Financial Derivatives Fundamentals

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1. Market Architecture and Structures 1.1 Exchange-Traded vs. Over-the-Counter (OTC) Exchange-Traded: Centralized marketplace (e.g., CME, Eurex). Exchange acts as Central Counterparty (CCP) , eliminating credit risk. Contracts are highly standardized. Over-the-Counter (OTC): Decentralized network of direct trading (bilateral). Terms are non-standard and negotiable. Participants face counterparty risk . Clearinghouse/Exchange Trader A Trader B Trader C Exchange Bank A Bank B Client C OTC 2. Linear Derivatives: Forwards and Futures 2.1 Forward Contracts Agreement to buy/sell an asset at a future time for a set price $K$. Venue: OTC Market. Notation: $K$ (Delivery Price), $S_T$ (Spot Price at Maturity $T$). Payoff at Maturity: Long Position (Buyer): $S_T - K$ Short Position (Seller): $K - S_T$ Costs zero to enter ($V_0 = 0$). 2.2 Futures Contracts Standardized forward contract traded on an exchange. Venue: Exchange-Traded. Standardized asset quality, contract size, delivery. 2.3 The Critical Difference: Settlement Forwards: Single cash flow at time $T$. Futures: Settled daily via marking to market . 3. Mechanics of Futures Markets (Detailed) 3.1 The Margin System Initial Margin: Amount deposited to open a position. Maintenance Margin: Minimum balance allowed. Variation Margin: Funds deposited if balance falls below maintenance. Algorithm for Margin Problems Calculate daily gain/loss: $\Delta V = (\text{Close Price}_{\text{today}} - \text{Close Price}_{\text{yesterday}}) \times \text{Contract Size} \times N_{\text{contracts}}$. Adjust Margin Balance: $\text{Balance}_{\text{new}} = \text{Balance}_{\text{old}} + \Delta V$. If $\text{Balance}_{\text{new}} Margin Call occurs. Deposit required: $\text{Initial Margin} - \text{Balance}_{\text{new}}$. 3.2 Convergence As a futures contract approaches maturity, $F_T$ must converge to $S_T$. $F_T = S_T$ at maturity due to arbitrage. Time ($t$) Price ($P$) $t=0$ $t=T$ Spot Price Futures Price Basis 4. Non-Linear Derivatives: Options 4.1 Types of Options Call Option: Right to buy asset at strike $K$. Put Option: Right to sell asset at strike $K$. 4.2 Payoff Formulas Long Call Payoff: $\max(S_T - K, 0)$ Long Put Payoff: $\max(K - S_T, 0)$ Net Profit: Payoff - Option Premium $S_T$ Payoff $K$ Long Call $S_T$ Payoff $K$ Short Call $S_T$ Payoff $K$ Long Put $S_T$ Payoff $K$ Short Put 5. Market Participants 5.1 Hedgers Use derivatives to reduce existing risk. Short Hedge: Own asset, sell futures to lock in price. Long Hedge: Need to buy asset, buy futures to lock in price. 5.2 Speculators Bet on price movements without underlying exposure. Utilize leverage from margin. 5.3 Arbitrageurs Exploit price discrepancies for risk-free profit. Enforce the Law of One Price . 6. Synthesis: Building Problem-Solving Skills 6.1 The "No-Arbitrage" Argument Pricing formulas assume no risk-free profit. Look for opportunities to buy cheap, sell expensive. 6.2 Zero-Sum Game (excluding commissions) For every dollar gained by long, dollar lost by short. $\sum \text{Profits} + \sum \text{Losses} = 0$ 6.3 Time Lines Visualize cash flows and events over time. $t=0$: Contract inception. $t $t=T$: Final settlement. 6.4 Summary of Key Equations Instrument Long Payoff at $T$ Short Payoff at $T$ Margin Required? Forward $S_T - K$ $K - S_T$ No (usually) Future $S_T - K$ $K - S_T$ Yes (Daily) Call Option $\max(S_T - K, 0)$ $-\max(S_T - K, 0)$ Short side only Put Option $\max(K - S_T, 0)$ $-\max(K - S_T, 0)$ Short side only