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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
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
Options -- Contract
  • Calls and Puts
  • Underlying Security (Number of Units)
  • Exercise or Strike Price
  • Expiration date
  • Option Premium
  • American, European, Asian, etc.
options markets
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
options basic strategies
Options -- Basic Strategies
  • Buy Call
  • Sell (write) Call
  • Buy Put
  • Sell (write) Put
options advanced strategies
Options -- Advanced Strategies
  • Straddle
  • Strips and Straps
  • Vertical Spreads
    • Bullish
    • Bearish
options determinants of value
Options - Determinants of Value
  • Value of Underlying Asset
  • Exercise Price
  • Time to Expiration
  • VOLATILITY
  • Interest Rates
  • Dividends
options black scholes option pricing model
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
futures contract
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
futures contract marking to market
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
index futures market

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.
index futures market12

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.
index futures market13
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.
hedging with derivatives
Hedging with Derivatives
  • Basic option strategies
    • Covered call
    • Protective put
    • Synthetic short
  • Basic futures strategies
    • Using interest rate futures to reduce risk
covered call
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.
protective put
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
synthetic short
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.
delta hedging with options
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!

delta hedging puts
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!

delta hedging with options22
Delta Hedging with Options
  • Delta changes over time!
    • S changes
    • Time declines
    • Other factors (r, s) may change
true delta hedging adjust hedge when s changes
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)

true delta hedging adjust hedge when t changes
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!
true delta hedging adjust hedge when s changes25
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)

true delta hedging adjust hedge when t changes26
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!
delta hedging with options27
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.
a true protective put
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!
hedging with futures example from may 2001
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 !
adjusting systematic risk with futures
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.
adjusting systematic risk with futures33
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
adjusting systematic risk with futures34
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.
hedging with interest rate futures
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
hedging with interest rate futures36
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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