Lecture Notesin

Financial Economics andRisk Management

Dr. Mario J. MirandaAndersons Professor of Agricultural Finance and Risk Management

Department of Agricultural, Environmental, and Development EconomicsThe Ohio State University

Winter 2008

6 Derivatives

6.1 Introduction

A derivative is a financial instrument whose value is determined by the valueof some basic underlying variable. The underlying variable can be the marketprice of a commodity, equity, bond, or foreign currency. The underlyingvariable, however, can also be an objectively measured index, such as thevalue of the S&P 500 index or the losses experienced by property casualtyinsurers in Florida due to major hurricanes.

Derivative contracts include forward contracts, futures contracts, optionscontracts, and swaps. Forward contracts, custom option contracts, and swapsare traded regularly between corporations and their clients in “over-the-counter” trades involving only the two contracting parties. However, thebulk of trading in derivatives takes place in the form of standardized futuresand option contracts traded on one of the more than 80 organized exchangescurrently operating worldwide.

The top ten derivative exchanges, in terms of number of contracts traded,are presented in Table 1. The Korea Exchange, located in Seoul, Korea, iscurrently the largest exchange in the world in terms of contract volume. Theoverwhelming majority of trades undertaken at the exchange involve KoreanStock Price Index (KOSPI 200) futures. Eurex, located in Frankfurt, Ger-many, trades over twenty derivative products, including futures and optionson country-specific equity indices and German government bond rates. TheChicago Mercantile Exchange trades futures and options on foreign curren-cies, agricultural products (pork, cattle, butter, milk), equity indices (NAS-DAQ, S&P 500), U.S. and foreign government bond rates, and weather in-dices. The Chicago Board of Trade, the world’s oldest derivatives exchange,trades over 50 futures and options contracts, mostly on agricultural com-modities (corn, soybeans, soy products, wheat), metals (gold, silver), U.S.Treasury bond rates, and equity indices (Dow Jones Industrial Average).

Futures contracts were introduced by the Chicago Board of Trade in 1865.Contract trading was initially limited to agricultural commodities, but was

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Table 1: Top Ten Derivative Exchanges by Volume, in Millions of ContractsTraded, January-June 2006.

Korea Exchange (Seoul) 1,241Eurex (Frankfurt) 824Chicago Mercantile Exchange 705Chicago Board of Trade 401Euronext.liffe (London) 387Chicago Board Options Exchange 338International Securities Exchange (New York) 300Mexican Derivatives Exchange (Mexico City) 149Bovespa (Sao Paulo) 133Bolsa de Mercadorias & Futuros (Sao Paulo) 132

Source: Futures Industry Association

later expanded to include metals. Contracts on financial instruments such ascurrency and currency indexes, equities and equity indexes, and governmentinterest rates were introduced in the mid 1970s by the Chicago MercantileExchange. Financial derivatives quickly overtook commodities derivativesin terms of trading volume and now account for over 90% of the contractstraded worldwide. Global trading volumes by class of derivative are presentedin Table 2.

6.2 Forward Contracts

A forward contract is the simplest example of a derivative. It is an agreementbetween two parties to exchange a specified asset (or quantity of a commod-ity), at a specified future date, at a specified price. In a forward contract,one party assumes a long position, agreeing to buy the underlying asset. Theother party assumes a short position, agreeing to sell the underlying asset.The date on which the exchange is to take place is called the delivery dateand the price for which the asset will be exchanged is called the delivery

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Table 2: Global Futures and Options Volume, in Millions of ContractsTraded, by Sector

Jan-Jun Jan-Jun PercentSector 2006 2005 Change

Equity Index 2,252 1,780 27%Interest Rate 1,637 1,320 24%Individual Equity 1,463 1,139 29%Agriculture 205 164 25%Energy 172 131 31%Currencies 116 75 55%Metals 99 72 39%Other 1 1 16%Total 5,944 4,681 27%

Source: Futures Industry Association

price.

An example of a forward contract is when a farmer, at planting time,agrees to deliver a specified quantity of grain at harvest time to a grainelevator, which in turn agrees to pay the farmer a specified price for thegrain upon delivery. Forward contracts are common among firms of all sizesthat have long-standing working relationships. However, the majority offorward trades in terms of value take place among large financial institutionsand corporations. Forward contracts are said to be traded over-the-counter,that is, without the involvement of a third party or an organized exchange.

Forward contracts are settled by delivery on the delivery date. The payofffrom a forward contract at delivery will depend on the spot price of the asseton the delivery date, that is, the price at which the asset can be purchasedor sold for immediate delivery on that date. The payoffs from long and shortforward positions at delivery, as functions of the spot price on the deliverydate sT and the delivery price K, are illustrated in Figure 17. In general,

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the payoff at delivery from a long forward position is ST − K: if ST > K,the long investor gains because he is entitled to purchase an asset that isworth ST at a lower price K; if ST < K, the long investor loses because heis obligated to purchase an asset that is worth ST at a higher price K. Thepayoff at delivery from a short forward position is the exact opposite of thatfrom a long position. In general, the payoff from a short forward position isK − ST : if ST > K, the short investor loses because he is obligated to sellan asset that is worth ST at a lower price K; if ST < K, the short investorgains because he is entitled to sell an asset that is worth ST at a higher priceK.

ST

K

Payoff

0

Short

ST

K

Payoff

0

Long

Figure 17: Payoff From Forward Contract

6.3 Spot-Forward Price Relationships

At any point in time, the forward price of an asset for delivery at a futuredate is the delivery price at which a forward contract for that delivery datecan be consummated. The forward price of an asset for delivery at a specificdelivery date will vary over time as the delivery date approaches. The deliveryprice of any forward contract, however, becomes fixed when the contract isconsummated. Below, we derive some relationships between spot and forwardprices that must hold at any given point in time.

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6.3.1 Commodities

Consider a storable commodity, say, gold. Assume that one ounce of goldcurrently trades at a spot price s0; one ounce of gold can be stored for aperiod of T years at a cost kT ; the forward price of one ounce of gold fordelivery in T years is fT ; and the annual yield on a risk-free bond maturingin T years is r. Let δ = (1+r)−1 denote the annual discount factor for bonds.

Consider the following three transactions: 1) sell one ounce of gold for-ward for delivery at time T at the current forward price fT ; 2) buy one ounceof gold at the current spot price s0 and pay kT to store it for T years withthe intent to deliver the gold under the terms of the forward contract in Tyears; and 3) sell a risk-free bond with face value fT that matures in T years.The net cash flows at time 0 and time T generated by these transactions are:

Cash FlowTransaction t=0 t=T1) Sell Gold Forward 0 fT

2) Buy Gold Spot −s0 − kT 03) Sell Bond δT fT −fT

Net δT fT − s0 − kT 0

These transaction, as a whole, create a portfolio that provides a surepayoff of zero at time T . In order to avoid arbitrage, the net cash flow fromthese transactions at time 0 must be zero, implying

fT = (1 + r)T (s0 + kT ).

In the real world, of course, market imperfections such as transactioncosts, differential borrowing and lending rates, and restrictions on short sell-ing can limit arbitrage opportunities. Thus, the theoretical relationship be-tween forward and spot prices will need not be observed exactly by marketprices. The deviations from theory, however, should be relatively small ifmarkets are efficient and transaction costs are small as a percentage of thevalue of the underlying asset.

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6.3.2 Equities

Consider a share of common stock that does not pay dividends. Assume thatone share of stock currently trades at a spot price s0; the forward price ofone share of stock for delivery in T years is fT ; and the annual yield on arisk-free bond maturing in T years is r. Let δ = (1 + r)−1 denote the annualdiscount factor for bonds.

Consider the following three transactions: 1) sell one share of stock for-ward for delivery at time T at the current forward price fT ; 2) buy one shareof stock at the current spot price s0 and hold it for eventual delivery in Tyears; and 3) sell a risk-free bond with face value fT that matures in T years.The net cash flows at time 0 and time T generated by these transactions are:

Cash FlowTransaction t=0 t=T1) Sell Stock Forward 0 fT

2) Buy Stock Spot −s0 03) Sell Bond δT fT −fT

Net δT fT − s0 0

These transaction, as a whole, create a portfolio that provides a surepayoff of zero at time T . In order to avoid arbitrage, the net cash flow fromthese transactions at time 0 must be zero, implying

fT = (1 + r)T s0.

6.3.3 Currencies

Consider two currencies, dollars ($) and pounds (£). Assume that the currentexchange rate is s0 dollars per pound; the forward price of one pound fordelivery in T years is fT dollars; the annual yield on a risk-free bond maturingin T years denominated in dollars is r$; and the annual yield on a risk-free

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bond maturing in T years denominated in pounds is r£. Let δ$ = (1 + r$)−1

and δ£ = (1 + r£)−1 denote the annual discount factors for dollar bonds andpound bonds, respectively.

Consider the following four transactions: 1) sell one pound forward fordelivery at time T at the current forward price of fT dollars; 2) exchangeδT£s0 dollars for δT

£ pounds on the spot foreign exchange market; 3) sell arisk-free bond with face value of fT dollars that matures in T years; and 4)buy a risk-free bond with face value of 1 pound that matures in T years. Thenet cash flows at time 0 and time T associated with these transactions are:

Cash Flow t=0 Cash Flow t=TTransaction Dollars Pounds Dollars PoundsSell Pounds Forward 0 0 fT -1Buy Pounds Spot −δT

£s0 δT£ 0 0

Sell Dollar Bond δT$ fT 0 −fT 0

Buy Pound Bond 0 −δT£ 0 1

Net δT$ fT − δT

£s0 0 0 0

These transaction, as a whole, create a portfolio that provides a surepayoff of zero at time T . In order to avoid arbitrage, the net cash flow fromthese transactions at time 0 must be zero, implying

fT =(

1 + r$

1 + r£

)T

s0.

6.4 Futures Contracts

Like a forward contract, a futures contract is a binding, legal agreement tobuy (take delivery of) or sell (make delivery of) a specified quantity of acommodity or asset, at a specified future date, at a specified price.

There are, however, some significant differences between forward andfutures contracts. First, while a forward contract is a custom agreement

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between two parties, futures contracts are standardized with regard to un-derlying asset or commodity, quantity, delivery date, and delivery location,and are traded on organized exchanges under strict rules enforced by the ex-change. The exchange is a party to all the futures contracts sold and boughtat the exchange. As a matter of law, a futures contract is a contract betweenan investor and the exchange. The exchange maintains a neutral financialposition in the market by transacting purchases and sales of futures con-tracts simultaneously. In particular, the exchange sells a futures contract toan investor only if there is another investor who is willing to buy a futurescontract from the exchange for the same quantity, delivery date, and price.The investors, however, are mutually anonymous and assume no contrac-tual obligations with each another. The contractual obligations are entirelybetween the investor and the exchange.

Second, the default risk of futures contracts is much lower than that offorward contracts. With a forward contract there is always some risk that,when the delivery date arrives, either the seller will be unable to deliver thecommodity or the buyer will be unable to pay for it. With a futures contract,however, the exchange, as the intermediary, guarantees the performance ofbuyers and sellers of futures contracts through the use of performance bondscalled margin accounts. Whenever a trader buys or sells a futures contract,he is required to deposit a determined amount of cash in a margin accountheld by his broker. The size of the margin account is typically 5-15% ofcontract value, approximately equal to the maximum probable daily losson a contract. A trader’s margin account is subsequently adjusted dailyaccording to the profit or loss on the futures position on that day. Thispractice is known as marking-to-market. If the mark to market results in abalance that is less than the margin requirement established by the exchange,then the trader is issued a margin call, requiring the trader to deposit morecash in his margin account to bring it back up to the required level. If atrader fails to comply with a margin call, the broker is required to close thetrader’s position, thereby limiting the possibility of a loss.

When discussing futures contracts, several technical terms must be clearlyunderstood. The spot price refers to the price of the commodity or asset forimmediate delivery on a given date. The futures price is the price of thecommodity or asset for delivery at a future date, called the delivery date.

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At any time, futures contracts for commodities and assets are available formultiple delivery dates. Thus, while there is only one spot price at any giventime, there are generally more than one futures price corresponding to thedelivery dates of the various futures contracts traded on the exchange. Aninvestor can take one of two positions in a futures contract. An investor whotakes a long position agrees to buy the commodity or asset at the deliverydate at the stated futures price. An investor who takes a short position agreesto sell the commodity or asset at the delivery date at the stated futures price.

Users of forward and futures contracts include hedgers and speculators.A hedger buys or sells in the futures market to secure the future price of acommodity they must buy or sell at a later date in the cash market. Thishelps them protect against price risk. Speculators, on the other hand, do notaim to own the commodity in question. Rather, they buy and sell futurescontracts seeking to profit by anticipating increases or decreases in the priceof the underlying asset. Speculators serve an important function in futuresmarkets because they provide liquidity to the market.

In practice, most futures contracts (over 99%) are settled, not by makingor accepting delivery, but rather by taking an opposite position in the market.Thus, a farmer that sells a futures contract, obligating him to deliver acertain amount of grain at a location on a specified delivery date, will insteadsubsequently buy a futures contract for the same amount and delivery date,creating a net position in which the farmer needs neither make nor acceptdelivery of the commodity at the location and on the date specified in thefutures contract. The farmer will sell his grain on the spot market. However,as the following example illustrates, the net result of the cash sale and thefutures position guarantees the farmer’s income.

Example 6.1 One June 1, a farmer takes short position in the futures mar-ket, selling 10,000 bushels of November 1 wheat at the then futures price of$2 per bushel. On the same date, a baker takes a long position in the futuresmarket, buying 10,000 bushels of November 1 wheat at $2.00 per bushel. OnNovember 1, the farmer and the baker liquidate their futures market positionand, respectively, buy and sell wheat on the spot market. The net cash flowsto the farmer and baker on November 1 are illustrated in the following tables.

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Farmer Wheat Price on November 1$1.50 $2.00 $2.50

Wheat Sales $15,000 $20,000 $25,000Futures Payoff $5,000 $0 -$5,000Net Cash Flow $20,000 $20,000 $20,000

Baker Wheat Price on November 1$1.50 $2.00 $2.50

Wheat Purchases -$15,000 -$20,000 -$25,000Futures Payoff -$5,000 $0 $5,000Net Cash Flow -$20,000 -$20,000 -$20,000

A futures contract is highly standardized. As an example, consider thespecification of a Chicago Board of Trade Corn Futures Contract:

• Contract Size. 5,000 bushels

• Deliverable Grades. No. 2 Yellow at par, No. 1 yellow at 6 cents perbushel over contract price and No. 3 yellow at 6 cents per bushel undercontract price.

• Price Quote. Cents per bushel in increments of 1/4 cent/bu ($12.50/con-tract).

• Contract Months. Sep, Nov, Jan, Mar, May, Jul, Aug

• Last Trading Day. The business day prior to the 15th calendar day ofthe contract month.

• Last Delivery Day. Second business day following the last trading dayof the delivery month.

• Trading Hours. Open Auction: 9:30 a.m. - 1:15 p.m. Central Time,Mon-Fri. Electronic: 6:31 p.m. - 6:00 a.m. and 9:30 a.m. - 1:15 p.m.Central Time, Sun.-Fri. Trading in expiring contracts closes at noonon the last trading day.

• Ticker Symbols. Open Auction: S. Electronic: ZS.

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• Daily Price Limit. 50 cents/bu ($2,500/contract) above or below theprevious day’s settlement price. No limit in the spot month (limits arelifted beginning on First Position Day).

• Margin Information. See CBOT website.

6.5 Option Contracts

An option is a contract that gives the owner the right, but not the obligation,to buy or sell a specified asset at a specified price, on or by a specified date.A put option confers to the owner the right to sell the asset. A call optionconfers to the owner the right to buy the asset. Put and call options aretraded for many assets, including corporate stocks, major currencies, andmajor stock indices. The majority of options traded on exchanges, however,are options on futures contracts traded on the exchanges.

An option involves two parties. The owner or purchaser of a put (call)option obtains the right to sell (buy) an asset at a specified price by payinga premium to the writer or seller of the option, who assumes the collateralobligation to buy (sell) the asset, should the owner of the option choose toexercise it. The owner is said to take a long position in the option; the writeris said to take a short position in the option.

When discussing option contracts, several other technical terms must beclearly understood. The strike price or exercise price is the price at whichthe commodity or asset may be bought or sold by the owner of the optionunder the terms of the option contract. The expiration date, exercise date,or maturity refer to the date at which the option expires or, equivalently, thelast date on which the owner may exercise his option. An American optionmay be exercised at any time up to expiration date. A European option onthe other hand, may be exercised only on the expiration date. An optionis said to be in the money if its immediate exercise would produce positivecash flow. Thus, a put option is in the money if the strike price exceeds thespot price of the underlying asset and a call option is in the money if thespot price of the underlying asset exceeds the strike price. An option that isnot in the money is said to be out of the money.

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As seen in Figure 18, the payoff from a put option at the expiration date isa function of the strike price K and the spot price sT of the underlying asseton the delivery date. From this diagram we conclude that if a put option isheld until expiration (which must be so for a European option, but not anAmerican option) then the option will be exercised if, and only if, sT < K,in which case the owner of the option will realize a net payoff K − sT > 0and the writer of the option will realize a net payoff sT −K < 0.

ST

K

Payoff

0

Short

ST

K

Payoff

0

Long

Figure 18: Payoff From Put Option

As seen in Figure 19, the payoff from a call option at the expiration date isa function of the strike price K and the spot price sT of the underlying asseton the delivery date. From this diagram we conclude that if a call option isheld until expiration (which must be so for a European option, but not anAmerican option) then the option will be exercised if, and only if, sT > K,in which case the owner of the option will realize a net payoff sT − K > 0and the writer of the option will realize a net payoff K − sT < 0.

In interpreting the payoff diagrams for put and call options, one mustkeep in mind that options are bought and sold at a premium. Althoughthe cash flow at expiration for an owner of an option is nonnegative, theowner paid a premium to acquire the option initially. Similarly, although thecash flow at expiration for the writer of an option is nonpositive, the writerreceived a premium to write the option initially. Thus, the net cash flow

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ST

K

Payoff

0

Short

ST

K

Payoff

0

Long

Figure 19: Payoff From Call Option

from an option over time can be either positive or negative for the owner orwriter alike.

The premiums P and C, respectively, on European put and call optionswith common strike price K and expiration date T years must satisfy a aso-called put-call parity relation in order to avoid arbitrage. To derive thisrelation, assume that the underlying asset currently trades at a spot prices0 and the annual yield on a risk-free bond maturing in T years is r. Letδ = (1 + r)−1 denote the annual discount factor for bonds.

Consider now the following four transactions: 1) buy the underlying assetat the spot price s0 with the intent to sell it at time T ; 2) sell a risk-free bondwith face value K; 3) buy a put option with strike price K and expirationdate T , paying the put premium P ; and 4) write a call option with strikeprice K and expiration date T , receiving the call premium C. The cash flowfrom these transactions at time T will depend on the spot price sT of theunderlying asset at time T :

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Cash FlowsTransaction t = 0 t = T, sT < K t = T, sT > KBuy Asset −s0 sT sT

Sell Bond δT K −K −KBuy Put −P K − sT 0Write Call C 0 K − sT

Net −s0 + δT K − P + C 0 0

These transaction, as a whole, create a portfolio that provides a surepayoff of zero at time T . In order to avoid arbitrage, the net cash flow fromthese transactions at time 0 must be zero, implying

s0 + P = (1 + r)−T K + C.

6.6 Models of Asset Prices

The market premium for an option or other derivative can be derived only ifassumptions are made regarding how the price of the underlying asset behavesover time. The simplest, and the most common, assumption made regard-ing the behavior of an asset price is that it is a continuous-time stochasticprocesses that exhibits simple geometric Brownian motion. In particular, itis assumed that the rates of return on the asset over infinitesimally small in-crements of time ∆t are independently normally distributed with mean µ∆tand variance σ2∆t. That is,

St+∆t − St

St

∼ N(µ∆t, σ2∆t)

where St is the price of the asset at time t. In discussing asset price processes,we adopt the convention of measuring time in years and refer to µ as the driftof the asset price process and to σ as the volatility of the asset price process.

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Using a theorem from Stochastic Calculus6, it can be shown that assumingthat the asset price follows simple geometric Brownian motion is equivalentto assuming that, over infinitesimally small increments of time ∆t, changes inthe log of the asset price are independently normally distributed with mean(µ− 0.5σ2)∆t and variance σ2∆t. That is,

log St+∆t − log St ∼ N((µ− 0.5σ2)∆t, σ2∆t).

Another particularly simple model for the behavior of asset prices is that,over infinitesimally small increments of time ∆t, the price either goes up bya factor u with probability p or goes down by a factor of 1/u with probability1− p, where u > 1. That is

St+∆t =

Stu with probability p

St/u with probability 1− p

or, equivalently, that

log St+∆t =

log St + log u with probability p

log St − log u with probability 1− p.

This model is known as the binomial model of asset prices, since the distri-bution of the log of the asset price St at a date t, conditional on the price ofthe asset S0 at time 0, is simply a sum of simple Bernoulli trials, and thuspossesses a binomial distribution.

A remarkable fact is that the simple geometric Brownian motion modeland the binomial model are equivalent in that they imply the same asset price

6The theorem is known as Ito’s Lemma, a discussion of which is beyond the scope ofthese lecture notes.

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dynamics as ∆t approaches 0, provided that the parameters of the binomialmodel are chosen to be

u = eσ√

∆t

and

p =eµ∆t − 1/u

u− 1/u

This is a very convenient fact, since in deriving theoretical results or devisingnumerical techniques for the pricing of options or other derivatives, one isfree to choose whichever of the two representations of asset price dynamicsis most convenient.

Both the simple geometric Brownian motion model and the Binomialmodel imply that the conditional distribution of the asset price St at time t,given the asset price S0 at time 0, is lognormal. In particular, the distributionof St/S0 is lognormal with parameters (µ− 0.5σ2)t and σ2t; this is the sameas saying that the conditional distribution of log St − log S0 is normal withmean (µ− 0.5σ2)t and variance σ2t. Using standard results from probabilitytheory, it follows that

E(St/S0) = eµt

V (St/S0) = e2µt(eσ2t − 1)

Numerically simulating the path of an asset price that follows geometricBrownian motion with drift µ and volatility σ over an interval of time [0, T ] isstraightforward. First, one divides the time interval into into n subintervalsof equal length ∆t = T/n. Next, one generate a sequence zi of n indepen-dent standard normal variates using a numerical random number generator(included in Matlab and most other numerical programs). Given the price of

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the asset S0 at time 0, a representative sequence of asset prices Si at timest = i∆t, for i = 1, 2, 3, . . . , n, can then be generated recursively as follows:

log Si = log Si−1 + (µ− 0.5σ2)∆t + ziσ√

∆t

Three simulated paths for an asset price with µ = 0.1, σ = 0.1, T = 1, andS0 = 1 are illustrated in Figure 20. In each case, n = 10, 000.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

Time in Years

Ass

et P

rice

Figure 20: Simulated Geometric Brownian Motion

In order to replicate the behavior of the price of a given asset, the drift µand volatility σ must be specified. This is typically done by estimating theseparameters from historical return data for the asset. Suppose you have n+1observations, s0, s1, s2, . . . , sn, on a stock price taken at equal intervals of time∆t = T/n over a period of T years. If ui = log(si/si−1) for i = 1, 2, 3, . . . , n,then an unbiased estimate of the volatility of the stock price is provided by

σ =

√√√√∑n

i=1(ui − u)2

(n− 1)∆t

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where u is the mean of the ui. An estimate of the drift term µ is providedby

µ =1

Tlog

(sn

s0

).

Estimates of the drift term, however, as we shall see, are not needed for thevaluation of derivatives.

6.7 Risk Neutral Valuation

The principle of risk neutral valuation, introduced in Chapter 4, maintainsthat, in a complete, arbitrage-free market, the price of a security at any pointin time equals the expected future payoff of the security, computed using therisk-neutral probabilities, discounted at the risk-free rate. This result willprovide us with the key to valuing options and other derivatives.

Consider a risky asset whose price follows simple geometric Brownianmotion with drift µ and volatility σ. Using the binomial representation ofthe asset price process, if the price the risky asset at time t is St then itsprice in time t + ∆t will either equal Stu with probability p or St/u withprobability 1− p, where

u = eσ√

∆t

and

p =eµ∆t − 1/u

u− 1/u

The possible movements in the asset price between time t and time t + ∆tare illustrated in Figure 21.

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·

·

·

p

1−p

St

Stu

St/u

t t+∆ t

Figure 21: Binomial Model of Asset Price

Suppose now that there exists a risk-free asset with continuously com-pounded rate of return r, so that a dollar invested in the asset at time t willproduce a payoff of er∆t at time t+∆t. Then the risk-free asset and the riskyasset form a complete market with respect to the two possible payoffs of therisky asset at time t + ∆t.7 In particular, using the notation of Chapter 4,the price vector at time t for the two assets is (1, St)

′ and the payoff matrixat time t + ∆t is

A =

[er∆t Stuer∆t St/u

].

From the results presented in Chapter 4, the risk-neutral probability thatthe price of the risky asset will rise in ∆t is

π =er∆t − 1/u

u− 1/u.

7All this is required is that ∆t < σ2/r2.

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The probability that the price of the risky asset will fall is

1− π =u− er∆t

u− 1/u.

Thus, the risk-neutral probabilities of an up or down movement in price areidentical to the probabilities of an up or down movement in the underlyingasset price, except that the drift term µ is replaced by the continuous risk-freerate r. The importance of this result cannot be overstated: to the risk-neutralinvestor, the asset follows simple geometric Brownian motion with drift r.

Consider now a derivative asset, whose value is determined by the price ofthe risky asset. Since the risky asset can assume only two possible values nextperiod, the derivative asset also can assume only two values. Let fu denotethe value of the derivative next period if the price of the risky asset rises toStu and let fd denote the value of the derivative next period if the price ofthe risky asset falls to St/u. The, according to the principle of risk-neutralvaluation, the price of the derivative at time t is simply the expected value ofthe derivative at time t + ∆t, computed using the risk-neutral probabilities,discounted at the risk-free rate. That is, if f is the value of the derivative attime t, then

f = e−r∆t (πfu + (1− π)fd) .

6.8 Black-Scholes Formula

In the preceding section we found that if an asset follows geometric Brownianmotion with drift µ and volatility σ, then the risk-neutral investor sees theasset price as following geometric Brownian motion with drift r and volatilityσ, where r is the continuously compounded annual risk-free rate of return.This implies that, to the risk-neutral investor, the distribution of the assetprice St at time t, given the asset price S0 at time 0, is lognormal. Inparticular, the risk-neutral distribution of St/S0 is lognormal with parameters(r − 0.5σ2)t and σ2t.

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Consider then, a European call option that expires in T years. Thepremium one must pay for this option at time t = 0 will be the expectedpayoff of the option, computed using the risk-neutral distribution, discountedto the present at the risk-free rate. In particular, the premium is

C = e−rT E0 max{St −K, 0}

where K is the strike price of the option and E0 is the expectation takenwith respect to the risk-neutral distribution conditional on the informationknown at time t = 0.

Computing the expectation of the call option payoff with respect to therisk-neutral distribution is an exercise in the Calculus, which will be omittedhere. We simply present the result:

C = SN(d1)− e−rT KN(d2)

where

d1 =log(S/K) + (r + 0.5σ2)T

σ√

T.

d2 = d1 − σ√

T .

Here, S is current price, K is strike price, σ is volatility, T is time to expi-ration, r is the continuously-computed annual risk-free rate of return, andN is cumulative distribution function for a standard normal variate. This isthe Black-Scholes formula for the premium of a call option of an Europeanoption on a assuming that the asset pays no dividends and is costless tostore. Variants of the formula to allow for dividends and storage costs arewell-known, but will not be discussed here.

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By a similar set of arguments, the premium for a European put optionthat expires in time T and has strike price K is

P = e−rT KN(−d2)− SN(−d1)

where d1 and d2 are defined as above and N is the cumulative distributionfunction for a standard normal variate.

Example 6.2 A call option with an exercise price of $50 has three monthsto expiration. The continuously compounded annual risk-free rate is 4%; thestock currently trades for $45; and the volatility of the asset price is 0.4.Then T = 0.25, K = 50, S = 45, σ = 0.4, and r = 0.04. According to theBlack-Scholes formula,

d1 =log(45/50) + (0.04 + 0.5 · 0.42) · 0.25

0.4√

0.25= −0.3768.

d2 = d1 − 0.4 ·√

0.25 = −0.5768.

and the price of the European call option is

C = 45 ·N(−0.3768)− e−0.04·0.2550 ·N(−0.5768) = 1.9307.

6.9 Valuing American Options

Because American options have an early exercise feature, they cannot bepriced using the Black-Scholes formula. The price of an American option,however, can be derived numerically using a recursive procedure in which thetime to expiration of the option is divided into many smaller subintervals.

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We now show how this procedure would be implemented, assuming thattime to expiration is divided into four sub-intervals, keeping in mind thatfor the procedure to provide accurate answers, one would have to dividethe time interval into a much larger number of subintervals. The procedureis relatively easy to implement in Matlab, so computing the American putoption premium using many subintervals, which would be nearly impossibleby hand, is easily handled by a computer.

By way of example, let us price an American option under the followingassumptions:

Stock price S 40Strike price K 45Interest rate r 5%Expiration T 0.75Volatility σ 40%Time periods N 3Time interval ∆t 0.25

Under these assumptions, the binomial asset price model approximationwould be to assume that the asset price either goes up by a factor u or downby a factor 1/u in each subperiod, where

Up factor u = eσ√

∆t = 1.2214

Up probability π = er∆t−1/uu−1/u

= 0.4814

Discount factor δ = e−r∆t = 0.9876

The possible states of the asset price over time are represented graphicallyin Figure 22. The asset prices are written above the nodes; the price of theoption is written below the node.

6.10 Exercises

1. Oscar stock currently trades at $92 per share. One year from now,Oscar is expected to pay a dividend of $1.70 per share. If the annual

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·

·

·

·

·

·

·

·

·

·

π

1−π

π

1−π

π

1−π

π

1−π

π

1−π

π

1−π

40.00

48.86

32.75

59.68

40.00

26.81

72.88

48.86

32.75

21.95

7.83

3.21

12.30

0

6.27

18.19

0

0

12.25

23.05

t=0 t=∆ t t=2∆ t t=3∆ t=T

Figure 22: Binomial Tree

yield on a risk-free bond maturing in one year is 7%, what is the forwardprice of one share of the stock for delivery in one year? In lecture,I covered futures-spot pricing relations only for non-dividend payingstocks. Thus, you will have to apply your analytical skills to extendthe relationship to accommodate dividend the payment. Assume that,under the terms of the forward contract, the stock is to be deliveredafter the dividend is paid.

2. A stock currently trades at $50 per share. At the end of six months,the stock price will be either $45 or $55. The annual yield on a risk-freebond maturing in six months is 10%. What are the risk-neutral prob-abilities of a price increase and price decrease? Use these probabilitiesto infer the current prices of European call and put options, both withstrike price $50 and expiring in 6 months.

3. A non dividend paying stock that currently trades at $200 per share willbe worth either $200 or $230 one year from today with equal probability.

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A European put option on one share of the stock that expires one yearfrom today and with a strike price of $211.50 currently sells for $4.What is the annual yield on a risk-free bond maturing in one year?

4. The stock of Smith Ltd. currently trades for $500 per share. A Eu-ropean call option on one share of stock that expires in one year withexercise price $200 currently sells for $400 and a European put op-tion on one share of that expires in one year with exercise price $200currently sells for $84.57.

(a) Infer the current yield on a one-year zero-coupon U.S. governmentbond.

(b) If the current actual yield on a one-year zero-coupon U.S. gov-ernment bond is 9%, construct a portfolio that that whose payofftomorrow will be zero with certainty, but which generates a certainprofit today.

5. Using the put-call parity relationship for European stock options as aguide, construct a replicating portfolio for a European call option onone share of stock with strike price K and expiration one year fromtoday. Your replicating portfolio may include bonds, the stock, and aput option on the stock. The yield on a one year T-bill is r.

6. Consider a non-dividend paying stock with expected annual return 15%and annual volatility 35%. The stock currently sells for $69 per shareand the continuously compounded risk-free annual rate of return is 5%.

(a) Using the Black-Scholes formula, compute the premium on a Eu-ropean put option with strike price 70$ that expires four monthsfrom today.

(b) Using the Matlab function BinomialOption, compute the pre-mium for the option in part (a).

(c) Using the Matlab function BinomialOption, compute the pre-mium for the option in part (a), assuming that it is an Americanput option that can be exercised early.

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