CFA Level I: Derivative Pricing and Valuation

Derivative pricing is the engine behind trillions in global financial transactions, from hedging commodity risks to speculating on equity movements. For you as a CFA candidate or finance professional, a firm grasp of these principles is non-negotiable; it allows you to value contracts accurately, identify arbitrage opportunities, and make informed investment and risk management decisions.

The No-Arbitrage Foundation

All modern derivative pricing rests on the principle of no-arbitrage, which states that two assets with identical future cash flows must have the same price today. If this were not true, a risk-free profit—an arbitrage opportunity—would exist, and traders would exploit it until prices align. This principle is not an assumption about market efficiency but a logical force that dictates prices in liquid markets. For example, if a stock is trading at $100inNewYorkand$102 in London, simultaneous buying and selling creates risk-free profit, forcing the prices to converge. In derivative pricing, we construct "replicating portfolios" of basic assets (like the underlying and a risk-free bond) that mimic the derivative's payoffs. The derivative's price must equal the cost of this replicating portfolio; otherwise, arbitrage emerges. This logic is the bedrock for every model that follows.

Pricing Forward Contracts: The Cost of Carry Model

A forward contract is an agreement to buy or sell an asset at a predetermined price on a future date. Its fair price is not a forecast but a calculated value that prevents arbitrage, derived using the cost of carry model. This model accounts for the costs and benefits of holding the underlying asset until the forward's delivery.

The basic formula for the forward price $F_0$ of an asset with no interim income (like dividends) is: $$F_0=S_0{e}^{rT}$$ Where $S_0$ is the current spot price, $r$ is the risk-free interest rate, and $T$ is the time to delivery. The term $e^{rT}$ represents the future value of financing the purchase of the asset today. If the asset provides income (e.g., dividends or yield), it reduces the cost of carry. For a stock index with a continuous dividend yield $q$, the formula adjusts to: $$F_0=S_0{e}^{(r-q)T}$$ Consider a practical scenario: a stock trades at $50,paysa2$F0 = 50 \times e^{(0.05-0.02) \times 1} = 50 \times e^{0.03} \approx 51.52MATHINLINE14_53, you could arbitrage by shorting the forward, borrowing $50 to buy the stock, collecting dividends, and delivering the stock in one year for a risk-free profit.

Futures Versus Forwards: Understanding Price Differences

While futures and forwards are both binding agreements for future delivery, their pricing can diverge due to institutional differences. The key distinction is that futures contracts are standardized, exchange-traded, and marked-to-market daily, while forwards are customized over-the-counter contracts. This daily settlement of gains and losses on futures introduces funding implications. If interest rates are constant and predictable, futures and forward prices for the same asset and maturity are theoretically identical. However, when interest rates fluctuate, a correlation between asset prices and interest rates creates a price difference.

If the underlying asset price is positively correlated with interest rates (e.g., stock indices), rising prices lead to daily gains on a long futures position. These gains can be reinvested at higher rates, making futures slightly more attractive than forwards. Consequently, futures prices will tend to be higher than forward prices in this scenario. Conversely, with negative correlation, forward prices may be higher. For most exam and practical purposes, especially for short-dated contracts, the difference is often minimal, but you must recognize it as a source of potential valuation discrepancy.

Option Valuation: From Parity to Pricing Models

Options grant the right, but not the obligation, to buy (call) or sell (put) an asset at a set strike price. Their valuation is more complex than forwards, as it involves probability and volatility.

First, put-call parity establishes a no-arbitrage relationship between European call and put options (with the same strike and expiry) and their underlying asset. The formula is: $$C+PV(K)=P+S_0$$ Where $C$ is the call price, $P$ is the put price, $S_0$ is the stock price, and $PV(K)$ is the present value of the strike price $K$. This parity shows that a protective put (stock plus put) is equivalent to a fiduciary call (call plus bond). If this equation is violated, arbitrage is possible. For instance, if $C+PV(K)>P+S_0$, you would short the call, borrow $PV(K)$, and buy the put and the stock, locking in a risk-free profit.

Several factors affect an option's premium: the underlying price, strike price, time to expiration, risk-free rate, volatility, and dividends. Volatility is paramount—it measures the uncertainty of the underlying's return and directly increases an option's value due to the asymmetric payoff (limited loss, unlimited gain potential).

To price options numerically, we use models. The binomial option pricing model is a discrete-time, intuitive approach. It constructs a lattice of possible future asset prices over time. At each node, the option value is calculated based on the probability-weighted expected future value, discounted at the risk-free rate. This model is versatile and can handle American options (which allow early exercise). A one-period binomial model involves calculating an up-factor $u$, down-factor $d$, and risk-neutral probability $p=\frac{e^{rT}-d}{u-d}$. The option price is then $e^{-rT}[p\times V_u+(1-p)\times V_d]$.

The Black-Scholes-Merton model is a continuous-time, closed-form solution for European options on non-dividend-paying stocks. Its famous formula for a call option is: $$C=S_{0}N(d_1)-K{e}^{-rT}N(d_2)$$ where $$d_1=\frac{\ln (S_0/K)+(r+{\sigma }^2/2)T}{\sigma \sqrt{T}}$$ and $$d_2=d_1-\sigma \sqrt{T}$$ Here, $N()$ is the cumulative normal distribution, and $\sigma$ is volatility. The model assumes constant volatility and interest rates, no dividends, and lognormally distributed asset prices. It revolutionized finance by providing a practical way to estimate option values, with $N(d_2)$ representing the risk-neutral probability of exercise.

Swap Valuation: A Portfolio of Forward Contracts

A swap is a series of exchanges of cash flows, such as exchanging fixed for floating interest payments. The key to valuation is recognizing that a swap can be decomposed into a portfolio of forward contracts. For a plain vanilla interest rate swap, the fixed rate is set so that the swap's initial value is zero. To value it after inception, you calculate the net present value (NPV) of the remaining cash flows.

Specifically, you value the floating leg by noting that immediately after a payment, its value equals the notional principal (it "resets to par"). The fixed leg is valued by discounting its known future payments. The swap value to the fixed-rate payer is: Value = PV(floating leg) - PV(fixed leg). Since each exchange can be viewed as a forward contract on interest rates, the swap is essentially a bundle of forwards, and its valuation relies on the same no-arbitrage, present value logic applied repeatedly over multiple periods. For a currency swap, the principle is similar, but it involves exchanging principal amounts in different currencies, treated as a portfolio of foreign exchange forwards.

Common Pitfalls

1. Misapplying the Cost of Carry Model: A frequent error is using the wrong rate or forgetting to adjust for income. For instance, using a risky discount rate instead of the risk-free rate in the forward price formula violates no-arbitrage. Always remember: the cost of carry uses the risk-free rate because the replicating portfolio is risk-free when held to hedge.

1. Confusing Futures and Forward Prices: Assuming futures and forward prices are always identical can lead to valuation mistakes in scenarios with significant interest rate volatility. On the exam, if a question highlights changing interest rates and correlation with the asset, consider that futures prices may diverge from forward prices.

1. Misunderstanding Put-Call Parity: Students often misremember the parity equation or apply it to American options (where it doesn't hold exactly due to early exercise). Remember, put-call parity is for European options. A classic trap is to think a protective put is always cheaper than a fiduciary call; parity shows they are equivalent in present value terms.

1. Overlooking Assumptions in Black-Scholes-Merton: Applying the BSM model without considering its assumptions—like constant volatility—can yield inaccurate prices in real markets. While the model is foundational, recognize that implied volatility often varies, leading to the "volatility smile" observed in practice.

Summary

• No-arbitrage is the core principle: Derivative prices are determined by constructing replicating portfolios that preclude risk-free profits, not by forecasting future prices.
• Forward prices are set by the cost of carry: They equal the spot price compounded at the risk-free rate, adjusted for any income from the underlying asset.
• Futures and forward prices can differ due to daily settlement and the correlation between interest rates and asset prices, though the difference is often small.
• Put-call parity links option prices: For European options, calls, puts, the underlying, and the risk-free bond are interconnected through a no-arbitrage identity.
• Option values are driven by multiple factors, with volatility being critically important. Pricing models like the binomial tree and Black-Scholes-Merton provide quantitative estimates based on no-arbitrage and risk-neutral valuation.
• A swap is valued as a series of forwards: Its value is the net present value of the remaining cash flows, leveraging the same discounting principles used for forward contracts.