Nemeth Lecture Notes

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MSM3G11/MSM4G11Mathematical Finance

These lecture notes follow the recommended text of Wilmott, Howison and Dewynne. It

is recommended that students purchase this textbook. This book contains many moredetails and examples of great interest, aimed at the audience of this module.

As already noted this module assumes no previous knowledge of mathematical finance.The objective is for students to gain a fundamental understanding of the concepts un-derpinning mathematical finance and to be able to apply the appropriate mathematicaltools derived in this course.

1 Introduction

It is virtually impossible not to notice that the current financial world is in absoluteturmoil: major banks, insurance companies, investment houses, etc., are collapsing, re-structuring, struggling, merging . . . . The projected job losses in the finance sector in theUK are estimated to be in the 10s to 100s of thousands. What has gone wrong?

This course is meant to be an introduction to mathematical finance and the simplestpossible financial models will be discussed. The recent troubles in the worlds financialmarkets are beyond the scope of this module, and indeed, beyond the scope of mostadvanced financial theory. Research into more sophisticated models of mathematicalfinance is likely to be a significant growth area in the future.

1.1 Stocks, shares and equitiesThe terms stocks, shares and equities essentially mean the same thing and are usedinterchangeably. Suppose company XYZ wishes to raise money to build a new factory.XYZ thinks that it will be able to make a profit by building the new factory (and thenselling whatever it is that is produced in the factory). XYZ then sells shares in itselfto investors, in order to finance the new factory. The investors, or shareholders, thenshare the ownership of the company, based upon the number of shares that each investorowns. If XYZ then makes a profit, it can choose to pay a dividend to shareholders, or itmay choose to invest the profit in some other way. If company XYZ is then subsequentlysold, the proceeds from the sale is then split amongst the shareholders (according to the

number of shares that they own). If XYZ is sold its net worth is equal to the sum of all ofits assets (e.g.. property, inventory, IPR, etc.) minus the sum of all of its liabilities (e.g..outstanding loans, deferred tax payments, pension plan commitments, etc.). If there aren outstanding shares in the company XYZ then the value V of share can be approximatedas

V 1n

(A L) + D

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where A represents the sum of all assets of XYZ, L represents the sum of all liabilitiesof XYZ and D represents a contribution due to the expected dividend per share thatwill be paid out by XYZ. Clearly V should increase if the expected dividends from XYZincrease. Similarly, V should increase if the difference (A L) increases. The difficultywith actually assigning a value to V is that different people may have different opinionsabout the values of A, L and D. (Why might this be the case?).

The actual value of a share is determined when the buyer of a share, and the seller ofa share, agree a sale price. (In reality most shares are bought and sold by an intermediaryknown as a stockbroker. The seller agrees to sell at the bid price, and the buyer agreesto buy at the ask price. The ask price is always greater than the bid price, and thestockbroker makes a commission of ask price - bid price for each share sold. Duringthe first term of this module we will ignore the effects of commission charges and assumethat the ask price and the bid price are equal. The effects of commission will be discussedin the second term of this module). The buyer of a share hopes that the value of theshare will increase and/or to receive dividend payments such that their investment is

worthwhile. The seller of a share wishes to release the value of their investment, eitherto make further investments, or to fund other expenditures (e.g. a new car, a holiday,retirement, education, etc.). The value of a share is only determined when a buyer and aseller agree a sale price. The value of stocks listed in the news media and online reflectthe most recent sale price agreed between a buyer and a seller, or, the last sale pricedagreed at the end of a business day.

1.2 Supply and Demand

Ultimately, the value of shares are determined by the laws of supply and demand. Suppose

there are a large number of investors who feel that purchasing shares in company XYZis a good investment (i.e. demand is high), but there are relatively few people willing tosell these shares (i.e. supply is low). In this case the sellers are at liberty to increase theprice at which they are willing to sell their shares since the shares are in high demand.Conversely if a large number of investors feel that shares in XYZ are a poor investmentand they all attempt to sell their shares then supply will be high. If there are relativelyfew investors wanting to buy these shares (i.e. demand is low) then buyers can agree tobuy these shares at a lower price. Since the value of a share is only determined when it isbought and sold, the value will be determined when there is a balance between supply anddemand. In particular the share price must be sufficiently high that the seller is willing

to sell while simultaneously being sufficient low that the buyer is willing to buy.

1.3 Long selling and short selling

The term long is used to refer to assets that you actually own. For example if you are long10 shares in company XYZ you own 10 shares in this company. The term short refersto assets that you do not actually own, but that you are willing to trade in. The practiceof short selling has frequently been in the news in the recent months and weeks, and it

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refers to selling shares that you do not actually own. Laws which allow for short sellingare currently being examined very closely. When you short sell, you sell (at the currentmarket value) shares which you do not actually own. At some specified, or unspecifiedtime in the future, you are then obliged to purchase the shares at the new market value.When you short sell, you are taking the position that the value of the share will fall in thefuture, and you will thus make a profit. If the value of the share increases in the futureand you are obliged to purchase the shares which you originally short sold, you will makea loss.

The practice of short selling relies on the existence of a stockbroker with many clients.Suppose client A wishes to short sell 10 shares in company XYZ, and client B owns 10shares in this company. The stockbroker, assuming he trusts client A, can then sell clientBs shares in XYZ subject to client A agreeing to return these shares to client B at somepoint in the future. Any dividends that are paid out on shares in XYZ in the interveningperiod must be paid directly from client A to client B. If at any time client B actuallywishes to sell his shares in XYZ, then client A is immediately obliged to purchase the

shares that he originally short sold. In reality the stockbroker will have many clientswith shares in company XYZ, and if client B wishes to sell his shares in XYZ that wereoriginally sold by client A, then client A can then borrow shares in XYZ from client C,and so on. If the stockbroker runs out of additional clients with shares in XYZ, then clientA is short-squeezed and he must immediately buy the shares at the prevailing marketvalue.

The reason short selling has been in the news recently is directly related to the lawsof supply and demand. People short sell shares because they feel that the price of theshare will fall in the future. If a large number of shares are short sold (thus artificiallyincreasing the supply of shares) it is possible that the market will respond by decreasing

the value of the shares and subsequently yield a profit to the short seller. Of course ifdemand outstrips supply, than the short seller will make a loss.

(Question: Do you think short selling caused the collapse of HBOS? No doubt thiswill be the topic of many PhD theses and research papers in the future.)

1.4 Arbitrage

One of the most important concepts in Mathematical Finance is that of arbitrage. Thisterm is used to mean that it is impossible to make an instantaneous risk-free profit.More accurately, the concept of arbitrage demands that opportunities to make risk-free

profits will vanish quickly due to the market forces of supply and demand. As an example,suppose that the shares of company XYZ are traded in pounds sterling on the LondonStock Exchange and in American dollars on the New York Stock Exchange and that theshares traded on these two stock exchanges are identical (i.e. offer the same dividends,same ownership rights, same voting rights, etc., and can be sold freely on either marketregardless of where they were purchased from). Further suppose that the exchange ratebetween pound sterling and the American dollar is 1 = $2. What would happen if theprice of a share in XYZ was $2.00 in New York, and 1.10 in London? Ignoring the effects

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of commission charges and any tax implications, the prudent investor would buy sharesin XYZ in New York and then immediately sell them in London gaining an instantaneous10 pence profit per share. Presumably there are many prudent investors out there andthey will buy as many shares in XYZ in New York as possible, and then instantly sellthem in London. The large demand to buy XYZ on the New York stock exchange willdrive the price of XYZ up in New York. The large supply of XYZ which are being sold inLondon will drive down the price of XYZ in London. Very quickly, the laws of supply anddemand will ensure that the effective price of XYZ is the same on both stock markets.An important point is that the financial institutions with access to the lowest commissioncosts, are also the ones trying to take advantage of arbitrage opportunities. The more thatthese institutions compete with each other to take advantage of an arbitrage opportunity,the faster that opportunity vanishes.

1.5 Risk

In mathematical finance the term risk specifically refers to uncertainty. The value ofa financial investment may go up as well as down (as we are constantly reminded bythe Financial Services Authority). Each individual investor must evaluate the risks ofeach individual investment and make their investment decisions accordingly. High riskinvestments are expected to have the potential to earn a great deal of money for theinvestor, but they also have the potential to lose a great deal of money for the investor.Conversely low risk investments are expected to earn a lower amount of money, but theyalso have a lower potential of losing money. Nothing is certain in the financial market(certainly within the current financial market) and what constitutes a high risk or a lowrisk opportunity is ultimately reflected in the price agreed to between the buyer and seller

of any financial product.In financial theory there are different types of risk. The term specific risk refers

to the risk associated with investing in a specific company. The term non-specific riskrefers to the risk associated with investing in a particular market as a whole. For example,there is a non-specific risk at the moment associated with investing in the Aviation sector(on a global scale all Airlines are faced with high fuel costs at present, and there is noparticular reason to assume any one airline should perform better or worse than any otherairline ... there is a risk associated with investing in airlines, but that risk is faced bythe sector as a whole). On the other hand, there may be a specific risk associated withinvesting in Quantus Airlines (in the past couple of months they have been involved in

several serious safety incidents). The Aviation sector seems risky on the whole at themoment, but, if you choose to invest in an airline that out performs its competitors inthe current financial situation, than you are likely to make a profit in the long term.Current investors in Quantus may be willing to sell on the cheap at the moment becauseof future safety concerns (or more likely the view of the market on future safety concerns).Ultimately, market forces will reflect the value of a share in a company quoted on a stockexchange. People who feel that there is a strong risk that the company will decrease invalue will sell their shares and drive the value of the share down. People who feel that it

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is worth the risk to invest in the company will buy shares and then drive the value of theshare up. Market forces ultimately will determine the value of a share.

1.6 Risk-free investments and arbitrage

Most financial models assume the (theoretical) existence of risk-free investments whichgive a guaranteed return. In this course we will assume that money deposited in a soundbank which earns interest at a rate r is a risk-free investment. (Is this true in the currentfinancial climate?) IfI represents the amount of money deposited in the bank, r representsthe interest rate (which is continuously compounded) and t represents time, then the valueof the risk free investment varies as

dI

dt= rI . (1)

This is a separable first order differential equation and it can easily be solved according

to the appropriate initial conditions. Suppose that at time t = to an amount I = Io isinvested and further suppose that the interest rate r is constant. We thus obtain

I(t) = Ioer(tto) (2)

At a later time t = T the value of our investment is simply I(T) = Ioer(Tto). The increase

in the value of our risk-free investment (i.e. our risk free profit) is

Prf = I(T) I(to) = Ioer(Tto) Io = Io

er(Tto) 1 (3)Note, if this investment is risk-free to the investor depositing the money, it is also risk-

free to the bank. The risk-free cost (to the bank) of borrowing an amount Io at timet0 and repaying the loan at time T is simply Prf above. It is a risk-free cost since thebank knows its liabilities at time t = T exactly, when it agrees to the investment attime t = to. In reality, depositing money in a bank is not risk-free (What happens if thebank goes bankrupt? How much of your investment is secure?) Banks would also claimthat loaning money is never risk-free (what happens if the client cannot repay the loan?).Nevertheless, the concept of a risk-free investment is central to Financial Mathematicalmodelling. In an ideal economy, market forces will determine the appropriate rates ofinterest for borrowing and depositing.

Now that we have defined a risk-free investment, we can re-examine the concept of

arbitrage which states that it is impossible to make an instantaneously risk free profit.Interest earned on a bank deposit is risk free but it is not instantaneous, since interest isearned over the time period T to. Now suppose there exists an investment opportunitywhich costs C = Io at time t = to and is guaranteed (so that there is no risk whatsoever)to have a return at time t = T of IT so the net profit Phypothetical from this hypotheticalinvestment is

Phypothetical = IT Io (4)

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The relevant question is whether or not you should invest in this hypothetical, risk-freeinvestment. If

Phypothetical < Prf (5)

you should definitely not invest, since you can make more money simply by depositing

your money in the bank. IfPhypothetical > Prf (6)

you should definitely invest. The optimal strategy would be to borrow the amount Io froma bank, at a known risk-free cost ofPrf and then instantaneously purchase the hypotheticalinvestment to yield an instantaneous risk-free profit of Phypothetical Prf. This profit willonly be realized at time t = T, but you know instantaneously at time t = t0 that youwill yield this profit (with no risk whatsoever). Clearly you, and all sensible investorswill try to borrow as much money as possible, so that you can immediately invest in ourhypothetical investment. According to the laws of supply and demand this will increasethe rate of interest and thus increase your cost Prf of borrowing, and it will also increase

the cost of the hypothetical investment, thus decreasing the return on your investmentPhypothetical. Ultimately market forces will ensure that the guaranteed profit from ourhypothetical investment will balance the profit that could be obtained by depositing ourmoney in the bank.

The principle of arbitrage is effected by friction factors such as commission costs, thefact that banks have different interest rates for borrowing and lending, and possible taximplications of different investment opportunities, but in general it is sound. Marketforces are such that possibilities to make instantaneous risk free profits can only exist fora very short time. Remember, there are a great deal of investors who are continuouslysearching for opportunities to make risk-free profits.

1.7 Hedging

Other than depositing money in a bank (or investing in similar products) the conceptof arbitrage argues that other investment opportunities cannot be risk-free. There arecertainly investment opportunities that have the potential to earn a greater possible returnthan by simply depositing your money in a bank, but there is also the risk that you willhave a smaller return or even lose your money altogether. The term hedging is usedto refer to investment strategies that are designed to reduce or minimise the risk to aportfolio of investments. Hedging may result in smaller overall profits, but hopefully itwill also reduce the likelihood of large losses.

One hedging strategy is to invest in negatively correlated market sectors. Supposeyou believe that investing in shares in the aviation sector is risky because of high fuelcosts. One possible strategy might be to also invest in shares in the Oil sector. If one ofyour investments falls in value, it is likely that the other investment will rise in value. Ifyou have the right balance of shares in the two sectors, it is possible to remove a largecomponent of the risk in your total portfolio. Determining the optimal balance of sharesin various market sectors is obviously the challenge.

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There also hedging strategies that are based on investing in financial products intrin-sically related to the same shares. These products, known as financial options or financialderivatives, are the primary subject of this course. We shall learn how to assign a valueto these financial products, and how to combine them to reduce the level of risk

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2 What are Financial Options?

Financial options (also known as financial derivatives) are investment products which arebased on a contract between two parties which places options and obligations to each ofthe parties to either buy or sell shares in a specified company at a specified price in the

future. Financial options were originally (and can still be) contracts agreed to betweentwo interested parties (often negotiated through an intermediary). Today options areopenly traded on various stock markets and can be bought and sold by interested investors.Options can be used to either speculate (i.e. bet on the future performance of a particularstock) or as a tool for hedging (reducing the risk associated with a financial portfolio).There are many different types of options and we shall begin by looking at the simplest.

2.1 European Call Option

The simplest type of option is the European Call Option (often called a vanilla call or

European vanilla call ... why vanilla? This option is as simple as they come!). This is acontract with the following conditions:

At a certain time in the future, called the expiry date or expiration date, the holderof the option may choose to purchase a certain specified asset, called the underlying assetor the underlying (e.g. a share in a company, or currency, or a commodity etc), for acertain price known as the exercise price or strike price. The holder of the option has theright to buy the underlying on the expiry date, but they do not have to, which is why itis called an option. The other party to the contract, called the writer, has an obligation:they must sell the asset to the holder for the strike price on the expiry date if the holder

chooses to buy it. When the contract is started, the holder pays the writer a fee for thiscontract (usually via some middleman at the futures and options market) i.e. the holderbuys the option from the writer. Moreover, the options contract can be traded at anypoint up to the expiry date e.g. if you buy the option from the holder then you becomethe holder. Throughout this term we will assume that there are no commission chargesassociated with the buying or selling of options, or the subsequent purchase and sale ofthe underlying.

Note: if the underlying is a share of a company, the option has nothing todo with that company. The option is a contract between two parties (oftenlarge companies, investment firms, banks, etc.) about the option of buying ashare in that company. If the option is bought/sold on the open market thanthe holder need not know the identity of the writer. The volume of tradingin options in an underlying share can often exceed the volume of trading inthe underlying shares themself.

Further Note: European options are traded all over the world and thedesignation European specifically refers to the options as described abovewhere the option/obligation to buy/sell shares is based on the value of the

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underlying share at a specific time (which is typically just after the close ofthe markets on the expiry date). There are also American options whichwill be described later on in this term, and Asian options which will bediscussed during term 2 of this module. Again the terminology is used torefer to specific aspects of the contract between the writer and holder of theoption, and these options can be traded worldwide. Presumably the termsEuropean, American and Asian refer to where these types of options werefirst introduced, or, where they are most popularly traded.

Example: Today the share price of Johnson Matthey PLC is 1677 pence. I buy aEuropean call option today where the underlying is one share of Johnson Matthey PLC.The expiry date is 1st December and the exercise price is 1700 pence. It costs me 20pence to buy the option from the writer. I am the holder of the option.

(a) Suppose on 1st December that the share price is 1742 pence. I then as the holder

of the option have the right to buy a share of Johnson Matthey PLC for 1700 pence whichis actually worth 1742 pence. So I may buy the share from the writer for 1700 pence (thisis called exercising the option), and if I choose, I could immediately sell that share on thestock exchange for 1742 pence, making an immediate profit of 1742-1700=42 pence. Butit cost me 20 pence to buy the option in the first place, hence I make an overall profit of42-20=22 pence.

(b) Suppose instead on 1st December that the share price is 1689 pence. I then asthe holder of the option have the right to buy a share of Johnson Matthey PLC for 1700pence which is actually worth only 1689 pence. Hence I choose not to use the option

to buy the share, and hence the option has become worthless. Hence I make an overallloss on this deal of 20 pence, which was the price that the option originally cost me to buy.

Notice that the overall profit or loss is a large percentage of the original cost of enter-ing into the contract in the first place. So in case (a), I make an overall profit of 22/20 x100% = 110% on my original investment. In case (b), I lose all my original investment,and so I make a 100% loss on my original investment.

In contrast, suppose I didnt buy the option today, but I instead bought one shareof Johnson Matthey PLC today. That would have cost me 1677 pence. In case (a), the

share would have risen to 1742 pence, giving me an overall profit of 1742-1677=65 penceby the 1st December. This is an overall profit of 65/1677 x 100%=3.88%. In case (b), theshare would have risen to 1689 pence, giving me an overall profit of 1689-1677=12 pence.This is an overall profit of 12/1677 x 100%=0.72% of my original investment.

Hence in case (a), if I had bought the option I get a 110% profit on my investment,while if I had bought the share I would have instead received a 3.88% profit on my in-vestment.

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In case (b), if I had bought the option, I would have had a 100% loss on my originalinvestment, while if I had bought the share then I would have instead received a 0.72%profit on my investment.

This effect is called gearing. It means if you invest a certain sum of money in options,then the potential profits or losses are much greater than if you invested the same amountof money on shares. Imagine this effect when we are not dealing with pence, but insteadwith hundreds of millions of pounds. Throughout this course you must try to thinkof things in terms of investments made by large banks / investment firms who will beinvesting large quantities of money and who have the capacity to expose themselves tolarger levels of risk. The laws of supply and demand will not be effected by one individualinvestor buying one call option for one share. It is the large investors who ultimatelyinfluence the market.

Example: Consider the previous example from the point of view of the writer of theoption instead. Today the writer receives 20 pence when writing the option. In case (a),the holder forces the writer to sell a share for 1700 pence, which is actually worth 1742pence, on 1st December. Hence the writer makes a loss at that point of 1742-1700=42pence. But the writer originally received 20 pence when writing the option, and hencethe writer makes an overall loss of 42-20=22 pence.

In case (b), the holder does not exercise the option, and hence the writer has madean overall profit of 20 pence.

Note: The above examples have introduced the mechanics of buying/sellingand choosing or otherwise to exercise a European vanilla call option, but notethat the cost/value of the option was specified. One of the primary ob jectivesof this course is to examine how the cost/value of an option can be determinedso that it correctly reflects the risk which is being assumed by both parties tothe contract.

2.2 European Put Option

The next simplest type of option is the European Put Option (often called a vanilla putor European vanilla put). This is a contract with the following conditions:

At a certain time in the future, called the expiry date or expiration date, the holderof the option may choose to sell a certain specified asset, called the underlying asset orthe underlying (e.g. a share in a company, or currency, or a commodity etc), for a certainprice known as the exercise price or strike price. The holder of the option has the rightto sell the underlying on the expiry date, but they dont have to. The other party to the

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contract, called the writer, has an obligation: they must buy the asset from the holder forthe strike price on the expiry date if the holder chooses to sell it. When the contract isstarted, the holder pays the writer a fee for this contract i.e. the holder buys the optionfrom the writer. Moreover, the options contract can be traded at any point up to theexpiry date e.g. if you buy the option from the holder then you become the holder.

2.3 Exercising options

The holder of a call option is said to be in the money on the expiry date if the shareprice on the expiry date of the option is greater than the strike price. The holder can thenimmediately realize his/her profit by buying shares from the writer at the strike price andthen selling these shares on the open market. Alternatively, the holder may choose topurchase the shares at the strike price and then retain them as part of their investmentportfolio. Different investors will have different investment strategies depending on their

current feelings of the financial market. If the holder of a call option is in the money,it is sometimes possible for the writer to pay the holder the net difference between theshare price at expiry and the strike price, instead of the holder actually going throughthe act of purchasing and immediately selling shares. This of course will depend on thefiner details of the option contract.

2.4 What is the value/cost of an option?

Suppose you are interested in buying a call option. How much should you pay? Thereare certain things that are known. You purchase a call option because you expect the

share price S on the expiry date to be greater than the exercise price E. This definitelymeans that the cost of a call option should be inversely related to the exercise price (i.e.the lower the exercise price, the higher the cost of the option). The share price S onthe expiry date is not known, but, the share price at the time the option is purchased isknown, and this should presumably influence the price of the option (a call option to buya share with a current market value of 1000 pence should presumably be more expensivethan a share with a current market value of 100 pence). Further, the price of the optionshould depend on the current (and projected) rate of interest (instead of investing in theoption, the holder can choose to make a risk free profit by depositing money in the bank),as well as the time until the expiry date (which not only effects the amount of risk freeinterest that can be earned, but, also influences the risk/uncertainty associated with theunknown share price at the expiry date).

Consider the following scenarios. On January 1st you purchase a European vanillacall option for shares in XYZ at a cost of 100 pence with an expiry date of September 1stand an exercise price of 1000 pence. Todays date is August 1st (one month before theexpiry date). What do you do if:

(a) The current share price is 1200 pence and another investor offers to buy your calloption for 10, 50, 100, 200, 300, . . . pence, or,

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(b) The current share price is 800 pence and another investor offers to buy your calloption for 5, 10, 50, 100, 200, . . . pence?

In case (a) you are in the money if the share price of XYZ is the same on the expirydate as it is on August 1. In case (b) you are out of the money if the share price of XYZ

is the same on the expiry date as it is on August 1. In either case the option can only beexercised (if at all) on the expiry date. The value of XYZ might rise or fall in value inthe next month and different investors might have different views on this. In case (a) youmay be willing to accept an offer of 200 pence for your option since this would guaranteeyou a profit of 100 pence on your original investment. This might be a good decision ifyou think the value of XYZ is going to fall in the next month. Alternatively, you maystill think that the price of XYZ is going to rise, but, you choose to sell on August 1stsince you are happy to settle for a known profit (you may need this money immediatelyto pay off an imminent liability, or you may feel that you can use this money to invest inan even more lucrative venture). In case (b) you can choose to hold on to your option in

the hope that the value of XYZ will rise above the exercise price, or, you choose to sellthe option to minimise your losses. The purchaser of the option in case (b) may well betaking a large risk, so, he/she is unlikely to offer too high of a price for the option.

The key point is that both the buyer and seller of an option are subject to risk since thevalue of XYZ on the expiry date is not known. The buyer and the seller of an option haveto agree a sale price for the option, so, the sale price is to a certain extent determined bythe laws of supply and demand. Nevertheless, we have to assume that both the buyer andthe seller are prudent. Using the theory of mathematical finance it is possible to determine(estimate) the cost/value of an option that fairly reflects the risk to both parties. Thebuyer and the seller may have different estimates, but, at the very least, mathematical

finance can be used as a starting point for the negotiation of the sale price.

2.5 Pay-off functions and pay-off diagrams

So far we have dealt with hypothetical data with regards to the cost of options and theprofits that can be realized on the expiry date of the option. Let us consider a real worldexample. This example is based on the FT-SE index on 4 February 1993 as reportedin the recommended text of Wilmott, Howison and Dewynne (1995). The FT-SE indexrepresents a weighted average of the London Stock exchange and holders/writers of call

Exercise Price 2650 2700 2750 2800 2850 2900 2950 3000Cost of Call 233 183 135 89 50 24 9 3

Table 1: The cost of European Call Options on the FT-SE index on 4 February 1993 withan expiry date of 14 February 1993. The index value on 4 February 1993 was 2872 pence.

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options on the FT-SE index either hope that the index will go above/below the exerciseprice of the call option respectively. On 4 February 1993, the value of one share in theFT-SE index was 2872 pence. Table 1 lists the cost of purchasing European call optionson February 4 for different exercise prices with a known expiry date of 14 February 1993.The option price is plotted versus the exercise price in Figure 1. Also plotted in Figure1, as a solid line, is the hypothetical value of the option (share value at expiry minusexercise price) assuming that the share value on the expiry date is the same as it is on 4February 1993. Observe that the cost of purchasing a call option today exceeds the returnthat would be achieved if the underlying value of the share in question has the same valueon the expiry as it has on the day the call option was sold. This is perfectly consistentwith the presumption that purchasers of call options feel that the share price will increasein value (or more specifically, above the strike price) by the expiry date. Note that calloptions can be bought and sold for strike prices above or below the current value of theunderlying share and the buyer of a call option is specifically making an investment basedon a specified exercise price.

2650 2700 2750 2800 2850 2900 2950 3000

Excercise Price

0

100

200

300

CallPrice

Figure 1: The price of FT-SE index call options versus exercise price on 4 February 1993.

The data plotted in Figure 1 is based on information that is known today as opposed

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to information that is known at the expiry date. In particular the price of an option isplotted versus the exercise price. Once an investor has purchased an option the exerciseprice is fixed, and it is more important to know the value of the option as a function ofthe unknown value of the underlying at the expiry date.

European vanilla call

The European vanilla call option can be characterised by its pay-off function. Thisis the function, of the underlying share price S, which gives a formula for the immediateprofit to the holder of the option at the expiry date. For a European vanilla call, we haveseen that the holder only chooses to exercise the option (i.e. to buy the underlying sharefrom the writer) if the share price S of the underlying is greater than the strike price Eon the expiry date. If this is the case (i.e. S > E) then it is possible for the holder to buythe underlying from the writer at price E (the strike price) and to sell it immediately onthe stock exchange for the share price S, making an immediate profit of S

E.

However, if S < E, then the holder will choose not to exercise the option (i.e. whybuy a share for price E when the holder can get it more cheaply on the stock exchange forprice S). Therefore, if S < E, then the immediate profit is 0 to the holder of the optionsince the option has become worthless.

So, ignoring the fee paid by the holder for the call option in the first place, theimmediate profit on the expiry date for the holder is equal to S E if S > E, and equalto 0 if S < E. This is called the pay-off function of the option. So the pay-off functionfor a European vanilla call is

S E, when S > E,

0, when S < E.This can be written in a simplified form as

max(S E, 0),

where max is a defined as the function which chooses the maximum of the two numbersin the brackets.

Therefore, we say that the pay-off function of a European vanilla call is equal tomax(S E, 0). If we sketch a graph of max(S E, 0) on the vertical axis, and the shareprice S on the horizontal axis, then this is called the pay-off diagram. (See Figure 2.)

The overall profit on the deal for the holder of the option equals the pay-off minus theoriginal cost that the holder paid for the option.

European vanilla put

We can repeat the above to work out the pay-off function for a European vanilla put.For this option, if S > E on the expiry date, then the holder may sell the underlying to

14


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0 S

C

E

Figure 2: The pay-off diagram for a European call option.

the writer for price E when in fact the underlying is worth S. The holder would haveto be crazy to do so, and hence the option is not exercised if S > E and the immediateprofit, or pay-off, to the holder equals 0.

If S < E on the expiry date, then the holder may sell the underlying to the writerfor price E when in fact the underlying is worth S. This is a good deal for the holder,and hence the option will be exercised. Then the immediate profit for the holder (i.e. the

pay-off) equals E S.So the pay-off function for a European vanilla put is

0, when S > E,E S, when S < E.

This can be written in a simplified form as

max(E S, 0).

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Again a pay-off diagram can be sketched. (See Figure 3.)

0

P

SE

Figure 3: The pay-off diagram for a European put option.

Other European options (binaries or digitals)

Investors buy and sell call and put options because they have a certain view on thefuture price of the underlying. Vanilla calls and puts represent the simplest possible typeof options. Options can be used to hedge against uncertainty or to speculate, and wecan now define some of the most common types of European options by using the pay-offfunction.

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Cash-or-nothing call: pay-off equals

BH(S E),

where B is a constant (a sum of money). The function H is the Heaviside step function.

It is defined as

H(x) = 1 if x > 0,H(x) = 0, if x < 0.

Therefore, the cash-or-nothing call has pay-off equal to

B if S > E,0 if S < E.

Bullish vertical spread: pay-off equals

max(S E1, 0) max(S E2, 0),

where E2 > E1, so that there are two different strike prices on this option (E1 and E2).

Supershare: pay-off equals

1

d(H(S E) H(S E d)) ,

where d is some constant.

Asset-or-nothing call: pay-off equals

SH(S E).

Straddle: pay-off equals

max(S E, 0) + max(E S, 0).

From these definitions you should be able to draw the pay-off function for these moreintricate options. Also, you may wish to look at the various recommended textbooks fordefinitions of other common European options.

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3 Random walk model of asset prices

The term asset is used to collectively refer to stocks, commodities (such as gold, oil,wheat, etc.), options, currency, etc. The goal of this course is not to explicitly predicthow asset prices will vary as a function of time. If this were possible, the wise investor

would keep it a secret, and happily make as much money as they could regardless of theoverall market performance. In reality, it is not possible to predict with certainty, howasset prices will vary as a function of time. Figure 4 shows the closing value of the FT-SE100 stock index on trading days between 1 April 2008 and 21 October 2008. Over thistime period the general market trend is downwards, but there is considerable variability:on some days the market goes up on some days it goes down. The overall trend in thedata in Figure 4 is for the stock index to go downwards, but on any given day it seemsimpossible to predict whether the index is going to go up or go down. In essence there arerandom variations which combine with the overall market trend to determine the indexas a function of time. One data point is plotted for each trading day in Figure 4. Similar

fluctuations would also be seen if we plotted the index value with a much shorter timeinterval (for example once every minute over the course of a trading day). These randomfluctuations appear to be an inherent characteristic of asset values. In section 1.2 it wasargued that the value of an asset is determined by the market forces of supply and demandand that the ultimate value is the price agreed by the buyer and the seller. The randomfluctuations are the results of different buyer/seller pairs having slight differences in theirview of the the asset value.

The model we will use in this course is given by the following stochastic differentialequation

dS = S(dX + dt) . (7)

Here dS is the change in the asset price (i.e. share price) in a small time step dt. Also, Sis the current asset price, is a measure of the average rate of growth of the asset price(known as the drift), is called the volatility of the share (which measures the standarddeviation of any random variation in the share price), and dX is a random number takenfrom a normal distribution. The mean of the normal distribution that dX is taken fromis equal to zero and the variance of the normal distribution is equal to dt. Since thevariance is equal to dt, then the standard deviation is equal to

dt. Therefore, we can

write dX =

dt where is a random number taken from a normal distribution withmean zero and standard deviation equal to one.

Therefore, the above model describes how the asset price S, changes from S to S+ dS,

during a time step from t to t + dt. We will ultimately examine this model in the limitdt 0.Note that if = 0 then the above equation simplifies to

dS = Sdt. (8)

Therefore, using separation of variables,dS

S=

dt.

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0 50 100 150

Time (days since 1 April 2008)

3000

4000

5000

6000

7000

FTSE

100Index

Figure 4: Closing values of the FT-SE 100 index call between 1 April 2008 and 21 October 2008

Therefore,

ln S = t + c,

where c is a constant of integration. Therefore,

S = Soe(tto),

where the initial condition S = So at time t = to is used. This equation is formally

equivalent to the case of money earning continuously compounded interest in a bank.For the case of bank interest, however, the interest rate r is always positive. The driftcoefficient in equation (8) can be either positive, negative, or even zero. Therefore, wesee that measures the non-random increase in the share price if > 0 (or non-randomdecrease in the share price if < 0). Of course, the case with = 0 is not realistic, sincethe share price S varies randomly.

Example: Suppose S = 10, t = 1, = 0.1, = 0.01 and dt = 0.01. Then dXmust be a random number selected from a normal distribution of mean 0 and standard

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deviation equal to 0.1. Of course dX is random and cannot be predicted, but suppose itwas equal to 0.03. (Note that dX could be anything since it is random; this is just anexample of a number it could be!) Then dS = S(dX + dt), which in this case givesdS = 0.031. So in this time step, t increases from 1 to 1 + dt = 1.01, and S increases from10 to 10 + dS = 10.031.

For the next time step, we now have S = 10.031, t = 1.01, = 0.1, = 0.01 and dt =0.01. Again dX is a random number selected from a normal distribution of mean 0 andstandard deviation equal to 0.1. Once again dX cannot be predicted because it is random.But suppose the next value for dX was equal to 0.02. Then dS = S(dX + dt),which in this case gives dS = 0.0190589. So in this time step, t increases from 1.01 to1 + dt = 1.02, and S changes from 10.031 to 10.031 + dS = 10.0119411. etc.

The repeated iteration of equation (7) to calculate the value S(tnew) = S(told) + dSwhere tnew = told+dt is known as a random walk. Since dX is a random variable, repeatedindependent walks starting from the same initial position will likely give different predictedvalues of S at any later time. Thus, equation (7) cannot be used to make predictions

about future asset values. Nevertheless by performing many repeated random walks, itis possible to make some qualitative statements. Consider the following simple MatLABprogram:

s(1) = 100

mu = 0.1

sigma = 0.01

dt = 1/365

for i = 1:365

dX = randn*sqrt(t);

dS = s(i)*(mu*dt + sigma*dX);s(i+1) = s(i)+dS;

end

plot(s)

In the above we are assuming that time is measured in years and dt = 1/365 representsone day. The value of and are specified, as is the value of S on day 1. We haveexplicitly assumed that dX =

t where is a normal random variable with mean zero

and standard deviation one. In MatLAB we use the in-built function randn to calculatethe required values of at each iteration. After 365 days (one year) we then plot S as afunction of time. Using this program we can then easily perform repeated random walks

to determine the influence of varying the various parameters.Figures 5, 6 and 7 show repeated random walks for different values of . In each case

we start with S = 100 and we perform 365 time steps with dt = 1/365. The drift rate isheld fixed at = 0.1. In Figure 5 we have set = 0.01, in Figure 6 we have set = 0.1and in Figure 7 we have set = 0.3. The solid lines in these figures represent the repeatedrandom walk. The dotted line (which is the same in all figures) represents the exponentialgrowth that would be expected if = 0. Observe in Figure 5 that when is relativelysmall (in this case = 0.01) the different realizations of the random walk all lie close to

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0.0 100.0 200.0 300.0 400.0

Time (days)

50.0

75.0

100.0

125.0

150.0

S

Figure 5: Random walks using equation (10) with = 0.01.

exponential growth curve. As is increased to 0.1 in Figure 6 and to 0.3 in Figure 7 wesee that the different realizations of the random walk start to strongly diverge from eachother. In some cases the random walk greatly exceeds the expected exponential growthdue to drift along, and in some cases the random walk falls well below. The volatility ofan aspect is a direct measure of risk. The larger the volatility, the greater the uncertaintyof the investment. From these figures it is clear that high risk investments can lead tomuch greater earnings than low risk investments, but they can also lead to much greaterlosses. These figures certainly indicate that when the volatility is large, it is impossibleto make predictions about asset values in the future.

This module assumes that asset values can be modelled using the stochastic differentialequation (7). The results of figures 5-7 clearly show the effects of randomness, thusindicating that this equation cannot be used to make predictions about future asset values.Why then is this equation so important? First, in a statistical sense, it has been shownthat equation (7) generally does a very good job of modelling the historical data of mostasset prices (better agreement is achieved for stocks, options and commodities, than forcurrencies). Second, and more significantly, we can use equation (7) as the foundation for

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0.0 100.0 200.0 300.0 400.0

Time (days)

50.0

75.0

100.0

125.0

150.0

S


more sophisticated models, and these models can be used to completely eliminate risk.The formation of these models is the primary goal of this module. Before, we proceed toexamining these more sophisticated models, we first note that equation (7) requires thespecification of two fundamental parameters and . These parameters will depend onthe particular asset which is being modelled, and they can be estimated by looking at thehistoric data of the asset prices. In particular, suppose that we know the value of S atn + 1 discrete, and equally spaced, points in time, which we denote by So, S1, S2, . . . , S nwith the subscripts denoting sequential points in time. Assuming that dX is normallydistributed then we estimate

m = 1ndt

n1i=0

Si+1 SiSi

and

2 2 = 1(n 1)dt

n1i=0

Si+1 Si

Si mdt

2(9)

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0.0 100.0 200.0 300.0 400.0

Time (days)

50.0

75.0

100.0

125.0

150.0

S


The accuracy of these estimates will depend on the number of points in the time series,as well as the size of the time step dt. It has been assumed that and in equation (7)are constant, but, in reality these parameters will be slowly varying functions of time.

23


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3.1 The size of random fluctuations

The starting point for our modelling of asset prices is the stochastic differential equation

dS = S(dX + dt) . (10)

The drift term dt corresponds to the deterministic rise or fall in the value of S withtime in the absence of any random fluctuations. The dX term accounts for the influenceof random fluctuations. The relevant question is Which is more important?: the deter-ministic drift of S with respect to time, or the influence of random fluctuations. Thisquestion is relatively straightforward to answer. Ifdt >> dX then deterministic forceswill dominate. If dX >> dt then random fluctuations will dominate. If the termsdt and dX are approximately of the same magnitude, then, the value of the asset Swill depend on both deterministic and random forces. Historical evidence suggests thatstock prices are indeed determined by a combination of deterministic and random forcesand thus we would like to have an appropriate balance in the sizes of the dt term andthe dX term. Moreover, we would like the parameter to uniquely characterise thedeterministic drift (growth or fall) of S and to uniquely characterise the influence ofrandom fluctuations. Ultimately this means that we will need to comment on the relativesize of dt versus dX in order to ensure that equation (10) is a realistic model of the timeevolution of asset prices.

In the previous section we made the statement that dX can be modelled as a normallydistributed random variable with zero mean and variance equal to dt (i.e. dX =

dt

where is a random variable from a standardised normal distribution with mean 0 andvariance 1). This is indeed the appropriate choice of scaling for dX as a function of dt,but it is instructive to try to convince ourselves, why this scaling is appropriate. In this

section we will use computer simulations to justify this scaling. In the following sectionwe will use more rigorous mathematical methods to justify this scaling. The key pointis that if we can successfully argue that dX is proportional to

dt, then we can convert

the stochastic differential equation (10) for S into a fully deterministic equation (i.e. theBlack Scholes equation).

Our objective is to use equation (10) to perform a large sequence of random walksand then to compare these random walks to each other in order to see if any meaningfulinformation can be obtained. In particular, we shall keep and fixed and iterateequation (10) from some initial time t = t0 until some final time t = t1 by performingn iterations with fixed dt = (t1 t0)/n. Because equation (10) can be seen as a leadingorder Taylor series expansion for S as a function of time, we expect that our results willbecome most meaningful in the limit as dt 0 (or n ). In equation (10) we will set

dX = dtk (11)

where is a random variable from a standardised normal distribution with mean 0 andvariance 1, and we will then examine the influence of varying k for values of k > 0.

The following MatLAB program can be easily modified to perform our comparison:

24


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s(1) = 100

mu = 0.1

sigma = 0.3

n = 100000

dt = 1/n

k = 0.5

for j = 1:500 % perform 500 random walks

for i = 1:n

dX = randn*dt^k;

dS = s(i)*(mu*dt + sigma*dX);

s(i+1) = s(i)+dS;

end

ss(j) = s(n+1) % save the final value of each random walk

end

plot(ss)

The above program was used to integrate from t = 0 to t = 1 using a time-step ofdt = 105 with = 0.1 and = 0.3 fixed for different values of k. Figures 8, 9 and 10show results for k = 1, k = 1/2 and k = 1/4 respectively. In each case, 500 separaterandom walks were performed assuming an initial value of S(0) = 100, and the value ofS at time t = 1 is then plotted. Since we are interested in the limit where dt


26/90

0.0 100.0 200.0 300.0 400.0 500.0

Walk number

109.0

110.0

111.0

112.0

S

Figure 8: A sequence of random walks using equation (10) with dX= dt.

equation (10) to meaningfully model the variation of stock prices.

3.2 Black-Scholes equation: derivation (part 1)

The rest of this module is concerned with how to work out how much to pay for an option,or how much to sell it for if it is to be traded prior to the expiry date. We shall derive theBlack-Scholes equation, which will be the basis of all such calculations. We will derivethe fair price for an option, though a trader will always try to do better than that! Thisequation gives the basis for all trading in options and is used extensively in all Investment

Banks.

In the derivation of the Black-Scholes equation we will consider holding a portfoliomade up of one option (it can be a call or a put, or in fact any type of option) which hasvalue V, plus lots of the underlying share S. Thus the portfolio that we hold has totalvalue where

= V + S,

26


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0.0 100.0 200.0 300.0 400.0 500.0

Walk number

0.0

100.0

200.0

300.0

S

Figure 9: A sequence of random walks using equation (10) with dX= dt1/2.

i.e. equals the total value of the portfolio, V equals the value of the option, S equalsthe value of the underlying share, and is the number of shares in the portfolio. e.g.a portfolio like this might consist of one European Call Option where the underlying isRolls Royce plus lots of Rolls Royce shares where is a constant.

The value of the option V will vary with time t and also with the underlying shareprice S. Thus V = V(S, t). The equation that we will derive below will be valid for anyfunction V which is a function ofS where S is determined by the lognormal random walkgiven by equation (10) and time t. Strictly speaking V does not have to represent thevalue of an option, but this is a helpful starting point.

The following derivation is mathematically informal. See the recommended textbooksfor notes on how to make the following mathematical argument more rigorous.

We consider a small time step dt over which the time varies from t to t + dt, and theunderlying share price varies from S to S+ dS. During this small time interval, the valueof the option will vary from V to V + dV, where

dV = V(S+ dS,t + dt) V(S, t).

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0.0 100.0 200.0 300.0 400.0 500.0

Walk number

0.0

5000.0

10000.0

15000.0

20000.0

S

Figure 10: A sequence of random walks using equation (10) with dX= dt1/4.

We take to be constant over the time step t to t + dt, but may vary from one timestep to another.

We can use Taylors theorem (it is called Itos Lemma in the case of using an expansionwith a random variable) to find

V(S+ dS,t + dt) = V(S, t) + dSV

S+ dt

V

t+

1

2dS2

2V

S2+ dSdt

2V

St+

1

2dt2

2V

t2+ ...

where the derivatives are all evaluated at (S, t). This is just the standard formula for

Taylors theorem in two variables. Rearranging slightly gives

V(S+ dS,t + dt) V(S, t) = dSVS

+ dtV

t+

1

2dS2

2V

S2+ dSdt

2V

St+

1

2dt2

2V

t2+ ...

But dV = V(S+ dS,t + dt) V(S, t), therefore

dV = dSV

S+ dt

V

t+

1

2dS2

2V

S2+ dSdt

2V

St+

1

2dt2

2V

t2+ ...

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But dS = S(dX + dt). Substituting this into the above equation gives

dV = S(dX + dt)V

S+ dt

V

t+

1

2(S(dX + dt))2

2V

S2

+S(dX + dt) dt2V

St+

1

2dt2

2V

t2+ ...

Rearranging gives

dV = SdXV

S+ dt

S

V

S+

V

t

+

1

2(dX)2 S22

2V

S2

+dtdXS2

2V

S2+ S

2V

St +1

2dt2S

222V

S2+ 2S

2V

St+

2V

t2 + ...But from the previous section, we know we can write dX =

dt where is a random

number taken from a normal distribution with mean zero and standard deviation equalto one. Hence, for dt small, even though dX is random, we know the size of dX will be ofthe order of

dt i.e. dX is proportional to

dt. Using this we can examine the relative

size of terms in the above equation when dt is small. Retaining terms only up to the sizeof the order ofdt, gives

dV = SdXV

S+ dt

S

V

S+

V

t

+

1

2(dX)2 S22

2V

S2+ ...

All higher-order terms (denoted by +...) will be smaller and hence we will neglect. (Theseneglected terms are of size (dt)3/2 and smaller, where dt is small.)

But = V + S. Therefore

d = dV + dS.

Using the above formula for dV, as well as dS = S(dX + dt), gives

d = SdXV

S+ dt

S

V

S+

V

t

+

1

2(dX)2 S22

2V

S2+ ... + S(dX + dt)

Rearranging gives

d = SdX

V

S+

+ dt

S

V

S+

V

t+ S

+

1

2(dX)2 S22

2V

S2+ ...

Justification that (dX)2 can be replaced by dt

29


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As a rule of thumb, we can replace (dX)2 by dt in all such equations whenever we seeit, and this will give the correct result when used. We now give an informal demonstrationof this fact. We consider a typical time interval that we wish to solve the option problemfor and denote this as a time interval from 0 to t. We break this time interval up into nequal time steps each of time dt, where n is some integer. We consider the time at eachtime step to be tj where

tj =jt

n,

where j is an integer which varies from 1 to n. Note that when j = n then jt/n = t i.e.the end of that time interval. We choose

dt =t

n,

so that each time step is of length dt as required. i.e.

t1 = tn , t2 = 2tn = t1 + dt, t3 = 3tn = t2 + dt, ..., tn = t.

We let dX(tj) correspond to the value of the random variable at time tj. We let dX(tj)be a random variable selected from a normal distribution with mean zero and varianceequal to dt (i.e. standard deviation equal to

dt).

If dX is a random variable selected from a normal distribution with mean zero andvariance equal to dt, then we define the quantity expectation of the random functiong (dX) as

E[g(dX)] =

g(x)f(x)dx,

where

f(x) =1

2dtexp

x

2

2dt

.

(Those students who have taken modules in Statistics will recognise this, but it doesntmatter if you dont. We dont need to consider at all what the expectation correspondsto - we may think of it as just being an integral. In fact, f(x) is the probability densityfunction of the normal distribution with mean zero and variance equal to dt, and theexpectation tells us the mean of the resulting function.)

We use this definition to calculate

E

nj=1

(dX(tj))2 t

2 .

This equals

E

n

j=1

(dX(tj))2

2 2t

nj=1

(dX(tj))2 + t2

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by multiplying out the brackets. This equals

nE

(dX)4

+ n (n 1) E(dX)22 2tnE(dX)2 + t2E[1] .This is because each dX(tj) comes from the same normal distribution for every value

of j i.e. E

(dX(tj))2

= E

(dX(ti))2

for all integers i and j; also E

(dX(tj))4

=E

(dX(ti))4

for all integers i and j.The first two terms in the above equation come from the term

nj=1

(dX(tj))2

2.

When squaring this and expanding out the brackets, we get n terms of the form

(dX(tj))4 ,

which has given us the nE

(dX)4

term above, and we get n (n 1) terms of the form

(dX(tj))2 (dX(ti))

2

where i = j, which gives us the n (n 1) E(dX)22 term in the above equation.In doing this we have written E

(dX)4

= E

(dX(tj))

4

for all j, and E

(dX)2

=

E

(dX(tj))2

for all j. Also note that we have used

E(dX(tj))2 (dX(ti))

2

= E(dX(tj))2

E(dX(ti))2

= E(dX)2

2

which can be proved using the fact the the random numbers dX(ti) and dX(tj) are selectedat different times (i.e. at times ti and tj) and hence are independent from each other.

To explain the above steps in more detail, we will consider the following simple casewhen n = 3. Then

E

3

j=1

(dX(tj))2

2 2t

3j=1

(dX(tj))2 + t2

equals

E

(dX(t1))2 + (dX(t2))

2 + (dX(t3))22 2t (dX(t1))2 + (dX(t2))2 + (dX(t3))2 + t2 .

This equals

E

(dX(t1))4 + (dX(t2))

4 + (dX(t3))4 + 2(dX(t1))

2(dX(t2))2 + 2(dX(t1))

2(dX(t3))2

+2(dX(t2))2(dX(t3))

2 2t (dX(t1))2 + (dX(t2))2 + (dX(t3))2 + t2 .

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This equals

3E

(dX)4

+ 6

E

(dX)22 6tE(dX)2 + t2E[1] .

We note that this is of the correct form for n = 3. This can then be generalised for all n

in this manner.We now evaluate our expression

nE

(dX)4

+ n (n 1) E(dX)22 2tnE(dX)2 + t2E[1] .To do this we note that

E[1] =

f(x)dx = 1,

E(dX)2 =

x2f(x)dx = dt,

and

E

(dX)4

=

x4f(x)dx = 3 (dt)2 ,

using integration by parts.Substituting these results into the above formula gives that

En

j=1

(dX(tj))2

t

2

= n 3 (dt)2 + n (n 1) (dt)2

2tn (dt) + t2.

But dt = t/n, so

E

nj=1

(dX(tj))2 t

2 = n

3

t

n

2+ n (n 1)

t

n

2 2tn

t

n

+ t2 =

2t2

n.

We let n in the above expression. Therefore

En

j=1

(dX(tj))2

t

2

= 2t2

n 0

as n . Since the left-hand side of the above expression corresponds to an integralover the square of a quantity, the only way that this integral can be zero, in the limit ofn , is if

nj=1

(dX(tj))2 = t

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as n . This is called the mean square limit.We now define the stochastic integral as

t

0

h(t)dX = limn

n

j=1h(tj)dX(tj)

for a function h(t). While this looks complicated, it isnt. It just says that we can thinkof the integral on the left-hand side as an infinite sum of lots of very small areas underthe curve.

Therefore,

limn

nj=1

(dX(tj))2 =

t0

(dX)2,

by choosing h(t) to be equal to dX(t). Therefore,

t

0

(dX)2 = t.

We can rewrite this equation as t0

(dX)2 =

t0

dt.

Therefore, t0

(dX)2 dt = 0.

Therefore, (dX)2 is equal to dt when written under an integral.We now return to the equation

d = SdX

V

S+

+ dt

S

V

S+

V

t+ S

+

1

2(dX)2 S22

2V

S2+ ...

and integrate throughout from 0 to t givingt0

d =

t0

SdX

V

S+

+ dt

S

V

S+

V

t+ S

+

1

2(dX)2 S22

2V

S2+ ...

We can replace (dX)2 in this integral with dt. Therefore,t0

d =

t0

SdX

V

S+

+ dt

S

V

S+

V

t+ S

+

1

2dtS22

2V

S2+ ...

Hence, removing the integral, we get

d = SdX

V

S+

+ dt

S

V

S+

V

t+ S

+

1

2dtS22

2V

S2+ ...

Thus we have simply replaced (dX)2 in the equation by dt. The above justification ofthis has been rather informal. For more details see Paul Wilmott Introduces QuantitativeFinance pages 119 to 137.

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3.3 Black-Scholes equation: derivation (part 2)

So we can simply replace (dX)2 by dt, as demonstrated in the previous section.Therefore, by collecting together the dt terms, we get

d = SdX

VS

+

+ dt

S VS

+ Vt

+ S + 12

S222

VS2

+ ...

So far we have not chosen . If we choose

= VS

,

then

d = dt

V

t+

1

2S22

2V

S2

+ ...

Remember that the neglected terms, denoted by +..., are of size (dt)3/2 or smaller.Dividing through by dt gives

d

dt=

V

t+

1

2S22

2V

S2+ ...

Therefore, the neglected terms, denoted by +..., are now of size (dt)1/2 or smaller.Taking the limit dt 0, we see that all the extra terms, denoted by +... in the

previous equation, all tend to zero, since the largest of them are proportional to (dt)1/2,

and (dt)1/2

0 as dt

0. Hence

d

dt=

V

t+

1

2S22

2V

S2.

This is remarkable, since we have eliminated all the random dXs from the above equa-tion. Thus if we know the right-hand side at any time step, then varies in a non-randomway according to the above equation over that time step.

Before we proceed further, lets summarise what we have shown so far. We have shownthat a portfolio = V + S has a non-random evolution over a time-step dt given by

ddt

= 12

2S22VS2

+ Vt

where the value of the option is V (S, t), S is the value of the underlying and t is time.We also take E to be the strike price, to be the volatility, r to be the interest rate, andT to be the expiry time. The above equation requires us to choose

= VS

.

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Note that this is often instead re-written as = V S with

=V

S,

so that

= .Either way, this is commonly called a Delta Hedging strategy.

Now if the money is instead invested in a bank then it would increase by

d

dt= r

over the same time period, according to our interest rate equation. If these two equationsfor d/dt are not equal to each other, then it would be possible to make a risk free profit

i.e. it would be possible to borrow from the bank and invest in the portfolio, or sell theportfolio and put the money in the bank (depending on which expression for d/dt is thegreater), making a risk free profit which is not subject to any random effects. This is notconsidered possible, since the stock market would automatically adjust the value of theunderlying S by increasing or decreasing its value to remove this possibility of risk freeinvestment. This is called the principle of no arbitrage. This says that there is no suchthing as a risk free profit; or that the stock exchange acts by continuously repositioningthe price of shares S to remove any risk free profit. There are traders on the StockExchange who look for arbitrage opportunities, where risk free profits can be made, butthese opportunities close up very quickly if they ever form. Prices on the stock marketeffectively move continuously to prevent arbitrage (i.e. preventing risk free profit).

Therefore, by the principle of no arbitrage, these two expressions for d/dt must beequal to each other, and hence we have

d

dt=

1

22S2

2V

S2+

V

t= r.

But = V S. Therefore1

22S2

2V

S2+

V

t= r (V S) ,

where

= VS

.

Rearranging this gives the Black-Scholes equation

1

22S2

2V

S2+

V

t rV + rSV

S= 0.

This is a partial differential equation for the value of the option V (S, t).

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3.4 Black Scholes equation: brief discussion

Note that even though we have considered a Delta Hedging strategy (i.e. = V Swhere = V/S) in the derivation of the Black-Scholes equation, the Black-Scholesequation is independent of . Therefore it is a valid equation for V even if the portfolio

you own is not this Delta Hedging portfolio e.g. this equation for V is valid even if youjust own one option of value V and do not own any shares in the underlying; or, in fact,if you own any other portfolio containing V. This equation will always describe V(S, t).

The Black-Scholes equation can be solved to find the value of the option V at timet when the value of the underlying S is known. The value of S varies randomly though,and V is dependent on S. But since we always know the value of the underlying S at thecurrent time, this equation tells us the value of the option V at that time. The equationalso tells us how future values of V will vary with the share price S of the underlying.

As a bi-product of the previous derivation of the Black-Scholes equation, we have alsoshown that the Delta-Hedging portfolio = V S is a risk free investment, with valuevarying according to

d

dt= r.

For this we must repeatedly change so that it equals V/S, by buying or selling sharesin the underlying. Note that needs to be a constant over each time-step (since wasassumed constant over a time step in the derivation and = ). Therefore, you wouldtend to update the value of on a regular basis e.g. on a daily basis. This is calleddynamic hedging. Also, note that is called the Delta of the portfolio.

Also note that the Black-Scholes equation is independent of the drift , and also in-dependent of the random variable dX.

3.5 Solution of the Black-Scholes equation for European options

The Black-Scholes equation can be simplified. Substitute

V = Eu (x, )exp

(k 1) x

2 (k + 1)

2

4

(where S = Eex, t = T 2 /2 and k = 2r/2) into the Black-Scholes equation. Thiscan then be rearranged to show that u (x, ) satisfies the diffusion equation i.e.

u

=

2u

x2.

(Try this: Problem sheet 4. Question 1. Warning: very lengthy calculation!)Note that the share price 0 S < . Therefore < x < since S = Eex. Also,

the Black-Scholes equation is valid for t T, since we only want to solve the equation for

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the times before the expiry date (t < T) and at the expiry date (t = T). Therefore wewish to solve the diffusion equation for 0 since t = T 2 /2.

The diffusion equation has the solution

u =

1

2

u0 (s)exp

1

4 (x s)2

ds

where u = u0(x) at = 0. (Show this: Problem sheet 4. Question 3.) Therefore the Black-Scholes equation can be solved by just doing an integral! (But only for European options:later in this module we will look at American options which are more complicated.)Usually this integral must be solved numerically (e.g. trapezium rule or Simpsons rule).

The function u = u0(s) is found from the pay-off function for the European option.(See Problem sheet 4. Question 2.) This can be done as follows:

If the pay-off function for an option is g(S) where S is the value of the underlying,then V = g(S) when t = T since T is the time of expiry (the expiry date) i.e. the pay-offfunction is just the value of the option at the expiry time. But when t = T then = 0since t = T 2 /2. Therefore the pay-off function g(S) satisfies the equation

V(S, T) = g(S) = Eu(x, 0) exp

1

2(k 1)x

since

V(S, t) = Eu (x, )exp

(k 1) x

2 (k + 1)

2

4

(i.e. we have put = 0 and t = T into the above formula).

But u(x, 0) = u0(x) since u = u0(x) at = 0. Therefore,

g(S) = Eu0(x)exp

1

2(k 1)x

.

Therefore, rearranging this expression gives

u0(x) =g(S = Eex)

Eexp

1

2(k 1)x

where g(S) is the pay-off function, and since S = Eex. Therefore

u0(s) = g(S = Ees

)E

exp

12

(k 1)s .Example.For a European vanilla call g(S) = max(S E, 0) since this is the pay-off function.

Therefore for a European vanilla call

u0(x) =max(Eex E, 0)

Eexp

1

2(k 1)x

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since S = Eex. Therefore, replacing x with s gives

u0(s) =max(Ees E, 0)

Eexp

1

2(k 1)s

.

Simplifying gives

u0(s) = max(es 1, 0) exp

1

2(k 1)s

,

or equivalently

u0(s) = max

exp

1

2(k + 1)s

exp

1

2(k 1)s

, 0

by multiplying through by the exponential.

Therefore the value of a European vanilla call can be found from calculating

u =1

2

max

exp

1

2(k + 1)s

exp

1

2(k 1)s

, 0

exp

1

4(x s)2

ds

(which can be solved numerically), and then substituting that result for u into

V = Eu (x, )exp

(k 1) x

2 (k + 1)

2

4

gives the value of the option V.Alternately the above integral can be rearranged into the form

V = SN(d1) Eexp(r(T t))N(d2) (see problem sheet 4, question 3) where

N(q) =12

q

exp

1

2s2

ds,

d1 =log(S/E) + (r + 2/2)(T t)

T t

and

d2 =log(S/E) + (r 2/2)(T t)

T t .

Here log is the natural logarithm, or equivalently ln.The above expression in terms of N(d1) and N(d2) is a common way of rewriting the

integral.

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4 Solving The Black-Scholes Equation

The Black-Scholes equation can be used to model the value of a portfolio of the form

(t) = V(S, t) + (t)S(t). (12)

Here, V(S, t) refers to the value of any financial product that depends only on the currentvalue of some underlying stock S, where it is assumed that S follows a lognormal walk ofthe form described in section 3. The Black-Scholes Equation is given by

1

22S2

2V

S2+

V

t rV + rSV

S= 0. (13)

This is a partial differential equation for the value of the option V (S, t). It has beenderived under the assumption that there are no commission costs and that a portfolioat time t will have the same expected rate of return as investing your money in a bank

with interest rate r at time t if you instantaneously choose the value of = VSt. The

correct choice of to ensure that a portfolio of the form equation (12) is risk-free is knownas delta-hedging. Equation (13) is valid for modelling the current value of V(S, t) evenif delta-hedging is not used: if this were not true prudent investors could make a risk freeprofit by investing/trading in V. Market forces will ensure that equation (13) representsthe current value of V(S, t).

Most financial products are sold via an intermediary. If we assume that the value of afinancial product V is correctly identified via the Black Scholes equations, the intermediarywill purchase from the seller at a price of V Cseller and then sell to the ultimate buyerfor a price of V + Cbuyer. The intermediary makes a small (or large) profit from both

the buyer and the seller. The commission need not be the same for either the buyeror the seller, but irregardless, the difference between what you pay/receive to buy/sellV, and the solution to equation (13) can be thought of as commission. Equation (13)is appropriate since there are large financial institutions that can buy and/or sell largequantities of V and S for

It must be stressed that equation (13) is strictly valid if

(a) the interest rate r is assumed to be constant,

(b) the underlying stock S does not pay out dividends, and

(c) the volatility is assumed to be constant and known.

Equation (13) can be modified to account for variations of r as a function of time and toaccount for the payments of dividends (as will be shown later in this module). Equation(13) can also be modified to account for variations of the volatility with time (as willalso be shown later in this module). Of assumptions (a)-(c) above, it is assumption (c)which is perhaps the most most challenging. The volatility can be estimated today usinghistorical data via equation (9) from Section 3. Different investors will choose to estimate using different values of n and dt and these differences will effect the prediction of .

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Thus different investors may agree that equation (13) is the appropriate mathematicalmodel for V but they may disagree on the actual value of V if they cannot agree on theappropriate value of . As already noted, the drift coefficient does not appear in theBlack-Scholes equation (13). Correctly estimating the current and future value of isthus incredibly important.

4.1 Boundary/final conditions

Equation (13) is a second order partial differential equation which can be used to model Vas a function ofS and t. This equation involves a second derivative with respect to S anda first derivative with respect to t. Without going into the full details of the propertiesof partial differential equations we must provide two independent boundary conditions(specifying either V or V

dSeither as constants or a function of t at the boundary points

a = a(t) and b = b(t), and one condition of the form V(S, T) = V(S) for some specified (orto be determined) value ofT. The Black-Scholes equations are used to give the value of a

financial product at the current time t given some known information at some future timeT. The Black-Scholes equations are then integrated backwards in time from the specifiedfinal conditions. For European vanilla calls and puts, the exact solution to the Black-Scholes equation is known as previously discussed. For more complicated options, theBlack-Scholes equations will have to be solved numerically, and the numerical solutionwill depend vitally on the appropriate boundary/final conditions. When developing anumerical method it is clear that the semi-analytical solutions for European calls andputs should be considered as appropriate test cases.

4.2 Boundary and final conditions for European vanilla call andput options

We have seen that at expiry (i.e. at t = T), the value of the option V equals the pay-offfunction.

In this section we derive boundary conditions on the European vanilla call and Euro-pean vanilla put which will be useful later on in the module.

European vanilla call

From now on, we write the value of the European vanilla call as C(S, t) instead ofV(S, t). Therefore, C(S, t) satisfies the Black-Scholes equation, namely

1

22S2

2C

S2+

C

t rC + rSC

S= 0.

Also, at expiry, at t = T, C(S, t) equals the pay-off function. Therefore,

C(S, T) = max(S E, 0).

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We also need to work out boundary conditions at S = 0 and also for S .We note that when the underlying share becomes worthless (i.e. S = 0) then the

stochastic differential equation dS = S(dX + dt) implies that dS = 0. If dS = 0 thenS = constant. Hence once S = 0 then S becomes equal to a constant for all subsequenttimes. Hence once S = 0 then Sremains equal to zero for all subsequent times. Therefore,once S = 0 then the underlying remains at zero for all subsequent times and hence is equalto zero at expiry, and so the pay-off equals zero. The call option is worthless since it givesthe holder of the option the right to buy a worthless share for the strike price E, whichclearly would not be exercised. Therefore the call option has zero value. In other words,

C(0, t) = 0.

As the price of the underlying tends to infinity (i.e. as S ), then the Europeancall option will certainly be exercised. This of course gives the right for the holder tobuy the underlying for the strike price E when the underlying is actually worth S .Therefore the price of the option is just equal to the price of the underlying since theoption will certainly be exercised. Therefore,

C S

as S .

European vanilla put

From now on, we write the value of the European vanilla put as P(S, t) instead ofV(S, t). Therefore, P(S, t) satisfies the Black-Scholes equation, namely

1

22S2

2P

S2+

P

t rP + rSP

S= 0.

Also, at expiry, at t = T, P(S, t) equals the pay-off function. Therefore,

P(S, T) = max(E S, 0).

We also need to work out boundary conditions at S = 0 and also for S .We note that when the underlying share becomes worthless (i.e. S = 0) then the

stochastic differential equation dS = S(dX + dt) implies that dS = 0. If dS = 0 then

S = constant. Hence once S = 0 then S becomes equal to a constant for all subsequenttimes. Hence once S = 0 then Sremains equal to zero for all subsequent times. Therefore,once S = 0 then the underlying remains at zero for all subsequent times and hence is equalto zero at expiry, and so the pay-off equals E (since the pay-off equals max(E S, 0)).The put option gives the holder of the option the right to sell a worthless share for thestrike price E, which clearly would be exercised since this is an excellent deal for theholder of the option, making an immediate profit of E. Therefore, if ever S = 0 then weknow that the pay-off for the put is certainly going to be equal to E at time T. To find

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the value ofP for times t < T for S = 0 then we need to solve the Black-Scholes equationfor S = 0 subject to this condition at t = T. Substituting S = 0 into the Black-Scholesequation gives

P

t rP = 0.This is just the same equation that we solved for money growing in a bank with constantinterest rates. It has the general solution

P = Aert,

where A is a constant of integration. But P = E when t = T, and so

E = AerT.

Solving this gives

A = EerT.

Therefore, when S = 0,

P = Eer(tT).

Or equivalently,

P(0, t) = Eer(Tt),

as it is often written. This is the boundary condition on P at S = 0 for all t T.Finally, as S , the option will certainly not be exercised (why sell a share that isworth for price E?), and so obviously,

P(S, t) 0

as S .It is important that you know these boundary conditions for C and P at t = T, at

S = 0 and at S = .

4.3 Put-call parity for European options

Consider the portfolio V = S+ P C where S is the underlying, P is a European vanillaput and C is a European vanilla call. (This is often called long one asset, long one putand short one call - the long refers to a +, the short refers to a -.)

At expiry, the pay-off equals S+ max(E S, 0) max(S E, 0). If S E then thispay-off function equals S+ 0 (S E) = E. If S E then this pay-off function equalsS+ (E S) 0 = E. Therefore, the pay-off equals E for this portfolio for all values of S.

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Therefore the value of this portfolio is independent of S. Let the value of this portfolioequal V. Then V = V(t) since its value has been shown to be independent of S.

Substitute V(t) into the Black-Scholes equation. This gives

dV

dt rV = 0.Using separation of variables this has the solution

V = Aert,

where A is a constant. But when t = T (at expiry), V = E (from the above argumentthat the pay-off always equals E). Therefore substituting this into the solution of thedifferential equation gives E = AerT. Therefore A = EerT. Therefore the value of thisportfolio is

V = Eer(t

T).

Therefore

S+ P C = Eer(tT). (14)

Therefore this combination of a share, put and (short) a call produces a guaranteed risk-free investment (i.e. hedging). This formula is called put-call parity.

Compare this form of hedging, with a guaranteed risk free investment, to the deltahedging strategy discussed earlier. Note in both cases the overall value of the portfoliogrows according to the differential equation d/dt = r, or equivalently dV/dt = rV

(since we have used different notation for the total value of the portfolio in each case i.e. or V). With delta hedging the number of shares in the underlying must continuallybe updated to ensure the portfolio is risk free. Using put-call parity the portfolio istheoretically risk free from the instant it is purchased.

It must be stressed that put-call parity and delta hedging are not practical investmentstrategies: if you want a guaranteed return equivalent to the rate of interest (withouthaving to pay commission charges) then you should put your money in the bank. Thesestrategies may however become practical if you think you can predict future changes in rand more accurately than other investors.

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4.4 Dividends

If a company makes an overall profit it can choose to invest some/all of this money for thefuture (perhaps by building a new factory so that the company can expand, or by puttingthe money in the bank or some other type of investment, to cover potential losses in the

future) or it can pay some/all of this money to its shareholders in the form of dividends.Some investors purchase shares in a company in the hope of making a steady income inthe form of dividends. Indeed if a company is known to regularly pay out dividends, thenit is likely that more people will want to purchases shares in that company, and by thelaws of supply and demand, this should in the long term boost the value of the share price.When a company pays out dividends, this will have an immediate effect on the underlyingshare price. Let S(t) represent the value of the underlying share price and suppose thecompany pays a dividend D at time t = td. Then by using arbitrage considerations

S(t+d ) = S(

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