing to buy the asset in the future, and
• implicitly borrowing to finance the purchase between today and maturity.

You “lock in” a future buying price, but no money changes hands at initiation (ignoring transaction costs and margin for futures).

There are two key notions you need to distinguish:

• Forward price: the fixed delivery price agreed at initiation and used for settlement.
• Forward value: the contract’s economic value (gain or loss) at a given time, which is zero at initiation under no‑arbitrage but generally non‑zero afterward.

At inception, the forward price is chosen so that the forward’s initial value is zero to both parties. As time passes and the spot price and interest rates change, the value to long and short diverges, even though the original forward price remains fixed. This distinction is examined repeatedly in the Level 1 curriculum.

Pricing a Forward Contract: Non-Dividend Asset

For a non‑dividend‑paying stock (or other asset with no income or storage costs), the fair value of a forward contract, initiated at time $t = 0$ and settling at time $T$, is derived from a simple replication argument.

Assume:

• constant risk‑free rate $r$ over the period;
• no transaction costs, taxes, or default risk;
• borrowing and lending are possible at the same risk‑free rate;
• the asset pays no income (no dividends, coupons, or other benefits).

Compare two strategies that deliver one unit of the asset at time $T$:

• Strategy A: Buy the asset today at $S_0$ and finance it by borrowing at the risk‑free rate.
• Strategy B: Enter a long forward to buy the asset at $F_{0,T}$ at time $T$.

Under Strategy A, the time‑$T$ cash flows are:

• you own the asset (worth $S_T$);
• you must repay the loan: the future value of $S_0$ at the risk‑free rate.

With continuous compounding at rate $r$, the repayment is:

With simple annual compounding (if $T$ is in years), it is:

Under Strategy B, at time $T$:

• you pay $F_{0,T}$ and receive the asset (worth $S_T$).

To rule out arbitrage, the net cost of Strategy A at maturity must equal the net cost of Strategy B. Because both strategies deliver the same asset at time $T$, the law of one price implies:

• with continuous compounding:

• with annual compounding (as is more common in Level 1 questions):

The exam will tell you which compounding convention to use and how $T$ is measured (e.g., $T = 0.5$ for six months if using annualized rates).

If the actual market forward price differs from this no‑arbitrage level, arbitrageurs can lock in a risk-free profit by combining positions in the forward, the asset, and borrowing/lending.

Key Term: cash‑and‑carry arbitrage
An arbitrage strategy in which you buy the asset in the spot market, finance it (borrow), and simultaneously sell a forward when the forward price is too high.

Key Term: reverse cash‑and‑carry arbitrage
An arbitrage strategy in which you short‑sell the asset, invest the proceeds, and simultaneously buy a forward when the forward price is too low.

Worked Example 1.1

A stock trades at $100. The one-year risk-free rate is 4% (continuously compounded), and there are no dividends. What is the no-arbitrage 1-year forward price? How would you exploit mispricing if the actual quoted forward price is $108?

Answer:
The forward price is:

$$F_{0,1} = 100 \times e^{0.04} \approx 100 \times 1.0408 = \$104.08$$

Because the quoted forward price ($108) is above $104.08, the forward is overpriced relative to the no-arbitrage level.

To exploit this:

• At $t = 0$:
  • Borrow $100 at 4% (continuously compounded).
  • Use the $100 to buy the stock in the spot market.
  • Enter a short forward at $108.
• At $T = 1$:
  • Deliver the stock into the forward and receive $108.
  • Repay the loan: 100 \times e^{0.04} = \104.08$.
  • Risk‑free profit: 108 - 104.08 = \3.92$ per share.

This is a cash‑and‑carry arbitrage: you “carry” the asset after buying it in the spot market and have locked in the selling price via the forward.

If instead the market forward price were below $104.08 (too low), you would reverse the trade:

• Short the stock today and invest the proceeds at the risk‑free rate.
• Go long the underpriced forward.
• At maturity, use the forward to obtain the stock at the cheap forward price and close the short sale, locking in a risk‑free profit.

This is called a reverse cash‑and‑carry arbitrage.

Being able to describe the step‑by‑step arbitrage strategy (timing, direction of trades, and cash flows) is important for conceptual exam questions even when numbers are simple.

Adjustments for Income and Carrying Costs

Many reference assets generate income (e.g., dividends on equities, coupons on bonds) or involve costs (e.g., storage, insurance for commodities). These affect the fair forward price because the spot holder earns or pays these flows, whereas the forward holder typically does not.

Key Term: cost of carry
The net cost of holding an asset over the life of a derivative, including financing costs plus storage and other out‑of‑pocket expenses, minus any benefits such as income or convenience yield.

Key Term: convenience yield
A non‑monetary benefit from physically holding a commodity (e.g., security of supply), which effectively reduces the net cost of carry.

The general idea:

• Any cash flows paid by the asset (dividends, coupons) reduce the fair forward price because spot holders receive them but forward buyers do not.
• Any costs of holding the asset (storage, insurance) increase the fair forward price because spot holders pay them.

For a stock with known discrete cash dividends during the life of the forward, the discrete‑compounding formula becomes:

where $PV(\text{dividends})$ is the present value (discounted at appropriate risk‑free rates) of all dividends to be paid before time $T$.

If instead the stock pays a continuous dividend yield at rate $q$ (as is sometimes assumed for stock indexes), the continuous‑compounding formula is:

Key Term: dividend yield
The ratio of expected annual dividends over the current price for an equity or equity index, often denoted $q$ in forward pricing formulas.

Dividends lower the fair forward price because they are received only by the spot holder, not by the buyer of the forward.

Worked Example 1.2

A stock is priced at $50 today and is expected to pay a $1 dividend in 6 months. The 1‑year risk‑free rate (simple annual compounding) is 5%, and the 6‑month risk-free rate is 4%. What is the no‑arbitrage 1‑year forward price?

Answer:
First find the present value of the 6‑month dividend:

$$PV(\text{dividend}) = \frac{1}{1 + 0.04 \times 0.5} = \frac{1}{1.02} \approx 0.9804$$

Adjust the spot price for the present value of the dividend:

$$S_0 - PV(\text{div}) = 50 - 0.9804 = 49.0196$$

Grow this amount at the 1‑year risk‑free rate:

$$F_{0,1} = 49.0196 \times (1 + 0.05) = 49.0196 \times 1.05 \approx \$51.47$$

A quoted forward price significantly above or below $51.47 would imply an arbitrage opportunity using a combination of stock, forward, and borrowing/lending.

Currency Forwards and Interest Rate Differentials

Forward pricing also applies to currencies. Here, both currencies have interest rates, so holding one currency instead of another involves an opportunity cost or benefit.

For a forward on a foreign currency (domestic currency = D, foreign = F), with no transaction costs, the no‑arbitrage forward exchange rate (domestic price of 1 unit of foreign currency) is:

where:

• $S_0$ is the current spot exchange rate (D per 1 F),
• $r_D$ is the domestic risk‑free rate,
• $r_F$ is the foreign risk‑free rate,
• $T$ is the time (in years).

Intuition:

• Borrow foreign currency and convert to domestic today versus
• Borrow domestic currency and buy foreign on a forward.

These strategies must yield the same domestic amount at $T$, otherwise arbitrage exists. The foreign interest rate acts like a dividend yield (it reduces the forward price), because holding foreign currency earns the foreign risk‑free rate.

Worked Example 1.3

Suppose the spot exchange rate is 1.2000 USD/EUR. The 1‑year USD risk‑free rate is 3% and the 1‑year EUR risk‑free rate is 1% (simple annual compounding). What is the 1‑year no‑arbitrage forward rate USD/EUR?

Answer:
Apply the currency forward formula:

$$F_{0,1} = 1.2000 \times \frac{1.03}{1.01}$$ $$F_{0,1} = 1.2000 \times 1.01980198 \approx 1.2238 \ \text{USD/EUR}$$

A quoted forward rate above 1.2238 would be too high, and a reverse cash‑and‑carry strategy using borrowing and lending in the two currencies would generate arbitrage profits. A forward rate below 1.2238 would be too low and would justify the opposite arbitrage direction.

While Level 1 will not ask you to build the full multi‑step currency arbitrage in detail, you must recognize how interest rate differentials affect fair currency forward rates.

No-Arbitrage, the Law of One Price, and Forward Valuation over Time

So far we have focused on the forward price at initiation. The value of a specific forward contract changes over time as the spot price and interest rates move.

At initiation (time 0), the contract is set so that its value is zero to both parties:

Later, at time $t < T$, the current spot price is $S_t$, but the contract still obliges the long to pay the fixed delivery price $F_{0,T}$ at time $T$. To find the contract’s value at time $t$ to the long, consider:

• Buying the asset today for $S_t$, and
• Financing the future payment $F_{0,T}$ by borrowing its present value.

With simple annual compounding at risk‑free rate $r$, the present value at time $t$ of the future payment $F_{0,T}$ is:

With continuous compounding:

The value of the existing forward to the long at time $t$ is then:

• with simple compounding:

• with continuous compounding:

Intuition:

• $S_t$ is what the asset costs now.
• The present value term is what the long is effectively committed to pay at maturity.
• If that commitment is cheap relative to the current spot price (i.e., $S_t$ is high), the contract has positive value to the long; if it is expensive, the contract has negative value.

The value to the short is simply the negative of the above, because the long’s gain is the short’s loss:

At maturity, $t = T$, discounting disappears and:

which is just the payoff formula introduced earlier.

The same no‑arbitrage reasoning that linked the original forward price to the initial spot price and interest rate also links the current spot price to the current fair forward price $F_{t,T}$ at time $t$ (for a new contract starting at $t$). For a non‑dividend‑paying asset:

• with annual compounding:

If the existing contract’s delivery price $F_{0,T}$ differs from this current theoretical forward price, the contract’s value $V_t(T)$ will be non‑zero.

Worked Example 1.4

A non‑dividend‑paying stock currently trades at $52. Six months ago you entered into a 1‑year long forward contract at a forward price of $54. The annual risk‑free rate is 5% with simple annual compounding. What is the value of your forward position today?

Answer:
Here:

• $T = 1$ year,
• $t = 0.5$ years have passed since initiation,
• $S_t = 52$,
• $F_{0,T} = 54$,
• $r = 5\%$.

The time remaining is $T - t = 0.5$ years. The present value at time $t$ of the delivery price is:

$$PV_t(F_{0,T}) = \frac{54}{(1 + 0.05)^{0.5}} \approx \frac{54}{1.0247} \approx 52.73$$

The value to the long is:

$$V_t(T) = S_t - PV_t(F_{0,T}) = 52 - 52.73 \approx -0.73$$

So the forward has a value of about –$0.73 to the long (and +$0.73 to the short). If the two parties wished to settle the contract today, the long would have to pay approximately $0.73 per share to the short.

The exam may ask you both to compute fair forward prices and to value an existing contract at an intermediate date.

No-Arbitrage, the Law of One Price, and Option Pricing

Options also rely on no‑arbitrage and replication arguments, although the payoffs are asymmetric and state dependent.

Key Term: option
A derivative granting the right, but not the obligation, to buy or sell an associated asset at a preset price before or at expiration.

Key Term: call option
An option that gives the holder the right to buy the reference asset at a specified exercise (strike) price.

Key Term: put option
An option that gives the holder the right to sell the reference asset at a specified exercise (strike) price.

Key Term: European option
An option that can be exercised only at expiration.

Key Term: American option
An option that can be exercised at any time up to and including expiration.

Key Term: exercise (strike) price
The fixed price at which the option holder may buy (call) or sell (put) the reference asset if the option is exercised.

Key Term: exercise value
The immediate exercise payoff of an option: for a call, $\max(0, S_0 - K)$; for a put, $\max(0, K - S_0)$.

Key Term: time value
The portion of an option’s price above its exercise value, reflecting the possibility of favorable future price movements before expiration.

Key Term: moneyness
A description of the relationship between the current spot price and the strike price of an option (e.g., in the money, at the money, out of the money).

• A call is in the money if $S_0 > K$.
• A put is in the money if $S_0 < K$.

Arbitrage‑free pricing for options also rests on the law of one price: if two positions provide identical future cash flows in all states of the world, they must cost the same today. The simplest and most important relationship of this type in the Level 1 curriculum is put–call parity.

Key Term: put–call parity
A no‑arbitrage relationship linking the prices of European calls and puts with the same strike and maturity, the reference asset price, and the present value of the strike.

Put–Call Parity (Non-Dividend-Paying Stock)

For a non‑dividend‑paying stock, European call and put options with the same strike price $K$ and maturity $T$ satisfy:

where:

• $c_0$ = current price of the European call;
• $p_0$ = current price of the European put;
• $S_0$ = current stock price;
• $K/(1 + r)^T$ = present value of paying the strike $K$ at maturity (assuming simple annual compounding).

Equivalently:

Interpretation: two portfolios with the same payoff at $T$:

• Portfolio A: Long call + risk‑free bond that will grow to $K$ at time $T$.
• Portfolio B: Long put + long stock.

At expiration:

• If $S_T > K$:
  • Portfolio A: call pays $S_T - K$; bond pays $K$ → total $S_T$.
  • Portfolio B: put expires worthless; stock is worth $S_T$ → total $S_T$.
• If $S_T \leq K$:
  • Portfolio A: call expires worthless; bond pays $K$ → total $K$.
  • Portfolio B: put pays $K - S_T$; stock worth $S_T$ → total $K$.

Since the portfolios have identical payoffs in all possible states, the law of one price requires that their current costs be equal. Any violation of put–call parity implies an arbitrage opportunity.

Key Term: synthetic forward
A position created using options and/or other instruments that replicates the payoff of a forward contract.

Put–call parity is one of the most frequently tested arbitrage relationships and is central to understanding synthetic forwards.

Worked Example 1.5

A non‑dividend‑paying stock trades at $40. The risk‑free rate is 5% (simple annual compounding). A 1‑year European call with strike $42 is priced at $3, and the corresponding 1‑year European put with the same strike is priced at $5. Are the options correctly priced according to put–call parity?

Answer:
Compute the present value of the strike:

$$PV(K) = \frac{42}{1.05} = \$40$$

Put–call parity requires:

$$c_0 - p_0 = S_0 - PV(K)$$

Left‑hand side: $c_0 - p_0 = 3 - 5 = -2$
Right‑hand side: $S_0 - PV(K) = 40 - 40 = 0$

The two sides are not equal, so parity is violated. The put is too expensive relative to the call (or equivalently, the call is too cheap relative to the put), providing an arbitrage opportunity.

An arbitrageur could:

• Sell (write) the overpriced put.
• Buy the underpriced call.
• Take appropriate positions in the stock and the risk‑free bond so that the net position replicates one side of the parity equation.

In an exam question, you would typically be asked to identify which option is mispriced and in which direction (overpriced or underpriced) and to specify whether to buy or sell it.

Adjustments to Put–Call Parity for Dividends

If the stock pays known dividends before option expiration, put–call parity must be adjusted to reflect the present value of those dividends (because the long stock in Portfolio B receives them, but the call and put do not):

or:

If the stock or index has a continuous dividend yield $q$ (and using continuous compounding), parity can be written as:

Level 1 questions will clearly state whether there are dividends and whether they are to be modeled via discrete cash flows or a continuous yield.

Options and Synthetic Forwards

Put–call parity also implies a useful relationship between options and forwards. Rearranging the basic parity formula for a non‑dividend‑paying stock:

The right‑hand side is the present value of a long forward contract with delivery price $K$: it equals the value today of being long the stock and short a risk‑free bond that will pay $K$ at maturity.

Therefore:

• Long call + short put (same $K, T$) replicates a long forward.
• Short call + long put (same $K, T$) replicates a short forward.

This leads to an important consistency requirement: the market price of an actual forward must equal the cost of the synthetic forward constructed from options and bonds. If not, arbitrage opportunities exist.

Worked Example 1.6

A non‑dividend‑paying stock is at $100. The 1‑year risk‑free rate is 5% (simple annual compounding). A 1‑year forward on the stock trades at $106. A 1‑year European call and put, both with strike $105, are priced at $7 and $6, respectively. Is there an arbitrage opportunity between the forward and the synthetic forward created from options?

Answer:
First, compute the theoretical no‑arbitrage forward price:

$$F_{0,1}^{\text{theoretical}} = 100 \times 1.05 = \$105$$

The actual forward trades at $106, slightly above the theoretical level.

Now compute $c_0 - p_0$:

$$c_0 - p_0 = 7 - 6 = \$1$$

Compute $S_0 - PV(K)$:

$$PV(K) = \frac{105}{1.05} = 100 \quad \Rightarrow \quad S_0 - PV(K) = 100 - 100 = 0$$

Put–call parity requires $c_0 - p_0 = S_0 - PV(K) = 0$. Instead, $c_0 - p_0 = 1$, so the combination “long call + short put” (synthetic forward) has a value of $1, rather than 0. The synthetic forward is overpriced relative to the stock plus financing.

An arbitrageur could:

• Sell the overpriced synthetic forward (short call, long put).
• Buy the cheaper equivalent (long stock and borrowing/lending position).

If the actual traded forward is also misaligned with the synthetic forward, a three‑way arbitrage involving the forward, options, and stock is possible. In the Level 1 exam, you are typically expected to recognize that an inconsistency exists and to identify which side (actual forward or synthetic forward) is relatively expensive or cheap.

Conceptual Arbitrage with Options

Key Term: mark‑to‑market
The process of adjusting the stated value of a derivative or other position to reflect its current market value.

Under the no-arbitrage condition, an option must be fairly priced relative to any portfolio that replicates its payoff. If the option price deviates, arbitrageurs can trade the option against its replicating portfolio.

Worked Example 1.7

Suppose a one-period European call option on a non-dividend‑paying stock trades at a price inconsistent with its replicating portfolio of stock and risk-free bond. What does this imply, and how could an arbitrageur respond?

Answer:
It implies that an arbitrage opportunity exists.

• If the option is overpriced relative to the replicating portfolio:
  • Sell (write) the call.
  • Buy the cheaper replicating portfolio (a specific combination of stock and risk‑free bond that matches the call’s payoff).
  • The initial difference between the option premium received and the cost of the replicating portfolio is locked‑in profit; future payoffs offset exactly.
• If the option is underpriced:
  • Buy the call.
  • Short‑sell the stock and adjust borrowing or lending so that the overall position replicates the reverse of the call’s payoff.
  • Again, the mismatch between the option price and the replicating portfolio’s cost produces risk‑free profit.

Because the option and the replicating portfolio have identical payoffs in all states, the law of one price requires them to have the same cost. Any mismatch signals mispricing and hence arbitrage.

In the Level 1 curriculum, you are not required to derive option‑pricing models in detail, but you must understand this logic qualitatively and apply put–call parity numerically.

Option Price Bounds and No-Arbitrage Inequalities

Put–call parity leads to bounds that any arbitrage‑free option price must satisfy. For a non‑dividend‑paying stock:

• A European call must satisfy:

• A European put must satisfy:

These inequalities state that:

• A call cannot be worth less than its exercise value, and cannot be worth less than $S_0 - PV(K)$.
• A put cannot be worth less than its exercise value, and cannot be worth less than $PV(K) - S_0$.

If actual option prices violate these bounds, simple arbitrage strategies (buying or selling the under‑ or over‑priced option and hedging with the stock and bond) can be used to lock in risk‑free profits.

Worked Example 1.8

A non‑dividend‑paying stock trades at $60. The risk‑free rate is 5% (simple annual compounding). A 1‑year European call with strike $55 is quoted at $2. Is this consistent with the lower bound for the call price?

Answer:
Compute the present value of the strike:

$$PV(K) = \frac{55}{1.05} \approx 52.38$$

Lower bound from parity:

$$S_0 - PV(K) = 60 - 52.38 = 7.62$$

Exercise value (if exercised immediately, hypothetically):

$$\max(0, S_0 - K) = \max(0, 60 - 55) = 5$$

The theoretical lower bound is the maximum of these two:

$$\text{Lower bound} = \max(5, 7.62) = 7.62$$

But the call is trading at $2, far below $7.62. This is inconsistent with no‑arbitrage.

An arbitrageur could:

• Buy the underpriced call for $2.
• Short‑sell the stock at $60.
• Invest the net proceeds so that at maturity the portfolio’s payoff matches the call’s payoff but the initial cash flow is positive.

The exact hedge proportions are beyond Level 1 expectations, but recognizing that the call is too cheap relative to its lower bound is sufficient for exam purposes.

Exam Warning: Contract Details and Common Pitfalls

Key Term: futures contract
A standardized forward contract traded on an exchange, marked to market daily through a clearinghouse.

Ignoring contract details is a frequent source of mistakes. When applying no‑arbitrage pricing:

• Adjust for dividends, coupons, or other income the reference asset pays before expiration. Forgetting to subtract the present value of income will overstate the forward or futures price.
• Incorporate borrowing and lending costs correctly; check whether rates are simple or continuously compounded and how time to maturity is measured.
• Distinguish clearly between forward price (delivery price) and forward value (economic gain or loss). The forward price is fixed at contract initiation, but the contract value changes over time.
• For options, check whether the question states European or American exercise. Put–call parity holds exactly for European options on non‑dividend‑paying stocks. For American options, parity provides inequalities rather than a strict equality.
• Remember that futures are similar to forwards in terms of pricing under Level 1 assumptions (ignoring daily settlement effects), but in practice their daily mark‑to‑market can cause small differences when interest rates are uncertain.

Careful reading of the question stem and attention to what is given (and what is not) is essential in typical CFA Level 1 items.

Summary

The no-arbitrage principle dictates that derivative prices be set so as to rule out the possibility of guaranteed, riskless profit without net investment. By constructing replicating portfolios, the fair values for forwards, futures, and options can be derived and checked for consistency.

For non‑dividend‑paying assets, the fair forward price equals the spot price grown at the risk‑free rate to maturity. Income, storage costs, and convenience yields are incorporated via the cost‑of‑carry framework. The difference between a forward contract’s price and its value over time is critical: the contract is priced so its initial value is zero, but its value then fluctuates with the asset price and interest rates.

For options, the law of one price is captured succinctly in put–call parity, which links call and put prices to the reference asset and the risk‑free rate. Violations of parity or inconsistencies between actual and synthetic forwards created from options reveal arbitrage opportunities. Option price bounds and exercise value relationships provide quick tests for mispricing.

For the CFA exam, you must be able to:

• recognize arbitrage situations;
• apply replication logic;
• use the standard forward pricing formulas (with and without income);
• apply put–call parity (with or without dividends); and
• compute arbitrage‑free prices and values for basic derivative contracts.

Key Point Checklist

This article has covered the following key knowledge points:

• Define arbitrage and explain the no-arbitrage condition as central to derivative pricing.
• Explain the law of one price and how it supports replication and arbitrage-free valuation.
• Distinguish between the price and value of a forward contract over its life.
• Calculate fair forward and futures prices on non‑dividend and dividend‑paying assets, and on currencies.
• Value an existing forward contract at an intermediate time using current spot prices and interest rates.
• Interpret and apply put–call parity to check option prices for consistency and to construct synthetic forwards.
• Use exercise value and lower/upper bounds to detect obvious mispricing in option markets.
• Identify arbitrage opportunities when forwards or options are mispriced relative to their replicating portfolios.
• Outline cash‑and‑carry and reverse cash‑and‑carry strategies to exploit forward mispricing in exam scenarios.
• Recognize the role of dividends, carrying costs, and contract features (European vs American, maturity, strike) in no‑arbitrage relationships.

Key Terms and Concepts

• derivative
• reference asset
• spot price
• arbitrage
• no-arbitrage condition
• law of one price
• replication
• long position
• short position
• payoff
• arbitrage-free price
• forward contract
• forward price
• forward value
• futures contract
• cash‑and‑carry arbitrage
• reverse cash‑and‑carry arbitrage
• cost of carry
• convenience yield
• dividend yield
• option
• call option
• put option
• European option
• American option
• exercise (strike) price
• exercise value
• time value
• moneyness
• put–call parity
• synthetic forward
• mark‑to‑market