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Hedging risk with Derivatives

Hedging risk with Derivatives. Review of equity options Review of financial futures Using options and futures to hedge portfolio risk Introduction to Hedge Funds. Options -- Contract. Calls and Puts Underlying Security (Number of Units) Exercise or Strike Price Expiration date

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Hedging risk with Derivatives

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  1. Hedging risk with Derivatives • Review of equity options • Review of financial futures • Using options and futures to hedge portfolio risk • Introduction to Hedge Funds

  2. Options -- Contract • Calls and Puts • Underlying Security (Number of Units) • Exercise or Strike Price • Expiration date • Option Premium • American, European, Asian, etc.

  3. Options -- Markets • 1 Buyer + 1 Seller (writer) = 1 Contract • Examples of Price Quotations • Premium = Intrinsic Value + Time Prem • Options available on • Equities • Indicies • Foreign Currencies • Futures

  4. Options -- Basic Strategies • Buy Call • Sell (write) Call • Buy Put • Sell (write) Put

  5. Options -- Advanced Strategies • Straddle • Strips and Straps • Vertical Spreads • Bullish • Bearish

  6. Options - Determinants of Value • Value of Underlying Asset • Exercise Price • Time to Expiration • VOLATILITY • Interest Rates • Dividends

  7. Options -- Black Scholes Option Pricing Model • C = SN(d1) - Xe-rTN(d2) ln(S/X) +(r+s2/2)T d1 = ---------------------------sT1/2d2 = d1 - sT1/2 • Put-Call Parity: P = C + Xe-rT - S

  8. Futures Contract • Agreement to make (sell) or take (buy) delivery of a prespecified quantity of an asset at an agreed upon price at a specific future date. • ex. S&P 500 Index Futures: • Price: 1126.10; Delilvery month: June • Buyer agrees to purchase a portfolio representing the S&P 500 (or its cash equivalent) for $1126.10 x 250 = $281,525 on Thursday prior to 3rd Friday in June. (Buyer is locking in the purchase price for the portfolio.) • Seller agrees to deliver the portfolio described above. • Note: since this is a cash settled contract, if the price was 1116.10 on the delivery date, the buyer would pay the seller $2,500 (= 10 x 250). If the price was 1136.10, the seller would pay the buyer $2,500

  9. Futures Contract: Marking to Market • Marking to market: • Price of Futures contract is reset every day • Gains/Losses versus previous day are posted to buyer and seller margin accounts • Futures = a bundle of consecutive 1-day forward contracts • If futures held to expiration, effective delivery price is same as when contract initiated

  10. Futures Contract: Marking to Market example (C$ contract)

  11. April 4 June 17 1. Contract to sell S&P @ 1126.1 ($281,525) on June 17. 2. Buy S&P @ 1106.1 ($276,525) on spot market and deliver @ 1126.1 3. Profit = $5,000. Index Futures Market • Speculators often sell index futures when they expect the underlying index to depreciate, and vice versa.

  12. April 4 June 17 1.Contract to sell S&P @ 1126.1 ($281,525) on June 17. 2. Market falls to 1106.1.Gain =$5000 3. Gain offsets (approx.) loss of $5000 on securities held Index Futures Market • Index futures may be sold by investors to hedge risk associated with securities held.

  13. Index Futures Market • Most index futures contracts are closed out before their settlement dates (99%). • Brokers who fulfill orders to buy or sell futures contracts earn a transaction or brokerage fee in the form of the bid/ask spread.

  14. Hedging with Derivatives • Basic option strategies • Covered call • Protective put • Synthetic short • Basic futures strategies • Using interest rate futures to reduce risk

  15. Covered Call • Sell call on stock you own. (Long stock, short call) • Good: • As value of stock falls, loss is partially offset by premium received on calls sold. • Essentially costless since hedge generates a cash inflow • Bad: • Maximum inflow from call = premium; Hedge is less effective for large drop in stock price • If stock price rises, call will be exercised; Investor transfers gains on stock to holder of call.

  16. Protective Put • Buy put on stock you own. (Long stock, long put) • Good: • As value of stock falls, loss is partially offset by gain in value of put. Gain from put continues to grow as stock price falls. • If stock price rises, maximum loss on put = premium; Investor keeps all stock gains less fixed put premium. • Bad: • More expensive to hedge with put

  17. Synthetic Short • Sell call and buy put on stock you own. (Long stock, short call, long put) • Good: • As value of stock falls, loss is offset by gain in value of put. Gain from put continues to grow as stock price falls. • If stock price rises, gain is offset by loss on call. Loss from call continues to grow as stock price rises. • Very effective hedging device • Can be self-financing (premium received on put sold offsets premium paid on call purchased) • Bad: • Often more expensive than simply shorting the stock itself.

  18. Delta Hedging with Options • Call Delta = DC= dC/dS • From Black-Scholes model, DC = N(d1) Ex.: If S=74.49, X=75, r=1.67%, s =38.4%, t=0.1589 yrs. Then, C = 4.40 and N(d1) = 0.5197 If S increases by $1, C increases by $0.5197 Hedge Ratio = H = 1/DC = 1/0.5197 = 1.924 Sell 1.924 calls per share of stock held to hedge!

  19. Example of Call Hedge – Held to Expiration, 1000 share stock position

  20. Delta Hedging - Puts • Put Delta = DP= dP/dS • From Black-Scholes model and Put-Call Parity, DP= DC – 1 =N(d1) - 1 Ex.: If S=74.49, X=75, r=1.67%, s =38.4%, t=0.1589 yrs. Then, C = 4.40, P = 4.71, N(d1) = 0.5197, and N(d1) -1 = -0.4803 If S increases by $1, P decreases by $0.4803 Hedge Ratio = H = 1/D = 1/0.4803 = 2.082 Buy 2.082 puts per share of stock held to hedge!

  21. Example of Put Hedge – Held to Expiration, 1000 share stock position

  22. Delta Hedging with Options • Delta changes over time! • S changes • Time declines • Other factors (r, s) may change

  23. True Delta Hedging – Adjust hedge when S changes • Scenarios 1 & 2: • IBM stock drops by $1 to $73.49 ==> Loss of $1000 • Call options also drop by $0.5197 ==> Gain of $1037.97 ==>Net change $37.97 • IBM stock rises by $1 to $75.49 ==> Gain of $1000 • Call options also rise by $0.5193 ==> Loss of $1037.97 ==> Net change ($37.97)

  24. True Delta Hedging – Adjust hedge when t changes • Scenario 3: • One week passes, IBM stock at $71.49 ==> Loss of $3000 • Call options now worth $2.73 ==> Gain of $3173 ==>Net change $173 • New call delta = 0.4029 • New hedge ratio = 1/0.4029 = 2.482 ==> Sell 5 more contracts! • Scenario 4: • One week passes, IBM stock at $77.49 ==> Gain of $3000 • Call options now worth $5.82 ==> Loss of $2698 ==> Net change ($302) • New call delta = 0.6238 • New hedge ratio = 1/0.6238 = 1.603 ==> Buy 3 contracts!

  25. True Delta Hedging – Adjust hedge when S changes • Scenarios 1 & 2: • IBM stock drops by $1 to $73.49 ==> Loss of $1000 • Put options also rise by $0.4803 ==> Gain of $1008.63 ==>Net change $8.63 • IBM stock rises by $1 to $75.49 ==> Gain of $1000 • Put options also fall by $0.4803 ==> Loss of $1008.63 ==> Net change ($8.63)

  26. True Delta Hedging – Adjust hedge when t changes • Scenario 3: • One week passes, IBM stock at $71.49 ==> Loss of $3000 • Put options now worth $6.06 ==> Gain of $2835 ==>Net change ($165) • New put delta = 0.4028 – 1 = -0.5972 • New hedge ratio = 1/0.5972 = 1.674 ==> Sell 4 contracts! • Scenario 4: • One week passes, IBM stock at $77.49 ==> Gain of $3000 • Put options now worth $3.15 ==> Loss of $3276 ==> Net change ($276) • New put delta = 0.6238 – 1 = -0.3762 • New hedge ratio = 1/0.3762 = 2.658 ==> Buy 5 more contracts!

  27. Delta Hedging with options • Delta represents response of call (or put) price with change in the stock price • Delta changes as stock price, time to expiration, interest rates, volatility change • It is too expensive to hedge individual stock positions with matching options. It is more common to hedge a portfolio with index options (cross hedging) • Most managers monitor delta itself to decide when to rebalance.

  28. A True Protective Put • Puts can be used to build a floor under the value of a long position • Buy 1 put per long share • Ex.: Long 1000 shares of IBM at $74.49 • Buy 1000 puts at $4.71 • Puts guarantee a value of $75 per share • This is insurance, not a hedge!

  29. A True Protective Put

  30. Hedging with Futures (example from May 2001) • There are futures on the S&P500. Suppose I have a portfolio that is currently worth $1,117,672. The portfolio has a beta of 1.3. • June S&P500 futures are at 1430.70 • ==> contract is worth 500 x 1430.70 = $715,350 • Hedge ratio = • (Value of portfolio / Value of Futures contract)(Portfolio Beta) • = (1,117,672/715,350)(1.3) = 2.031 ==> Sell 2 Contracts !

  31. Hedging with Futures (example from May 2001)

  32. Adjusting Systematic Risk with Futures • PM may choose to adjust systematic exposure up or down to reflect • investor desires • expectations of market movements • About index futures: • Represents contract to make/take delivery of a portfolio represented by the index • Since index itself may be non-investable, most index futures contracts are cash-settled • example: • S&P500 futures CME contract value = 250 x index • Initial margin: $6K for spec, $2.5K for hedgers.

  33. Adjusting Systematic Risk with Futures • I have an $11 million stock portfolio with b=1.05. I want to increase b to 1.2. • Value of Futures = 1314.50 x 250 = $328,625 • bf = 1.0. • Target b = contribution from portfolio + contribution from futures • 1.2 = (1.0)(1.05) + [(F x 328,625)/$11,000,000](1.0) • F = (bT - Wsbs)(Vs/VF) • F = 5.02 => buy 5 contracts • What have we done? • Used futures contracts to leverage holdings and increase exposure to market risk

  34. Adjusting Systematic Risk with Futures • Suppose target b = .90 • 0.90 = (1.0)(1.05) + [(F x 328,625)/$11,000,000](1.0) • F = (.90 - 1.05)(33.4728)(1.0) = -5.02 contracts (sell) • We have shorted futures to reduce systematic exposure.

  35. Hedging with Interest Rate Futures • How do you reduce duration for a bond portfolio? • Sell high D, buy low D • Sell bonds, buy Tbills • Sell interest rate futures • Interest rate futures: agreement to make/take delivery of a fixed income asset on a particular date for an agreed upon price • ex: Sept Tbond futures contract • $100K FV US Treas bonds with 15-years to maturity and 8% coupon (what if they don't exist?) • Price: 99-27 = 99 27/32 % of $100,000 = $998,437.50 • (Tick = $31.25) D = 8.64 years

  36. Hedging with Interest Rate Futures • I own an $11,000,000 face value portfolio of high grade US corporate bonds with an aggregate value of 101-08 (or $11,137,500) and a duration of 7.7 years. • I expect rates to rise. How can I immunize my portfolio? • Target D = contribution of bond port + contribution of fut. • 0 = (1.0)(7.7) + [(F x 998,437.50)/11,137,500](8.64) • F = (0.0 - (1.0)(7.7))(11,137,500/998,437.50)/8.64 • F = -9.94 contracts => short 10 Tbond futures contracts • This is the weighted average duration approach

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