Name Graph Description Payoff Profit Comments

Long Forward Commitment to purchase ST - F ST – F  No premium commodity at some point  Asset price contingency: Always in the future at a pre-  Maximum Loss: -F specified price  Maximum Gain: Unlimited

Short See above Commitment to sell F - ST F - ST  No premium Forward commodity at some point  Asset price contingency: Always in the future at a pre-  Maximum Loss: Unlimited specified price  Maximum Gain: F

Long Call Right, but not obligation, Max[0, ST – K] Max[0, ST –  Premium paid

(Purchased to buy a commodity at K] – FV(PC)  Asset price contingency: ST>K Call) some future date  Maximum Loss: - FV(PC)  Maximum Gain: Unlimited  COB: Call is an Option to Buy  “Call me up”: Call purchaser benefits if price of underlying asset rises

Short Call Commitment to sell a - Max[0, ST – -Max[0, ST –  Premium received

(Written Call) commodity at some K] K] + FV(PC)  Asset price contingency: ST>K future date if the  Maximum Loss: FV(PC) purchaser exercises the  Maximum Gain: FV(PC) option

Long Put Right, but not obligation, Max[0, K - ST] Max[0, K - ST]  Premium paid

(Purchased to sell a commodity at - FV(PP)  Asset price contingency: K>ST Put) some future date  Maximum Loss: - FV(PP)  Maximum Gain: K - FV(PP)  POS: Put is an Option to Sell  “Put me down”: Put purchaser benefits if price of underlying asset falls  Short with respect to underlying asset but long with respect to derivative 1

Short Put Commitment to buy a -Max[0, K - -Max[0, K -  Premium received

(Written Put) commodity at some ST] ST] + FV(PP)  Asset price contingency: K>ST future date if the  Maximum Loss: -K + FV(PP) purchaser exercises the  Maximum Gain: FV(PP) option  Long with respect to underlying asset but short with respect to derivative

Floor Long Position in Asset +  Used to insure a long position against Purchased Put price decreases  Profit graph is identical to that of a purchased call  Payoff graphs can be made identical by adding a zero-coupon bond to the purchased call

Cap Short Position in Asset +  Used to insure a short position against Purchased Call price increases  Profit graph is identical to that of a purchased put  Payoff graphs can be made identical by adding a zero-coupon bond to the purchased put

Covered call Long Position in Asset + Long Index Payoff  Graph similar to that of a written put writing Sell a Call Option + {-max[0, ST – K] + FV(PC)}

Covered put Short Position in Asset + - Long Index Payoff  Graph similar to that of a written call writing Write a Put Option + {-max[0, K - ST] + FV(PP)}

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Synthetic Purchase Call Option + {max[0, ST – K] –  Mimics long forward position, but Forward Write Put Option with FV(PC)} + involves premiums and uses “strike SAME Strike Price and {-max[0, K - ST] + price” rather than “forward price” Expiration Date FV(PP)}  Put-call parity: Call(K,T) – Put(K,T) = PV(F0,T – K)

Bull Spread Purchase Call Option with Strike {max[0, ST – K1] –  Investor speculates that stock price Price K1 and Sell Call Option with FV(PC1)} + will rise Strike Price K2, where K2>K1 {-max[0, ST – K2] +  Although investor gives up a portion of OR FV(PC2)} his profit on the purchased call, this is Purchase Put Option with Strike offset by the premium received for Price K1 and Sell Put Option with selling the call Strike Price K , where K >K 2 2 1

Bear Spread Sell Call Option with Strike Price K1 {-max[0, ST – K1] +  Investor speculates that stock price and Purchase Call Option with FV(PC1)} + will fall Strike Price K2, where K2>K1 {max[0, ST – K2] -  Graph is reflection of that of a bull OR FV(PC2)} spread about the horizontal axis Sell Put Option with Strike Price K1 and Purchase Put Option with Strike Price K , where K >K 2 2 1

Box Spread Bull Call Spread Bear Put Consists of 4 Options and creates a  Guarantees cash flow into the future Spread Synthetic Long Forward at one  Purely a means of borrowing or Synthetic Long Buy Call at K1 Sell Put at K1 price and a synthetic short forward lending money Forward at a different price  Costly in terms of premiums but has Synthetic Short Sell Call at K2 Buy Put at K2 no stock price risk Forward

Ratio Spread Buy m calls at strike price K1 and –  Enables spreads with 0 premium sell n calls at strike price K2  Useful for paylater strategies OR Buy m puts at strike price K1 and - sell n puts at strike price K2

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Purchased Buy at-the-money Put Option  Collar width: Collar with strike price K1 + Sell out-of- K2 - K1 the-money Call Option with strike price K2, where K2>K1

Written Sell at-the-money Put Option Collar with strike price K1 + Buy out-of- the-money Call Option with strike price K2, where K2>K1 Collared Buy index + Buy at-the-money K1-  Purchased Put insures the index Stock strike put option + sell out-of-the-  Written Call reduces cost of insurance money K2 strike call option, where K2>K1

Zero-cost Buy at-the-money Put + Sell out-  For any given stock, there is an infinite number of zero-cost collar of-the-money Call with the same collars premium  If you try to insure against all losses on the stock (including interest), then a zero-cost collar will have zero width

Straddle Buy a Call + Buy a Put with the  This is a bet that volatility is really greater than the market same strike price, expiration assessment of volatility, as reflected in option prices time, and underlying asset  High premium since it involves purchasing two options

 Guaranteed payoff as long as ST is different than K  Profit = |ST – K| – FV(PC) – FV(PP)

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Strangle Buy an out-of-the-money Call +  Reduces high premium cost of straddles Buy an out-of-the money Put  Reduces maximum loss but also reduces maximum profit with the same expiration time and underlying asset

Written Sell a Call + Sell a Put with the  Bet that volatility is lower than the market’s assessment Straddle same strike price, expiration time, and underlying asset

Butterfly Sell a K2-strike Call + Sell a K2-  Combination of a written straddle and insurance against Spread strike Put extreme negative outcomes AND  Out of the Money Put insures against extreme price Buy out-of-the-money K3-strike decreases Put  Out of the Money Call insures against extreme price AND increases Buy out-of-the-money K1-strike Call

K1< K2< K3 Asymmetric λ = K3 - K2 Butterfly K3 – K1 Spread Buy λ K1-strike calls Buy (1 – λ) K3-strike calls K2 = λK1 + (1 – λ)K3 K1< K2< K3

Cash-and- Buy Underlying Asset +  No Risk carry Short the Offsetting Forward  Payoff = ST + (F0,T – ST) = F0,T Contract  Cost of carry: r – δ

Cash-and- Buy Underlying Asset +  Can be created if a forward price F0,T is available such that (r – δ)T carry Sell it forward F0,T > S0e arbitrage 5

Reverse cash- Cash Flow at t=0 Cash Flow at t=T Short Underlying Asset +  Payoff = -ST + (ST - F0,T) = -F0,T -δT and-carry Short-tailed S0e -ST Long the Offsetting Forward position in stock, Contract -δT receiving S0e -δT -δT (r – δ)T Lent S0e -S0e S0e Long Forward 0 ST – F0,T (r – δ)T Total 0 S0e – F0,T

Reverse cash-  Can be created if a forward price F0,T is available such that (r – δ)T and-carry F0,T < S0e arbitrage

If… THEN If Volatility ↑ If Unsure about Direction of Volatility Change If Volatility ↓ Price ↓ Buy puts Sell underlying asset Sell calls Unsure about Direction of Buy straddle No action Write straddle Price Change Price↑ Buy calls Buy underlying asset Sell Puts

Reasons to hedge Reasons NOT to hedge 1. Taxes Transaction costs (commissions, bid-ask spread) 2. Bankruptcy and distress costs Cost-benefit analysis may require costly expertise 3. Costly external financing Must monitor transactions to prevent unauthorized trading 4. Increase debt capacity (amount a firm can borrow) Tax and accounting consequences of transactions may complicate reporting 5. Managerial risk aversion 6. Nonfinancial risk management

Method of purchasing stock Pay at time Receive security at time Payment At time

Outright Purchase 0 0 S0 t=0 rT Fully-leveraged purchase T 0 S0e t=0 -δT Prepaid Forward Contract 0 T S0e t=T (r - δ)T Forward Contract T T S0e t=T

r = Continuously-compounded interest rate δ = Annualized daily compounded dividend yield rate

α = Annualized Dividend Yield: (1 ÷ T) × ln(F0,T ÷ S0)

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P Pricing Prepaid Forward and Forward Contracts: Prepaid Forward Contract F 0,T Forward Contract F0,T

rT No Dividends S0 S0e rT r(T – t) Discrete Dividends S0 - ∑PV0,t(Dt) S0e - ∑e ×Dt -δT (r - δ)T Continuous Dividends S0e S0e P Initial Premium Initial Premium = Price = F 0,T Initial Premium = 0 P Price = F0,T = FV(F 0,T)

Forwards Futures

Obligation to buy or sell underlying asset at specified price on expiration Same date Contracts tailored to the needs of each party Contracts are standardized (in terms of expiration dates, size, etc.) Not “marked to market”; settlement made on expiration date only “Marked to market” and settled daily Relatively illiquid Liquid Traded over-the-counter and handled by dealers/brokers Exchange-traded and marked to market Risk that one party will not fulfill obligation to buy or sell (credit risk) Marked to market and daily settlement minimize credit risk Price limits are not applicable Complicated price limits

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