MATH39032 Mathematical Modelling of Finance

Lecture notes prepared by Professor Peter Duck

LECTURE TIMES: Tuesday 3pm Rutherford TheatreWednesday 9 am Rutherford Theatre

CONTACT DETAILS: Office - 2.141 Turing BuildingE-mail - sergei.fedotov@manchester.ac.uk

EXAMPLES SHEETS:

There will be 8 examples sheets to accompany the course. The firstfeedback/examples class will be in Week 2. I would strongly encourage youto attend these classes - they will be a useful forum for feedback. I would alsoencourage you to attempt the questions on the sheets prior to the classes.

ASSESSMENT:

MATH 39032

This module is entirely assessed by a final examination (2 hours). Thepast three years exam papers will provide you with a good overview of thestandard of questions and also the topics which will be covered.

1

RECOMMENDED TEXTS:

Text books:

• The best book for this course is still:

Wilmott, P., Howison, S., Dewynne, J., 1995: The Mathe-matics of Financial Derivatives, Cambridge U.P. ISBN: 0521497892

• Alternatively, as an introductory text to the area:

Wilmott, P., 2001: Paul Wilmott Introduces Quantitative Finance,2nd Edition, Wiley. ISBN: 0471498629.

• For a very detailed (and expensive) look at mathematical finance:

Wilmott, P., 2000: Paul Wilmott on Quantitative Finance, Wiley.ISBN: 0471874388

• There are more probabilistic ways of approaching the area (as consid-ered in other modules) and for those seeking to obtain a full knowledgeof the area, including more on stochastic processes and Martingale the-ory, these courses are highly recommended. Some introductory booksfor stochastic calculus as applied to finance are:

Etheridge, A., 2002: A Course in Financial Calculus, Cambridge U.P.ISBN: 0521890772

Neftci, S. N., 2000: An Introduction to the Mathematics of FinancialDerivatives, 2nd Ed., Academic Press. ISBN: 0125153929

• For a more financial look at options and derivatives the following isexcellent and is the course text for finance students (usually MBA orPhD) studying derivatives:

Hull, J. C., 2002: Options, Futures and other Derivatives, 5th edition,Prentice Hall. ISBN: 0130465925.

• For a readable book on Stochastic Finance:

Higham, D.J. 2004: An introduction to financial option valuation.Cambridge University Press. ISBN 0521 54757 1 for paperback andISBN 0521 83884 3 for hardback.

General interest books:

• One description (not the best, as the best one is, sadly, out of print)of how the Nobel prize winning academics (whose work underpins thiscourse) tried to make money from their theories is:

Lowenstein, F., 2002: When Genius Failed: The Rise and Fall of LongTerm Capital Management, Fourth Estate. ISBN: 1841155047.

2

• For a very readable discussion about investment banks using and abus-ing derivatives:

Partnoy, F., 1998: F.I.A.S.C.O: Guns, Booze and Bloodlust: the TruthAbout High Finance, Profile Books. ISBN: 1861970773.

Partnoy, F., 2004: Infectious Greed, Profile books. ISBN: 1861974736.

• The original description of what it’s really like working and makingmoney on Wall Street was the following:

Lewis, M., 1999: Liar’s Poker, Coronet. ISBN: 0340767006.

• For those of you who are interested in the history of modern financetheory and the major players, I thoroughly recommend:

Bernstein, P., 1995: Capital Ideas: The Improbable Origins of WallStreet, The Free Press. ISBN: 0029030129.

Finally, Peter Bernstein has also written an excellent book on risk andits origins:

Bernstein, P., 1998, Against the Gods: The Remarkable Story of Risk,Wiley. ISBN: 0471295639.

3

1 Introduction

1.1 Introduction

Mathematics has myriad applications in the world of finance and assuch the title of this module may be a little broader than what is actuallystudied. The main focus of this course is on the financial instruments knownas options and, most importantly, how to calculate their value.

This proved to be a remarkably interesting problem both mathematicallyand financially and one which took centuries to satisfactorily solve. Theearliest known use of options was by the Greek philosopher Thales in 600B.C. who used them to make money from his predictions about the harvest,in this example the price of olives is the underlying asset. Since thenoptions have been traded the world over although rarely in a regulatedmanner and, until 1973, their values were primarily calculated by guesswork.

In 1973, Fischer Black and Myron Scholes, together with help from BobMerton derived the Black-Scholes partial differential equation whichdescribes the value of an option, V (which is dependent on the time since theoption had been sold, t, the value of the underlying asset, S, the interest rate,r, and the volatility of the underlying asset, σ) together with an appropriateset of boundary conditions, as follows

∂V

∂t+

1

2σ2S2∂

2V

∂S2+ rS

∂V

∂S− rV = 0. (1)

This equation changed the face of option pricing, not only did it earnNobel prizes for Merton and Scholes in 1997 (Black having died in 1995) butit paved the way for an explosion in the trading of options and other deriva-tive products (an option is a type of financial derivative and you’ll see moreabout these shortly). The first organised options exchange also opened inChicago in 1973 and the volume of trade in options has increased from 5.7mcontracts in 1974, to 673m in 2000, 3,899,068,670 in 2010, 4,111,275,659 in2013, 4,265,368,807 in 2014. This boom in the, occasionally mathematicallycomplex, derivatives markets has also led to many investment banks activelyrecruiting skilled mathematicians and physicists to help value such products.

1.2 Terminology

Derivatives

In a financial sense a derivative is a product whose value is derived fromthe price or value of another product. This is normally an underlying asset,such as a stock or share (Marks and Spencer shares, Parmalat shares etc.),a commodity (oil, gold, tin etc.), an exchange rate (Euro to Sterling etc.).The most common types of derivative products are forwards, futures andoptions.

Underlying assets

Throughout this course we will be considering options on underlyingassets, the value of which is denoted by S in the Black-Scholes equation

4

(equation (1) above). This underlying asset is usually assumed to be a shareprice, but can also be a commodity price or an exchange rate. Underlyingassets have an associated drift (µ) and volatility (σ), where the drift is theexpected percentage increase over a certain period of time and the volatilityis the measure of uncertainty of this return. For example, one would expecta fledgling technology share to have a higher volatility than a blue-chipcompany like AT&T.

Interest rates and the time value of money

The famous mantra from courses on economics and finance is ‘a dollartoday is worth more than a dollar tomorrow’, as it is possible to investyour dollar in a risk-free investment, like a US government bond, today andtomorrow it will be worth more than a dollar. There are a few possibleconventions as to how much money is worth after a certain amount of time.Assume a time scale of 1 year, given a yearly interest rate, r, then if A isinvested today, at the end of the year it will be worth A(1 + r). However, ifit is invested for only 6 months at the same quoted yearly rate and the newtotal is then invested for another 6 months we have a compounding processsuch that after 1 year A will be worth A(1 + r

2 )2. This can be extended to

the continuously compounded case, which is used throughout this course, inwhich the money is reinvested m times giving A(1+ r

m )m after one year. Asm→ ∞ then

(1 +r

m)m → er. (2)

Thus, an amount, A, invested at a continually compounded rate of r for tyears is worth Aert. This convention is used primarily because it makes themathematics far simpler than using the cumbersome discretely compoundedformula. This time value of money is mainly used in its inverted form todetermine what an expected amount in the future is worth today. Thisis known as discounting. For example, if an investor is going to receive$100 at some time in the future T then at an earlier time t it is worth$100e−r(T−t). For more realistic models, where the interest rate can be afunction of time, an amount A invested for t years is worth

Ae∫ t

0r(t)dt.

Interest rates and discounting is used extensively in developing option pric-ing techniques.

1.3 Forwards and Futures

Although it is always possible to buy a share or commodity today or toexchange currency at a particular rate, investors or companies often want toarrange a deal for some time in the future. A forward contract is one inwhich one party (in the long position) agrees to buy an underlying assetat a certain price (the delivery price F ) at a certain future time, T . Theother party (in the short position) agrees to sell the asset at time T atthis price F .

5

S

PAYOFF

F T

Figure 1: Payoff from the long position in a forward contract

A futures contract is a standardised forward contract in that partiescan enter into long or short positions on an exchange where the deliveryprices and dates are set by the exchange. In a futures contract the opposingparties (long and short) do not necessarily know each other and so theexchange ensures that the contracts are honoured.

Why would anyone want to use such a derivative?

Example 1.1 Suppose Ryanair know that on August 5th 2008 they willhave to pay an American supplier $1m and they want to hedge againstunfavourable exchange rate movements. On February 5th 2008 the bank isoffering a six month forward exchange rate of 0.8. Ryanair then take thelong position in the forward contract and on August 5th will buy $1m for800,000 Euro. The bank has taken the short position and has agreed to sell$1m for 800,000 Euro. Note that neither party pays anything to enterinto the contract.

Obviously in this example if in 6 months the dollar has strengthenedagainst the euro (i.e. the exchange rate is higher than 0.8) then Ryanair arepleased because they have locked in a lower exchange rate. Obviously if thereverse has happened (the exchange rate has dropped) then they will loseout.

In general, if the underlying asset has value St at time t then the payoffat the delivery time, T , to the party in the long position is

ST − F

where F is the delivery price. Similarly, for the party in the short positionthe payoff is

F − ST

It is simple to depict these payoffs graphically in figures 1 and 2.The key thing here is that we have not yet determined what would be

a suitable choice of value for F ; this is discussed shortly after the definitionof an option.

6

S

PAYOFF

F T

Figure 2: Payoff from the short position in a forward contract

1.4 Options

In Example 1.1 if the dollar weakens against the euro, then Ryanair loseout in the above forward contract. Ideally, they would like it so that if theexchange rate drops then they can walk away from the contract and buytheir $1m for less than 800,000 Euro. This is exactly the freedom whichan option would give them, the would have the option of whether to takethe delivery price or to walk away and take the favourable current price.Obviously, if the exchange rate has increased then they can still pay 800,000Euro for their $1m. Crucially, unlike in forward contracts the party whobuys the option must pay some premium to obtain the option. The mainthrust of this course is to determine how much she should pay.

There are two principal types of options:

Definition (Call options) A call option gives the holder the right, but notthe obligation to buy the underlying, S, at (or before) a certain date, T , fora certain price, known as the exercise (or strike) price, X.Definition (Put options) A put option gives the holder the right, but notthe obligation to sell the underlying, S, at (or before) a certain date, T , fora certain price, known as the exercise (or strike) price, X.

There are also two main genres of options:

Definition (European options) A European option can can only be ex-ercised at the expiration date T .Definition (American options) An American option can be exercised at

any time up to and including the expiry date, T .

There are also many exotic types of options such as Asian, Russian,Parisian, Bermudan, Lookback, Barrier etc. which have different exerciseconditions and are not considered fully in this course. This course mainlydeals with the valuation of European options.

7

1.4.1 European Call Options

Denote the value of a European call option by C(S, t) where S is thevalue of the underlying asset at time t. If the strike price of the option isX then at the expiry of the option, t = T , the holder of the option has theright, but not the obligation, to buy the underlying, of value S at t = T atthis price, X. Clearly if S > X then the holder of the option would exercisethe option and buy the underlying (worth S) for X. This would yield theholder of the option a profit of S −X. If S ≤ X then there is no point inexercising the option as the holder can buy the underlying on the marketfor less than X.

Hence at expiry (t = T ) the value of the call option is

C(S, T ) = max(S −X, 0).

1.4.2 European Put Options

In a similar way to call options, denote the value of a put option byP (S, t). Again the option has a strike price of X and at expiry the holderof the option has the right, but not the obligation, to sell the underlyingasset at this price. With a put option at expiry (t = T ) S < X then theholder of the option would exercise as she can sell the underlying for morethan she could on the market, and the option would then be worth X − S.If, however, S > X the the holder of the options could sell the underlyingfor more than X and thus it would not be worth exercising the option.

Hence, at t = T the value of a put option is

P (S, T ) = max(X − S, 0)

8

S95

PROFIT

−10

105

Figure 3: Profit/loss from purchasing one call option as in Example 1.2

Example 1.2 (Option profit) An investor buys a European Call option tobuy 100 Hewlett Packard shares with a strike price, X, of $95. The currentstock price is $100, the expiration date is 6 months and the cost of the calloption to buy one option is $10. Plot a graph of profit from buying theoption against underlying asset value in 6 months time.

If, at expiry, S < 95 the investor would choose not to exercise the optionas there is no point in buying the share for $95 when you can buy it on themarket for less. In this case the investor will have lost the cost of the options($1000). If, at t = T , S > 95 then the options will be exercised, yieldinga profit of (S − 95) × 100 − 1000. The diagram below shows the profit atexpiry for different levels of the underlying S. Note that if 95 < S < 105the investor has exercised the options but has still made a loss overall. Seefigure 3 for the profit from one option.

See Examples 1 for drawing profit diagrams.

1.5 Why options?

Why is the options market such a big deal. Options appeal to three maintypes of investors - hedgers, speculators and arbitrageurs.

1.5.1 Options for hedging

This was how we introduced the idea of forward and option contracts.If a company or investor requires a certain amount of goods or currencyin a certain amount of time then options provide insurance for cases wherethere are adverse market moves. It is sometimes possible to hedge againstmovements in a market which will affect your business. As an example, ifjet fuel goes up then it costs BA more money to run their aircraft, so if theybuy call options in jet fuel then if the price goes up then they make money tooffset their operating losses. If the price has gone down then they’re happybecause their operating costs are low.

9

1.5.2 Options for speculating

If an investor has a hunch about which way a market is moving then hecan obtain more leverage by using options. Consider the following example:an investor feels that Barclays is likely to increase in value over the nextthree months and has $5000 to invest. The current stock price is $20 andcall options with a strike price of $25 are available for three months at thecost of $1. Consider two possible alternatives, the stock price goes up to $35or down to $15.

Table 1.1: profit and loss from the two dif-ferent strategies when speculating on Barclaysstock

Stock price at expiryStrategy $15 $35

Buy shares -$1250 $3750Buy call options -$5000 $45000

Investing the $5000 in shares, enables the investor to buy 250 sharesat $20, so if the price drops to $15 then the shares are now worth $3750,realising a loss of $1250, and if it increases to $35 then they’re worth $8750,a profit of $3750.

However, the options provide far more leverage in that a small increaseor decrease in the underlying can realise big profits or losses. In this casethe $5000 buys 5000 call options with a strike price of $25. If the price goesdown to $15 then none of these would be exercised and the investor wouldhave lost their entire $5000. If, on the other hand, the price has gone upto $35 then each of the options enables the investor to make a $10 profit,meaning they would make 5000 × 10 = $50000 less the initial outlay of$5000 which is a massive profit of $45000.

1.5.3 Options for arbitrage

The principle of arbitrage is an important one in option pricing theoryand will be defined and expanded more fully later. However, an arbitrageopportunity is one in which it is possible to lock in a risk free profit. Anarbitrageur will look for anomalies in the market, which by definition, existfor only short periods of time and lock in these profits. A good example is inspread betting, if one book has the spread 8-10 and another 12-14 then youcan buy at 10 in the first and sell at 12 in the second and make a risk-freeprofit.

1.6 No arbitrage principle

Definition (Arbitrage opportunity) An arbitrage opportunity is one inwhich it is possible to make an instantaneous, risk free profit.

10

1.6.1 Introduction

Although we introduced the idea of making money from arbitrage op-portunities one of the principles required for most of the option pricingmethodology of this course is that arbitrage opportunities do not exist, oronly appear for a very short time. Also associated with the concept of noinstantaneous risk-free profit is the idea of a risk-free rate. This is the rateof return (or interest rate) that an investor receives upon making a risk-freeinvestment, such as investing in US treasury bonds.

1.6.2 Determining forward prices

To determine the correct delivery or forward price on a forward contractit is necessary to invoke the ideas of no arbitrage and the risk-free rate.Consider a forward contract to purchase IBM stock, which pays no divi-dends, in three months time. Suppose that the current share price is $40and the current risk-free rate is 5%, also assume that the current deliveryprice is $43. Now if an arbitrageur is sharp then she can spot an arbitrageopportunity here, she can borrow the $40 to buy a share today and go shortin the forward contract. In three months time she will sell the share for $43.She can use this to pay off the loan which will have increased to just

40e0.05×3/12 = 40.5.

Hence, whatever happens she will have made a profit of

43− 40.5 = 2.5.

NB. This collection of one or more products (in this case short on a forwardcontract and owning a share) is known as a portfolio.

The principle of no arbitrage says that as soon as this opportunity arisesthen people like our investor here will rush in to go short on forwards.However, very few people will be willing to go long on forwards with sucha high delivery price, so very quickly the price will drop to a fair level andarbitrage opportunities will vanish.

Finding this ‘fair price’ is simple for forward contracts. Let the currentunderlying asset price be S, the risk-free rate be r, the time to expiry beT and the delivery price F . We have shown that there are arbitrage op-portunities if F > SerT and similarly (from Examples 1) it is possible toshow that there are arbitrage opportunities if F < SerT . Hence the correctdelivery price on a forward contract is

F = SerT

Note that this derivation does not assume or predict anything about themovement of the underlying asset S but is able to predict a correct valuefor the forward price. Unfortunately for options it is not this simple.

11

1.6.3 The put-call parity

There are a few relationships between option prices which we can de-termine from basic no arbitrage arguments. One of the most useful is theput-call parity which will eventually enable you to calculate the value of aEuropean put using the value of the call. Consider, two portfolios, A and Bwhich at t = 0 consist of the following:

• Portfolio A: A European call option, C(S, t), with exercise price Xand expiry date T ; and an amount of cash Xe−rT .

• Portfolio B: A European put option, P (S, t), with exercise price Xand expiry date T ; and one share in the underlying S.

At expiry, t = T , the portfolios both have value max(S,X) (for example inportfolio A at expiry Π = max(S −X, 0) +X = max(S,X)) and so as it isimpossible to exercise early they must have the same value throughout thelifetime of the option. Hence we have the following relationship

C(S, t) +Xe−r(T−t) = P (S, t) + S. (3)

where t is the current time. It is possible to determine some fairly loosebounds for European options using a similar approach (see examples sheet2) but for accurate valuation it is necessary to model the movements of theunderlying asset.

2 Model of stock price movements

In order to value more complex products than forward and futures con-tracts we will have to use stochastic processes in an attempt to accuratelymirror the real life movements of underlying asset prices. Fortunately, al-though dealing with stochastic variables it is often possible to transform theproblem into a deterministic one. In this case it is achieved through the em-ployment of Ito’s lemma, which can be seen to be the analogue of Taylor’stheorem for stochastic calculus.

2.1 Efficient markets and Markovian processes

Most of modern option pricing theory is based on the efficient mar-ket hypothesis. The hypothesis states that all the data available about aparticular company or commodity is reflected in the current price. So itis impossible to gain an edge by having studied the historical data or byexamining in intricate detail company reports or the financial press. Thismeans that any movements in underlying price will be unpredictable (i.e.random and without memory).

Obviously to come to such a conclusion empirical experiments will haveto be made to check that increments in stock prices are random. Unsurpris-ingly this is the case although there is a lot of academic dispute about thetrue efficiency of markets. In any case, for this course we will assume thatthey are efficient.

12

The idea of a process randomly evolving without memory fits very nicelyinto the theory of basic random processes. Markov processes are processeswhich have no memory, in that whatever movement or information has oc-curred before a certain time in the process, has no impact on where its nextmovement will be.

Example 2.1 One of the simplest random processes, which also happensto be Markovian, is the simple symmetric discrete random walk. Consider astochastic process, St starting from S0 = 0, at each point in time the changein position of S call it ∆Si is given by

∆Si =

{

+1 with probability 12

−1 with probability 12

After n steps in this walk then the position of S is given by

Sn∆t =

n∑

i=1

∆Si

This is Markov because each movement is independent of what has occurredbefore. Also, it so happens that, in this case, by the Central Limit TheoremSn∆t is Normally distributed with a mean of 0 and a variance of n∆t (thusa standard deviation of

√n∆t). In general if the variance of each of the

increments ∆Si is σ2 then the variance of Sn∆t is σ

2n∆t.

13

2.2 Brownian motion and the model of stock price movement

This discrete random walk is one possible way of modelling stock pricemovements. It is, however, very simplistic and only works for discrete time.The latter of these problems can be easily overcome by using its continu-ous time analogue: Brownian motion, or as it’s sometimes referred to, theWiener process.

The stochastic model of Brownian motion was, obviously, defined tomirror the movement of tiny particles in water but has applications in morefields than that, one being in option pricing theory. There is a lot of rig-orous mathematics surrounding such processes but as regards this course aheuristic overview will be provided.

Definition (Brownian motion) A real valued stochastic process Wt is aBrownian motion (or Wiener process) under a probability measure P if

1. For each t ≥ 0 and s > 0 the random variableWt+s−Wt (often termeddW ) is distributed Normally with mean 0 and variance s.

2. For each n and for any times 0 ≤ t0 ≤ t1 ≤ · · · ≤ tn the randomvariables {Wtr −Wtr−1

} are independent.

3. W0 = 0 (this is merely a convention, it can start from any point).

4. Wt is continuous in t ≥ 0.

This is basically just an extension of the discrete simple random walk tocontinuous time. The change Wt+s −Wt over a very small period of timedt is often denoted by dW and obviously is distributed accordingly(meanof zero and variance of dt). Brownian motions obviously have very strangepaths and, in fact, the expected length of path followed by W in a any timeinterval is infinite, this will make calculus difficult on Brownian motions (seesection 3)

One way of understanding dW is to see it as ǫ√dt where ǫ is distributed

normally with a mean of 0 and variance of 1. The standard Brownian motionWt will not model stock prices very well for two main reasons:

• The general trend of stock prices is upwards whereas the expectedmovement of Brownian motion is to stay at the same level.

• Stock prices cannot drop below 0 whereas Brownian motion can takeany real value.

2.2.1 Generalised Brownian motion

The first of these concerns can be over come easily. On top of the randomincrements generating by the Brownian motion term it is possible to addin deterministic terms. When dealing with stock prices there is a generalupward drift, call this µ and so if the stock price is denoted by St then wehave the following stochastic differential equation

St+dt − St = dS = µdt+ σdW (4)

14

where σ is just scaling the effect of Brownian motion. So, in this case over aperiod of time dt the stock price increases from St by an amount µdt plus anunknown amount σdW where dW is a Brownian motion. The distributionof the stock price increases, dS, is slightly different in this case

E[dS] = E[µdt+ σdW ]= µdt+ σE[dW ]= µdt

and similarly the variance is σ2dt as before. This is an improvement onBrownian motion but still has the problem that there is nothing to preventSt from dropping below zero.

2.2.2 Geometric Brownian motion

To over come this adapt the above process ever so slightly to

dS = µSdt+ σSdW. (5)

This is saying that both the deterministic and random terms are scaleddepending on the size of S at time t. The larger S is the bigger, on average,its movements are. This makes sense as a share with price $3 is more likelyto move by a cent than one worth 2c. More importantly, S ≥ 0 as as soonas S = 0 then the process remains there as dS ≡ 0

This process does not give rise to increments which are distributed Nor-mally but rather ones which are distributed lognormally a point seen onExamples 2.

3 Basics of Stochastic calculus and Ito’s lemma

The usual way of approximating derivatives is to use a Taylor expansion.Consider a function of the stock price f(S) and look at the change in valueof f over a small change in S, δS

f(S + δS) = f(S) + δSdf

dS+

1

2(δS)2

d2f

dS2+O((δS)3). (6)

Usually, as δS → 0 then the (δS)2 term disappears enabling the usualrepresentation of df

dS as

limδS→0

f(S + δS) − f(S)

δS

However in this case we have

dS2 = µ2S2dt2 + 2µσS2dtdW + σ2S2dW 2

but as dW is a random variable then it clearly has some variance henceE[dW 2] ≥ 0, in fact E[dW 2] = dt and so this term will not disappear asdt → 0. This means that it is not possible to perform calculus on stochas-tic variables in the same way as it is for deterministic variables. In order

15

to overcome this we need to refer to the work of Japanese mathematicianKiyoshi Ito.

There is a huge amount of theory behind Ito calculus but we shall referonly to the main results and most of the explanation will, hence, be heuris-tic. For a better treatment see the books by Neftci or, better, Etheridge,alternatively attend other (probability) modules.

Ito’s Lemma: If we have the standard stochastic differential equation

dS = a(S, t)dt+ b(S, t)dW

and F = f(S, t) then if f is twice continuously differentiable on [0,∞) ×Rthen F is also a stochastic process given by

dF =

[

a(S, t)∂f

∂S+∂f

∂t+

1

2b2(S, t)

∂2f

∂S2

]

dt+ b(S, t)∂f

∂SdW (7)

(Note: the process for S can also be described in its integral form

S = S0 +

∫ t

0a(S, s)ds +

∫ t

0b(S, s)dW

where again the problem in evaluation comes with the random dW term,a problem which can be overcome by defining the Ito integral which is,obviously, closely linked to Ito’s lemma.)

It is worth noting that if one assumes that as dt → 0, dW 2 → dt anddtdW = o(dt) it is possible to obtain the above result from performing aTaylor series in two dimensions (see examples sheet 2). So, as a rule ofthumb to arrive at Ito’s lemma assume that dW 2 → dt as dt → 0 a resultwhich does not take a great leap of faith to assume to be correct as we knowthat E[dW 2] = dt.

Example 2.2 If dS = adt + bdW where a and b are constants then whatprocess is followed by G = S2?

Well from Ito’s lemma

dG =

[

a∂g

∂S+∂g

∂t+

1

2b2∂2g

∂S2

]

dt+ b∂g

∂SdW

and in this case f(S, t) = S2 and so ∂f∂S = 2S, ∂

2f∂S2 = 2 and ∂f

∂t = 0 thus theprocess followed by G is as follows

dG = (2aS + b2)dt+ 2bSdW

In our particular case of stock price movements we have the particularcase where a(S, t) = µS and b(S, t) = σS and so the process followed by anyfunction (satisfying the Ito conditions), F = f(S, t) will be as follows:

dF =

[

µS∂f

∂S+∂f

∂t+

1

2σ2S2 ∂

2f

∂S2

]

dt+ σS∂f

∂SdW (8)

16

X − X

X X

12

1 2

Π

S

Figure 4: Payoff from a bull spread for underlying asset, S.

What is the importance of this result? Well it is clear that the optionprice, or any derivative price, is a function of the underlying asset and time.This above notation has enabled us to define the process followed by anyfunction of these variables (within very broad constraints). This is a crucialbuilding block for the derivation of the Black-Scholes partial differentialequation.

3.1 Aside - portfolios of options

It is possible, see the many examples on Examples 3, to combine differentoptions to achieve a desired payoff.

Example (Bull spread): The portfolio consists of buying (long) one calloption with exercise price X1 and writing (short) another call option withthe same expiry but a larger exercise price X2. Thus, the portfolio, Π is ofvalue

Π = C(S, t;X1)− C(S, t;X2)

and the value of the portfolio (the payoff) at expiry will be

Π = max(S −X1, 0) −max(S −X2, 0)

and so the portfolio pays nothing for S < X1, S − X1 for X1 < S < X2

and X2 − X1 for S > X2. This will be used by an investor who thinksthat the underlying asset will increase but is happy to take a known amount(X2 −X1) if the increase is substantial - note that this makes the portfoliocheaper than just a call option with exercise price X1. See figure 4.

17

4 The Black-Scholes analysis

4.1 Converting a stochastic process to a deterministic one

In the previous section we have defined a particular model for the move-ment of stock prices. This is by no means the only possible process used forunderlying assets but is the one which is used for the Black-Scholes analysis,which still remains the most popular model for practitioners. From here wenow proceed to derive the Black-Scholes PDE.

The main problem with the process followed by the function of S, F , isthat there is still a random term present which makes constructing a PDEsomewhat problematic. The solution to this is to create a new function gwhich is completely deterministic. Consider a function

g = f −∆S

where ∆ is an as yet unknown parameter which is constant across a timeperiod dt. In which case the change in the value of g over this period is

dg = df −∆dS

and by substituting in the expressions for df and dS from equations (8) and(5) we obtain

dg =

[

µS ∂f∂S + ∂f

∂t +12σ

2S2 ∂2f∂S2

]

dt+ σS ∂f∂SdW −∆[µSdt+ σSdW ]

= σS

[

∂f∂S −∆

]

dW +

[

µS

(

∂f∂S −∆

)

+ ∂f∂t +

12σ

2S2 ∂2f∂S2

]

dt

Thus, if we choose

∆ =∂f

∂S

then the equation reduces to one which has only deterministic variables.This is the basis of the technique employed by Black and Scholes to derivetheir PDE

4.2 The Black-Scholes PDE

Notation:

• S is the current value of the underlying asset, can also be denoted bySt especially in SDEs but the t is usually dropped.

• t is the time elapsed since the option was created and the option expiresat time T .

• V (S, t) is the value of either a call or a put option.

• C(S, t) is the value of a call option.

• P (S, t) is the value of a put option.

• X is the exercise price of the option.

18

• σ is the volatility of the underlying asset or a measure of the uncer-tainty of its movements. For example, a telecommunications startupcompany’s shares will have a higher volatility than Tesco’s shares.

• µ is the drift of the underlying asset.

• r is the risk-free interest rate, the return that you would receive froma risk-free investment such as a government bond.

Black-Scholes assumptions:

• The underlying asset follows geometric Brownian motion (dS = µSdt+σSdW ) with constant drift, µ and volatility σ. It is possible to havethe volatility dependent on time but more complicated models willprovide much more challenging problems.

• It is permitted to short sell the underlying asset, i.e. sell an asset thatyou don’t actually own.

• There are no transaction costs, all securities are perfectly divisible andtrading takes place continuously.

• There are no dividends, or equivalent, paid out during the lifetime ofthe option (this will be relaxed at a later date).

• There are no riskless arbitrage opportunities. Any that do exist existonly for a very short period of time.

• The risk free rate r is constant. This can also be trivially relaxedto let r be a function of time. In practice, especially for long-termderivatives, the interest rate is itself modelled stochastically.

As the option price V (S, t) depends on the underlying asset, S, which followsgeometric Brownian motion

dS = µSdt+ σSdW (9)

and by Ito’s lemma we have

dV =

[

µS∂V

∂S+∂V

∂t+

1

2σ2S2∂

2V

∂S2

]

dt+ σS∂V

∂SdW (10)

Now construct a portfolio which consists of an option and short in ∆ of theunderlying. Π is defined to be the value of the portfolio where

Π = V −∆S. (11)

Assume, across a time period dt, that the value of ∆ is held fixed giving

dΠ = dV −∆dS, (12)

and so, on substituting in the expressions for dV and dS in equations (9)and (10) we get

dΠ = σS

[

∂V

∂S−∆

]

dW +

[

µS

(

∂V

∂S−∆

)

+∂V

∂t+

1

2σ2S2∂

2V

∂S2

]

dt (13)

19

The amount of the underlying which the holder of the portfolio is shortselling, ∆, has not yet been set. However, if ∆ is selected, as before, suchthat

∆ =∂V

∂S, (14)

then the stochastic differential equation for dΠ becomes deterministic, asthe coefficient of the dW term is now identically zero. Thus this portfolio isperfectly hedged as it provides a guaranteed return over a designated timeperiod. Obviously, this assumes that it is possible to change the value of∆ continuously, because as time evolves the value of ∂V

∂S is changing. Withthis continuous rebalancing of the portfolio the expression for dΠ is now

dΠ =

(

∂V

∂t+

1

2σ2S2∂

2V

∂S2

)

dt. (15)

However, this portfolio is perfectly hedged, in that it yields a risk-less valueafter any period of time t and, as such, should return the risk-free rate.Assuming no arbitrage then over a period of time, dt, and a constant risk-free interest rate, r, the change in the portfolio is

dΠ = rΠdt.

If it were the case that dΠ 6= rΠdt then one could make a risk-free profit byeither borrowing Π from the bank and investing in the portfolio (dΠ > rΠdt),or shorting the portfolio and investing the money in the bank (dΠ < rΠdt).On replacing Π by its definition, equation (11), equation (15) is now

r

(

V − S∂V

∂S

)

dt =

(

∂V

∂t+

1

2σ2S2∂

2V

∂S2

)

dt. (16)

On dividing equation (16) by dt one obtains

∂V

∂t+

1

2σ2S2∂

2V

∂S2+ rS

∂V

∂S− rV = 0. (17)

which is the Nobel prize winning Black-Scholes partial differential equation.Remarks:

• This equation defines the price of any derivative claim on an under-lying asset which follows geometric Brownian motion. The boundaryconditions will determine which type of derivative we are evaluating.

• This is a backwards parabolic partial differential equation, a class ofequations about which a lot more will be said below.

• Notice that by setting up the portfolio Π using what is known as theDelta Hedge the Black Scholes equation does not depend on the driftterm µ in any way. The only parameter which needs to be empiricallyestimated is σ.

• The Delta (∆) which is the rate of change of the derivative with respectto the underlying asset is a very important value.

20

• The linear operator

LBS =∂

∂t+

1

2σ2S2 ∂

2

∂S2+ rS

∂

∂S− r

is a measure of the difference between the return on the hedged port-folio (Π) which are the first two terms (see equation (15)) and thereturn on a bank deposit which are the last two terms. For a Euro-pean option these will be the same, though they are not necessarilyfor an American option.

• For many types of options it is not possible to obtain closed-formanalytic values but more often than not numerical procedures mustbe employed. In this lecture course, though, emphasis will remain onanalytic solutions.

21

4.3 Formulating the mathematical problem

4.3.1 Classifying the PDE

For there to be no arbitrage, the option value obtained from the Black-Scholes PDE must provide a unique option price. Later it will be shownthat, given suitable boundary conditions, this is indeed the case. First, inorder to determine the type of boundary conditions required it is necessaryto find out some general information about the PDE itself.

We know that in general a PDE with solution u(x, t) of the form

auxx + buxt + cutt + dux + eut + fu = g (18)

is classified depending on the sign of b2 − 4ac as follows:

• If b2 − 4ac < 0 then the equation is elliptic.

• If b2 − 4ac = 0 then the equation is parabolic.

• If b2 − 4ac > 0 then the equation is hyperbolic.

The most commonly seen parabolic equation is the diffusion or heat equation

∂2u

∂x2=∂u

∂t

which typically models the evolution of heat along a bar. As they are secondorder in x and only first order in t parabolic equations usually require twoboundary conditions in x (or S in the Black-Scholes case) and just the onein t. It is important to notice here that in the heat conduction equationthe ∂u/∂t term is of a different sign from that in the Black-Scholes equation(17). This is because the heat conduction equation is a forwards parabolicequation whilst the Black-Scholes equation is backwards parabolic. Thedifference between the two types is that forwards equations require initial

conditions, whilst backwards equations require final conditions.Note how these requirements are consistent with the individual nature

of the problems. When valuing options, we know the value at expiry (orthe final time) and so it makes sense that this problem gives rise to a back-wards parabolic type. The heat conduction (or diffusion) equation requiresa known distribution of heat on a bar (or equivalent system) at t = 0 andthen models how the heat distribution evolves as time moves forwards. Assuch the system requires initial conditions - thus is a forwards parabolictype.

It is essential to always solve parabolic equations ‘in the correct direction’.

4.3.2 Characteristics

The classification of PDEs in the above section is closely related to thenotion of characteristics. Characteristics are families of curves along whichinformation moves or across which discontinuities may occur. The trick isto attempt to write the derivative terms in the PDE in terms of directional

22

derivatives reducing the equation to one which behaves like an ODE alongthese characteristic curves.

Definition (Characteristic curve) A curve Γ is a characteristic for ageneral second order PDE if, for a general PDE in x and t,

∂t

∂x− b±

√b2 − 4ac

2a= 0

along Γ.

Clearly the value of b2 − 4ac will be important in determining the char-acteristic curves. In the parabolic case there is just one real valued solutiongiving

∂t

∂x=

b

2a.

In the case of the heat conduction equation where b = 0 then this reducesto

∂t

∂x= 0

giving characteristic curves along t = C where C is a constant.

4.3.3 Boundary conditions for the Black-Scholes equation

Returning to the Black-Scholes equation, for each particular type of op-tion we will require the following boundary conditions:

V (S, t) = Va(t) on S = aV (S, t) = Vb(t) on S = bV (S, t) = VT (S) on t = T

where Va(t) and Vb(t) are known functions of time and VT (S) is, correspond-ingly, a known function of the underlying asset price. To demonstrate how todo this for different types of options we’ll consider three cases: the standardEuropean call and put options and a cash-or-nothing call option.

European call option, C(S,t):

The most straightforward of the conditions to determine is the finalcondition C(S, t = T ) as this is the known payoff for the call option,(max(S −X, 0)), hence

C(S, T ) = max(S −X, 0). (19)

The conditions for specific values of S are also reasonably straightforward.Note that from the process followed by S, namely

dS = µSdt+ σSdW

if S = 0 then dS = 0 and , hence, the underlying asset remains at 0 fromthen on. Hence for a call option, however small the strike price X is, thisscenario will always result in the option being worthless, hence

C(0, t) = 0 (20)

23

For large S the situation is not as clear and there are three standard conven-tions (of which two are provided here for brevity). As S → ∞ then clearlythe call option is more and more likely to be exercised and in comparison tothe size of S, X will be small and so one can simply use

C(S, t) → S as S → ∞.

However, the S boundary conditions are more important when dealing withnumerical procedures where a large, but finite, limit is put on S (Smax say).In which case, more accurate conditions are required. One possibility is toassume that the option will be exercised at expiry, receiving S plus whateverelse contributes to the option’s value as time moves backwards. In this waywrite the option price for a particular high value of S as

C(S, t) = S + f(t)

on substituting into the Black-Scholes equation (17) we’re left with

dfdt + rS +−r(S + f(t)) = 0

dfdt = rf(t)

which on solving givesf(t) = Aert

substituting in the known time constraint from (19) we get

A = −Xe−rT

and so the boundary condition for large S is

C(S, t) → S −Xe−r(T−t) as S → ∞. (21)

European put option, P (S, t):

The case for a put option is far more straightforward. Again determiningthe final condition is trivial as a result of the discussion in Chapter 1, so wehave

P (S, T ) = max(X − S, 0). (22)

The conditions for particular values of S are extensions of the abovearguments for calls, only more routine. When S = 0 at a particular timethen by the nature of the underlying process then it will stay at 0 untilexpiry. Hence the put option will definitely be exercised and thus worthX − 0 = X at expiry. A guaranteed amount of money, in this case X, to bereceived at time T is worth Xe−r(T−t) at time t and hence

P (0, t) = Xe−r(T−t) (23)

As S becomes very large then the put options will certainly not be exercisedas S will be much larger than the exercise price X and so

P (S, t) → 0 as S → ∞. (24)

24

As before the most important conditions are the final ones, but the otherconditions are essential for numerical schemes as well as giving us moreinformation about the option prices.

Cash-or-nothing/binary options:

Cash-or-nothing call (put) options (denoted CC(S, t) or CP (S, t)) areoptions where, at expiry, if the underlying asset price is above (below) a cer-tain strike price, X, then the holder receives a pre-designated cash amountA, whereas if it is below (above) this amount the holder receives nothing.Hence at expiry, t = T , the final condition for a cash-or-nothing call is

CC(S, T ) = AH(S −X)

where H(.) is known as the Heaviside function. The Heaviside functionis defined as follows

H(x) =

{

0 if x < 01 if x ≥ 0

and will be important when solving PDEs later in the course. Cash-or-nothing options are a special type of option in that their payoff is completelydiscontinuous yet it is still possible to find an option value for them.

4.4 Analytic solutions to the Black-Scholes equation

The next chapter of the course will deal with solving the heat conductionor diffusion equation and how to adapt these techniques to solve the Black-Scholes equation for some standard option pricing problems. Before doingthat we will study the analytic solutions to the valuation problems and afew more key features of options.

The Black-Scholes formulae for the price of European call and putoptions are as follows:

C(S, t) = SN(d1)−Xe−r(T−t)N(d2) (25)

P (S, t) = Xe−r(T−t)N(−d2)− SN(−d1) (26)

where

d1 =log(S/X) + (r + 1

2σ2)(T − t)

σ√T − t

d2 =log(S/X) + (r − 1

2σ2)(T − t)

σ√T − t

.

(27)

and

N(x) =1√2π

∫ x

−∞e−

1

2s2ds (28)

which we recognise as the cumulative distribution function for a Normaldistribution. Note that these expressions satisfy the put call parity and soby calculating one it is routine to calculate the other, also note that theboundary conditions at S = 0 and S → ∞ are satisfied.

25

For those students interested in probability it may be worth noting thatN(d2) is the probability that the option will be exercised, i.e. S > X atexpiry. SN(d1) is the current value of a variable that equals ST at t = T if

ST > X and is zero otherwise.So, what does a graph of underlying asset against option price looks

like as time moves backwards from expiry? As one would expect from aPDE which is a close relative of the diffusion equation, the payoff functionmax(S − X, 0) gradually diffuses out as time moves backwards. The sameis also true for a cash or nothing option even though the payoff is in factdiscontinuous.

Example

The price of an asset (today) is £5. Find the value of a put and a calloption, both with an exercise price of £6, and both with expiration dates in9 months time. The risk-free interest rate is 3% per annum (fixed) and the

volatility (constant) is 10% per (annum)1

2 .

Solution

r = .03, T − t = 0.75, σ = .1, S = 5, X = 6.Using the formulae.d1 = −1.8021, d2 = −1.8888Then

N(d1) = N(−1.8021) = N(−1.80) − .21[N(−1.80) −N(−1.81)]

= 0.0359 − .21× (0.0359 − 0.0351)

= 0.0357

Similarly N(d2) = .0295Leads to C = .0060.Put can be calculated similarly - but best to use put-call parity:

P = C − S +Xe−r(T−t).

and this leads to P = 0.8725.

4.5 Delta hedging and the other hedge parameters

A tedious, yet straightforward, calculation (see example sheet 6) willshow that using the known expressions for the values of call and put options,that they have the following ∆’s

∆C =∂C

∂S= N(d1)

∆P =∂P

∂S= N(d1)− 1

What does this mean? During the lifetime of the option ∆ varies between 0for out of the money calls (puts) and 1 (−1) for in the money calls (puts) andvery close to T there is in fact a step function between these two extremes.

26

The ∆ simply approximates the rate of change of the option price wrt theunderlying asset and so any slight movement in the option price value willbe offset by a roughly equivalent movement in ∆ of the underlying. Clearlythe portfolio will have to be rebalanced as regularly as possible to have aperfect hedge. In practise the number of times a portfolio can be hedgedwill be limited by transaction costs.

For example, looking at the graph for the value of a cash-or-nothingcall option we immediately see a problem with the delta-hedging strategyunderlying the Black-Scholes analysis. If ∆ is ∂C/∂S then as t → T thenthe ∆ ranges from 0 away from S = X to approaching ∞ close to S = X.Thus as the underlying asset price moves, huge amounts of the underlyingwill have to be bought and sold to keep the portfolio properly hedged.

There are ways of hedging away other risks, not just those to do withthe movement of the asset price. There are hedge parameters (also knownas, somewhat loosely, as The Greeks) for each of the principle parametersin the Black-Scholes model, namely:

• The sensitivity to the decay of time of any option V is known as thetheta and is defined as

Θ =∂V

∂t

• The sensitivity to the volatility is known as the vega and is definedas

V =∂V

∂σ

• The sensitivity to interest rates is known as rho and, unsurprisinglyto be

ρ =∂V

∂r

• Finally, the sensitivity of the ∆ to the underlying asset is known asgamma and is defined as follows

Γ =∂2V

∂S2

Often these hedge parameters are used to see what would happen if therewas a small change in one of the parameters, this is important as both rand σ are not fixed or even time dependent in practice.

4.6 Implied volatility

One of the most important parameters, and the only one which is verydifficult to know for definite is the volatility, σ. There are several conventionsfor calculating the volatility of an underlying asset. One would perhapsassume that the best way is to look at the volatility of past returns and usethis as a decent guess as to what would happen in the future. However,another way is to assume that the Black-Scholes analysis is correct and usethe market prices for options to back-out the volatility, using a suitable

27

iterative procedure such as Newton-Raphson, the only unknown being σitself.

If one attempts this they will see a problem with the volatility. Depend-ing on how far in or out of the money the option is the volatility may wellnot be constant for a given r, S, and t. So, not only is it dependent ontime but also on the exercise and asset prices. Such a result is often termedthe volatility smile although many other shapes can be observed dependingon the market conditions such as a frown, wry smile etc. This is anotherexample of the faults in the Black-Scholes model.

5 Solving the heat conduction and Black-Scholes

equations

The PDE which defines the price of a derivative is now known to bea second-order parabolic equation, in the majority of cases this equationis also a linear one. This chapter is concerned with the nature of theseequations, focusing attention on the heat conduction equation and thenextending to the Black-Scholes equation itself.

5.1 Properties of the Heat conduction equation

The heat conduction equation takes the form

∂u

∂τ=∂2u

∂x2

where τ is the time and x is the spatial variable, it normally models the flowof heat or its diffusion and has been extensively studied over the years. Itsfundamental properties are as follows

• It is a second order linear PDE, as such if u1 and u2 are solutions thenso is a1u1 + a2u2 for any constants a1, a2

• It is a parabolic equation and it’s characteristics are simply along thelines τ = c (where c is a constant) which means that this is where infor-mation propagates along. So any change in the boundary conditionsis felt along these lines.

• The heat conduction equation generally has analytic solutions in x,technically in that for τ > 0, u(x, τ) has a convergent power series of(x− x0) for x0 6= x.

Crucially, the heat conduction (diffusion) equation is a smoothing outprocess, and as such discontinuities in the boundary or initial (final) condi-tions can be catered for. Recall that in the Black-Scholes equation the finalconditions are often discontinuous.

Example By way of demonstration consider the following initial value prob-

lem.∂u

∂τ=∂2u

∂x2

28

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

-10 -5 0 5 10

PSfrag

replacem

ents

uδ(x,τ)

τ = 0.1

x

Figure 5: A graphical representation of uδ(x, τ) for τ = 0.1, 0.2, 0.3, . . . , 1.

for τ > 0 and −∞ < x < ∞ where u(x, 0) = u0(x) and u → 0 as x→ ±∞.u(x, τ) is analytic for τ > 0. Consider a special solution, about which moreis said later

u(x, τ) = uδ(x, τ) =1

2√πτe−x

2/4τ (29)

for −∞ < x < ∞ and τ > 0. Now we verify that this indeed satisfies thePDE.

∂u

∂x=

−x4τ3/2

√πe−x

2/4τ

∂2u

∂x2=

−1

4τ3/2√πe−x

2/4τ +x2

8τ5/2√πe−x

2/4τ

∂u

∂τ=

−1

4τ3/2√πe−x

2/4τ +x2

8τ5/2√πe−x

2/4τ .

So, this is a solution which is well behaved except at one instance, the initialpoint in time τ = 0. At this point when x 6= 0 then uδ(x, 0) = 0 but atx = 0 it has infinite value. This clearly has discontinuous initial conditionsyet gives rise to a, reasonably, well behaved solution.

What more can we say about this special solution to the heat conductionequation? Well,

∫ ∞

−∞uδ(x, τ)dx = 1, ∀τ.

This function has all of the heat initially (τ = 0) concentrated at x = 0 andthen this immediately dissipates out as for any τ > 0, uδ(x, τ) > 0 for all

values of x.Finally note the close similarity between the probability density function

for the Normal distribution ( 1σ√2πe−(x−µ)2/2σ2) and the value of uδ(x, τ).

Clearly it is the same only with a mean(µ) of zero and a variance (σ2) of 2τ .As such it is possible to interpret this particular solution as the probabilitydensity function of the future position of a particle following a Brownianmotion (

√2dW ) along the x-axis, with the particle starting at the origin.

29

5.2 The Dirac delta function

The function uδ(x, τ) when τ = 0 is one representation of the (Dirac)delta function which is not a function in the normal sense but is known asa generalised function. It’s definition is as a linear map representing thelimit of a function whose effect is confined to a smaller and smaller intervalbut remains finite.

An informal definition is to consider a function

f(x) =

{

1/2ǫ, |x| ≤ ǫ0, |x| > ǫ

and as ǫ→ 0 the graph becomes taller and narrower but at all points

∫ ∞

−∞f(x)dx = 1

regardless of the value of ǫ although for all x 6= 0, f(x) → 0 as ǫ → 0. Ingeneral the delta function δ(x) is the limit as ǫ → 0 of any one-parameterfamily of functions δǫ with the following properties

• for each ǫ, δǫ(x) is piecewise smooth;

•∫∞−∞ δǫ(x)dx = 1;

• for each x 6= 0, limǫ→0 δǫ(x) = 0.

Note that the specific solution to the heat conduction equation uδ satisfiesthe above constraints with τ replaced by ǫ. The best way to look at thedelta function is to only consider its integral which we know to be 1 andwhich smooths out the function’s bad behaviour, especially when x = 0 andǫ → 0 (of τ → 0). When concentrating on the integral form we can see thedelta function as a test function, in that

∫ ∞

−∞δ(x)φ(x)dx = lim

ǫ→0

∫ ∞

−∞δǫ(x)φ(x)dx

= limǫ→0

{∫ −ǫ

−∞δǫ(x)φ(x)dx +

∫ ǫ

−ǫδǫ(x)φ(x)dx +

∫ ∞

ǫδǫ(x)φ(x)dx

}

= limǫ→0

{

φ(0)

∫ ǫ

−ǫδǫ(x)dx

}

= φ(0)

In fact, for any a, b > 0

∫ b

−aδ(x)φ(x)dx = φ(0)

and, as importantly, for any x0∫ ∞

−∞δ(x− x0)φ(x)dx = φ(x0)

30

ε

ε

2

21

x

Figure 6: The epsilon representation of δ(x) which is the limit as ǫ→ 0.

x

H(x)

H’(x) = 0

H’(x) = 0

1

H’(x) = 8

Figure 7: Demonstration that H ′(x) = δ(x).

and so integrating picks out the value of φ at x0, the reason why δ(x) is alsoknown as a test function.

Other properties concern its links with the Heaviside function as

∫ x

−∞δ(s)ds = H(x)

and conversely,H′(x) = δ(x)

where, as before

H(x) =

{

0 if x < 01 if x ≥ 0

31

5.3 Transforming the Black-Scholes equation

Consider the Black-Scholes equation

∂V

∂t+

1

2σ2S2∂

2V

∂S2+ rS

∂V

∂S− rV = 0

make the following three substitutions

S = Xex(or x = logS

X)

t = T − τ12σ

2(or τ =

σ2

2(T − t))

V = Xv(x, τ) (30)

thus

∂V

∂t= X

∂v

∂τ

dτ

dt= X

∂v

∂τ.− σ2

2= −Xσ

2

2

∂v

∂τ∂V

∂S= X

∂v

∂x

dx

dS= X

∂v

∂x

1

S= e−x

∂v

∂x∂2V

∂S2=

∂

∂S

(

∂V

∂S

)

=e−x

X

∂

∂x

(

e−x∂v

∂x

)

=e−x

X

(

e−x∂2v

∂x2− e−x

∂v

∂x

)

=e−2x

X

(

∂2v

∂x2− ∂v

∂x

)

which leads to∂v

∂τ=∂2v

∂x2+ (k − 1)

∂v

∂x− kv

wherek =

r12σ

2

Now attempt to remove the ∂v∂x and v terms by introducing the substitution

v(x, τ) = eαx+βτu(x, τ)

where α and β are constants to be determined, this gives

∂v

∂τ= βeαx+βτu+ eαx+βτ

∂u

∂τ∂v

∂x= αeαx+βτu+ eαx+βτ

∂u

∂x∂2v

∂x2= α2eαx+βτu+ 2αeαx+βτ

∂u

∂x+ eαx+βτ

∂2u

∂x2

which gives

βu+∂u

∂τ= α2u+ 2α

∂u

∂x+∂2u

∂x2+ (k − 1)

(

αu+∂u

∂x

)

− ku

to remove the ∂u∂x and u terms we require

α = −1

2(k − 1)

β = −1

4(k + 1)2.

32

Thus,

V (S, t) = Xe−1

2(k−1)x− 1

4(k+1)2τu(x, τ) (31)

and∂u

∂τ=∂2u

∂x2−∞ < x <∞τ > 0

To transform the final conditions, or the payoff from the option we have fora call option

V (S, T ) = max(S −X, 0)

so, from the definition of x, τ and v(x, τ) in (30)

Xv(x, 0) = max(Xex −X, 0)

orv(x, 0) = max(ex − 1, 0)

and so, from (31)

u(x, 0) = u0(x) = max

[

e1

2(k+1)x − e

1

2(k−1)x, 0

]

(32)

and similarly for a put option

u(x, 0) = u0(x) = max

[

e1

2(k−1)x − e

1

2(k+1)x, 0

]

(33)

As such the Black-Scholes equation has been converted to the heat conduc-tion equation for −∞ < x < ∞ and, for European call and put options,initial condition u0(x) from (32) and (33) above. If we can determine a pro-cedure for valuing the initial value problem for the heat conduction equationwe’ll be able to determine the correct values for call and put options.

33

5.4 Similarity solutions to the Heat conduction equation

Explanation is first by way of two examples

Example 5.1: Suppose that u(x, τ) satisfies the heat conduction equation

∂u

∂τ=∂2u

∂x2, x, τ > 0

with the following boundary conditions

u(x, τ = 0) = 0 (34)

u(x = 0, τ) = 1 (35)

u(x, τ) → 0 as x→ ∞ (36)

i.e. the bar initially has heat zero and then immediately the heat at one endis raised to 1 and kept there.

Seek a solution of the form u(x, τ) = U(ξ) where ξ = x/√τ on substitu-

tion∂u

∂τ=dU

dξ

∂ξ

∂τ= −1

2xτ−3/2 dU

dξ

∂u

∂x=dU

dξ

∂ξ

∂x= τ−1/2 dU

dξ

and∂2u

∂x2= τ−1/2 d

dξ

(

τ−1/2 dU

dξ

)

= τ−1 d2U

dξ2

and so, replacing x/√τ by ξ and multiplying by τ gives the ODE

d2U

dξ2+

1

2ξdU

dξ= 0

the boundary conditions become

U(0) = 1

andU(∞) = 0

with this second condition catering for both the initial condition and u(x, τ) →0 as x→ ∞. Integrating the ODE once gives

dU

dξ= Ce−ξ

2/4

(C constant) and on solving gives

U(ξ) = C

∫ ξ

0e−s

2/4ds+D

(D constant). Upon substituting the boundary conditions, first U(0) = 1gives

1 = D

34

and then U(∞) = 0 gives

0 = C

∫ ∞

0e−s

2/4ds+ 1

but we know that∫ ∞

0e−s

2/4ds =√π

thus−1 = C

√π

Thus,

U(ξ) = − 1√π

∫ ξ

0e−s

2/4ds+ 1

but∫ ξ

0=

∫ ∞

0−∫ ∞

ξ

hence

U(ξ) = − 1√π

(∫ ∞

0e−s

2/4ds−∫ ∞

ξe−s

2/4ds

)

+ 1

or

U(ξ) = − 1√π

(

−∫ ∞

ξe−s

2/4ds

)

− 1 + 1

so

U(ξ) =1√π

∫ ∞

ξe−s

2/4ds

and on replacing ξ by its definition we get

u(x, τ) =1√π

∫ ∞

x/√τe−s

2/4ds

The key trick being that to solve the equation we replace two variables (xand τ) by just one (ξ) and then the problem reduces to an ODE. Even moreuseful is the next example, for −∞ < x <∞.

Example 5.2: Consider the following equation for u(x, τ)

∂u

∂τ=∂2u

∂x2,

−∞ < x <∞τ > 0

where∫ ∞

−∞u(x, τ)dx = k,∀τ where k is a constant.

Choosing the normalised case where k = 1 we search for a solution of theform u(x, τ) = τ−1/2U(ξ) where ξ = x/

√τ . The other boundary condition

is a somewhat odd one but is that as |ξ| → ∞ then

U(ξ) = o(1/ξ)

35

which says that the solution must decay faster than 1/ξ as ξ gets very big (oralternatively u(x, τ) = o(1/x) as |x| → ∞). On transforming the derivativeswe get

∂u

∂τ= −1

2τ−3/2U + τ−1/2 dU

dξ.− 1

2xτ−3/2 = −1

2τ−3/2U − 1

2ξτ−3/2 dU

dξ

∂u

∂x= τ−1/2 dU

dξ

∂ξ

∂x= τ−1dU

dξ

and∂2u

∂x2= τ−1/2 d

dξ

(

τ−1dU

dξ

)

= τ−3/2 d2U

dξ2

which givesd2U

dξ2+

1

2ξdU

dξ+

1

2U = 0

ord2U

dξ2+

d

dξ

(

1

2ξU

)

= 0.

Integrating both sides wrt ξ gives

dU

dξ+

1

2ξU = C

where C is a constant. Now as ξ → ∞, U = o(1/ξ) so the LHS is o(1) thusthis constant C = 0. So then on solving the ODE

U(ξ) = Ae−ξ2/4,

where A is a constant. Putting in the condition we have

A

∫ ∞

−∞τ−1/2e−x

2/4τdx = 1

however, set x′ = x/√τ and we get dx =

√τdx′ and the equation becomes

A

∫ ∞

−∞e−x

′2/4dx′ = 1

and so using the usual result

2A√π = 1

thus

A =1

2√π

and so,

u(x, τ) = τ−1/2

(

1

2√πe−x

2/4τ

)

or

u(x, τ) =1

2√πτe−x

2/4τ

which is precisely the special solution uδ from section 5.1, equation 29.[Note: The derivation in Wilmott where he states that U(ξ) = Ce−ξ

2/4+Dis wrong .]

36

5.4.1 How similarity solutions work

The reason why the above similarity solution worked was because thegoverning equations and the boundary conditions do not change under thescalings x → λx and τ → λ2τ , where λ ∈ R. In particular consider newvariables x∗ = λx and τ∗ = λ2τ , these clearly satisfy the heat-conductionequation and in Example 5.1 the boundary conditions become u(x∗, 0) = 0and u(0, τ∗) = 1 for any λ.

Combining these two results to get a variable which is independent of λthe only possible combination is x/

√τ = x∗/

√τ∗. Hence the solution to the

problem must be a function of x/√τ only.

Similarity solutions only work in special cases where all the boundaryand initial conditions are invariant under the scaling transformation. It isalso possible to multiply U(ξ) by a function of τ as in Example 5.2 becauseas the heat-conduction equation is linear it is invariant under the scalingu→ µu.

In general with similarity solutions a good practical test to see if they’llwork is to search for a solution of the form u = ταU(xτβ) in the hope thatthe PDE will reduce to an ODE in ξ = xτβ and the boundary conditions willbe satisfied. For the heat conduction equation then in all cases β = −1/2but the value of α will be dependent on the specific boundary conditions.For example in 5.1 α = 0 because of the condition at x = 0 and, in Example5.2, α = −1/2 to remove τ from the integral condition.

5.5 General solution to the Heat-Conduction equation initial

value problem

Searching for a solution to the initial value problem in which we have tosolve

∂u

∂τ=∂2u

∂x2,

−∞ < x <∞τ > 0

with initial data u(x, 0) = u0(x) and there are suitable growth conditions at|x| → ∞ (usually lim|x|→∞ u(x, τ)e−ax

2

= 0 for a > 0 and τ > 0).The key to the formulation is the delta function, δ(x) as we can write

the initial conditions as

u0(x) =

∫ ∞

−∞u0(ξ)δ(ξ − x)dξ

we recall that the fundamental solution to the initial value problem from 5.2is

uδ(s, τ) =1

2√πτe−s

2/4τ

and has initial value uδ(s, 0) = δ(s). Noting that because uδ(s − x, τ) =uδ(x− s, τ) we have

uδ(s− x, τ) =1

2√πτe−(s−x)2/4τ

37

which is still a solution to the heat conduction equation with either s or xas the spatial independent variable and it has initial value

uδ(s− x, 0) = δ(s − x).

Now comes the important bit, hence, for each s the function

u0(s)uδ(s− x, τ)

as a function of x and τ with s held fixed, satisfies the heat conductionequation as u0(s) is simply a constant. Now using the fact that the diffusionequation is linear we can add together linear combinations of these solutionsfor any s all the way from −∞ to ∞ and obtain another solution to the heatconduction equation, namely

u(x, τ) =1

2√πτ

∫ ∞

−∞u0(s)e

−(x−s)2/4τds

and the initial data is

u(x, 0) =

∫ ∞

−∞u0(s)δ(s − x)ds = u0(x).

What does all this mean? Well, this solution satisfies the heat conductionequation for all x and for τ > 0 and is also satisfies the initial conditions forall initial conditions u0(x). It is also possible to show that this solution isunique (see Examples 5). Hence we have found the general solution.

5.6 Pricing European call and put options

We now know the general solution to the initial value problem for theheat conduction equation, where u(x, 0) = u0(x) for τ > 0 and −∞ < x <∞, namely

u(x, τ) =1

2√πτ

∫ ∞

−∞u0(s)e

−(x−s)2/4τds.

We start by valuing a European call option but the procedure is similar fora put option. In section 5.3 we transformed the European call option pricingproblem to the following system

∂u

∂τ=∂2u

∂x2,

−∞ < x <∞τ > 0

where

u(x, 0) = u0(x) = max

[

e1

2(k+1)x − e

1

2(k−1)x, 0

]

.

By using the known general solution to this problem we have

u(x, τ) =1

2√πτ

∫ ∞

−∞

{

max[e1

2(k+1)s − e

1

2(k−1)s, 0]e−(x−s)2/4τ

}

ds

but u0(x) = 0 for x < 0 hence

u(x, τ) =1

2√πτ

∫ ∞

0

{

[e1

2(k+1)s − e

1

2(k−1)s]e−(x−s)2/4τ

}

ds.

38

We make another change of variable, define

x′ =s− x√

2τ.

u(x, τ) =1√2π

{∫ ∞

−x/√2τe

1

2(k+1)(x′

√2τ+x)− 1

2x′2dx′−

∫ ∞

−x/√2τe

1

2(k−1)(x′

√2τ+x)− 1

2x′2dx′}.

Completing the square and removing the terms not dependent on x′ yields

u(x, τ) =e

1

2(k+1)x+ 1

4(k+1)2τ

√2π

∫ ∞

−x/√2τe−

1

2(x′− 1

2(k+1)

√2τ)2dx′

−e1

2(k−1)x+ 1

4(k−1)2τ

√2π

∫ ∞

−x/√2τe−

1

2(x′− 1

2(k−1)

√2τ)2dx′

= I1 − I2 (37)

Noting that the expression for the cumulative Normal distribution is asfollows

N(x) =1√2π

∫ x

−∞e−

1

2s2ds

we transform the dependent variable, x′, once again to

x1 = x′ − 1

2(k + 1)

√2τ

and

x2 = x′ − 1

2(k − 1)

√2τ

in I1 and I2 respectively and then

u(x, τ) = e1

2(k+1)x+ 1

4(k+1)2τN(d1)− e

1

2(k−1)x+ 1

4(k−1)2τN(d2)

where

d1 =x√2τ

+1

2(k + 1)

√2τ

d2 =x√2τ

+1

2(k − 1)

√2τ .

Transforming the variables back using the usual definitions

V (S, t) = Xe−1

2(k−1)x− 1

4(k+1)2τu(x, τ)

x = log

(

S

X

)

τ =σ2

2(T − t)

k =2r

σ2

gives the following expression for the value of the European call option

C(S, t) = V (S, t) = SN(d1)−Xe−r(T−t)N(d2),

39

where

d1 =log(S/X) + (r + 1

2σ2)(T − t)

σ√T − t

d2 =log(S/X) + (r − 1

2σ2)(T − t)

σ√T − t

.

The European put can be valued in a similar manner or, more easily, byuse of the put-call parity, equation. Either approach yields the followingexpression for its value, P (S, t)

P (S, t) = Xe−r(T−t)N(−d2)− SN(−d1).

(To use put-call parity note that N(x) +N(−x) = 1).

40

6 Options on assets paying dividends

6.1 Introduction

The majority of companies who have issued shares pay out dividends ofsome form another, fortunately it is relatively easy to incorporate dividendpayments into the option pricing methodology. Of even greater use is thatthe methods used for pricing options on dividend paying stocks can be rou-tinely extended to deal with other, analogous, problems such as options onforeign currency where the dividend becomes the foreign risk-free interestrate and options on commodities where the dividend becomes minus thecost of carry.

There are two main ways of modelling dividend payments: as continuousand as discrete.

6.2 Continuous constant dividend yield

This is the simplest payment structure, assume that over a period oftime dt the underlying asset pays out a dividend DSdt in that D is theproportion of the value of the asset paid out over this period of time. Dis considered to be constant and independent of t though the size of thedividend will obviously depend on S which is dependent on t.

How does this affect our model? By using arbitrage arguments (seeExamples 1) a payment of dividends results in the underlying asset pricedropping by the value of the dividend. Hence with a continuous dividendthe stochastic process is given by

dS = (µ−D)Sdt+ σSdW.

To derive the governing PDE a similar process is followed but although theportfolio is still

Π = V −∆S

in this casedΠ = dV −∆(dS +DSdt)

as the holder of the portfolio receives the dividend as well. Proceeding asfor the non-dividend case

dV =∂V

∂tdt+

∂V

∂SdS + 1

2σ2S2∂

2V

∂S2dt,

and so

dΠ =∂V

∂tdt+ (

∂V

∂S−∆) [(µ−D)Sdt+ σSdW ]−∆DSdt+ 1

2σ2S2 ∂

2V

∂S2dt.

Setting

∆ =∂V

∂S

41

leads to a deterministic result, to which we can apply the usual no-arbitrageargument, i.e.

dΠ =∂V

∂tdt+ 1

2σ2S2∂

2V

∂S2dt−DS

∂V

∂Sdt

= rΠdt

= r(V − S∂V

∂S)dt

which gives the following PDE

∂V

∂t+

1

2σ2S2 ∂

2V

∂S2+ (r −D)S

∂V

∂S− rV = 0. (38)

The standard Black-Scholes equation derived earlier in the course is just aspecial case of this equation for the case when D = 0. Valuing Europeancall and put options is reasonably straightforward, the main difference beingthat r is replaced by r − D but only in the coefficient of the ∂C/∂S. Toaccount for this slight difference introduce

V (S, t) = e−D(T−t)V1(S, t)

so that we now have

∂V1∂t

+1

2σ2S2∂

2V1∂S2

+ (r −D)S∂V1∂S

− (r −D)V1 = 0

which is the Black-Scholes equation only with r replaced by r−D and withthe same final conditions. As such

C(S, t) = e−D(T−t)SN(d10)−Xe−r(T−t)N(d20)

where

d10 =log(S/X) + (r −D + 1

2σ2)(T − t)

σ√T − t

d20 =log(S/X) + (r −D − 1

2σ2)(T − t)

σ√T − t

.

6.3 Discrete dividend payments

When considering options where the underlying is a stock then a morerealistic model is to treat dividends as being paid at discrete points in time.This is because most companies pay out their dividends periodically, everyquarter, every six months, every year etc.

Assume, as a starting point, that just one dividend payment is madeduring the lifetime of the option. Assume that this is paid at time td andcan be expressed as a percentage of the level of the underlying, i.e. as dySwhere 0 ≤ dy < 1. Thus the holder of the asset receives a payment of dyS attd where S is the asset price prior to the dividend payment. How does thisaffect the asset price? By the usual arbitrage arguments if t−d is the time

42

immediately before the dividend is paid and t+d is the time immediately afterwe have

S(t+d ) = S(t−d )− dyS(t−d )

= (1− dy)S(t−d )

where S(t) is the value of the underlying asset at time t. There is a jumpin the value of S, in that the value of the underlying asset is discontinuousacross the dividend date. What effect will this have on the option price?Again in order to eliminate any possible arbitrage opportunities, the valueof the option must be continuous as a function of time across the dividenddate. In which case the value of the option immediately before the dividendpayment must be the same as the value immediately after (recall that theowner of the option does not receive the dividend) thus

V (S(t−d ), t−d ) = V (S(t+d ), t

+d ).

This brings to light something interesting in the relationship between Sand t. In the Black-Scholes methodology S and t are considered to beindependent variables although S is clearly dependent on t, this is possibleas we consider every possible value of S at a particular point in time, ratherthan just one. This is because given the random movement of stock prices,S can take any value.

As S is not fixed across the dividend date, in fact we know that S(t+d ) =(1−dy)S(t−d ) then there is no contradiction in the above relationship betweenV (t−d ) and V (t+d ), as we have

V (S, t−d ) = V (S(1− dy), t+d ).

So the option value is continuous across the dividend date even if the valueof the underlying is discontinuous and the relationship is given above.

6.3.1 Example: pricing a European call option when there is onedividend payment

As usual we work back from the known conditions at expiry to derivethe option value at a previous time. Moving backwards from expiry to justafter the dividend payment time, namely t+d . At the dividend payment datewe implement the jump condition

C(S, t−d ) = C(S(1− dy), t+d ).

then value the option back to any desired time t using these option valuesas new final conditions. Essentially you have to solve the Black Scholesequation twice

• Once for T > t > td with C(S, T ) = max(S −X, 0).

• Once for td > t > 0 with C(S, td) = C(S(1− dy), t+d )

43

We can simplify the methodology slightly by the following procedure:

Let C(S, t) be the standard European call option and Cd(S, t) be anoption on an underlying asset paying discrete payments. If there is just onepayment at td then from above we have

Cd(S, t) = C(S, t;X), t+d ≤ t < T

Cd(S, t−d ) = Cd(S(1− dy), t

+d )

= C(S(1− dy), t+d ;X).

For t < t−d there is a shortcut to using the BSE. Prior to the dividendpayment the value of the call option is just subject to a scaling in S, i.eS 7→ S(1 − dy) as such C(S(1 − dy), t;X) still satisfies the Black-Scholesequation. As this is equal to the value of Cd(S, t) at td then the two are alsoequivalent for t < td. Thus if we can find the value of C(S(1 − dy), t;X)then we’ll know the value of Cd for t < td and hence for all t.

At expiry,

C(S(1− dy), T ;X) = max(S(1− dy)−X, 0)

= (1− dy)max(S − X

1− dy, 0)

which is the same as (1−dy) calls with an exercise price of X/(1−dy), hencewe now know the value of the call option for 0 ≤ t < td, which is

Cd(S, t) = (1− dy)C(S, t;X

1− dy).

In conclusion

Cd(S, t) =

{

(1− dy)C(S, t; X1−dy ) for 0 ≤ t < td

C(S, t;X) for td ≤ t < T.

which can be valued using the standard option pricing formulae.

Remark: Note that if the underlying asset pays a dividend then thisdecreases the value of the call option, since the holder of the the option doesnot receive the dividend yet a dividend payment reduces the value of theunderlying asset. Correspondingly the value of a put option increases whendividends are paid.

7 American Options

American options are options which can be exercised at any time toreceive S−X or X −S for call and put options respectively. Unfortunatelythis gives rise to a non-linear problem and as such it is not possible ingeneral to derive explicit formulae like those for European options.

44

7.1 American put options

The first problem is to decide at which values of S and t it is optimalto exercise. To consider the problem, treat the American put option as aEuropean put option with the extra early exercise feature. At expiry theearly exercise condition has no effect, as the value of the American put,P (S, t), is given by

P (S, T ) = max(X − S, 0).

Moving back from expiry there will, however, be certain values of S forwhich

X − S > PBS(S, t)

where PBS(S, t) is the value of the European put option derived from theBlack-Scholes PDE. In this case the holder of the option would exercise theirright and receive X − S. The major problem is to locate the value of S atwhich it becomes optimal to exercise the option, if we call this value Sf (t)then we have

P (S, t) =

{

X − S for S ≤ Sf (t)PBS(S, t) for S > Sf (t).

This is known as a free boundary problem and they are very difficult to solve.More formally when pricing American options the Black-Scholes equationbecomes an inequality, which is an equality when it is optimal to hold theoption:

Sf (t) < S <∞ : P > X − S,∂P

∂t+

1

2σ2S2∂

2P

∂S2+ rS

∂P

∂S− rP = 0,

and an inequality when it is optimal to exercise

0 ≤ S < Sf (t) : P = X − S,∂P

∂t+

1

2σ2S2 ∂

2P

∂S2+ rS

∂P

∂S− rP < 0.

The boundary conditions are as follows:

P (S, T ) = max(X − S, 0),

P (Sf (t), t) = X − Sf (t),

P (S, t) → 0 as S → ∞.

where the first and third are as for a European put option but the secondis one of the conditions on the free boundary, Sf (t). There is another, lessobvious condition at S = Sf (t), known as the smooth pasting conditionwhich ensures that the ∆ (= ∂P/∂S) is smooth across the early exerciseboundary, namely

∂P

∂S(Sf (t), t) = −1.

If this were not the case then there are arbitrage possibilities (see the expla-nation in Wilmott, Howison and Dewynne, 1995, p. 110-111)

In general, numerical methods must be used to price American put op-tions. There is one exception though and that is the perpetual case.

45

80

82

84

86

88

90

92

94

96

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

PSfrag replacements

T − t

S

P = X − S

P = PBS(S, t)

S = Sf (t)

Figure 8: The position of Sf (t) and the valuation regions for an Americanput option.

7.2 American call options

If the underlying asset pays no dividends then pricing an American calloption is remarkably simple. Recall that these options can be described asa European call with the added feature that it is possible to exercise at anytime to receive S −X. However, consider a portfolio

Π = S − C

where C is a European call option, so at expiry

Π = S −max(S −X, 0) ≤ X

hence, for t < TS − C ≤ Xe−r(T−t)

orC ≥ S −Xe−r(T−t) ≥ S −X

thus it is never optimal to early exercise this American call option andso the price is the same as for a European call option. This is not the casewhen the underlying asset is paying continuous dividends as one can observefrom the option profiles in figures 9 and 10. In the continuous dividend casethe problem becomes similar to that for the American put, with analogousboundary conditions.

0 < S < Sf (t) : C > S−X, ∂C

∂t+1

2σ2S2∂

2C

∂S2+(r−D)S

∂C

∂S−rC = 0,

and with the BSE being an inequality when it is optimal to exercise

Sf (t) < S <∞ : C = S−X, ∂C

∂t+1

2σ2S2∂

2C

∂S2+(r−D)S

∂C

∂S−rC < 0.

46

0

5

10

15

20

25

30

35

75 80 85 90 95 100 105 110 115 120 125

C(S

,t)

S

t = 0

t = T

Figure 9: The value of C(S, t) at t = 0, . . . and t = T on a non-dividendpaying asset - note how the value of C(S, t) does not drop below S −X.

The boundary conditions are as follows:

C(S, T ) = max(S −X, 0),

C(Sf (t), t) = Sf (t)−X,

C(0, t) = 0.

and there is also an equivalent smooth pasting condition:

∂C

∂S(Sf (t), t) = 1.

7.3 Perpetual options

These are options with an infinite life, corresponding to T → ∞. In thiscase we look for solutions (for American puts) of the form P (S) only. TheBlack-Scholes equation then becomes the following ODE:

12σ

2S2 d2P

dS2+ rS

dP

dS− rP = 0.

This is a form of Euler’s equation, and hence has solutions of the formP = ASα, where

12σ

2α(α − 1) + rα− r = 0,

and solving this (quadratic) equation for α yields two values, α = 1 orα = − 2r

σ2.

The conditions to be satisfied are that

P (S → ∞) → 0, P (S = Sf ) = X − Sf ,dP

dS(S = Sf ) = −1.

47

0

10

20

30

40

50

60

70

80

70 80 90 100 110 120 130 140 150 160 170 180

PSfrag replacements

C(S,t)

S

Figure 10: The value of C(S, t) at t = 0, . . . and t = T on a dividend payingasset - note how the value of C(S, t) can drop below S −X.

The first of these conditions indicates we can discard the α = 1 solution,and so

P = AS−2r/σ2 .

The smooth pasting conditions lead to

X − Sf = AS−2r/σ2

f ,

−1 = −2r

σ2AS

−2r/σ2−1f ,

which lead to the location of the free boundary

Sf =X

σ2

2r + 1.

Again it is possible to value a perpetual call options with dividends byusing simple ODE theory together with the relevant boundary conditions.

48

8 Interest rate models and bonds

So far we have assumed that interest rates are constant or at best knownfunctions of time; this is clearly not the case in reality. Although the effectsof interest-rate changes on option prices are generally small (because of theirshort lifetime), many other securities with much longer durations can be verysusceptible to interest rate changes.

8.1 Bonds

A bond is a contract, paid for up-front, that yields a known amount ona known date in the future, the maturity date, t = T . The bond may alsopay a known cash dividend (the coupon) at fixed times during the life ofthe contract. If there are no coupons, the bond is known as a zero-couponbond. Bonds may be issued by both governments and companies to raisecapital, and the up-front premium can be regarded as a loan.

A typical question related to this is: how much should I pay now to get

a guaranteed $1 in 10 years’ time?

In the simple case of a zero-coupon bond V (t) which pays Z at t = Twe may equate the return to that of a bank deposit, i.e.

dV = r(t)V dt,

with V (T ) = Z. If the interest rate is deterministic, then

V (r, t;T ) = Ze−∫ T

tr(τ)dτ .

IF the bond pays a single coupon (‘dividend’) amount Z1 at t = T1 < T ,then the net effect is that of an additional ‘mini’ bond maturing at t = T1,in addition to the main bond. The value overall for t < T1 is then modifiedas follows:

V (r, t < T1;T1;T ) = Ze−∫ T

tr(τ)dτ + Z1e

−∫ T1t r(τ)dτ ,

whilst for t > T1 the value is unaffected, i.e.

V (r, t > T1;T1;T ) = Ze−∫ T

tr(τ)dτ .

8.2 Stochastic interest rates

In the same way we developed a model for the asset price as a lognormalwalk, suppose that the interest rate r is governed by a stochastic differentialequation

dr = w(r, t)dX + u(r, t)dt.

The functional form of w(r, t) and u(r, t) determines the behaviour of thespot rate r.

49

8.3 The bond-pricing equation

Pricing a bond is trickier than pricing an option, since there is no un-derlying asset with which to hedge: we cannot go out and buy an interestrate of 5%. Instead, we hedge with bonds of different maturity dates.

We set up a portfolio comprising two bonds with different maturities T1and T2, namely V1 and V2 respectively. We hold one V1 bond and −∆ of V2bonds, and so

Π = V1 −∆V2.

Using the above stochastic differential equation for the interest rate, inconjunction with Ito’s Lemma, gives the change in this portfolio in a timedt:

dΠ =∂V1∂t

dt+∂V1∂r

dr + 12w

2 ∂2V1∂r2

dt

−∆(∂V2∂t

dt+∂V2∂r

dr + 12w

2 ∂2V2∂r2

dt).

From this we see that the choice

∆ =∂V1∂r

/∂V2∂r

eliminates the random component of dΠ. We then have

dΠ =

(

∂V1∂t

+ 12w

2∂2V1∂r2

− ∂V1∂r

/∂V2∂r

(∂V2∂t

+ 12w

2 ∂2V2∂r2

)

)

dt

= rΠdt

= r(V1 − V2∂V1∂r

/∂V2∂r

)dt,

where we have used arbitrage arguments to set the return on the portfolioto equal the risk-free (spot) rate.

Gathering all the V1 terms on the left-hand-side and all the V2 terms onthe right-hand-side yields

(∂V1∂t

+ 12w

2 ∂2V1∂r2

− rV1)/∂V1∂r

= (∂V2∂t

+ 12w

2 ∂2V2∂r2

− rV2)/∂V2∂r

This is one equation in two unknowns, however the left-hand-side is a func-tion of T1 but not T2, and the right-hand-side is a function of T2 but not T1.The only way that this is possible is for both sides to be independent of thematurity date. Thus

(∂V

∂t+ 1

2w2∂

2V

∂r2− rV )/

∂V

∂r= a(r, t)

for some function a(r, t). It is convenient to write

a(r, t) = w(r, t)λ(r, t) − u(r, t)

for given w(r, t) and u(r, t), but λ(r, t) unspecified.The zero-coupon bond pricing equation is therefore

∂V

∂t+ 1

2w2 ∂

2V

∂r2+ (u− λw)

∂V

∂r− rV = 0,

subject to the final condition V (r, T ) = Z, and generally V (r → ∞, t) → 0;the boundary condition on r = 0 is generally dependent on λ, u and w.

50

8.4 The market price of risk

Consider now in more detail the unknown function λ(r, t). In a timestepdt the bond V changes in value by

dV = w∂V

∂rdX + (

∂V

∂t+ 1

2w2 ∂

2V

∂r2+ u

∂V

∂r)dt.

From the PDE derived above for V we can rewrite the bracketed term, giving

dV = w∂V

∂rdX + (wλ

∂V

∂r+ rV )dt,

or

dV − rV dt = w∂V

∂r(dX + λdt).

The presence of dX indicates this is not a risk-less portfolio. The right-hand-side is the excess return above the risk-free rate for accepting a certain levelof risk. In return for taking the extra risk the portfolio profits by an extraλdt per unit of extra risk dX. The function λ is called the market priceof risk.

8.5 The Vasicek model

This takes the form

dr = (η − γr)dt+ β12 dX

The model is tractable - explicit formulae exist. For a zero-coupon bond,value is

eA(t;T )−rB(t;T )

Substituting into the PDE, and considering the O(r0) and O(r) termsseparately (see examples 8), yields

B =1

γ(1− e−γ(T−t))

A =1

γ2(B − T + t)(ηγ − λγβ

12 − 1

2β)−βB2

4γ

Model is mean reverting (which is good), but interest rates can go negative(which is bad).

8.6 Cox, Ingersoll, Ross Model

The CIR model takes the form

dr = (η − γr)dt+√αrdX

Spot rate is mean reverting, and remains positive if η > α/2. For a zero-coupon bond, value is again of the form

A(t;T )e−rB(t;T )

51

Substituting into the PDE, and considering the O(r0) and O(r) termsseparately, yields

dA

dt= ηA(r)B(t)

dB

dt= (γ + λ

√αB + 1

2αB2 − 1

with A(T ) = 1, B(T ) = 0The solution is given by

A(t) =

{

2ξe(ξ+ψ)(T−t)/2

(ξ + ψ)(eξ(T−t) − 1) + 2ξ

}2η/α

B(t) =2(eξ(T−t) − 1)

(ξ + ψ)(eξ(T−t) − 1) + 2ξ

where ψ = γ + λ√α, ξ =

√

ψ2 + 2α

52

9 Barrier options

Barrier options are path dependent options - they have a payoff that de-pends on the realised asset price via its level; certain aspects of the contractare triggered if the asset price becomes too high or too low.

Example: An up-and-out call option pays off the usual max(S − X, 0)at expiry unless at any time previously the underlying asset has traded at avalue Su or higher. If the asset reaches this level (obviously from below) thenit is ‘knocked out’, becoming worthless. As well as ‘out’ options, there arealso ‘in’ options which only receive a payoff if a level is reached, otherwisethey expire worthless.

Barrier options are useful for a number of reasons, including

(i) The purchaser has precise views about the direction of the market.

(ii) The purchaser wants the payoff from an option, but does not wantto pay for the upside potential, believing that the movement of theunderlying will be limited prior to expiry.

(iii) These options are cheaper than their corresponding vanilla ‘cousins’.

9.1 Pricing barrier options with PDEs

Although barrier options are path dependent, this dependency can bequite readily incorporated into the PDE methodology - we only need toknow whether or not the barrier has been triggered; we do not need anyother information about the path. This is in contrast to other more exotictypes of option, such as Asian options (where, for example, the payoff maydepend on the average value of the underlying during the lifetime of theoption contact).

Consider the value of a barrier contract before the barrier has been trig-gered. The value still satisfies the Black-Scholes equation

∂V

∂t+ 1

2σ2S2∂

2V

∂S2+ rS

∂V

∂S− rV = 0.

9.2 Out barriers

If the underlying reaches the barrier in an ‘out’ barrier option, then thecontract becomes worthless. This leads to the boundary condition

V (Su, t) = 0 for t < T,

for an up-and-out barrier option with the barrier level at S = Su. We mustsolve the Black-Scholes equation for 0 ≤ S ≤ Su with the above conditionon S = Su and the usual payoff condition if the barrier is not triggered.

If we have a down-and-out option with a barrier at Sd, then we solve forSd ≤ S <∞ with

V (Sd, t) = 0,

and the relevant final condition at expiry.

53

9.3 In barriers

An ‘in’ barrier option only has a payoff if the barrier is triggered. Ifthe barrier is not triggered, then the option expires worthless. The value inthe option is the potential to hit the barrier. If the option is an up-and-incontract then on the upper barrier the contract must have the same valueas a vanilla contract (say Vv(S, t)). We then have

V (Su, t) = Vv(Su, t) for t < T.

A similar boundary condition holds for a down-and-in option.The contract we receive when the barrier is triggered is a derivative itself,

and therefore the ‘in’ option is a second-order contract. We must thereforesolve for the vanilla option first, before solving for the value of the barrieroption.

9.4 Down-and-out call options

Consider the down-and-out call option with barrier level Sd below thestrike price X. The function Vv(S, t) is the Black-Scholes value of the cor-responding vanilla option.

It is easy (see examples 8) to show that

V = S1−2r/σ2Vv(A/S, t)

also satisfies the Black-Scholes equation for any A (constant).From this we can infer the value of a down-and-out call option, namely

V (S, t) = Vv(S, t)− (S

Sd)1−2r/σ2Vv(S

2d/S, t).

We can confirm this is the solution (from the above we know this will satisfythe Black-Scholes equation). If we substitute S = Sd, we find V (Sd, t) = 0.Since S2

d/S < X for S > Sd, the value of Vv(S2d/S, T ) is zero; thus the final

condition is satisfied.

9.5 Down-and-in call options

The relationship between an ‘in’ barrier option and an ‘out’ barrier op-tion (with the same payoff and barrier level) is very simple

in + out = vanilla.

If the ‘in’ barrier is triggered, then so is the ‘out’ barrier, so whether or notthe barrier is triggered, we still obtain the vanilla payoff at expiry. Thus thevalue of a down-and-in call option is

(S

Sd)1−2r/σ2Vv(S

2d/S, t).

54