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Key topics: Interest rate derivative instruments Credit analysis Case study: how to apply Altman’s Z score model to a company based on available financial data

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**Session 7 Interest Rate Derivative Instruments and Related Valuation**

Fixed Income Analysis Session 7 Interest Rate Derivative Instruments and Related Valuation

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**Interest Rate Derivative Instruments by Frank J. Fabozzi**

PowerPoint Slides by David S. Krause, Ph.D., Marquette University Copyright 2007 John Wiley & Sons, Inc. All rights reserved. Reproduction or translation of this work beyond that permitted in Section 117 of the 1976 United States Copyright Act without the express permission of the copyright owner is unlawful. Request for futher information should be addressed to the Permissions Department, John Wiley & Sons, Inc. The purchaser may make back-up copies for his/her own use only and not for distribution or resale. The Publisher assumes no responsibility for errors, omissions, or damages caused by the use of these programs or from the use of the information contained herein.

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**Chapter 13 Interest Rate Derivative Instruments**

Major learning outcomes: Understand financial contracts that are popularly referred to as interest rate derivative instruments Understand the basic features of futures, forwards, options, swaps, caps, and floors Understand the differences between derivatives and cash market instruments

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Key Learning Outcomes Discuss the characteristics of interest rate futures and forward contracts. Calculate the implied repo rate for an acceptable to-deliver bond for a Treasury futures contract and demonstrate how this rate is used to choose the cheapest-to-deliver issue. Contrast (1) interest rate options and interest rate futures, (2) exchange-traded-options and over-the-counter options, and (3) futures options on fixed income securities and options on fixed income securities. Characterize the change in the value of an interest rate swap for each counterparty when interest rates change.

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**Key Learning Outcomes Compare the position of:**

the counterparties in an interest rate swap to the counterparties in an interest rate futures and (2) the counterparties in an interest rate swap to the counterparties in a floating rate bond purchased by borrowing on a fixed-rate basis. Demonstrate how both a cap and a floor are packages of (1) options on interest rates and (2) options on fixed income instruments. Compute the payoff for a cap and a floor and explain how a collar is created.

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**Why Use Interest Rate Derivatives Rather than Cash Instruments?**

There are three principal reasons for doing this when there is a well-developed interest rate derivatives market for a particular cash market instrument. Typically it costs less to execute a transaction or a strategy in the interest rate derivatives market in order to alter the interest rate risk exposure of a portfolio than to make the adjustment in the corresponding cash market. Portfolio adjustments typically can be accomplished faster in the interest rate derivatives market than in the corresponding cash market. Interest rate derivatives may be able to absorb a greater dollar transaction amount without an adverse effect on the price of the derivative instrument compared to the price effect on the cash market instrument; that is, the interest rate derivative may be more liquid than the cash market.

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Interest Rate Futures A futures contract is an agreement that requires a party to the agreement either to buy or sell something at a designated future date at a predetermined price. Futures contracts are products created by exchanges. Futures contracts based on a financial instrument or a financial index are known as financial futures. Financial futures can be classified as (1) stock index futures, (2) interest rate futures, and (3) currency futures.

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**Mechanics of Futures Trading**

A futures contract is an agreement between a buyer (seller) and an established exchange or its clearinghouse in which the buyer (seller) agrees to take (make) delivery of something at a specified price at the end of a designated period of time. A forward contract is an agreement for the future delivery o something at a specified price at a designated time, but differs from a futures contract in that it is usually non-standardized and traded in the over-the-counter market. An investor who takes a long futures position realizes a gain when the futures price increases; an investor who takes a short futures position realizes a gain when the futures price decreases.

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**Mechanics of Futures Trading**

The parties to a futures contract are required to satisfy margin requirements. Parties to over-the-counter interest rate contracts are exposed to counterparty risk which is the risk that the counterparty will not satisfy its contractual obligations. When a position is first taken in a futures contract, the investor must deposit a minimum dollar amount per contract as specified by the exchange. This amount is called initial margin and is required as deposit for the contract.

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**Exchange-Traded Interest Rate Futures Contracts**

Interest rate futures contracts can be classified by the maturity of their underlying security. Short-term interest rate futures contracts have an underlying security that matures in less than one year. Examples of these are futures contracts in which the underlying is a 3-month Treasury bill and a 3-month Eurodollar certificate of deposit. The maturity of the underlying security of long-term futures contracts exceeds one year. Examples of these are futures contracts in which the underlying is a Treasury coupon security, a 10-year agency note, and a municipal bond index.

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**U.S. Treasury Security Interest Rate Futures Contracts**

The major focus is on futures contracts in which the underlying is a Treasury coupon security (a Treasury bond or a Treasury note). These contracts are the most widely used by managers of bond portfolios and it is necessary to understand the specifications of the Treasury bond futures contract. Also of interest is the agency note futures contracts. There are futures contracts on non-U.S. government securities traded throughout the world. Many of them are modeled after the U.S. Treasury futures contracts and consequently, the concepts discussed apply directly to those futures contracts.

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Treasury Bond Futures For the Treasury bond futures contract the underlying instrument is $100,000 par value of a hypothetical 20-year 6% coupon Treasury bond. Conversion factors are used to adjust the invoice price of a Treasury bond futures contract to make delivery equitable to both parties. The short in a Treasury bond futures contract has several delivery options: quality option (or swap option), timing option, and wildcard option.

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Treasury Bond Futures For all the issues that may be delivered to satisfy a Treasury futures contract, a rate of return can be computed in a cash and carry trade; the rate of return is called the implied repo rate. For all the issues that may be delivered to satisfy a Treasury futures contract, the cheapest-to-deliver issue is the one with the highest implied repo rate. By varying the yield on Treasury bonds, it can be determined which issue will become the new cheapest-to-deliver issue.

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**Treasury Note and Agency Futures**

The three Treasury note futures contracts are 10-year, 5-year, and 2-year note contracts. All three contracts are modeled after the Treasury bond futures contract and are traded on the CBOT. The underlying instrument for the 10-year Treasury note futures contract is $100,000 par value of a hypothetical 10-year, 6% Treasury note. Several acceptable Treasury issues may be delivered by the short. An issue is acceptable if the maturity is not less than 6.5 years and not greater than 10 years from the first day of the delivery month. Delivery options are granted to the short position. There are futures contracts in which the underlying is a Fannie Mae and Freddie Mac debenture.

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Interest Rate Options An option is a contract in which the writer of the option grants the buyer of the option the right, but not the obligation, to purchase from or sell to the writer something at a specified price within a specified period of time (or at a specified date). The option buyer pays the option writer (seller) a fee, called the option price (or premium). A call option allows the option buyer to purchase the underlying from the option writer at the strike price; a put option allows the option buyer to sell the underlying to the option writer at the strike price.

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Interest Rate Options Interest rate options include options on fixed income securities and options on interest rate futures contracts; the latter, called futures options, are the preferred exchange-traded vehicle for implementing investment strategies. Because of the difficulties of hedging particular fixed income securities, some institutional investors have found over-the-counter options more useful.

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Call Option (Long)

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Call Option (Short)

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Put Option (Long)

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Put Option (Short)

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**Differences Between Options and Futures Contracts**

Unlike a futures contract, one party to an option contract is not obligated to transact. Specifically, the option buyer has the right, but not the obligation, to transact. The option writer does have the obligation to perform. In the case of a futures contract, both buyer and seller are obligated to perform. Of course, a futures buyer does not pay the seller to accept the obligation, while an option buyer pays the option seller an option price.

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**Differences Between Risk/Reward Characteristics**

In the case of a futures contract, the buyer of the contract realizes a dollar-for-dollar gain when the price of the futures contract increases and suffers a dollar-for-dollar loss when the price of the futures contract drops. The opposite occurs for the seller of a futures contract. Options do not provide this symmetric risk/reward relationship. The most that the buyer of an option can lose is the option price. While the buyer of an option retains all the potential benefits, the gain is always reduced by the amount of the option price. The maximum profit that the writer may realize is the option price; this is compensation for accepting substantial downside risk.

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Interest Rate Swaps An interest rate swap is an agreement specifying that the parties exchange interest payments at designated times, with a generic or vanilla swap calling for one party to make fixed-rate payments and the other to make floating-rate payments based on a notional principal. The swap rate is the interest rate paid by the fixed-rate payer. The swap spread is the spread paid by the fixed-rate payer over the on-the-run Treasury rate with the same maturity as the swap agreement.

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Interest Rate Swaps The convention in quoting swaps is to quote the payments made by the floating-rate payer flat (that is, without a spread) and the fixed-rate payer payments as a spread to the on-the-run Treasury with the same maturity as the swap (the swap spread) A swap position can be interpreted as either a package of forward/futures contracts or a package of cash flows from buying and selling cash market instruments.

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Interest Rate Swaps Floating rate payer

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**Interest Rate Swap Caps**

An interest rate cap specifies that one party receive a payment if the reference rate is above the cap rate; an interest rate floor specifies that one party receive a payment if a reference rate is below the floor rate. The terms of a cap and floor set forth the reference rate, the strike rate, the length of the agreement, the frequency of reset, and the notional amount. In an interest rate cap and floor, the buyer pays an upfront fee, which represents the maximum amount that the buyer can lose and the maximum amount that the seller of the agreement can gain. Buying a cap is equivalent to buying a package of puts on a fixed income security and buying a floor is equivalent to buying a package of calls on a fixed income security.

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Interest Rate Caps If an option is viewed as one in which the underlying is an interest rate, then buying a cap is equivalent to buying a package of calls on interest rates and buying a floor is equivalent to buying a package of puts on interest rates. An interest collar is created by buying an interest rate cap and selling an interest rate floor. Forward contracts and swaps expose the parties to bilateral counterparty risk while buyers of OTC options, caps, and floors face unilateral counterparty risk.

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**Valuation of Interest Rate Derivative Instruments by Frank J. Fabozzi**

PowerPoint Slides by David S. Krause, Ph.D., Marquette University Copyright 2007 John Wiley & Sons, Inc. All rights reserved. Reproduction or translation of this work beyond that permitted in Section 117 of the 1976 United States Copyright Act without the express permission of the copyright owner is unlawful. Request for futher information should be addressed to the Permissions Department, John Wiley & Sons, Inc. The purchaser may make back-up copies for his/her own use only and not for distribution or resale. The Publisher assumes no responsibility for errors, omissions, or damages caused by the use of these programs or from the use of the information contained herein.

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**Chapter 14 Valuation of Interest Rate Derivative Instruments**

Major learning outcomes: Understand the valuation of interest rate derivative instruments - futures, forwards, options, swaps, caps, and floors. Understand the basic of using these tools to control interest rate risk.

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Key Learning Outcomes Compute the profit or loss generated in a cash and carry trade and in a reverse cash and carry trade using futures. Compute the theoretical price of an interest rate futures contract. Explain how the theoretical price of a Treasury bond futures contract is affected by the delivery options. Explain the complications in extending the standard arbitrage pricing model to the valuation of Treasury bond and Treasury note futures contracts.

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Key Learning Outcomes Compute the floating-rate payments in an interest rate swap given the futures price of a Eurodollar CD futures contract. Justify the appropriate interest rate to use in calculating the present value of the payments in an interest rate swap. Calculate the forward discount factor used to discount the swap payments given the forward rates. Explain how the swap rate and swap spread – and how the value of a swap is determined.

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Key Learning Outcomes Calculate the swap rate, swap spread, and value of a swap. Compute the new floating-rate payments and value for a swap if interest rates change. Discuss the factors that affect the value of an option, or options on futures, for a fixed income instrument. Explain the limitations of applying the Black-Scholes model to valuing options on bonds.

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Key Learning Outcomes Compute the value of an option on a bond using the arbitrage-free binomial model and discuss the Black model for valuing options on futures. Explain how to measure the sensitivity of an option to changes in the factors that affect its value. Compare the roles of delta and duration in approximating price changes. Compute the value of each caplet and the value of a cap and a floor given a binomial interest rate tree.

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**Cash and Carry Bond Trade**

A cash and carry trade and a reverse cash and carry trade can be used to determine the arbitrage profit available from a futures strategy. A cash and carry trade and a reverse cash and carry trade can be used to determine the theoretical price of a futures contract. The theoretical price of a futures contract is equal to the cash or spot price plus the cost of carry.

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**Cash and Carry Bond Trade**

The cost of carry is equal to the cost of financing the position less the cash yield on the underlying security. The shape of the yield curve affects the cost of carry. The ‘‘cash and carry’’ arbitrage model must be modified to take into consideration the nuances of a particular futures contract. For a Treasury bond futures contract, the delivery options granted to the seller reduce the theoretical futures price below the theoretical futures price suggested by the ‘‘cash and carry’’ arbitrage model.

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**Cash and Carry Bond Trade**

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Interest Rate Swaps The dollar amount of the interest payments exchanged is based on the notional principal. In the most common type of swap, there is a fixed-rate payer and a fixed-rate receiver. The convention for quoting swap rates is that a swap dealer sets the floating rate equal to the reference rate (i.e., the interest rate used to determine the floating-rate in a swap) and then quotes the fixed rate that will apply.

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Interest Rate Swaps To compute the payments for both parties to an interest rate swap, the number of days in the payment period must be determined. The first floating-rate swap payment is determined by the current value of the reference rate. In a swap where the reference rate is 3-month LIBOR, the Eurodollar CD futures contract provides the forward rate for locking in future floating-rate payments, as well as the forward rates that should be used for discounting all swap payments. In determining the present value of swap payments, care must be exercised in determining exactly when the payments will occur.

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**Eurodollar CD Futures Contract**

A swap position can be interpreted as a package of forward/futures contracts or a package of cash flows from buying and selling cash market instruments. It is the former interpretation that is used as the basis for valuing a swap. Eurodollar certificates of deposit (CDs) are denominated in dollars but represent the liabilities of banks outside the United States.

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**Eurodollar CD Futures Contract**

The contracts are traded on both the International Monetary Market of the Chicago Mercantile Exchange and the London International Financial Futures Exchange. The rate paid on Eurodollar CDs is LIBOR. The forward rates obtained from Eurodollar CD futures contracts are used to compute the forward discount factor. The forward discount factor for a period multiplied by the swap payment for a period determines the present value of the swap payment.

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**Calculation of the Swap Rate**

At the inception of a swap, the present value of the floating-rate payments must equal the present value of the fixed-rate payments to prevent arbitrage. The swap rate is the rate that will produce fixed-rate payments such that the present value of these payments is equal to the present value of the floating-rate payments. The swap spread is the difference between the swap rate and the rate on a selected benchmark. When interest rates change in the market, the future floating-rate payments will change, but the fixed-rate payments do not change.

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**Calculation of the Swap Rate**

When interest rates change in the market, the forward rates change and therefore the present value of the swap payments changes. The value of a swap is the difference in the present value of the swap payments for a party to a swap—that is, the difference between the present value of the payments to be received and the present value of the payments to be paid.

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Options An option grants the buyer of the option the right, but not the obligation, to purchase from or sell to the contract writer an asset (the underlying) at a specified price (the strike price) within a specified period of time (or at a specified date). The compensation that the option buyer pays to acquire the option from the option writer is the option price. (The option price is also referred to as the option premium.) A call option grants the buyer the right to purchase the underlying from the writer (seller); a put option gives the buyer the right to sell the underlying to the writer.

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Options An American option allows the buyer to exercise the option at any time up to and including the expiration date. A European option allows the buyer to exercise the option only on the expiration date. The maximum amount that an option buyer can lose is the option price. The maximum profit that the option writer can realize is the option price. The option buyer has substantial upside return potential, while the option writer has substantial downside risk.

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**Components of the Option Price**

The option price can be decomposed into two parts: the intrinsic value and the time value. Intrinsic Value The option value is a reflection of the option’s intrinsic value and its time value. The intrinsic value of an option is its economic value if it is exercised immediately. If no positive economic value would result from exercising the option immediately, then the intrinsic value is zero.

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**Intrinsic Value of an Option**

When an option has intrinsic value, it is said to be in the money. When the strike price of a call option exceeds the current price of the security, the call option is said to be out of the money; it has no intrinsic value. An option for which the strike price is equal to the current price of the security is said to be at the money. Both at-the-money and out-of-the-money options have an intrinsic value of zero because they are not profitable to exercise.

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**Components of the Option Price**

The option price can be decomposed into two parts: the intrinsic value and the time value. Time Value The time value of an option is the amount by which the option price exceeds its intrinsic value. The option buyer hopes that at some time up to the expiration date, changes in the market price of the underlying security will increase the value of the rights conveyed by the option. For this prospect, the option buyer is willing to pay a premium above the intrinsic value.

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**Factors Affecting an Option’s Price**

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Value of an Option The value of an option is equal to its intrinsic value plus its time value. The six factors that affect the value of an option are: the price of the underlying security, the strike price of the option, the time to expiration of the option, the expected interest rate volatility over the life of the option, the short-term risk-free interest rate over the life of the option, and the coupon interest payment over the life of the option.

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**Factors Influencing Option Value (When the Underlying Security is a Fixed Income Instrument)**

Price of the Underlying Security The option price changes as the price of the underlying security changes. For a call option, as the price of the underlying security increases (holding all other factors constant), the option price increases. This is because the intrinsic value of a call option increases when the price of the underlying security increases. The opposite holds for a put option: as the price of the underlying security increases, the price of a put option decreases. This is because the intrinsic value of a put option decreases when the price of the underlying security increases.

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**Factors Influencing Option Value (When the Underlying Security is a Fixed Income Instrument)**

Strike Price All other factors equal, the lower the strike price, the higher the price of a call option. For put options, the higher the strike price, the higher the option price.

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**Factors Influencing Option Value (When the Underlying Security is a Fixed Income Instrument)**

Time to Expiration of the Option An option is a ‘‘wasting asset.’’ That is, after the expiration date passes the option has no value. Holding all other factors equal, the longer the time to expiration of the option, the greater the option price. As the time to expiration decreases, less time remains for the underlying security’s price to rise (for a call buyer) or to fall (for a put buyer)—to compensate the option buyer for any time value paid—and, therefore, the probability of a favorable price movement decreases. Consequently, for American options, as the time remaining until expiration decreases, the option price approaches its intrinsic value.

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**Factors Influencing Option Value (When the Underlying Security is a Fixed Income Instrument)**

Expected Interest Rate Volatility Over the Life of the Option All other factors equal, the greater the expected interest rate volatility or yield volatility, the more an investor would be willing to pay for the option, and the more an option writer would demand for it. This is because the greater the volatility, the greater the probability that the price of the underlying security will move in favor of the option buyer at some time before expiration.

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**Factors Influencing Option Value (When the Underlying Security is a Fixed Income Instrument)**

Short-Term Risk-Free Rate Over the Life of the Option Buying the underlying security ties up one’s money. Buying an option on the dollar amount of the underlying security makes available for investment the difference between the security price and the option price at the risk-free rate. All other factors constant, the higher the short-term risk-free rate, the greater the cost of buying the underlying security and carrying it to the expiration date of the call option. Hence, the higher the short-term risk-free rate, the more attractive the call option is relative to the direct purchase of the underlying security.

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**Factors Influencing Option Value (When the Underlying Security is a Fixed Income Instrument)**

Coupon Payments Over the Life of the Option Coupon interest payments on the underlying security tend to decrease the price of a call option because they make it more attractive to hold the underlying security than to hold the option. That is, the owner of the security receives the coupon payments but the buyer of the call option does not. The higher the coupon payment received by the owner of the security, the more attractive it is to own the security and the less attractive it is to own the call option. So, the value of a call option declines the higher the coupon payment.

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**Factors that Influence the Value of a Futures Option**

There are five factors that influence the value of an option in which the underlying is a futures contract: 1. current futures price 2. strike price of the option 3. time to expiration of the option 4. expected interest rate volatility over the life of the option 5. short-term risk-free rate over the life of the option

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Option Pricing Models Several assumptions underlying the Black-Scholes model limit its use in pricing options on bonds. The arbitrage-free binomial model is the proper model to value options on bonds since it takes into account the yield curve. The most common option pricing model for bonds is the Black-Derman-Toy model. The Black model is the most common model for valuing options on bond futures. Money managers need to know how sensitive an option’s value is to changes in the factors that affect the value of an option.

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**Option Pricing Model Greeks**

The delta of an option measures how sensitive the option price is to changes in the price of the underlying bond and varies from minus one (for put options deep in the money) to zero (for call options deep out of the money) to one (for call options deep in the money). The gamma of an option measures the rate of change of delta as the price of the underlying bond changes. The theta of an option measures the change in the option price as the time to expiration decreases. The kappa of an option measures the change in the price of the option for a 1% change in expected interest rate volatility.

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**Black-Scholes Option Pricing Model**

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