Derivatives

Derivatives are financial instruments whose value is derived from an underlying asset. They are categorized as forward commitments (forwards, futures, swaps), which are obligations, or contingent claims (options), which provide a right. They are used for risk management, speculation, and arbitrage, with pricing based on no-arbitrage principles.

Study Guide

Derivative Instrument and Derivative Market Features

Derivative: A financial contract whose value is derived from the performance of an underlying asset, index, or interest rate.
Underlying Asset: Can be a physical asset (e.g., commodities like oil or gold) or a financial asset (e.g., stocks, bonds, currencies, or interest rates).
Exchange-Traded Derivatives (ETDs):
- Standardized contracts (size, maturity, quality).
- Traded on an organized exchange (e.g., Chicago Mercantile Exchange).
- Cleared through a central clearinghouse, which mitigates counterparty risk.
- Examples: Futures, most options.
Over-the-Counter (OTC) Derivatives:
- Customized contracts negotiated directly between two parties.
- Not traded on an exchange.
- Higher counterparty risk (risk that the other party will default).
- Examples: Forwards, swaps, exotic options.

Forward Commitment and Contingent Claim Features and Instruments

Forward Commitments: An obligation to buy or sell the underlying asset at a predetermined price at a future date.
- Forward Contract: A customized OTC contract between two parties to buy or sell an asset at a specified price on a future date. High counterparty risk.
- Futures Contract: A standardized forward contract traded on an exchange. Counterparty risk is managed by the clearinghouse through daily settlement (marking-to-market).
- Swap: An OTC agreement to exchange a series of cash flows over a period of time. A common example is an interest rate swap where one party exchanges fixed-rate interest payments for floating-rate payments.
Contingent Claims: A contract that gives the holder the right, but not the obligation, to buy or sell an underlying asset. The payoff is contingent on the underlying's value.
- Option: The primary type of contingent claim.
  - Call Option: Right to buy the underlying at a specified price (strike price) by a certain date (expiration).
  - Put Option: Right to sell the underlying at a specified price by a certain date.
  - The option buyer (holder) pays a premium to the option seller (writer).

Feature	Forward/Futures	Option
Right/Obligation	Both parties are obligated	Buyer has the right, Seller has the obligation
Upfront Cost	Typically none (except margin for futures)	Buyer pays a premium
Payoff Profile	Linear, symmetric	Asymmetric
Risk for Buyer	Symmetric (potential for large loss or gain)	Limited to the premium paid
Risk for Seller	Symmetric (potential for large loss or gain)	Limited gain (premium), unlimited or large loss potential

Derivative Benefits, Risks, and Issuer and Investor Uses

Benefits:
- Risk Management (Hedging): Transfer risk from one party to another.
- Price Discovery: Derivative prices reflect expectations about future prices of the underlying.
- Market Efficiency: Reduce transaction costs and facilitate arbitrage, keeping asset prices in line.
- Leverage: Control a large position with a small initial investment.
Risks:
- Counterparty Risk: The risk that the other party in an OTC contract will default on its obligation.
- Leverage Risk: Small price changes in the underlying can lead to large gains or losses.
- Liquidity Risk: The risk of not being able to sell a derivative position quickly at a fair price.
Uses:
- Hedging: To reduce or eliminate risk. A farmer might sell a futures contract to lock in a price for their crop.
- Speculation: To bet on the future direction of an asset's price.
- Arbitrage: To exploit mispricings between a derivative and its underlying asset to earn a risk-free profit.

Arbitrage, Replication, and the Cost of Carry in Pricing Derivatives

Law of One Price: Two assets with identical future cash flows should have the same price.
Arbitrage: A transaction that generates a risk-free profit without any net investment of capital. The existence of arbitrage opportunities forces prices to align with the Law of One Price.
Replication: Creating a synthetic asset or portfolio with the same cash flows as another asset. For example, a portfolio of the underlying asset and a risk-free bond can replicate the payoff of a forward contract.
Cost of Carry Model: Used for pricing forward and futures contracts. It states that the forward price must equal the spot price plus the cost of "carrying" the asset until maturity.
- Formula: $F_0(T) = S_0(1+r)^T$
- For assets with cash flows (e.g., dividends D or foreign interest r_f): $F_0(T) = (S_0 - PV(\text{benefits}))(1+r)^T$
- For assets with storage costs (C): $F_0(T) = (S_0 + PV(\text{costs}))(1+r)^T$

Pricing and Valuation of Forward Contracts and for an Underlying with Varying Maturities

Pricing (at initiation, t=0): Determining the forward price ( $F_0(T)$ $F_{0} (T)$ ) that makes the contract value zero.
- Formula (no income/costs): $F_0(T) = S_0(1+r)^T$
- Where:
  - $S_0$ = Spot price of the underlying at t=0
  - $r$ = Risk-free rate
  - $T$ = Time to maturity (in years)
Valuation (during life of contract, t): Determining the market value of an existing forward contract.
- Formula (long position): $V_t(T) = S_t - F_0(T)(1+r)^{-(T-t)}$
- Where:
  - $V_t(T)$ = Value of the contract at time t
  - $S_t$ = Spot price of the underlying at time t
  - $F_0(T)$ = Forward price agreed upon at initiation
  - $(T-t)$ = Remaining time to maturity

Pricing and Valuation of Interest Rates and Other Swaps

Interest Rate Swap: An agreement to exchange fixed-rate interest payments for floating-rate interest payments on a notional principal amount.
Pricing (at initiation): The goal is to find the fixed swap rate ( $r_{FIX}$ $r_{F I X}$ ) that makes the present value of the fixed payments equal to the present value of the expected floating payments. At initiation, the value of the swap is zero to both parties.
- The fixed rate is a weighted average of the implied forward rates.
Valuation (during life of swap): The value of the swap is the difference between the present value of the remaining cash flows.
- $V_{swap} = PV(\text{Floating Payments}) - PV(\text{Fixed Payments})$
- As interest rates change, the value of the swap becomes positive for one party and negative for the other. For the fixed-rate payer, the swap value increases if interest rates fall.

Pricing and Valuation of Options

Key Terminology:
- Strike Price (X): The price at which the option can be exercised.
- Expiration Date (T): The last date the option can be exercised.
- Moneyness:
  - In-the-money (ITM): Exercising the option would be profitable (ignoring premium). Call: $S > X$ . Put: $S < X$ .
  - At-the-money (ATM): $S = X$ .
  - Out-of-the-money (OTM): Exercising would result in a loss. Call: $S < X$ . Put: $S > X$ .
Option Value:
- Intrinsic Value: The value if exercised immediately. Max(0, S-X) for a call; Max(0, X-S) for a put.
- Time Value: The amount by which the option premium exceeds its intrinsic value. Reflects the probability that the option's value will increase before expiration.
- Option Premium = Intrinsic Value + Time Value
Factors Affecting Option Prices:

Increase In...	Call Option Value	Put Option Value
Underlying Price (S)	Increases	Decreases
Strike Price (X)	Decreases	Increases
Time to Expiration (T)	Increases	Increases
Volatility (σ)	Increases	Increases
Risk-Free Rate (r)	Increases	Decreases
Dividends/Income	Decreases	Increases

Option Replication Using Put–Call Parity

Put-Call Parity: A no-arbitrage relationship for European options that links the prices of a call option, a put option, the underlying stock, and a risk-free bond.
Formula: $c_0 + X(1+r)^{-T} = p_0 + S_0$ $c_{0} + X (1 + r)^{- T} = p_{0} + S_{0}$
- $c_0$ = Price of a European call option
- $p_0$ = Price of a European put option
- $S_0$ = Price of the underlying asset
- $X$ = Strike price
- $X(1+r)^{-T}$ = Present value of a zero-coupon bond that pays X at maturity T
Interpretation: A Fiduciary Call (long call + long bond) has the same payoff as a Protective Put (long put + long stock). If the equation does not hold, an arbitrage opportunity exists.
Synthetic Positions: The formula can be rearranged to create synthetic assets:
- Synthetic Long Stock: $S_0 = c_0 - p_0 + X(1+r)^{-T}$
- Synthetic Long Call: $c_0 = p_0 + S_0 - X(1+r)^{-T}$

Valuing a Derivative Using a One-Period Binomial Model

The binomial model assumes that over a single period, the price of the underlying will move to one of two possible values (up or down).

Steps for Valuation:
1. Define the model:
  - $S_0$ : Current stock price
  - $S_u$ : Up-state price = $S_0 \times u$ (where u > 1)
  - $S_d$ : Down-state price = $S_0 \times d$ (where d < 1)
2. Calculate derivative payoffs at expiration for both the up-state ( $C_u$ ) and down-state ( $C_d$ ).
3. Calculate the risk-neutral probability of an up-move ( $\pi$ $π$ ):
  - $\pi = \frac{(1+r) - d}{u - d}$
4. Calculate the value of the derivative by finding the expected future payoff using risk-neutral probabilities and discounting it back to today at the risk-free rate.
  - $C_0 = \frac{[\pi \times C_u + (1-\pi) \times C_d]}{(1+r)}$

Pricing and Valuation of Futures Contracts

Key Difference from Forwards: Futures are marked-to-market daily.
- At the end of each trading day, gains and losses on the contract are settled. The party with a loss pays the party with a gain.
- This daily settlement process resets the value of the futures contract to zero at the end of each day.
Pricing: The futures price is determined by the same cost of carry model as the forward price.
- $F_0(T) \approx S_0(1+r)^T$
- The prices can differ slightly if interest rates are correlated with the underlying asset's price, but for the Level 1 exam, they are generally treated as being equal.
Valuation:
- The value of a futures contract is zero at the end of each day after the marking-to-market process.
- The price of a futures contract changes continuously throughout the day.