Derivatives & Risk Mgmt

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### Course Introduction - **Course No.:** FIN F311 / ECON F354 - **Course Title:** Derivatives and Risk Management - **Objective:** Familiarize students with instruments for risk management, including options, futures, swaps, and credit derivatives. Discuss pricing and hedging principles, and introduce complex instruments and risk management frameworks. - **Textbook:** John C. Hull & Basu Sankarshan, *Options, Futures and Other Derivatives*, 8th Edition, Pearson Education. ### Introduction to Derivatives - **Definition:** A financial instrument whose value depends on, or is derived from, the value of another asset. - **Purpose:** - **Hedging:** Reducing risk exposure to price fluctuations. - **Speculation:** Betting on future price movements for profit. - **Arbitrage:** Exploiting price discrepancies for risk-free profit. - **Key Characteristics:** Leverage, Volatility, Contractual Agreement. - **Types of Underlying Assets:** Equities, Fixed Income, Commodities, Currencies. - **Trading:** - **Exchanges:** Eg., Chicago Board Options Exchange (CBOE), CME Group. - **Over-the-Counter (OTC) Market:** Direct trading between financial institutions. - **OTC Market Post-2008:** - Became regulated to reduce systemic risk and increase transparency. - Standardized products traded on Swap Execution Facilities (SEFs). - Central Clearing Parties (CCPs) used for standardized transactions to mitigate counterparty risk. ### Forward Contracts - **Definition:** Customized contract between two parties to buy or sell an asset at a specified price on a future date. - **Characteristics:** - **OTC:** Customized, but high counterparty risk. - **No Upfront Payment:** Typically no premium paid. - **Obligation:** Both parties are obligated. - **Positions:** - **Long Position:** Agreed to buy. - **Short Position:** Agreed to sell. - **Profit from Long Forward (Buyer):** $S_T - K$ - **Profit from Short Forward (Seller):** $K - S_T$ - $S_T$: Spot price of underlying at maturity. - $K$: Delivery price (forward price) specified in contract. - **Arbitrage Opportunity:** If the market forward price deviates from the theoretical forward price, an arbitrage opportunity exists. - **Forward Price ($F_0$) (Non-Dividend Paying Stock):** $F_0 = S_0 e^{rT}$ - $S_0$: Current spot price. - $r$: Risk-free interest rate (continuously compounded). - $T$: Time to maturity (in years). - **Forward Price (Investment Asset with Known Income $I$):** $F_0 = (S_0 - PV(I))e^{rT}$ (where $PV(I)$ is present value of income). - **Forward Price (Investment Asset with Known Yield $q$):** $F_0 = S_0 e^{(r-q)T}$ - **Valuing a Forward Contract:** Value of a long forward contract, $f = (F_0 - K)e^{-rT}$. ### Futures Contracts: Mechanics - **Definition:** Standardized forward contracts traded on an organized exchange. - **Characteristics:** - **Standardized:** Predetermined contract size, expiration dates, quality. - **Exchange-Traded:** Provides liquidity, reduces counterparty risk via clearing house. - **Marking-to-Market:** Daily settlement of gains/losses. - **Margin Requirements:** Initial and maintenance margins. - **Clearing House:** Acts as intermediary, guarantees performance, eliminates counterparty risk. - **Margin Account:** - **Initial Margin:** Deposit when opening position. - **Maintenance Margin:** Minimum margin level. - **Margin Call:** If margin falls below maintenance, additional funds required. - **Convergence:** Futures price converges to spot price at maturity. - **Open Interest:** Total number of contracts outstanding. - **Settlement Price:** Price just before final bell, used for daily settlement. - **Terminology:** - **Backwardation:** Futures price expected future spot price. - **Delivery:** If not closed out, usually settled by delivering underlying asset. Short position chooses what, where, when. Some contracts (stock indices, Eurodollars) are cash settled. - **Types of Orders:** - **Limit Order:** Executed at a specified price or more favorable. - **Stop Order (Stop-Loss):** Becomes a market order when price hits specified level. - **Stop-Limit Order:** Combination of stop and limit. - **Market-if-Touched (MIT):** Becomes market order when price hits specified level. - **Discretionary Order:** Broker can delay execution for better price. - **Time-of-Day Order:** Executed within specific period. - **Open Order (Good-till-Canceled):** In effect until executed or contract ends. - **Fill-or-Kill Order:** Executed immediately or not at all. - **Regulation:** - **US:** Commodity Futures and Trading Commission (CFTC). - **India:** SEBI. - **Accounting & Tax:** Hedging profits/losses recognized with underlying; speculation on mark-to-market basis. ### Hedging Strategies Using Futures - **Basic Assumption:** Static hedge (no adjustment once placed). - **Long Futures Hedge:** Appropriate when purchasing an asset in the future (e.g., bread manufacturer hedging wheat prices). - **Short Futures Hedge:** Appropriate when selling an asset in the future (e.g., wheat producer hedging wheat prices). - **Arguments for Hedging:** Focus on core business, minimize market risks. - **Arguments Against Hedging:** Shareholders diversified, competitors may not hedge, difficult to explain losses on hedge when underlying gains. - **Basis Risk:** Risk that spot and futures prices do not move in perfect tandem. - **Basis:** Spot Price - Futures Price. - **Strengthening Basis:** Increase in basis. - **Weakening Basis:** Decrease in basis. - **Choice of Contract:** - Delivery month: As close as possible to, but later than, hedge end. - Cross Hedging: If no direct contract, choose one with high correlation. - **Optimal Hedge Ratio ($h^*$):** $h^* = \rho \frac{\sigma_S}{\sigma_F}$ - $\rho$: Correlation between spot and futures price changes. - $\sigma_S$: Std dev of spot price changes. - $\sigma_F$: Std dev of futures price changes. - **Optimal Number of Contracts:** - For hedging value $V_A$: $N = h^* \frac{V_A}{V_F}$ (where $V_F$ is value of one futures contract). - **Hedging Using Index Futures:** - Number of contracts to short for portfolio hedging: $N = \beta \frac{V_A}{V_F}$ - $V_A$: Value of portfolio. - $\beta$: Beta of portfolio. - $V_F$: Value of one futures contract. - **Changing Beta:** Adjust number of contracts to achieve desired portfolio beta. - **Stack and Roll:** Rolling futures contracts forward to hedge future exposures by closing out near-maturity contracts and opening new ones. - **Capital Asset Pricing Model (CAPM):** $Expected Return = R_F + \beta(R_M - R_F)$ - $R_F$: Risk-free rate. - $R_M$: Market return. - $\beta$: Systematic risk. ### Interest Rates - **Importance:** Factor in valuation of virtually all derivatives. - **Types of Rates:** - **Treasury Rates:** Rates on government-issued instruments (risk-free in own currency). - **LIBOR (London Interbank Offered Rate):** Rate at which banks lend to each other (historically, being phased out). - **MIBOR (Mumbai Interbank Offered Rate):** Indian inter-bank market rate. - **Repo Rates:** Rate in repurchase agreements (effectively collateralized loans). In India, RBI's benchmark rate. - **Risk-Free Rate:** Traditionally LIBOR, increasingly Overnight Indexed Swap (OIS) rate due to post-2007 crisis. - **Day Count Convention:** Defines how interest accrues (e.g., Actual/Actual, 30/360, Actual/360). - **Measuring Interest Rates:** - **Compounding Frequency:** Unit of measurement (e.g., quarterly, annually, continuously). - **Continuous Compounding:** $A(T) = A_0 e^{rT}$. Rates in option pricing often continuous. - **Zero Rates (Spot Rates):** Interest rate for an investment with a single payoff at time T. - **Bond Pricing:** Discount each cash flow at the appropriate zero rate. - **Bond Yield:** Discount rate that equates bond's present value of cash flows to its market price. - **Par Yield:** Coupon rate that makes bond price equal its face value. - **Bootstrap Method:** Deriving zero rates from market prices of Treasury bills and coupon bonds. - **Forward Rates:** Future zero rate implied by today's term structure. $R_2 = \frac{R_1 T_1 + F(T_1, T_2)(T_2 - T_1)}{T_2}$. - **Duration:** Measure of bond's price sensitivity to interest rate changes. - **Key Duration Relationship:** $\Delta P \approx -D P \frac{\Delta y}{1+y/m}$ (for non-continuous compounding). - **Modified Duration:** $D/(1+y/m)$. - **Bond Portfolios:** Weighted average duration. - **Convexity:** Measures curvature of bond price-yield relationship, providing more accurate price change estimates for larger yield shifts. - **Theories of Term Structure:** - **Expectations Theory:** Forward rates equal expected future zero rates. - **Market Segmentation:** Short, medium, long rates determined independently. - **Liquidity Preference Theory:** Forward rates higher than expected future zero rates. ### Interest Rate Futures - **Treasury Bill Prices (US):** Quoted as $100 - (\frac{100-P}{100}) \times \frac{360}{n}$ - **Treasury Bond Futures:** - **Cash Price:** Quoted price + Accrued Interest. - **Price Received by Short:** (Settlement Price $\times$ Conversion Factor) + Accrued Interest. - **Conversion Factor:** Value of bond assuming flat 6% yield curve. - **Cheapest-to-Deliver (CTD) Bond:** Bond that minimizes (Quoted Bond Price - (Settlement Price $\times$ Conversion Factor)). Short position chooses CTD. - **Eurodollar Futures:** - Futures on 3-month Eurodollar deposit rate (LIBOR). - Contract on $1 million. - 1 basis point change = $25 change in contract price. - Cash settled; final settlement price = $100 - actual 3-month Eurodollar deposit rate. - **Forward Rates vs. Eurodollar Futures:** - Futures settled daily, forward once. - Futures settled at start of period, FRA at end. - **Convexity Adjustment:** Forward Rate $\approx$ Futures Rate $- 0.5 \sigma^2 T_1 T_2$ - **Extending LIBOR Zero Curve:** Eurodollar futures can be used to bootstrap the zero curve by deriving forward rates. - **Duration-Based Hedging:** Matching durations of assets and liabilities to hedge against interest rate risk. - **Hedge Ratio:** $-\frac{P_P D_P}{P_F D_F}$ - **Limitations:** Assumes parallel shifts, small changes, no change in CTD bond. ### Mechanics of Options Markets - **Definition:** Right, but not obligation, to buy/sell underlying at specified price (strike) on/before specified date (expiration). - **Types:** - **Call Option:** Right to buy. - **Put Option:** Right to sell. - **Expiration Styles:** - **European Option:** Exercised only at maturity. - **American Option:** Exercised any time up to and including maturity. - **Positions:** Long Call, Short Call, Long Put, Short Put. - **Payoffs:** - **Long Call:** $\max(0, S_T - K)$ - **Short Call:** $-\max(0, S_T - K)$ - **Long Put:** $\max(0, K - S_T)$ - **Short Put:** $-\max(0, K - S_T)$ - **Assets Underlying Exchange-Traded Options:** Stocks, Foreign Currency, Stock Indices, Futures. - **Moneyness:** - **In-the-Money (ITM):** Call ($S_T > K$), Put ($S_T K$). - **Intrinsic Value:** Value if exercised immediately. - **Time Value:** Option premium - intrinsic value. - **Dividends & Stock Splits:** - No adjustment for cash dividends. - N-for-M stock split: Strike reduced to $mK/n$, no. of options increased to $nN/m$. - **Market Makers:** Facilitate trading by quoting bid/ask prices. - **Margins (for written options):** Required to cover potential losses. Specific rules for naked calls/puts. - **Warrants:** Options issued by corporation/financial institution. - **Employee Stock Options:** Remuneration, usually ATM when issued. - **Convertible Bonds:** Bonds exchangeable for equity. ### Properties of Stock Options - **Factors Affecting Option Prices:** | Variable | Call (c/C) | Put (p/P) | |----------|------------|-----------| | $S_0$ | + | - | | K | - | + | | T | ? | ? | | $\sigma$ | + | + | | r | + | - | | D | - | + | - **American vs. European:** American option $\ge$ European option ($C \ge c, P \ge p$). - **Lower Bounds (European, No Dividends):** - Call: $c \ge S_0 - K e^{-rT}$ - Put: $p \ge K e^{-rT} - S_0$ - **Put-Call Parity (European, No Dividends):** $c + K e^{-rT} = p + S_0$ - Portfolio A: Long European call + Zero-coupon bond (pays K at T). - Portfolio C: Long European put + Stock. - Both portfolios have value $\max(S_T, K)$ at maturity. - **Early Exercise (American Options):** - **Call (Non-dividend paying stock):** Never exercised early due to: 1. No income sacrificed. 2. Delay paying strike price. 3. Insurance against price fall. - **Put:** Can be optimal to exercise early if ITM significantly, especially with low interest rates. - **Impact of Dividends on Lower Bounds:** - Call: $c \ge S_0 - D - K e^{-rT}$ - Put: $p \ge K e^{-rT} + D - S_0$ - $D$: Present value of dividends during option life. - **Extensions of Put-Call Parity:** - American (No Divs): $S_0 - K ### Trading Strategies Involving Options - **Principal Protected Note:** Allows risky position without risking principal (e.g., zero-coupon bond + call option). - **Positions in an Option & Underlying:** - **Covered Call:** Long stock + Short call. Generates income, limits upside, reduces effective purchase price. - **Protective Put:** Long stock + Long put. Limits downside risk, retains upside (like insurance). - **Spreads (Two or more options of same type):** - **Bull Spread (Call):** Buy call (low strike) + Sell call (high strike). Profits from moderate price increase. - **Bull Spread (Put):** Buy put (low strike) + Sell put (high strike). Profits from moderate price increase. - **Bear Spread (Call):** Buy call (high strike) + Sell call (low strike). Profits from moderate price decrease. - **Bear Spread (Put):** Buy put (high strike) + Sell put (low strike). Profits from moderate price decrease. - **Box Spread:** Combination of bull call spread and bear put spread. - **Butterfly Spread (Calls or Puts):** Long 1 (lowest strike), Short 2 (middle strike), Long 1 (highest strike). Profits if price stays in narrow range. - **Calendar Spread (Calls or Puts):** Buy long-term option, Sell short-term option (same strike). Profits from time decay of short-term option. - **Combinations (Mix of Calls and Puts):** - **Straddle (Long):** Long call + Long put (same strike, same expiration). Profits from large price movement in either direction. - **Straddle (Short):** Short call + Short put (same strike, same expiration). Profits if price stays in narrow range. - **Strips:** Long 1 call + Long 2 puts (same strike, same expiration). Bullish on volatility, more bearish on direction. - **Straps:** Long 2 calls + Long 1 put (same strike, same expiration). Bullish on volatility, more bullish on direction. - **Strangle (Long):** Long OTM call + Long OTM put (different strikes, same expiration). Cheaper version of straddle, requires larger movement. - **Strangle (Short):** Short OTM call + Short OTM put. Profits if price stays in narrow range, less risk than short straddle. - **Zero Collar Strategy:** Long put ($K_1$) + Short call ($K_2$) where $K_1 ### Binomial Trees - **Concept:** Discretizes time, asset price moves to two values (up or down) per step. - **No-Arbitrage Argument:** Construct a riskless portfolio of stock and derivative. - **Risk-Neutral Valuation:** - Assume stock price earns risk-free rate. - Discount expected payoff at risk-free rate using risk-neutral probabilities. - **Risk-Neutral Probability of Up Move ($p$):** $p = \frac{e^{r\Delta t} - d}{u - d}$ - $u$: Up factor, $d$: Down factor. - $\Delta t$: Time step. - **Delta ($\Delta$):** Ratio of change in option price to change in underlying price. - For a portfolio of long $\Delta$ shares and short 1 derivative: $\Delta = \frac{f_u - f_d}{S_0 u - S_0 d}$ - **Choosing $u$ and $d$:** Often based on volatility $\sigma$: $u = e^{\sigma\sqrt{\Delta t}}$, $d = e^{-\sigma\sqrt{\Delta t}}$. - **American Options:** At each node, compare exercise value with value from continuing to hold (take max). - **Girsanov's Theorem:** Volatility is the same in real and risk-neutral worlds, allowing real-world volatility to be used for risk-neutral tree construction. ### Black-Scholes-Merton (BSM) Model - **Assumptions:** - Stock price follows Geometric Brownian Motion (constant $\mu, \sigma$). - Short selling permitted, no transaction costs/taxes, perfect divisibility. - No dividends (or continuous dividend yield). - No riskless arbitrage. - Continuous trading. - Constant risk-free rate ($r$) for all maturities. - European options. - **Call Option Price ($C_0$):** $C_0 = S_0 N(d_1) - K e^{-rT} N(d_2)$ - **Put Option Price ($P_0$):** $P_0 = K e^{-rT} N(-d_2) - S_0 N(-d_1)$ - **Where:** - $N(x)$: Cumulative standard normal distribution function. - $d_1 = \frac{\ln(S_0/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}$ - $d_2 = d_1 - \sigma\sqrt{T}$ - $S_0$: Current stock price. - $K$: Strike price. - $r$: Risk-free interest rate. - $T$: Time to expiration. - $\sigma$: Volatility of underlying asset. - **Implied Volatility:** Volatility for which BSM price equals market price. Traders often quote implied volatilities. ### The Greeks - **Definition:** Measures of an option's sensitivity to various factors. Used for hedging and risk management. - **Delta ($\Delta$):** - **Sensitivity:** Option price to change in underlying asset's price. - Call $\Delta = N(d_1)$ (0 to 1). - Put $\Delta = N(d_1) - 1$ (-1 to 0). - **Delta Hedging:** Create zero-delta portfolio, immune to small price changes. Requires rebalancing. - **Gamma ($\Gamma$):** - **Sensitivity:** Delta to change in underlying asset's price (second derivative). - **Interpretation:** High gamma means delta changes rapidly, requiring more frequent rebalancing. - **Gamma Hedging:** Create zero-gamma (and zero-delta) portfolio. - **Vega ($\mathcal{V}$):** - **Sensitivity:** Option price to change in volatility. - **Interpretation:** Long vega benefits from increasing volatility. - **Theta ($\Theta$):** - **Sensitivity:** Option price to passage of time (time decay). - **Interpretation:** Negative for long options (value decreases as expiration approaches). - **Rho ($\rho$):** - **Sensitivity:** Option price to change in risk-free interest rate. - **Interpretation:** Calls benefit from higher rates (positive rho), puts from lower (negative rho). ### Risk Management Concepts - **Value at Risk (VaR):** Max potential loss over time horizon at confidence level. - **Methods:** Historical, Parametric, Monte Carlo. - **Limitations:** Doesn't measure "tail risk," assumes normal distribution for parametric. - **Expected Shortfall (ES) / Conditional VaR (CVaR):** Expected loss given loss exceeds VaR. Measures tail risk. - **Stress Testing:** Impact of extreme but plausible market scenarios. - **Scenario Analysis:** Impact of specific hypothetical events. - **Hedging Strategies:** - **Static Hedge:** Implemented once, not adjusted. - **Dynamic Hedge:** Requires continuous adjustment (e.g., delta hedging). - **Risk Aggregation:** Combining different risk types for firm-wide view. ### Risk Management Framework - **Identification:** Identifying all relevant risks (market, credit, operational, liquidity, legal, reputational). - **Measurement:** Quantifying risks (VaR, ES, stress testing). - **Monitoring:** Continuously tracking risk exposures. - **Mitigation:** Strategies to reduce/transfer risk (hedging, diversification). - **Reporting:** Communicating risk information. - **Governance:** Establishing policies and responsibilities. ### Types of Risk - **Market Risk:** Losses due to changes in market prices (interest rates, FX, equity, commodity). - **Credit Risk:** Counterparty default. - **Default Risk:** Failure to pay. - **Downgrade Risk:** Rating lowered. - **Settlement Risk:** One party pays, other defaults before delivering. - **Liquidity Risk:** - **Market Liquidity Risk:** Inability to buy/sell quickly at fair price. - **Funding Liquidity Risk:** Inability to meet short-term obligations. - **Operational Risk:** Losses from inadequate processes, people, systems, or external events. - **Legal Risk:** Loss due to unenforceable contracts. - **Systemic Risk:** Collapse of entire financial system. ### Regulatory Aspects - **Dodd-Frank Act (US):** Regulates financial markets, OTC derivatives, reduces systemic risk. - **Basel Accords (International):** Capital requirements for banks, including market/credit risk from derivatives. - **EMIR (European Market Infrastructure Regulation):** EU regulation for OTC derivatives, central clearing, reporting. - **Central Clearing:** Many OTC derivatives now centrally cleared to reduce counterparty risk.