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Options UNIVERSITE D’AUVERGNE Drake DRAKE UNIVERSITY Options UNIVERSITE D’AUVERGNE Drake DRAKE UNIVERSITY

Option Terminology UNIVERSITE D’AUVERGNE Drake University Call Option – the right to buy an Option Terminology UNIVERSITE D’AUVERGNE Drake University Call Option – the right to buy an asset at some point in the future for a designated price. Put Option – the right to sell an asset at some point in the future at a given price

Review of Option Terminology UNIVERSITE D’AUVERGNE Drake University Expiration Date The last day the Review of Option Terminology UNIVERSITE D’AUVERGNE Drake University Expiration Date The last day the option can be exercised (American Option) also called the strike date, maturity, and exercise date Exercise Price The price specified in the contract American Option Can be exercised at any time up to the expiration date European Option Can be exercised only on the expiration date

Review of Option Terminology UNIVERSITE D’AUVERGNE Drake University Long position: Buying an option Long Review of Option Terminology UNIVERSITE D’AUVERGNE Drake University Long position: Buying an option Long Call: Bought the right to buy the asset Long Put: Bought the right to sell the asset Short Position: Writing or Selling the option Short Call Agreed to sell the other party the right to buy the underlying asset, if the other party exercises the option you deliver the asset. Short Put - Agreed to buy the underlying asset from the other party if they decide to exercise the option.

Review of Terminology UNIVERSITE D’AUVERGNE Drake University In - the - money options when Review of Terminology UNIVERSITE D’AUVERGNE Drake University In - the - money options when the spot price of the underlying asset for a call (put) is greater (less) than the exercise price Out - of - the - money options when the spot price of the underlying asset for a call (put) is less (greater) than the exercise price At the money options when the exercise price and spot price are equal.

Interest Rate Options UNIVERSITE D’AUVERGNE Drake University Traded on Chicago Board of Options Exchange Interest Rate Options UNIVERSITE D’AUVERGNE Drake University Traded on Chicago Board of Options Exchange (CBOE) Interest rate Options are traded on 13 Week, 5 year, 10 year and 30 year treasury securities

Options on Futures UNIVERSITE D’AUVERGNE Drake University Options on futures are as popular or Options on Futures UNIVERSITE D’AUVERGNE Drake University Options on futures are as popular or even more popular than on the actual asset. Options on futures do not require payments for accrued interest. The likelihood of delivery squeezes is less. Current prices for futures are readily available, they are more difficult to find for bonds.

Futures Options UNIVERSITE D’AUVERGNE Drake University Call option holder will own a long futures Futures Options UNIVERSITE D’AUVERGNE Drake University Call option holder will own a long futures position if the option is exercised. The writer of the call option accepts the corresponding short position at the exercise price.

Mechanics of Options on Futures UNIVERSITE D’AUVERGNE Drake University Call Option Example Exercise price Mechanics of Options on Futures UNIVERSITE D’AUVERGNE Drake University Call Option Example Exercise price = $85 Current Futures price = $95 Upon exercise both the long position and the short owned by the writer of the short option is set to $85. When marked to market the holder of the long makes $10, the holder of the short looses $10. The holders of the short and long position then face the same risks as any other holder of the futures contract.

Buyer Margin Requirements on Futures Options UNIVERSITE D’AUVERGNE Drake University The buyer of the Buyer Margin Requirements on Futures Options UNIVERSITE D’AUVERGNE Drake University The buyer of the call option is not required to place any margin deposits. The most that could be lost is the cost of the option.

Call Option Writer Margin Requirements UNIVERSITE D’AUVERGNE Drake University The writer of the call Call Option Writer Margin Requirements UNIVERSITE D’AUVERGNE Drake University The writer of the call option accepts all of the risk since the buyer will not exercise if there would be a loss. The writer is required to deposit the original margin that would be required on the futures contract and the option price that is received for writing the option. The writer is also required to deposit variation margin as the contract is marked to market.

Call Option Profit UNIVERSITE D’AUVERGNE Drake University Call option – as the price of Call Option Profit UNIVERSITE D’AUVERGNE Drake University Call option – as the price of the asset increases the option is more profitable. Once the price is above the exercise price (strike price) the option will be exercised If the price of the underlying asset is below the exercise price it won’t be exercised – you only loose the cost of the option. The Profit earned is equal to the gain or loss on the option minus the initial cost.

Profit Diagram Call Option (Long Call Position) Profit S-X-C S Cost X Spot Price Profit Diagram Call Option (Long Call Position) Profit S-X-C S Cost X Spot Price UNIVERSITE D’AUVERGNE Drake University

Call Option Intrinsic Value UNIVERSITE D’AUVERGNE Drake University The intrinsic value of a call Call Option Intrinsic Value UNIVERSITE D’AUVERGNE Drake University The intrinsic value of a call option is equal to the current value of the underlying asset minus the exercise price if exercised or 0 if not exercised. In other words, it is the payoff to the investor at that point in time (ignoring the initial cost) the intrinsic value is equal to max(0, S-X)

UNIVERSITE D’AUVERGNE Payoff Diagram Call Option Payoff S-X X S Spot Price Drake University UNIVERSITE D’AUVERGNE Payoff Diagram Call Option Payoff S-X X S Spot Price Drake University

Example: Naked Call Option UNIVERSITE D’AUVERGNE Drake University Assume that you can purchase a Example: Naked Call Option UNIVERSITE D’AUVERGNE Drake University Assume that you can purchase a call option on an 8% coupon bond with a par value of $100 and 20 years to maturity. The option expires in one month and has an exercise price of $100. Assume that the option is currently at the money (the bond is selling at par) and selling for $3. What are the possible payoffs if you bought the bond and held it until maturity of the option?

Five possible results UNIVERSITE D’AUVERGNE Drake University The price of the bond at maturity Five possible results UNIVERSITE D’AUVERGNE Drake University The price of the bond at maturity of the option is $100. The buyer looses the entire purchase price, no reason to exercise. The price of the bond at maturity is less than $100 (the YTM is > 8%). The buyer looses the $3 option price and does not exercise the option.

Five Possible Results continued UNIVERSITE D’AUVERGNE Drake University The price of the bond at Five Possible Results continued UNIVERSITE D’AUVERGNE Drake University The price of the bond at maturity is greater than $100, but less than $103. The buyer will exercise the option and recover a portion of the option cost. The price of the bond is equal to $103. The buyer will exercise the option and recover the cost of the option. The price of the bond is greater than $103. The buyer will make a profit of S-$100 -$3.

Profit Diagram Call Option (Long Call Position) Profit S-100 -3 103 -3 100 S Profit Diagram Call Option (Long Call Position) Profit S-100 -3 103 -3 100 S Spot Price UNIVERSITE D’AUVERGNE Drake University

Price vs. Rate UNIVERSITE D’AUVERGNE Drake University Note buying a call on the price Price vs. Rate UNIVERSITE D’AUVERGNE Drake University Note buying a call on the price of the bond is equivalent to buying a put on the interest rate paid by the bond. As the rate decreases, the price increases because of the time value of money.

Profit Diagram Call Option (Short Call Position) Profit X C+X-S S Spot Price UNIVERSITE Profit Diagram Call Option (Short Call Position) Profit X C+X-S S Spot Price UNIVERSITE D’AUVERGNE Drake University

Put option payoffs UNIVERSITE D’AUVERGNE Drake University The writer of the put option will Put option payoffs UNIVERSITE D’AUVERGNE Drake University The writer of the put option will profit if the option is not exercised or if it is exercised and the spot price is less than the exercise price plus cost of the option. In the previous example the writer will profit as long as the spot price is less than $103. What if the spot price is equal to $103?

Put Option Profits UNIVERSITE D’AUVERGNE Drake University Put option – as the price of Put Option Profits UNIVERSITE D’AUVERGNE Drake University Put option – as the price of the asset decreases the option is more profitable. Once the price is below the exercise price (strike price) the option will be exercised If the price of the underlying asset is above the exercise price it won’t be exercised – you only loose the cost of the option.

Profit Diagram Put Option UNIVERSITE D’AUVERGNE Drake University Profit X-S-C Spot Price S Cost Profit Diagram Put Option UNIVERSITE D’AUVERGNE Drake University Profit X-S-C Spot Price S Cost X

Put Option Intrinsic Value UNIVERSITE D’AUVERGNE Drake University The intrinsic value of a put Put Option Intrinsic Value UNIVERSITE D’AUVERGNE Drake University The intrinsic value of a put option is equal to exercise price minus the current value of the underlying asset if exercised or 0 if not exercised. In other words, it is the payoff to the investor at that point in time (ignoring the initial cost) the intrinsic value is equal to max(X-S, 0)

Payoff Diagram Put Option UNIVERSITE D’AUVERGNE Drake University Profit X-S S Cost X Spot Payoff Diagram Put Option UNIVERSITE D’AUVERGNE Drake University Profit X-S S Cost X Spot Price

Profit Diagram Put Option Short Put Profit S S-X+C X Spot Price UNIVERSITE D’AUVERGNE Profit Diagram Put Option Short Put Profit S S-X+C X Spot Price UNIVERSITE D’AUVERGNE Drake University

Pricing an Option UNIVERSITE D’AUVERGNE Drake University Arbitrage arguments Black Scholes Binomial Tree Models Pricing an Option UNIVERSITE D’AUVERGNE Drake University Arbitrage arguments Black Scholes Binomial Tree Models

PV and FV in continuous time UNIVERSITE D’AUVERGNE Drake University e = 2. 71828 PV and FV in continuous time UNIVERSITE D’AUVERGNE Drake University e = 2. 71828 y = lnx x = ey FV = PV (1+k)n for yearly compounding FV = PV(1+k/m)nm for m compounding periods per year As m increases this becomes FV = PVern =PVert let t =n rearranging for PV PV = FVe-rt

Lower Bound of Call Option Price UNIVERSITE D’AUVERGNE Drake University Assume that you have Lower Bound of Call Option Price UNIVERSITE D’AUVERGNE Drake University Assume that you have an asset that does not pay a cash income (A non dividend paying stock for example) Consider the case of an option as it expires. In this case, regardless of whether it is an American or European option it will be worth its intrinsic value (max(S-X, 0)). Assuming a positive value the lower bound is given by: S-X

Formal Argument UNIVERSITE D’AUVERGNE Drake University Consider two portfolios A: One European call option Formal Argument UNIVERSITE D’AUVERGNE Drake University Consider two portfolios A: One European call option on the stock of Widget Inc. plus cash equal to Xe-r. T B: One share of stock in Widget Inc. Note: If the cash in portfolio A is invested at r, it will grow to be worth X at time T.

Portfolio A UNIVERSITE D’AUVERGNE Drake University There are two possible outcomes at time T Portfolio A UNIVERSITE D’AUVERGNE Drake University There are two possible outcomes at time T depending upon the value of S at time T ST > X Exercise the option and purchase the asset with a current value of ST (The value of portfolio A at time T is ST ). ST < X Do not exercise the option, The portfolio is then worth the value of the cash, X. Therefore the portfolio is worth: max(ST, X)

Portfolio B UNIVERSITE D’AUVERGNE Drake University The value of portfolio B is simply the Portfolio B UNIVERSITE D’AUVERGNE Drake University The value of portfolio B is simply the value of the stock at time T, ST.

Comparing A to B UNIVERSITE D’AUVERGNE Drake University Combining the two results it is Comparing A to B UNIVERSITE D’AUVERGNE Drake University Combining the two results it is easy to see that portfolio A (the option and the cash) is always worth at least as much as portfolio B (owning the stock), and sometimes it is worth more than B. Without arbitrage, the same relationship should be true today as well as at time T in the future.

Equal value of the portfolios today UNIVERSITE D’AUVERGNE Drake University Let c be the Equal value of the portfolios today UNIVERSITE D’AUVERGNE Drake University Let c be the call price (value of option) today. Then the value of portfolio A is: c + Xe-r. T The value of portfolio B is: S Since the value of A is always worth as much as B and sometimes it is worth more: c + Xe-r. T >S or rearranging c > S - Xe-r. T

Final result UNIVERSITE D’AUVERGNE Drake University The worst outcome to buying a call option Final result UNIVERSITE D’AUVERGNE Drake University The worst outcome to buying a call option is that it expires worthless, so the option is worth either nothing or S-Xe-r. T Therefore: c > max(S - Xe-r. T, 0)

Put Option UNIVERSITE D’AUVERGNE Drake University Similar to a call option the put option Put Option UNIVERSITE D’AUVERGNE Drake University Similar to a call option the put option should always have a positive value. Considering the case of an option as it expires (either an American or European Option), the value of the option should be equal to its intrinsic value. The lower bound is therefore: X-S

European Put Option UNIVERSITE D’AUVERGNE Drake University Again, in the case of a European European Put Option UNIVERSITE D’AUVERGNE Drake University Again, in the case of a European option prior to maturity this equation will not hold and it is necessary to account for the time value of money. In this case the lower bound for the option is given by: Xe-r. T - S

Formal Argument: UNIVERSITE D’AUVERGNE Drake University Consider two Portfolios C: One European put option Formal Argument: UNIVERSITE D’AUVERGNE Drake University Consider two Portfolios C: One European put option plus one share and D: An amount of cash equal to Xe-r. T

Portfolio C: UNIVERSITE D’AUVERGNE Drake University There are two possibilities: ST < X Exercise Portfolio C: UNIVERSITE D’AUVERGNE Drake University There are two possibilities: ST < X Exercise the option at time T and the portfolio is worth X. ST > X The option expires and the portfolio is worth ST Portfolio C is therefore worth max(ST, X)

Portfolio D UNIVERSITE D’AUVERGNE Drake University Investing the amount at a rate equal to Portfolio D UNIVERSITE D’AUVERGNE Drake University Investing the amount at a rate equal to r, the portfolio will be worth X at Time T. Combining the two arguments it is easy to see that portfolio C is always worth at least the same amount as portfolio D and sometimes it is worth more.

Comparing the two UNIVERSITE D’AUVERGNE Drake University Let p = the value (price) of Comparing the two UNIVERSITE D’AUVERGNE Drake University Let p = the value (price) of the put option Without arbitrage opportunities: p + S > Xe-r. T or rearranging p > Xe-r. T - S the value of the put option is then given as p > max(Xe-r. T-S, 0)

Put Call Parity UNIVERSITE D’AUVERGNE Drake University Consider portfolio A and C above A: Put Call Parity UNIVERSITE D’AUVERGNE Drake University Consider portfolio A and C above A: One European call option plus an amount of cash equal to Xe-r. T C: One European put option plus one share Both portfolios are have a value of max(X, ST) at the expiration of the options. If no arbitrage opportunities exist, they should also have the same value today which implies: c + Xe-r. T = p + S

Put Call Parity UNIVERSITE D’AUVERGNE Drake University In other words, the value of a Put Call Parity UNIVERSITE D’AUVERGNE Drake University In other words, the value of a European call with a given exercise date can be deduced from the value of a European put with the same exercise date and exercise price.

Put call Parity UNIVERSITE D’AUVERGNE Drake University Without this condition arbitrage opportunities exist: Put-Call Put call Parity UNIVERSITE D’AUVERGNE Drake University Without this condition arbitrage opportunities exist: Put-Call Parity specifies that: c + Xe-r. T = p + S which rearranges to p = c + Xe-r. T - S

A Fixed Income Example UNIVERSITE D’AUVERGNE Drake University Previously we discussed had a call A Fixed Income Example UNIVERSITE D’AUVERGNE Drake University Previously we discussed had a call option on an 8% coupon bond with a par value of $100 and 20 years to maturity. The option expires in one month and has an exercise price of $100. The option is currently at the money (the bond is selling at par) and selling for $3. Assume we also have a put option on the same bond and the put option is selling for $2.

A Fixed Income Example UNIVERSITE D’AUVERGNE Drake University For now, ignore the coupon payments A Fixed Income Example UNIVERSITE D’AUVERGNE Drake University For now, ignore the coupon payments on the bond. Consider three possible strategies simultaneously Buy the bond in the spot market for $100 Enter into a short call position (sell the call) for $3 Buy a put at a price of $2

Possible Outcomes at expiration UNIVERSITE D’AUVERGNE Drake University Bond Price > $100 The call Possible Outcomes at expiration UNIVERSITE D’AUVERGNE Drake University Bond Price > $100 The call option is exercised so you are forced to sell the bond at a price equal to $100. The put option expires. You make $1 profit from the difference in the call and put prices Bond Price <$100 You exercise the Put option and sell the Bond for $100, which is the same price you paid. The Call expires worthless. You make a $1 from the difference in the price.

Arbitrage UNIVERSITE D’AUVERGNE Drake University Regardless of the price of the bond at expiration, Arbitrage UNIVERSITE D’AUVERGNE Drake University Regardless of the price of the bond at expiration, there was a $1 profit. Three possible things would eliminate arbitrage. An increase in the price of the bond today. A decrease in the call option price. An increase in the put option price.

Arbitrage continued UNIVERSITE D’AUVERGNE Drake University Assume that the price of the bond doesn’t Arbitrage continued UNIVERSITE D’AUVERGNE Drake University Assume that the price of the bond doesn’t change: There would be an increase in market participants attempting to short call options. To compete with each other they lower the price and the call price will decrease. There will be an increase in the number of market participants wanting to purchase long put options. To compete with each other they will offer higher prices increasing the price.

Put Call Parity Revisited p = c + Xe-r. T - S UNIVERSITE D’AUVERGNE Put Call Parity Revisited p = c + Xe-r. T - S UNIVERSITE D’AUVERGNE Drake University We have ignored the Time value of money so the relationship becomes: p=c+X-S In our example X=S which implies that p should equal c. If both the put and call price equaled each othere would be no arbitrage profits regardless of what happened to the bond price at maturity.

Put Call Extensions UNIVERSITE D’AUVERGNE Drake University We ignored the time value of money Put Call Extensions UNIVERSITE D’AUVERGNE Drake University We ignored the time value of money and the coupon payments paid by the bond. The coupon payment can be treated similar to the price. If you own the bond you will receive the cash payment in the future. The put call parity relationship for a coupon bond is simply: p=c+Xe-r. T+CPe-r. T-S where CP is the coupon payment received at time T

Put Call Parity and American Options UNIVERSITE D’AUVERGNE Drake University Put-Call Parity holds only Put Call Parity and American Options UNIVERSITE D’AUVERGNE Drake University Put-Call Parity holds only for European Options but it is possible to use the relationship to specify some generalizations concerning the relationship between American Puts and Calls.

American call Option vs. European Call Option UNIVERSITE D’AUVERGNE Drake University Should an American American call Option vs. European Call Option UNIVERSITE D’AUVERGNE Drake University Should an American call option on a non dividend paying stock be exercised prior to maturity? NO (assuming that the investor plans to hold the stock past the maturity date of the option. )

Should an American call Option be Exercised Early? UNIVERSITE D’AUVERGNE Drake University Assume that Should an American call Option be Exercised Early? UNIVERSITE D’AUVERGNE Drake University Assume that the option currently is deep in the money, The following possibilities exist 1) S > X The investor can earn interest amount of cash equal to X and then still pay X for the stock upon expiration of the option. 2) S < X The investor can then purchase the stock at the spot price and let the option expire. 3) S=X Again there is no reason to exercise the option, and the investor will let the option expire.

Exercising Call Options UNIVERSITE D’AUVERGNE Drake University Since it is never optimal to exercise Exercising Call Options UNIVERSITE D’AUVERGNE Drake University Since it is never optimal to exercise the call early, the value of the American Call (C) should be equal to the value of the European Call (c).

Exercising Put Options UNIVERSITE D’AUVERGNE Drake University Should an American put option on a Exercising Put Options UNIVERSITE D’AUVERGNE Drake University Should an American put option on a non dividend paying stock be exercised prior to maturity? Yes (if it is sufficiently in the money) The general argument is that the Put option serves as insurance and that early exercise is a good idea if the investor realizes a significant gain from the exercise of the option

Exercising Put Options UNIVERSITE D’AUVERGNE Drake University The price of an American put option Exercising Put Options UNIVERSITE D’AUVERGNE Drake University The price of an American put option should be above that of an Equivalent European option (P>p) The value of an American Call should equal the value of an European Call. Using the put call parity relationship and substituting generalizations can be made about American Options:

UNIVERSITE D’AUVERGNE Drake University P > p = c + Xe-r. T - S UNIVERSITE D’AUVERGNE Drake University P > p = c + Xe-r. T - S P > C + Xe-r. T - S Which rearranges to C - P < S - Xe-r. T It can also be shown that C - P > S-X Which combines with the above equation to prove: S - X < C - P < S - Xe-r. T

Black Scholes UNIVERSITE D’AUVERGNE Drake University The basic starting point for the actual pricing Black Scholes UNIVERSITE D’AUVERGNE Drake University The basic starting point for the actual pricing of an European option is the model developed by Fisher Black, Myron Scholes, and Robert Merton.

Black Scholes Assumptions UNIVERSITE D’AUVERGNE Drake University 1) 2) 3) 4) 5) 6) 7) Black Scholes Assumptions UNIVERSITE D’AUVERGNE Drake University 1) 2) 3) 4) 5) 6) 7) Stock prices follow a lognormal distribution with m and s constant. There are no transaction costs or taxes and all securities are perfectly divisible There are no dividend on the asset during the life of the option There are no riskless arbitrage opportunities Security trading is continuous Investors can borrow and lend at the same risk free rate The short term risk free rate is constant

Inputs you will need UNIVERSITE D’AUVERGNE Drake University S = Current value of underlying Inputs you will need UNIVERSITE D’AUVERGNE Drake University S = Current value of underlying asset X = Exercise price t = life until expiration of option r = riskless rate s 2 = variance

Black Scholes UNIVERSITE D’AUVERGNE Drake University Value of Call Option = SN(d 1)-Xe-rt. N(d Black Scholes UNIVERSITE D’AUVERGNE Drake University Value of Call Option = SN(d 1)-Xe-rt. N(d 2) S = Current value of underlying asset X = Exercise price t = life until expiration of option r = riskless rate s 2 = variance N(d ) = the cumulative normal distribution (the probability that a variable with a standard normal distribution will be less than d)

Black Scholes (Intuition) UNIVERSITE D’AUVERGNE Drake University Value of Call Option SN(d 1) The Black Scholes (Intuition) UNIVERSITE D’AUVERGNE Drake University Value of Call Option SN(d 1) The expected Value of S if S > X - Xe-rt N(d 2) PV of cost Risk Neutral of investment Probability of S>X

Black Scholes UNIVERSITE D’AUVERGNE Drake University Value of Call Option = SN(d 1)-Xe-rt. N(d Black Scholes UNIVERSITE D’AUVERGNE Drake University Value of Call Option = SN(d 1)-Xe-rt. N(d 2) Where:

Extending Black Scholes to Futures Options Black extended the original model to price options Extending Black Scholes to Futures Options Black extended the original model to price options on futures. UNIVERSITE D’AUVERGNE Drake University

Time Value of an Option UNIVERSITE D’AUVERGNE Drake University The time value of an Time Value of an Option UNIVERSITE D’AUVERGNE Drake University The time value of an option is the difference in theoretical price of the option and the intrinsic value. It represents the possibility that the value of the option will increase over the time it is owned.

UNIVERSITE D’AUVERGNE Time Value of Call Option Drake University Payoff S-X Time value of UNIVERSITE D’AUVERGNE Time Value of Call Option Drake University Payoff S-X Time value of option X S Spot Price

Delta of an option UNIVERSITE D’AUVERGNE Drake University The delta of the option shows Delta of an option UNIVERSITE D’AUVERGNE Drake University The delta of the option shows how theoretical price of the option will change with a small change in the underlying asset.

UNIVERSITE D’AUVERGNE Time Value of Call Option Drake University Payoff S-X Time value of UNIVERSITE D’AUVERGNE Time Value of Call Option Drake University Payoff S-X Time value of option X S Spot Price Delta is the slope of the tangent line at the given stock price

Delta of an option UNIVERSITE D’AUVERGNE Drake University Intuitively a higher stock price should Delta of an option UNIVERSITE D’AUVERGNE Drake University Intuitively a higher stock price should lead to a higher call price. The relationship between the call price and the stock price is expressed by a single variable, delta. The delta is the change in the call price for a very small change it the price of the underlying asset.

Delta UNIVERSITE D’AUVERGNE Drake University Delta can be found from the call price equation Delta UNIVERSITE D’AUVERGNE Drake University Delta can be found from the call price equation as: Using delta hedging for a short position in a European call option would require keeping a long position of N(d 1) shares at any given time. (and vice versa).

Delta explanation UNIVERSITE D’AUVERGNE Drake University Delta will be between 0 and 1. A Delta explanation UNIVERSITE D’AUVERGNE Drake University Delta will be between 0 and 1. A 1 cent change in the price of the underlying asset leads to a change of delta cents in the price of the option.

Delta UNIVERSITE D’AUVERGNE Drake University For deep in the money call options the delta Delta UNIVERSITE D’AUVERGNE Drake University For deep in the money call options the delta will be close to 1. For deep out of the money call options the delta will be close to zero.

Gamma UNIVERSITE D’AUVERGNE Drake University Gamma measures the curvature of theoretical call option price Gamma UNIVERSITE D’AUVERGNE Drake University Gamma measures the curvature of theoretical call option price line.

Gamma of an Option UNIVERSITE D’AUVERGNE Drake University The change in delta for a Gamma of an Option UNIVERSITE D’AUVERGNE Drake University The change in delta for a small change in the stock price is called the options gamma: Call gamma =

Other Measures UNIVERSITE D’AUVERGNE Drake University Other Measures UNIVERSITE D’AUVERGNE Drake University

Hedging UNIVERSITE D’AUVERGNE Drake University If a firm is worried about an increase in Hedging UNIVERSITE D’AUVERGNE Drake University If a firm is worried about an increase in borrowing costs it could buy a call option on the relevant interest rate. Any gains on the call options will offset the increased borrowing. Similarly if the firm is worried about a decline in rates decreasing income it could buy a put option on the interest rate. Any decline in income would be offset by the change in rates.

A Short Hedge UNIVERSITE D’AUVERGNE Drake University Agree to sell 10 Eurodollar future contracts A Short Hedge UNIVERSITE D’AUVERGNE Drake University Agree to sell 10 Eurodollar future contracts (each with an underlying value of $1 Million). We want to look at two results the spot market and the futures market. Assume you close out the futures position and that the futures price will converge to the spot at the end of the three months.

Rates increase to 8% UNIVERSITE D’AUVERGNE Drake University Spot position: Need to pay 8% Rates increase to 8% UNIVERSITE D’AUVERGNE Drake University Spot position: Need to pay 8% + 1% = 9% on $10 Million(. 09/4) = $225, 000 Futures Position: Fut Price = $92 interest rates increased by. 9% Close out futures position: profit = ($10 million)(. 009/4) = $22, 500

Rates Increase to 8% UNIVERSITE D’AUVERGNE Drake University Net interest paid $225, 000 - Rates Increase to 8% UNIVERSITE D’AUVERGNE Drake University Net interest paid $225, 000 - $22, 500 = $202, 500 $10 million(. 0810/4) = $202, 500

Rates decrease to 6% UNIVERSITE D’AUVERGNE Drake University Spot position: Need to pay 6% Rates decrease to 6% UNIVERSITE D’AUVERGNE Drake University Spot position: Need to pay 6% + 1% = 7% on $10 Million(. 07/4) = $175, 000 Futures Position: Fut Price = $94 interest rates decreased by 1. 1% Close out futures position: loss = ($10 million)(. 011/4) = $27, 500

Rates Decrease to 8% UNIVERSITE D’AUVERGNE Drake University Net interest paid $175, 000 + Rates Decrease to 8% UNIVERSITE D’AUVERGNE Drake University Net interest paid $175, 000 + $27, 500 = $202, 500 $10 million(. 0810/4) = $202, 500

Results of Hedge UNIVERSITE D’AUVERGNE Drake University Either way the final interest rate expense Results of Hedge UNIVERSITE D’AUVERGNE Drake University Either way the final interest rate expense was equal to 8. 10 % or 100 basis points above the initial futures rate of 7. 10% Should the position be hedged? It locks in the interest rate, but if rates had declined you were better off without the hedge.

Hedging with options UNIVERSITE D’AUVERGNE Drake University Assume that you can purchase a put Hedging with options UNIVERSITE D’AUVERGNE Drake University Assume that you can purchase a put option on the futures contract with a strike price of $93 (7%) and a cost of. 40 If interest rates increase to 8% the put guarantees that the worst case would be a rte of 7% +1% +. 4% = 8. 4% which implies a total interest cost of $210, 000 If rates decrease to 7% you only have the added cost of the option resulting in a total interest expense of 7. 4% or $185, 000

Hedging Interest Rate Risk UNIVERSITE D’AUVERGNE Drake University Previously we purchased a put option Hedging Interest Rate Risk UNIVERSITE D’AUVERGNE Drake University Previously we purchased a put option on a eurodollars futures contract to hedge against a change in interest rates. The result was that it limited the upper rate that might be paid and allowed a decrease in rates to decease the actual rate paid (the option wasn’t exercised). There is a (sometimes substantial) cost to entering into the option contracts to accomplish this.

Option position UNIVERSITE D’AUVERGNE Drake University In the example the option profited as the Option position UNIVERSITE D’AUVERGNE Drake University In the example the option profited as the level of interest rates increased (the price of bonds decreased)

Profit Diagram Put Option UNIVERSITE D’AUVERGNE Drake University Profit X-S-C Spot Price S Cost Profit Diagram Put Option UNIVERSITE D’AUVERGNE Drake University Profit X-S-C Spot Price S Cost X

Spot Position UNIVERSITE D’AUVERGNE Drake University The investor was attempting to hedge against an Spot Position UNIVERSITE D’AUVERGNE Drake University The investor was attempting to hedge against an increase in the level of interest rates, in other words, they paid a higher borrowing cost as rates increased. This is the same idea as saying they lost money as the price of the bonds declined. This was offset by the profit from the option, but you incur the cost of the option.

Diagramming the spot UNIVERSITE D’AUVERGNE Drake University The spot position could be represented by Diagramming the spot UNIVERSITE D’AUVERGNE Drake University The spot position could be represented by a straight line that represents the corresponding savings in interest rates. The line will also slope up to the right. As the price increases (rate decreases) there is a relative improvement since the rate decrease saves the investor money.

UNIVERSITE D’AUVERGNE Profit Diagram Spot Drake University Profit Spot Price Cost UNIVERSITE D’AUVERGNE Profit Diagram Spot Drake University Profit Spot Price Cost

UNIVERSITE D’AUVERGNE Drake University Assume that the current interest rate is just below the UNIVERSITE D’AUVERGNE Drake University Assume that the current interest rate is just below the rate implied by the strike price. The two positions could be represented on a single graph which explains the results of the hedge. At rates above the strike price the profit on the option cancels out the loss from the increased rates. At rates below the strike price you gain, but the gain is reduced by the cost of the option.

Profit Diagram Put Option UNIVERSITE D’AUVERGNE Drake University X Combined Position Profit Diagram Put Option UNIVERSITE D’AUVERGNE Drake University X Combined Position

“Selling off” benefits UNIVERSITE D’AUVERGNE Drake University It is possible to decrease the impact “Selling off” benefits UNIVERSITE D’AUVERGNE Drake University It is possible to decrease the impact of the option’s cost One approach would be buying a cap as before, but also selling at cap at a higher rate (lower price). The money received from selling the option offsets the initial cost of the other option position. The downside is that if rates increased above the second level, you are exposed to the interest rate change.

Selling off benefits – options position UNIVERSITE D’AUVERGNE Drake University However if the level Selling off benefits – options position UNIVERSITE D’AUVERGNE Drake University However if the level of interest rates increased too high the option that was sold would experience a loss offsetting the gains from the original position. Therefore the original position is no longer hedged. Assume that the two options are for the same expiration date and are both European

UNIVERSITE D’AUVERGNE Profit Diagram Put Options Drake University Short Put Spot Price Long Put UNIVERSITE D’AUVERGNE Profit Diagram Put Options Drake University Short Put Spot Price Long Put

Profit Diagram Bear Spread UNIVERSITE D’AUVERGNE Drake University Spot price Bear Spread Profit Diagram Bear Spread UNIVERSITE D’AUVERGNE Drake University Spot price Bear Spread

Bear Spread UNIVERSITE D’AUVERGNE Drake University The previous example was essentially buying an interest Bear Spread UNIVERSITE D’AUVERGNE Drake University The previous example was essentially buying an interest rate cap (buying put on price of bonds) and selling an interest rate cap (selling a put on the price of bonds). The position could also be thought of as buying and selling call options on the level of interest rates. Below the lower price (above the higher yield) the two options cancel each other out so the increased cost associated with the spot position is unhedged.

Adding the spot position UNIVERSITE D’AUVERGNE Drake University Again assume that the current level Adding the spot position UNIVERSITE D’AUVERGNE Drake University Again assume that the current level of interest rates is slightly below the lower of the two strike prices.

Profit Diagram Bear Spread UNIVERSITE D’AUVERGNE Drake University Spot price Hedged Position Bear Spread Profit Diagram Bear Spread UNIVERSITE D’AUVERGNE Drake University Spot price Hedged Position Bear Spread

Another Strategy UNIVERSITE D’AUVERGNE Drake University To avoid the downside of the previous example Another Strategy UNIVERSITE D’AUVERGNE Drake University To avoid the downside of the previous example you could buy the same interest rate cap, but sell an interest rate floor at a higher price (lower yield).

Interest Rate Floor UNIVERSITE D’AUVERGNE Drake University A call option on the price of Interest Rate Floor UNIVERSITE D’AUVERGNE Drake University A call option on the price of the bond can be represented as a floor on the level of interest rates. The option will be profitable if the price of the bond increases above the strike price (the interest rate decreases below the strike). This would offset a loss on an asset that is rate sensitive and effectively limit the loss.

Profit Diagram Call Option (Long Call Position) Profit S-X-C S Cost X Spot Price Profit Diagram Call Option (Long Call Position) Profit S-X-C S Cost X Spot Price UNIVERSITE D’AUVERGNE Drake University

Spot position UNIVERSITE D’AUVERGNE Drake University Since the option profits as the rates decrease Spot position UNIVERSITE D’AUVERGNE Drake University Since the option profits as the rates decrease (the price of a bond increases) this would offset lost income on an asset that is rate sensitive. In our new position we want to sell the option.

Profit Diagram Call Option (Short Call Position) Profit X C+X-S S Spot Price UNIVERSITE Profit Diagram Call Option (Short Call Position) Profit X C+X-S S Spot Price UNIVERSITE D’AUVERGNE Drake University

Another Strategy UNIVERSITE D’AUVERGNE Drake University The new strategy is to avoid the downside Another Strategy UNIVERSITE D’AUVERGNE Drake University The new strategy is to avoid the downside of the previous example you could buy the same interest rate cap, while selling an interest rate floor at a higher price (lower yield).

Profit Diagram Call Option (Short Call Position) Profit Long Put Short Call UNIVERSITE D’AUVERGNE Profit Diagram Call Option (Short Call Position) Profit Long Put Short Call UNIVERSITE D’AUVERGNE Drake University

UNIVERSITE D’AUVERGNE Profit Diagram Costless Collar Profit Drake University Long Put Combined Profit Short UNIVERSITE D’AUVERGNE Profit Diagram Costless Collar Profit Drake University Long Put Combined Profit Short Call

Combined with Spot UNIVERSITE D’AUVERGNE Drake University By using options selling at the same Combined with Spot UNIVERSITE D’AUVERGNE Drake University By using options selling at the same price the net cost is zero. At prices above the higher strike price (below the lower yield) the gain is offset by a loss in the option position. At prices below the lower strike price (above the higher yield) the loss is offset by gains in the option. You have limited both the gain and loss. Assume that the current interest rate is exactly between the two strike prices

Profit Diagram Costless Collar (Fence) UNIVERSITE D’AUVERGNE Drake University Profit Costless Collar Profit Diagram Costless Collar (Fence) UNIVERSITE D’AUVERGNE Drake University Profit Costless Collar

Complications UNIVERSITE D’AUVERGNE Drake University In the previous examples we ignored many real world Complications UNIVERSITE D’AUVERGNE Drake University In the previous examples we ignored many real world complications. This is especially true if you are buying options on futures (treasury bond futures for example). Many of these complications arise from the ideas of basis risk presented earlier.

New Example: Protective Put (From Fabozzi) UNIVERSITE D’AUVERGNE Drake University Assume that you own New Example: Protective Put (From Fabozzi) UNIVERSITE D’AUVERGNE Drake University Assume that you own a corporate bond and you are afraid that an increase in interest rates will decrease the value of the bond. It would be possible to use futures or futures options to lock in a future sale price for the bonds. Assume that the coupon on the bond is 11. 75% and they mature on April 19, 2023. Today is April 19, 1985 and you plan to sell the bond in June 1985.

The Hedge UNIVERSITE D’AUVERGNE Drake University To hedge against the possible increase in interest The Hedge UNIVERSITE D’AUVERGNE Drake University To hedge against the possible increase in interest rates you decide to buy a put option on the treasury bond futures contract. If interest rates increase, the price of the underlying bonds will decrease allowing you to own a short futures position with the higher futures price (equal to the strike price).

Determining the Strike Price UNIVERSITE D’AUVERGNE Drake University The strike price will effectively set Determining the Strike Price UNIVERSITE D’AUVERGNE Drake University The strike price will effectively set a cap on the level of interest rates since it rates increase above the rate corresponding to the strike price you profit from the option offsetting the loss in bond value. Assume that you do not want the price of the bond to drop below $87. 668.

Target Strike Price = 87. 668 UNIVERSITE D’AUVERGNE Drake University The problem is that Target Strike Price = 87. 668 UNIVERSITE D’AUVERGNE Drake University The problem is that the futures option is not for the bond which you own, it is for a treasury bond. You need to set a strike price for the Treasury bond that corresponds with a price of 87. 688 for the corporate bond.

Price Vs. Yield UNIVERSITE D’AUVERGNE Drake University Choosing the minimum price is equivalent to Price Vs. Yield UNIVERSITE D’AUVERGNE Drake University Choosing the minimum price is equivalent to selecting the maximum yield on the corporate bond. A price of 87. 668 implies that the corporate bond will be paying a yield of 13. 41% (since it is selling at a large discount the yield will be above the coupon rate)

Yield on CTD treasury UNIVERSITE D’AUVERGNE Drake University The futures contract underlying the option Yield on CTD treasury UNIVERSITE D’AUVERGNE Drake University The futures contract underlying the option has a large set of acceptable treasuries that can be delivered. You can find the cheapest to deliver at the current date. Assume that after finding the cheapest to deliver bond, you find that is has a current yield 90 basis points less than current yield on the corporate. Assume that the yield spread stays fixed.

Price of CTD Treasury UNIVERSITE D’AUVERGNE Drake University Given that the yield spread stays Price of CTD Treasury UNIVERSITE D’AUVERGNE Drake University Given that the yield spread stays fixed at 90 basis points and that the maximum acceptable yield on the corporate bond is 13. 41% it implies a yield of 12. 51% on the treasury. This implies a price of $63. 756 for the treasury that is currently CTD. The price used in the futures option will not be this price, it must be found using the conversion factor

Finding the Strike Price UNIVERSITE D’AUVERGNE Drake University Given the treasuries price of $63. Finding the Strike Price UNIVERSITE D’AUVERGNE Drake University Given the treasuries price of $63. 756 and the conversion factor for the treasury of. 9660, a futures price is then found to be: 63. 756/. 9660 = 66 Therefore a strike price of 66 on the treasury futures option contract would be used.

Hedge ratio UNIVERSITE D’AUVERGNE Drake University Hedging the position with just futures contract would Hedge ratio UNIVERSITE D’AUVERGNE Drake University Hedging the position with just futures contract would have required finding the hedge ratio, this still applies. Assuming that you found the hedge ratio to be 1. 24, you will need 1. 24 put futures options for each spot position.

Another approach: A Covered Call UNIVERSITE D’AUVERGNE Drake University The assumption is that the Another approach: A Covered Call UNIVERSITE D’AUVERGNE Drake University The assumption is that the change in interest rates will be small. To hedge against a possible decline in rates, the holder of a bond (or portfolio) sells out of the money calls. The income from the sale of the option provides income to offset a possible increase in rates that lowers the bond value.

The Maximum Effective Call Price UNIVERSITE D’AUVERGNE Drake University Assume that the maximum effective The Maximum Effective Call Price UNIVERSITE D’AUVERGNE Drake University Assume that the maximum effective call price you set is 102. 66 plus the premium from the call option. The price of 102. 66 corresponds to a yield of 11. 436% on the corporate bonds. Keeping the 90 basis point spread the yield on the CTD treasury should be 10. 536% which implies a 75. 348 price for the treasury

The strike price UNIVERSITE D’AUVERGNE Drake University Using the conversion factor, this implies a The strike price UNIVERSITE D’AUVERGNE Drake University Using the conversion factor, this implies a futures price of 75. 348/. 9960 = 78 which is also the strike price on the call option you sell.

Comparing Strategies UNIVERSITE D’AUVERGNE Drake University Comparing a basic futures position to the covered Comparing Strategies UNIVERSITE D’AUVERGNE Drake University Comparing a basic futures position to the covered call and protective put in the previous examples shows that each has its own advantages and disadvantages. The basic futures position sets the price (and yield) regardless of what happens to the level of interest rates in the economy. However the other two provide scenarios where you are better off than this ( and scenarios where you would have been worse off)

Comparing Strategies UNIVERSITE D’AUVERGNE Drake University The protective put does better if rates decrease Comparing Strategies UNIVERSITE D’AUVERGNE Drake University The protective put does better if rates decrease and the call option in the covered call option is exercised. The protective put also outperforms the basic futures option if as rates decline, but it is outperformed by the covered call. For extreme rate increases, the option strategies are both outperformed by the basic futures position.