MATH39032 Mathematical Modelling of Finance

Lecture notes prepared by Professor Peter Duck

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

CONTACT DETAILS: Office - 2.141 Turing Building E-mail - [email protected]

EXAMPLES SHEETS:

There will be 8 examples sheets to accompany the course. The first feedback/examples class will be in Week 2. I would strongly encourage you to attend these classes - they will be a useful forum for feedback. I would also encourage 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). The past three years exam papers will provide you with a good overview of the standard 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 knowledge of the area, including more on stochastic processes and Martingale the- ory, these courses are highly recommended. Some introductory books for 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 Financial Derivatives, 2nd Ed., Academic Press. ISBN: 0125153929

For a more financial look at options and derivatives the following is • excellent and is the course text for finance students (usually MBA or PhD) 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 and ISBN 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 this course) tried to make money from their theories is:

Lowenstein, F., 2002: When Genius Failed: The Rise and Fall of Long Term 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 Truth About 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 making • money 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 finance • theory and the major players, I thoroughly recommend:

Bernstein, P., 1995: Capital Ideas: The Improbable Origins of Wall Street, The Free Press. ISBN: 0029030129. Finally, Peter Bernstein has also written an excellent book on risk and its 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 as such the title of this module may be a little broader than what is actually studied. The main focus of this course is on the financial instruments known as options and, most importantly, how to calculate their value. This proved to be a remarkably interesting problem both mathematically and financially and one which took centuries to satisfactorily solve. The earliest known use of options was by the Greek philosopher Thales in 600 B.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 then options have been traded the world over although rarely in a regulated manner and, until 1973, their values were primarily calculated by guesswork. In 1973, Fischer Black and Myron Scholes, together with help from Bob Merton derived the Black-Scholes partial differential equation which describes the value of an option, V (which is dependent on the time since the option 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 appropriate set of boundary conditions, as follows

∂V 1 ∂2V ∂V + σ2S2 + rS rV = 0. (1) ∂t 2 ∂S2 ∂S − This equation changed the face of option pricing, not only did it earn Nobel prizes for Merton and Scholes in 1997 (Black having died in 1995) but it 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 more about these shortly). The first organised options exchange also opened in Chicago in 1973 and the volume of trade in options has increased from 5.7m contracts in 1974, to 673m in 2000, 3,899,068,670 in 2010, 4,111,275,659 in 2013, 4,265,368,807 in 2014. This boom in the, occasionally mathematically complex, derivatives markets has also led to many investment banks actively recruiting 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 from the 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 and options.

Underlying assets Throughout this course we will be considering options on underlying assets, 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 share price, but can also be a commodity price or an exchange rate. Underlying assets have an associated drift (µ) and volatility (σ), where the drift is the expected percentage increase over a certain period of time and the volatility is the measure of uncertainty of this return. For example, one would expect a fledgling technology share to have a higher volatility than a blue-chip company like AT&T.

Interest rates and the time value of money The famous mantra from courses on economics and finance is ‘a dollar today is worth more than a dollar tomorrow’, as it is possible to invest your dollar in a risk-free investment, like a US government bond, today and tomorrow it will be worth more than a dollar. There are a few possible conventions 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 is invested today, at the end of the year it will be worth A(1 + r). However, if it is invested for only 6 months at the same quoted yearly rate and the new total is then invested for another 6 months we have a compounding process r 2 such that after 1 year A will be worth A(1 + 2 ) . This can be extended to the continuously compounded case, which is used throughout this course, in r m which the money is reinvested m times giving A(1 + m ) after one year. As m then →∞ r (1 + )m er. (2) m → Thus, an amount, A, invested at a continually compounded rate of r for t years is worth Aert. This convention is used primarily because it makes the mathematics far simpler than using the cumbersome discretely compounded formula. This time value of money is mainly used in its inverted form to determine what an expected amount in the future is worth today. This is 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 r(T t) $100e− − . For more realistic models, where the interest rate can be a function of time, an amount A invested for t years is worth

t AeR0 r(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 to exchange currency at a particular rate, investors or companies often want to arrange a deal for some time in the future. A forward contract is one in which one party (in the long position) agrees to buy an underlying asset at a certain price (the delivery price F ) at a certain future time, T . The other party (in the short position) agrees to sell the asset at time T at this price F .

5 PAYOFF

F S T

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

A futures contract is a standardised forward contract in that parties can enter into long or short positions on an exchange where the delivery prices and dates are set by the exchange. In a futures contract the opposing parties (long and short) do not necessarily know each other and so the exchange 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 will have to pay an American supplier $1m and they want to hedge against unfavourable exchange rate movements. On February 5th 2008 the bank is offering a six month forward exchange rate of 0.8. Ryanair then take the long position in the forward contract and on August 5th will buy $1m for 800,000 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 enter into the contract. Obviously in this example if in 6 months the dollar has strengthened against the euro (i.e. the exchange rate is higher than 0.8) then Ryanair are pleased because they have locked in a lower exchange rate. Obviously if the reverse has happened (the exchange rate has dropped) then they will lose out.

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

S F T − where F is the delivery price. Similarly, for the party in the short position the payoff is F S − T 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 definition of an option.

6 PAYOFF

F S 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 lose out in the above forward contract. Ideally, they would like it so that if the exchange rate drops then they can walk away from the contract and buy their $1m for less than 800,000 Euro. This is exactly the freedom which an option would give them, the would have the option of whether to take the delivery price or to walk away and take the favourable current price. Obviously, if the exchange rate has increased then they can still pay 800,000 Euro for their $1m. Crucially, unlike in forward contracts the party who buys the option must pay some premium to obtain the option. The main thrust 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 not the obligation to buy the underlying, S, at (or before) a certain date, T , for a certain price, known as the exercise (or strike) price, X. Definition (Put options) A put option gives the holder the right, but not the obligation to sell the underlying, S, at (or before) a certain date, T , for a 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 exercise conditions and are not considered fully in this course. This course mainly deals 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 the value of the underlying asset at time t. If the strike price of the option is X then at the expiry of the option, t = T , the holder of the option has the right, but not the obligation, to buy the underlying, of value S at t = T at this price, X. Clearly if S > X then the holder of the option would exercise the option and buy the underlying (worth S) for X. This would yield the holder of the option a profit of S X. If S X then there is no point in − ≤ exercising the option as the holder can buy the underlying on the market for 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 by P (S,t). Again the option has a strike price of X and at expiry the holder of the option has the right, but not the obligation, to sell the underlying asset at this price. With a put option at expiry (t = T ) S < X then the holder of the option would exercise as she can sell the underlying for more than 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 underlying for 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 PROFIT

95 105 S −10

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 to buy 100 Hewlett Packard shares with a strike price, X, of $95. The current stock price is $100, the expiration date is 6 months and the cost of the call option to buy one option is $10. Plot a graph of profit from buying the option against underlying asset value in 6 months time. If, at expiry, S < 95 the investor would choose not to exercise the option as there is no point in buying the share for $95 when you can buy it on the market 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, yielding a profit of (S 95) 100 1000. The diagram below shows the profit at − × − expiry for different levels of the underlying S. Note that if 95

1.5 Why options? Why is the options market such a big deal. Options appeal to three main types 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 currency in a certain amount of time then options provide insurance for cases where there are adverse market moves. It is sometimes possible to hedge against movements in a market which will affect your business. As an example, if jet fuel goes up then it costs BA more money to run their aircraft, so if they buy call options in jet fuel then if the price goes up then they make money to offset their operating losses. If the price has gone down then they’re happy because 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 he can obtain more leverage by using options. Consider the following example: an investor feels that Barclays is likely to increase in value over the next three months and has $5000 to invest. The current stock price is $20 and call options with a strike price of $25 are available for three months at the cost of $1. Consider two possible alternatives, the stock price goes up to $35 or down to $15.

Table 1.1: profit and loss from the two dif- ferent strategies when speculating on Barclays stock Stock price at expiry Strategy $15 $35 Buy shares -$1250 $3750 Buy call options -$5000 $45000

Investing the $5000 in shares, enables the investor to buy 250 shares at $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 increase or decrease in the underlying can realise big profits or losses. In this case the $5000 buys 5000 call options with a strike price of $25. If the price goes down to $15 then none of these would be exercised and the investor would have lost their entire $5000. If, on the other hand, the price has gone up to $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 theory and will be defined and expanded more fully later. However, an arbitrage opportunity is one in which it is possible to lock in a risk free profit. An arbitrageur will look for anomalies in the market, which by definition, exist for only short periods of time and lock in these profits. A good example is in spread betting, if one book has the spread 8-10 and another 12-14 then you can buy at 10 in the first and sell at 12 in the second and make a risk-free profit.

1.6 No arbitrage principle Definition (Arbitrage opportunity) An arbitrage opportunity is one in which 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 pricing methodology of this course is that arbitrage opportunities do not exist, or only appear for a very short time. Also associated with the concept of no instantaneous risk-free profit is the idea of a risk-free rate. This is the rate of return (or interest rate) that an investor receives upon making a risk-free investment, such as investing in US treasury bonds.

1.6.2 Determining forward prices To determine the correct delivery or forward price on a forward contract it 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 $40 and the current risk-free rate is 5%, also assume that the current delivery price is $43. Now if an arbitrageur is sharp then she can spot an arbitrage opportunity here, she can borrow the $40 to buy a share today and go short in 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

0.05 3/12 40e × = 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 forward contract and owning a share) is known as a portfolio. The principle of no arbitrage says that as soon as this opportunity arises then 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 such a high delivery price, so very quickly the price will drop to a fair level and arbitrage opportunities will vanish. Finding this ‘fair price’ is simple for forward contracts. Let the current underlying asset price be S, the risk-free rate be r, the time to expiry be T 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 to show that there are arbitrage opportunities if F

F = SerT

Note that this derivation does not assume or predict anything about the movement of the underlying asset S but is able to predict a correct value for 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 the put-call parity which will eventually enable you to calculate the value of a European put using the value of the call. Consider, two portfolios, A and B which at t = 0 consist of the following:

Portfolio A: A European call option, C(S,t), with exercise price X • rT and expiry date T ; and an amount of cash Xe− . Portfolio B: A European put option, P (S,t), with exercise price X • and expiry date T ; and one share in the underlying S.

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

r(T t) C(S,t)+ Xe− − = P (S,t)+ S. (3) where t is the current time. It is possible to determine some fairly loose bounds for European options using a similar approach (see examples sheet 2) but for accurate valuation it is necessary to model the movements of the underlying 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 accurately mirror the real life movements of underlying asset prices. Fortunately, al- though dealing with stochastic variables it is often possible to transform the problem into a deterministic one. In this case it is achieved through the em- ployment of Itˆo’s lemma, which can be seen to be the analogue of Taylor’s theorem 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 a particular company or commodity is reflected in the current price. So it is impossible to gain an edge by having studied the historical data or by examining in intricate detail company reports or the financial press. This means 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 have to 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 the true efficiency of markets. In any case, for this course we will assume that they are efficient.

12 The idea of a process randomly evolving without memory fits very nicely into the theory of basic random processes. Markov processes are processes which 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 next movement will be. Example 2.1 One of the simplest random processes, which also happens to be Markovian, is the simple symmetric discrete random walk. Consider a stochastic process, St starting from S0 = 0, at each point in time the change in position of S call it ∆Si is given by

+1 with probability 1 ∆S = 2 i 1 with probability 1  − 2 After n steps in this walk then the position of S is given by

n Sn∆t = ∆Si Xi=1 This is Markov because each movement is independent of what has occurred before. Also, it so happens that, in this case, by the Central Limit Theorem Sn∆t is Normally distributed with a mean of 0 and a variance of n∆t (thus a standard deviation of √n∆t). In general if the variance of each of the 2 2 increments ∆Si is σ then the variance of Sn∆t is σ n∆t.

13 2.2 Brownian motion and the model of stock price movement This discrete random walk is one possible way of modelling stock price movements. 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, the Wiener process. The stochastic model of Brownian motion was, obviously, defined to mirror the movement of tiny particles in water but has applications in more fields than that, one being in option pricing theory. There is a lot of rig- orous mathematics surrounding such processes but as regards this course a heuristic overview will be provided.

Definition (Brownian motion) A real valued stochastic process Wt is a Brownian motion (or Wiener process) under a probability measure P if 1. For each t 0 and s> 0 the random variable W W (often termed ≥ t+s − t dW ) is distributed Normally with mean 0 and variance s.

2. For each n and for any times 0 t t t the random ≤ 0 ≤ 1 ≤ ··· ≤ n variables W W are independent. { tr − tr−1 } 3. W0 = 0 (this is merely a convention, it can start from any point). 4. W is continuous in t 0. t ≥ This is basically just an extension of the discrete simple random walk to continuous time. The change W W over a very small period of time t+s − t dt is often denoted by dW and obviously is distributed accordingly(mean of zero and variance of dt). Brownian motions obviously have very strange paths and, in fact, the expected length of path followed by W in a any time interval is infinite, this will make calculus difficult on Brownian motions (see section 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 motion Wt will not model stock prices very well for two main reasons: The general trend of stock prices is upwards whereas the expected • movement of Brownian motion is to stay at the same level.

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

2.2.1 Generalised Brownian motion The first of these concerns can be over come easily. On top of the random increments generating by the Brownian motion term it is possible to add in deterministic terms. When dealing with stock prices there is a general upward drift, call this µ and so if the stock price is denoted by St then we have the following stochastic differential equation

S S = dS = µdt + σdW (4) t+dt − t

14 where σ is just scaling the effect of Brownian motion. So, in this case over a period of time dt the stock price increases from St by an amount µdt plus an unknown amount σdW where dW is a Brownian motion. The distribution of 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 on Brownian motion but still has the problem that there is nothing to prevent St 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 scaled depending 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 likely to move by a cent than one worth 2c. More importantly, S 0 as as soon ≥ as 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 on Examples 2.

3 Basics of Stochastic calculus and Itˆo’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 value of f over a small change in S, δS

df 1 d2f f(S + δS)= f(S)+ δS + (δS)2 + O((δS)3). (6) dS 2 dS2 Usually, as δS 0 then the (δS)2 term disappears enabling the usual →df representation of dS as f(S + δS) f(S) lim − δS 0 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 hence E[dW 2] 0, in fact E[dW 2] = dt and so this term will not disappear as ≥ dt 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 mathematician Kiyoshi Itˆo. There is a huge amount of theory behind Itˆocalculus but we shall refer only 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. Itˆo’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, ) R ∞ × then F is also a stochastic process given by

∂f ∂f 1 ∂2f ∂f dF = a(S,t) + + b2(S,t) dt + b(S,t) dW (7) ∂S ∂t 2 ∂S2 ∂S   (Note: the process for S can also be described in its integral form

t t S = S0 + a(S,s)ds + b(S,s)dW Z0 Z0 where again the problem in evaluation comes with the random dW term, a problem which can be overcome by defining the Itˆointegral which is, obviously, closely linked to Itˆo’s lemma.) It is worth noting that if one assumes that as dt 0, dW 2 dt and → → dtdW = o(dt) it is possible to obtain the above result from performing a Taylor series in two dimensions (see examples sheet 2). So, as a rule of thumb to arrive at Itˆo’s lemma assume that dW 2 dt as dt 0 a result → → which does not take a great leap of faith to assume to be correct as we know that E[dW 2]= dt. Example 2.2 If dS = adt + bdW where a and b are constants then what process is followed by G = S2? Well from Itˆo’s lemma ∂g ∂g 1 ∂2g ∂g dG = a + + b2 dt + b dW ∂S ∂t 2 ∂S2 ∂S   2 2 ∂f ∂ f ∂f and in this case f(S,t)= S and so ∂S = 2S, ∂S2 = 2 and ∂t = 0 thus the process followed by G is as follows

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

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

∂f ∂f 1 ∂2f ∂f dF = µS + + σ2S2 dt + σS dW (8) ∂S ∂t 2 ∂S2 ∂S  

16 Π

X − X 12

XX1 2 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 option price, 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 any function of these variables (within very broad constraints). This is a crucial building block for the derivation of the Black-Scholes partial differential equation.

3.1 Aside - portfolios of options It is possible, see the many examples on Examples 3, to combine different options to achieve a desired payoff. Example (Bull spread): The portfolio consists of buying (long) one call option with exercise price X1 and writing (short) another call option with the same expiry but a larger exercise price X2. Thus, the portfolio, Π is of value Π= C(S,t; X ) C(S,t; X ) 1 − 2 and the value of the portfolio (the payoff) at expiry will be

Π = max(S X , 0) max(S X , 0) − 1 − − 2 and so the portfolio pays nothing for S < X , S X for X X . This will be used by an investor who thinks 2 − 1 2 that the underlying asset will increase but is happy to take a known amount (X X ) if the increase is substantial - note that this makes the portfolio 2 − 1 cheaper 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 for underlying assets but is the one which is used for the Black-Scholes analysis, which still remains the most popular model for practitioners. From here we now proceed to derive the Black-Scholes PDE. The main problem with the process followed by the function of S, F , is that there is still a random term present which makes constructing a PDE somewhat problematic. The solution to this is to create a new function g which is completely deterministic. Consider a function

g = f ∆S − where ∆ is an as yet unknown parameter which is constant across a time period 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

2 ∂f ∂f 1 2 2 ∂ f ∂f dg = µS + + σ S 2 dt + σS dW ∆[µSdt + σSdW ] ∂S ∂t 2 ∂S ∂S −   2 ∂f ∂f ∂f 1 2 2 ∂ f = σS ∆ dW + µS ∆ + + σ S 2 dt ∂S − ∂S − ∂t 2 ∂S       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 derive their PDE

4.2 The Black-Scholes PDE Notation:

S is the current value of the underlying asset, can also be denoted by • St especially in SDEs but the t is usually dropped. t is the time elapsed since the option was created and the option expires • at 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 startup company’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 from • a 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 have the volatility dependent on time but more complicated models will provide much more challenging problems. It is permitted to short sell the underlying asset, i.e. sell an asset that • you don’t actually own. There are no transaction costs, all securities are perfectly divisible and • trading takes place continuously. There are no dividends, or equivalent, paid out during the lifetime of • the option (this will be relaxed at a later date). There are no riskless arbitrage opportunities. Any that do exist exist • only for a very short period of time. The risk free rate r is constant. This can also be trivially relaxed • to let r be a function of time. In practice, especially for long-term derivatives, the interest rate is itself modelled stochastically. As the option price V (S,t) depends on the underlying asset, S, which follows geometric Brownian motion

dS = µSdt + σSdW (9) and by Itˆo’s lemma we have ∂V ∂V 1 ∂2V ∂V dV = µS + + σ2S2 dt + σS dW (10) ∂S ∂t 2 ∂S2 ∂S   Now construct a portfolio which consists of an option and short in ∆ of the underlying. 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 ∂V ∂V ∂V 1 ∂2V dΠ= σS ∆ dW + µS ∆ + + σ2S2 dt (13) ∂S − ∂S − ∂t 2 ∂S2       19 The amount of the underlying which the holder of the portfolio is short selling, ∆, has not yet been set. However, if ∆ is selected, as before, such that ∂V ∆= , (14) ∂S then the stochastic differential equation for dΠ becomes deterministic, as the coefficient of the dW term is now identically zero. Thus this portfolio is perfectly hedged as it provides a guaranteed return over a designated time period. Obviously, this assumes that it is possible to change the value of ∂V ∆ continuously, because as time evolves the value of ∂S is changing. With this continuous rebalancing of the portfolio the expression for dΠ is now

∂V 1 ∂2V dΠ= + σ2S2 dt. (15) ∂t 2 ∂S2   However, this portfolio is perfectly hedged, in that it yields a risk-less value after 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 = rΠdt then one could make a risk-free profit by 6 either 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

∂V ∂V 1 ∂2V r V S dt = + σ2S2 dt. (16) − ∂S ∂t 2 ∂S2     On dividing equation (16) by dt one obtains

∂V 1 ∂2V ∂V + σ2S2 + rS rV = 0. (17) ∂t 2 ∂S2 ∂S − 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 boundary conditions will determine which type of derivative we are evaluating.

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

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

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

20 The linear operator • ∂ 1 ∂2 ∂ = + σ2S2 + rS r LBS ∂t 2 ∂S2 ∂S − 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 the return on a bank deposit which are the last two terms. For a Euro- pean option these will be the same, though they are not necessarily for an American option.

For many types of options it is not possible to obtain closed-form • analytic values but more often than not numerical procedures must be employed. In this lecture course, though, emphasis will remain on analytic 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 shown that, given suitable boundary conditions, this is indeed the case. First, in order to determine the type of boundary conditions required it is necessary to 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 ∂u = ∂x2 ∂t which typically models the evolution of heat along a bar. As they are second order in x and only first order in t parabolic equations usually require two boundary conditions in x (or S in the Black-Scholes case) and just the one in t. It is important to notice here that in the heat conduction equation the ∂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 parabolic equation whilst the Black-Scholes equation is backwards parabolic. The difference 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 (or the final time) and so it makes sense that this problem gives rise to a back- wards parabolic type. The heat conduction (or diffusion) equation requires a known distribution of heat on a bar (or equivalent system) at t = 0 and then models how the heat distribution evolves as time moves forwards. As such the system requires initial conditions - thus is a forwards parabolic type. 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 the notion of characteristics. Characteristics are families of curves along which information moves or across which discontinuities may occur. The trick is to 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 along these characteristic curves. Definition (Characteristic curve) A curve Γ is a characteristic for a general second order PDE if, for a general PDE in x and t,

∂t b √b2 4ac ± − = 0 ∂x − 2a 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 solution giving ∂t b = . ∂x 2a In the case of the heat conduction equation where b = 0 then this reduces to ∂t = 0 ∂x 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 = a V (S,t) = Vb(t) on S = b V (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 to do this for different types of options we’ll consider three cases: the standard European 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 final condition 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 from then on. Hence for a call option, however small the strike price X is, this scenario 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 clearly → ∞ the call option is more and more likely to be exercised and in comparison to the 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 with numerical procedures where a large, but finite, limit is put on S (Smax say). In which case, more accurate conditions are required. One possibility is to assume that the option will be exercised at expiry, receiving S plus whatever else contributes to the option’s value as time moves backwards. In this way write 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

df dt + rS + r(S + f(t)) = 0 − df dt = rf(t) which on solving gives f(t)= Aert substituting in the known time constraint from (19) we get

rT A = Xe− − and so the boundary condition for large S is

r(T t) C(S,t) S Xe− − as S . (21) → − →∞

European put option, P (S,t): The case for a put option is far more straightforward. Again determining the final condition is trivial as a result of the discussion in Chapter 1, so we have P (S, T ) = max(X S, 0). (22) − The conditions for particular values of S are extensions of the above arguments for calls, only more routine. When S = 0 at a particular time then by the nature of the underlying process then it will stay at 0 until expiry. Hence the put option will definitely be exercised and thus worth X 0= X at expiry. A guaranteed amount of money, in this case X, to be − r(T t) received at time T is worth Xe− − at time t and hence

r(T t) P (0,t)= Xe− − (23)

As S becomes very large then the put options will certainly not be exercised as 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 other conditions are essential for numerical schemes as well as giving us more information about the option prices. Cash-or-nothing/binary options: Cash-or-nothing call (put) options (denoted CC(S,t) or CP (S,t)) are options 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 amount A, 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 )= A (S X) H − where (.) is known as the Heaviside function. The Heaviside function H is defined as follows 0 if x< 0 (x)= H 1 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 completely discontinuous 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 conduction or diffusion equation and how to adapt these techniques to solve the Black- Scholes equation for some standard option pricing problems. Before doing that we will study the analytic solutions to the valuation problems and a few more key features of options. The Black-Scholes formulae for the price of European call and put options are as follows:

r(T t) C(S,t)= SN(d ) Xe− − N(d ) (25) 1 − 2 r(T t) P (S,t)= Xe− − N( d ) SN( d ) (26) − 2 − − 1 where

1 2 log(S/X) + (r + 2 σ )(T t) d1 = − σ√T t −1 2 log(S/X) + (r 2 σ )(T t) d2 = − − . σ√T t − (27) and x 1 1 s2 N(x)= e− 2 ds (28) √2π Z−∞ which we recognise as the cumulative distribution function for a Normal distribution. Note that these expressions satisfy the put call parity and so by calculating one it is routine to calculate the other, also note that the boundary conditions at S = 0 and S are satisfied. →∞

25 For those students interested in probability it may be worth noting that N(d2) is the probability that the option will be exercised, i.e. S > X at expiry. 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 a PDE which is a close relative of the diffusion equation, the payoff function max(S X, 0) gradually diffuses out as time moves backwards. The same − is also true for a cash or nothing option even though the payoff is in fact discontinuous.

Example

The price of an asset (today) is £5. Find the value of a put and a call option, both with an exercise price of £6, and both with expiration dates in 9 months time. The risk-free interest rate is 3% per annum (fixed) and the 1 volatility (constant) is 10% per (annum) 2 .

Solution r = .03, T t = 0.75, σ = .1, S = 5, X = 6. − Using the formulae.d = 1.8021, d = 1.8888 1 − 2 − Then

N(d ) = N( 1.8021) = N( 1.80) .21[N( 1.80) N( 1.81)] 1 − − − − − − = 0.0359 .21 (0.0359 0.0351) − × − = 0.0357

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

r(T t) P = C S + Xe− − . − 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) will show that using the known expressions for the values of call and put options, that they have the following ∆’s ∂C ∆ = = N(d ) C ∂S 1 ∂P ∆ = = N(d ) 1 P ∂S 1 − What does this mean? During the lifetime of the option ∆ varies between 0 for out of the money calls (puts) and 1 ( 1) for in the money calls (puts) and − very 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 the underlying asset and so any slight movement in the option price value will be offset by a roughly equivalent movement in ∆ of the underlying. Clearly the portfolio will have to be rebalanced as regularly as possible to have a perfect hedge. In practise the number of times a portfolio can be hedged will be limited by transaction costs. For example, looking at the graph for the value of a cash-or-nothing call option we immediately see a problem with the delta-hedging strategy underlying the Black-Scholes analysis. If ∆ is ∂C/∂S then as t T then → the ∆ ranges from 0 away from S = X to approaching close to S = X. ∞ Thus as the underlying asset price moves, huge amounts of the underlying will 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 with the movement of the asset price. There are hedge parameters (also known as, somewhat loosely, as The Greeks) for each of the principle parameters in the Black-Scholes model, namely:

The sensitivity to the decay of time of any option V is known as the • theta and is defined as ∂V Θ= ∂t The sensitivity to the volatility is known as the vega and is defined • as ∂V = V ∂σ The sensitivity to interest rates is known as rho and, unsurprisingly • to be ∂V ρ = ∂r Finally, the sensitivity of the ∆ to the underlying asset is known as • gamma and is defined as follows

∂2V Γ= ∂S2

Often these hedge parameters are used to see what would happen if there was a small change in one of the parameters, this is important as both r and 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 very difficult to know for definite is the volatility, σ. There are several conventions for calculating the volatility of an underlying asset. One would perhaps assume that the best way is to look at the volatility of past returns and use this 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 use the market prices for options to back-out the volatility, using a suitable

27 iterative procedure such as Newton-Raphson, the only unknown being σ itself. 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 well not be constant for a given r, S, and t. So, not only is it dependent on time but also on the exercise and asset prices. Such a result is often termed the volatility smile although many other shapes can be observed depending on the market conditions such as a frown, wry smile etc. This is another example 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 be a second-order parabolic equation, in the majority of cases this equation is also a linear one. This chapter is concerned with the nature of these equations, focusing attention on the heat conduction equation and then extending to the Black-Scholes equation itself.

5.1 Properties of the Heat conduction equation The heat conduction equation takes the form

∂u ∂2u = ∂τ ∂x2 where τ is the time and x is the spatial variable, it normally models the flow of heat or its diffusion and has been extensively studied over the years. Its fundamental properties are as follows

It is a second order linear PDE, as such if u and u are solutions then • 1 2 so is a1u1 + a2u2 for any constants a1, a2 It is a parabolic equation and it’s characteristics are simply along the • lines τ = c (where c is a constant) which means that this is where infor- mation propagates along. So any change in the boundary conditions is 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 x ) for x = x. − 0 0 6 Crucially, the heat conduction (diffusion) equation is a smoothing out process, and as such discontinuities in the boundary or initial (final) condi- tions can be catered for. Recall that in the Black-Scholes equation the final conditions are often discontinuous. Example By way of demonstration consider the following initial value prob- lem. ∂u ∂2u = ∂τ ∂x2

28 Srgreplacements PSfrag

0.9

0.8 τ = 0.1

0.7

0.6

) 0.5 x,τ ( δ u

0.4

0.3

0.2

0.1

0 -10 -5 0 5 10 x

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

for τ > 0 and 0. Consider a special solution, about which more is said later 1 x2/4τ u(x,τ)= u (x,τ)= e− (29) δ 2√πτ for 0. Now we verify that this indeed satisfies the −∞ ∞ PDE. ∂u x x2/4τ = − e− ∂x 4τ 3/2√π 2 2 ∂ u 1 x2/4τ x x2/4τ = − e− + e− ∂x2 4τ 3/2√π 8τ 5/2√π 2 ∂u 1 x2/4τ x x2/4τ = − e− + e− . ∂τ 4τ 3/2√π 8τ 5/2√π So, this is a solution which is well behaved except at one instance, the initial point in time τ = 0. At this point when x = 0 then u (x, 0) = 0 but at 6 δ x = 0 it has infinite value. This clearly has discontinuous initial conditions yet gives rise to a, reasonably, well behaved solution. What more can we say about this special solution to the heat conduction equation? Well, ∞ u (x,τ)dx = 1, τ. δ ∀ Z−∞ This function has all of the heat initially (τ = 0) concentrated at x = 0 and then 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 2 2 for the Normal distribution ( 1 e (x µ) /2σ ) and the value of u (x,τ). σ√2π − − δ 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 probability density function of the future position of a particle following a Brownian motion (√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 as a generalised function. It’s definition is as a linear map representing the limit of a function whose effect is confined to a smaller and smaller interval but remains finite. An informal definition is to consider a function 1/2ǫ, x ǫ f(x)= | |≤ 0, x >ǫ  | | and as ǫ 0 the graph becomes taller and narrower but at all points → ∞ f(x)dx = 1 Z−∞ regardless of the value of ǫ although for all x = 0, f(x) 0 as ǫ 0. In 6 → → general the delta function δ(x) is the limit as ǫ 0 of any one-parameter → family of functions δǫ with the following properties for each ǫ, δ (x) is piecewise smooth; • ǫ ∞ δǫ(x)dx = 1; • −∞ forR each x = 0, limǫ 0 δǫ(x) = 0. • 6 → Note that the specific solution to the heat conduction equation uδ satisfies the above constraints with τ replaced by ǫ. The best way to look at the delta function is to only consider its integral which we know to be 1 and which smooths out the function’s bad behaviour, especially when x = 0 and ǫ 0 (of τ 0). When concentrating on the integral form we can see the → → delta function as a test function, in that

∞ ∞ δ(x)φ(x)dx = lim δǫ(x)φ(x)dx ǫ 0 Z → Z −∞ −∞ ǫ ǫ − ∞ = lim δǫ(x)φ(x)dx + δǫ(x)φ(x)dx + δǫ(x)φ(x)dx ǫ 0 → ǫ ǫ Z−∞ ǫ Z− Z  = lim φ(0) δǫ(x)dx ǫ 0 ǫ →  Z−  = φ(0)

In fact, for any a, b > 0

b δ(x)φ(x)dx = φ(0) a Z− and, as importantly, for any x0

∞ δ(x x )φ(x)dx = φ(x ) − 0 0 Z−∞

30 1 2ε

x 2ε

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

H(x)

H’(x) = 0 1

H’(x) = 8

x H’(x) = 0

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 also known as a test function. Other properties concern its links with the Heaviside function as

x δ(s)ds = (x) H Z−∞ and conversely, ′(x)= δ(x) H where, as before 0 if x< 0 (x)= H 1 if x 0  ≥

31 5.3 Transforming the Black-Scholes equation Consider the Black-Scholes equation ∂V 1 ∂2V ∂V + σ2S2 + rS rV = 0 ∂t 2 ∂S2 ∂S − make the following three substitutions S S = Xex(or x = log ) X τ σ2 t = T (or τ = (T t)) 1 2 − 2 σ 2 − V = Xv(x,τ) (30) thus ∂V ∂v dτ ∂v σ2 Xσ2 ∂v = X = X . = ∂t ∂τ dt ∂τ − 2 − 2 ∂τ ∂V ∂v dx ∂v 1 x ∂v = X = X = e− ∂S ∂x dS ∂x S ∂x 2 x x 2 2x 2 ∂ V ∂ ∂V e− ∂ x ∂v e− x ∂ v x ∂v e− ∂ v ∂v = = e− = e− e− = ∂S2 ∂S ∂S X ∂x ∂x X ∂x2 − ∂x X ∂x2 − ∂x         which leads to ∂v ∂2v ∂v = + (k 1) kv ∂τ ∂x2 − ∂x − where r k = 1 2 2 σ ∂v Now attempt to remove the ∂x and v terms by introducing the substitution v(x,τ)= eαx+βτ u(x,τ) where α and β are constants to be determined, this gives ∂v ∂u = βeαx+βτ u + eαx+βτ ∂τ ∂τ ∂v ∂u = αeαx+βτ u + eαx+βτ ∂x ∂x ∂2v ∂u ∂2u = α2eαx+βτ u + 2αeαx+βτ + eαx+βτ ∂x2 ∂x ∂x2 which gives ∂u ∂u ∂2u ∂u βu + = α2u + 2α + + (k 1) αu + ku ∂τ ∂x ∂x2 − ∂x −   ∂u to remove the ∂x and u terms we require 1 α = (k 1) −2 − 1 β = (k + 1)2. −4

32 Thus, 1 1 (k 1)x (k+1)2τ V (S,t)= Xe− 2 − − 4 u(x,τ) (31) and ∂u ∂2u 0 To transform the final conditions, or the payoff from the option we have for a 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) − or v(x, 0) = max(ex 1, 0) − and so, from (31)

1 1 (k+1)x (k 1)x u(x, 0) = u (x) = max e 2 e 2 − , 0 (32) 0 −   and similarly for a put option

1 1 (k 1)x (k+1)x u(x, 0) = u (x) = max e 2 − e 2 , 0 (33) 0 −   As such the Black-Scholes equation has been converted to the heat conduc- tion equation for

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 ∂2u = , x,τ> 0 ∂τ ∂x2 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 end is 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 dU ∂ξ 1 3/2 dU = = xτ − ∂τ dξ ∂τ −2 dξ

∂u dU ∂ξ 1/2 dU = = τ − ∂x dξ ∂x dξ and 2 2 ∂ u 1/2 d 1/2 dU 1 d U = τ − τ − = τ − ∂x2 dξ dξ dξ2   and so, replacing x/√τ by ξ and multiplying by τ gives the ODE

d2U 1 dU + ξ = 0 dξ2 2 dξ the boundary conditions become

U(0) = 1 and U( ) = 0 ∞ with this second condition catering for both the initial condition and u(x,τ) → 0 as x . Integrating the ODE once gives →∞ dU ξ2/4 = Ce− dξ

(C constant) and on solving gives

ξ s2/4 U(ξ)= C e− ds + D Z0 (D constant). Upon substituting the boundary conditions, first U(0) = 1 gives 1= D

34 and then U( ) = 0 gives ∞

∞ s2/4 0= C e− ds + 1 Z0 but we know that ∞ s2/4 e− ds = √π Z0 thus 1= C√π − Thus, ξ 1 s2/4 U(ξ)= e− ds + 1 −√π Z0 but ξ = ∞ ∞ − Z0 Z0 Zξ hence 1 ∞ s2/4 ∞ s2/4 U(ξ)= e− ds e− ds + 1 −√π − Z0 Zξ  or 1 ∞ s2/4 U(ξ)= e− ds 1 + 1 −√π − −  Zξ  so 1 ∞ s2/4 U(ξ)= e− ds √π Zξ and on replacing ξ by its definition we get

1 ∞ s2/4 u(x,τ)= e− ds √π Zx/√τ The key trick being that to solve the equation we replace two variables (x and τ) by just one (ξ) and then the problem reduces to an ODE. Even more useful is the next example, for

∂u ∂2u 0 where ∞ u(x,τ)dx = k, τ where k is a constant. ∀ Z−∞ Choosing the normalised case where k = 1 we search for a solution of the 1/2 form u(x,τ) = τ − U(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 (or alternatively u(x,τ)= o(1/x) as x ). On transforming the derivatives | |→∞ we get

∂u 1 3/2 1/2 dU 1 3/2 1 3/2 1 3/2 dU = τ − U + τ − . xτ − = τ − U ξτ − ∂τ −2 dξ − 2 −2 − 2 dξ

∂u 1/2 dU ∂ξ 1 dU = τ − = τ − ∂x dξ ∂x dξ and 2 2 ∂ u 1/2 d 1 dU 3/2 d U = τ − τ − = τ − ∂x2 dξ dξ dξ2   which gives d2U 1 dU 1 + ξ + U = 0 dξ2 2 dξ 2 or d2U d 1 + ξU = 0. dξ2 dξ 2   Integrating both sides wrt ξ gives dU 1 + ξU = C dξ 2 where C is a constant. Now as ξ , U = o(1/ξ) so the LHS is o(1) thus →∞ this constant C = 0. So then on solving the ODE ξ2/4 U(ξ)= Ae− , where A is a constant. Putting in the condition we have

∞ 1/2 x2/4τ A τ − e− dx = 1 Z−∞ however, set x′ = x/√τ and we get dx = √τdx′ and the equation becomes

∞ x′2/4 A e− dx′ = 1 Z−∞ and so using the usual result 2A√π = 1 thus 1 A = 2√π and so, 1/2 1 x2/4τ u(x,τ)= τ − e− 2√π   or 1 x2/4τ u(x,τ)= e− 2√πτ which is precisely the special solution uδ from section 5.1, equation 29. ξ2/4 [Note: The derivation in Wilmott where he states that U(ξ)= Ce− +D is wrong.]

36 5.4.1 How similarity solutions work The reason why the above similarity solution worked was because the governing equations and the boundary conditions do not change under the scalings x λx and τ λ2τ, where λ R. In particular consider new → → 2 ∈ variables x∗ = λx and τ ∗ = λ τ, these clearly satisfy the heat-conduction equation and in Example 5.1 the boundary conditions become u(x∗, 0) = 0 and u(0,τ ∗)=1 for any λ. Combining these two results to get a variable which is independent of λ the 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 boundary and initial conditions are invariant under the scaling transformation. It is also possible to multiply U(ξ) by a function of τ as in Example 5.2 because as the heat-conduction equation is linear it is invariant under the scaling u µu. → In general with similarity solutions a good practical test to see if they’ll work is to search for a solution of the form u = τ αU(xτ β) in the hope that the PDE will reduce to an ODE in ξ = xτ β and the boundary conditions will be satisfied. For the heat conduction equation then in all cases β = 1/2 − but 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 Example 5.2, 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 to solve ∂u ∂2u 0 with initial data u(x, 0) = u0(x) and there are suitable growth conditions at ax2 x (usually lim x u(x,τ)e− = 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

u (x)= ∞ u (ξ)δ(ξ x)dξ 0 0 − Z−∞ we recall that the fundamental solution to the initial value problem from 5.2 is 1 s2/4τ u (s,τ)= e− δ 2√πτ and has initial value u (s, 0) = δ(s). Noting that because u (s x,τ) = δ δ − u (x s,τ) we have δ −

1 (s x)2/4τ u (s x,τ)= e− − δ − 2√πτ

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

u (s x, 0) = δ(s x). δ − − Now comes the important bit, hence, for each s the function

u (s)u (s x,τ) 0 δ − as a function of x and τ with s held fixed, satisfies the heat conduction equation as u0(s) is simply a constant. Now using the fact that the diffusion equation is linear we can add together linear combinations of these solutions for any s all the way from to and obtain another solution to the heat −∞ ∞ conduction equation, namely

1 ∞ (x s)2/4τ u(x,τ)= u (s)e− − ds 2√πτ 0 Z−∞ and the initial data is

u(x, 0) = ∞ u (s)δ(s x)ds = u (x). 0 − 0 Z−∞ What does all this mean? Well, this solution satisfies the heat conduction equation for all x and for τ > 0 and is also satisfies the initial conditions for all initial conditions u0(x). It is also possible to show that this solution is unique (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 the heat conduction equation, where u(x, 0) = u (x) for τ > 0 and 0 where 1 1 (k+1)x (k 1)x u(x, 0) = u (x) = max e 2 e 2 − , 0 . 0 −   By using the known general solution to this problem we have

1 1 1 ∞ (k+1)s (k 1)s (x s)2/4τ u(x,τ)= max[e 2 e 2 − , 0]e− − ds 2√πτ − Z−∞  but u0(x)=0 for x< 0 hence

1 1 1 ∞ (k+1)s (k 1)s (x s)2/4τ u(x,τ)= [e 2 e 2 − ]e− − ds. 2√πτ − Z0   38 We make another change of variable, define s x x′ = − . √2τ

1 1 1 1 1 ∞ (k+1)(x′√2τ +x) x′2 ∞ (k 1)(x′√2τ +x) x′2 u(x,τ)= e 2 − 2 dx′ e 2 − − 2 dx′ . √2π { x/√2τ − x/√2τ } Z− Z− Completing the square and removing the terms not dependent on x′ yields

1 1 2 2 (k+1)x+ 4 (k+1) τ 1 1 e ∞ (x′ (k+1)√2τ)2 u(x,τ) = e− 2 − 2 dx′ √2π x/√2τ 1 1 2Z− 2 (k 1)x+ 4 (k 1) τ 1 1 e − − ∞ (x′ (k 1)√2τ)2 e− 2 − 2 − dx′ − √2π x/√2τ Z− = I I (37) 1 − 2 Noting that the expression for the cumulative Normal distribution is as follows x 1 1 s2 N(x)= e− 2 ds √2π Z−∞ we transform the dependent variable, x′, once again to 1 x = x′ (k + 1)√2τ 1 − 2 and 1 x = x′ (k 1)√2τ 2 − 2 − in I1 and I2 respectively and then

1 1 1 1 (k+1)x+ (k+1)2τ (k 1)x+ (k 1)2τ u(x,τ)= e 2 4 N(d ) e 2 − 4 − N(d ) 1 − 2 where x 1 d1 = + (k + 1)√2τ √2τ 2 x 1 d2 = + (k 1)√2τ. √2τ 2 − Transforming the variables back using the usual definitions

1 1 (k 1)x (k+1)2τ V (S,t) = Xe− 2 − − 4 u(x,τ) S x = log X   σ2 τ = (T t) 2 − 2r k = σ2 gives the following expression for the value of the European call option

r(T t) C(S,t)= V (S,t)= SN(d ) Xe− − N(d ), 1 − 2

39 where

1 2 log(S/X) + (r + 2 σ )(T t) d1 = − σ√T t −1 2 log(S/X) + (r 2 σ )(T t) d2 = − − . σ√T t − The European put can be valued in a similar manner or, more easily, by use of the put-call parity, equation. Either approach yields the following expression for its value, P (S,t)

r(T t) P (S,t)= Xe− − N( d ) SN( d ). − 2 − − 1 (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 of some form another, fortunately it is relatively easy to incorporate dividend payments into the option pricing methodology. Of even greater use is that the methods used for pricing options on dividend paying stocks can be rou- tinely extended to deal with other, analogous, problems such as options on foreign currency where the dividend becomes the foreign risk-free interest rate and options on commodities where the dividend becomes minus the cost of carry. There are two main ways of modelling dividend payments: as continuous and as discrete.

6.2 Continuous constant dividend yield This is the simplest payment structure, assume that over a period of time dt the underlying asset pays out a dividend DSdt in that D is the proportion of the value of the asset paid out over this period of time. D is considered to be constant and independent of t though the size of the dividend will obviously depend on S which is dependent on t. How does this affect our model? By using arbitrage arguments (see Examples 1) a payment of dividends results in the underlying asset price dropping by the value of the dividend. Hence with a continuous dividend the stochastic process is given by

dS = (µ D)Sdt + σSdW. − To derive the governing PDE a similar process is followed but although the portfolio is still Π= V ∆S − in this case dΠ= dV ∆(dS + DSdt) − as the holder of the portfolio receives the dividend as well. Proceeding as for the non-dividend case ∂V ∂V ∂2V dV = dt + dS + 1 σ2S2 dt, ∂t ∂S 2 ∂S2 and so ∂V ∂V ∂2V dΠ= dt + ( ∆)[(µ D)Sdt + σSdW ] ∆DSdt + 1 σ2S2 dt. ∂t ∂S − − − 2 ∂S2 Setting ∂V ∆= ∂S

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

∂V ∂2V ∂V dΠ = dt + 1 σ2S2 dt DS dt ∂t 2 ∂S2 − ∂S = rΠdt ∂V = r(V S )dt − ∂S which gives the following PDE

∂V 1 ∂2V ∂V + σ2S2 + (r D)S rV = 0. (38) ∂t 2 ∂S2 − ∂S − The standard Black-Scholes equation derived earlier in the course is just a special case of this equation for the case when D = 0. Valuing European call and put options is reasonably straightforward, the main difference being that r is replaced by r D but only in the coefficient of the ∂C/∂S. To − account for this slight difference introduce

D(T t) V (S,t)= e− − V1(S,t) so that we now have ∂V 1 ∂2V ∂V 1 + σ2S2 1 + (r D)S 1 (r D)V = 0 ∂t 2 ∂S2 − ∂S − − 1 which is the Black-Scholes equation only with r replaced by r D and with − the same final conditions. As such

D(T t) r(T t) C(S,t)= e− − SN(d ) Xe− − N(d ) 10 − 20 where

1 2 log(S/X) + (r D + 2 σ )(T t) d10 = − − σ√T t − 1 2 log(S/X) + (r D 2 σ )(T t) d20 = − − − . σ√T t − 6.3 Discrete dividend payments When considering options where the underlying is a stock then a more realistic model is to treat dividends as being paid at discrete points in time. This is because most companies pay out their dividends periodically, every quarter, every six months, every year etc. Assume, as a starting point, that just one dividend payment is made during the lifetime of the option. Assume that this is paid at time td and can be expressed as a percentage of the level of the underlying, i.e. as dyS where 0 d < 1. Thus the holder of the asset receives a payment of d S at ≤ y y td where S is the asset price prior to the dividend payment. How does this affect the asset price? By the usual arbitrage arguments if td− is the time

42 + immediately before the dividend is paid and td is the time immediately after we have

+ S(t ) = S(t−) d S(t−) d d − y d = (1 d )S(t−) − y d where S(t) is the value of the underlying asset at time t. There is a jump in the value of S, in that the value of the underlying asset is discontinuous across the dividend date. What effect will this have on the option price? Again in order to eliminate any possible arbitrage opportunities, the value of the option must be continuous as a function of time across the dividend date. In which case the value of the option immediately before the dividend payment must be the same as the value immediately after (recall that the owner of the option does not receive the dividend) thus

+ + V (S(td−),td−)= V (S(td ),td ). This brings to light something interesting in the relationship between S and t. In the Black-Scholes methodology S and t are considered to be independent variables although S is clearly dependent on t, this is possible as we consider every possible value of S at a particular point in time, rather than 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(td )= (1 dy)S(td−) then there is no contradiction in the above relationship between − + V (td−) and V (td ), as we have

+ V (S,t−)= V (S(1 d ),t ). d − y d So the option value is continuous across the dividend date even if the value of the underlying is discontinuous and the relationship is given above.

6.3.1 Example: pricing a European call option when there is one dividend payment As usual we work back from the known conditions at expiry to derive the option value at a previous time. Moving backwards from expiry to just + after the dividend payment time, namely td . At the dividend payment date we implement the jump condition

+ C(S,t−)= C(S(1 d ),t ). d − y d then value the option back to any desired time t using these option values as new final conditions. Essentially you have to solve the Black Scholes equation twice

Once for T>t>t with C(S, T ) = max(S X, 0). • d − Once for t >t> 0 with C(S,t )= C(S(1 d ),t+) • d d − y 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 an option on an underlying asset paying discrete payments. If there is just one payment at td then from above we have

+ Cd(S,t) = C(S,t; X), td t < T +≤ C (S,t−) = C (S(1 d ),t ) d d d − y d = C(S(1 d ),t+; X). − y d

For t

C(S(1 d ), T ; X) = max(S(1 d ) X, 0) − y − y − X = (1 d ) max(S , 0) − y − 1 d − y which is the same as (1 d ) calls with an exercise price of X/(1 d ), hence − y − y we now know the value of the call option for 0 t

X (1 dy)C(S,t; ) for 0 t

7 American Options

American options are options which can be exercised at any time to receive S X or X S for call and put options respectively. Unfortunately − − this gives rise to a non-linear problem and as such it is not possible in general 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 optimal to exercise. To consider the problem, treat the American put option as a European put option with the extra early exercise feature. At expiry the early 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 for which X S > P (S,t) − BS where PBS (S,t) is the value of the European put option derived from the Black-Scholes PDE. In this case the holder of the option would exercise their right and receive X S. The major problem is to locate the value of S at − which it becomes optimal to exercise the option, if we call this value Sf (t) then we have X S for S S (t) P (S,t)= − ≤ f P (S,t) for S>S (t).  BS f This is known as a free boundary problem and they are very difficult to solve. More formally when pricing American options the Black-Scholes equation becomes an inequality, which is an equality when it is optimal to hold the option: ∂P 1 ∂2P ∂P S (t) X S, + σ2S2 + rS rP = 0, f ∞ − ∂t 2 ∂S2 ∂S − and an inequality when it is optimal to exercise ∂P 1 ∂2P ∂P 0 S

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 96

94

92

90 P = PBS (S,t)

S 88

PSfrag replacements 86

84 S = Sf (t) 82 P = X S −

80 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 T t −

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

7.2 American call options If the underlying asset pays no dividends then pricing an American call option is remarkably simple. Recall that these options can be described as a European call with the added feature that it is possible to exercise at any time 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 < T r(T t) S C Xe− − − ≤ or r(T t) C S Xe− − S X ≥ − ≥ − thus it is never optimal to early exercise this American call option and so the price is the same as for a European call option. This is not the case when the underlying asset is paying continuous dividends as one can observe from the option profiles in figures 9 and 10. In the continuous dividend case the problem becomes similar to that for the American put, with analogous boundary conditions.

∂C 1 ∂2C ∂C 0 S X, + σ2S2 +(r D)S rC = 0, f − ∂t 2 ∂S2 − ∂S − and with the BSE being an inequality when it is optimal to exercise

∂C 1 ∂2C ∂C S (t)

46 35

30

25

20 C(S,t) 15

10 t = 0

5 t = T

0 75 80 85 90 95 100 105 110 115 120 125 S Figure 9: The value of C(S,t) at t = 0, . . . and t = T on a non-dividend paying 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(S (t),t) = S (t) X, f f − C(0,t) = 0. and there is also an equivalent smooth pasting condition: ∂C (S (t),t) = 1. ∂S f

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

d2P dP 1 σ2S2 + rS rP = 0. 2 dS2 dS − This is a form of Euler’s equation, and hence has solutions of the form P = ASα, where 1 σ2α(α 1) + rα r = 0, 2 − − and solving this (quadratic) equation for α yields two values, α = 1 or 2r α = 2 . − σ The conditions to be satisfied are that dP P (S ) 0, P (S = S )= X S , (S = S )= 1. →∞ → f − f dS f −

47 80

70

60 ) 50 S,t (

C 40

30

20

PSfrag replacements 10

0 70 80 90 100 110 120 130 140 150 160 170 180 S

Figure 10: The value of C(S,t) at t = 0, . . . and t = T on a dividend paying asset - 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 2r/σ2 P = AS− . The smooth pasting conditions lead to

2r/σ2 X S = AS− , − f f

2r 2r/σ2 1 1= AS− − , − −σ2 f which lead to the location of the free boundary X S = . f σ2 2r + 1 Again it is possible to value a perpetual call options with dividends by using 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 known functions of time; this is clearly not the case in reality. Although the effects of interest-rate changes on option prices are generally small (because of their short lifetime), many other securities with much longer durations can be very susceptible to interest rate changes.

8.1 Bonds A bond is a contract, paid for up-front, that yields a known amount on a known date in the future, the maturity date, t = T . The bond may also pay a known cash dividend (the coupon) at fixed times during the life of the contract. If there are no coupons, the bond is known as a zero-coupon bond. Bonds may be issued by both governments and companies to raise capital, 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 = T we 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

T R r(τ)dτ V (r, t; T )= Ze− t .

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 modified as follows:

T T1 R r(τ)dτ R r(τ)dτ V (r, t < T1; T1; T )= Ze− t + Z1e− t , whilst for t > T1 the value is unaffected, i.e.

T R r(τ)dτ V (r, t > T1; T1; T )= Ze− t .

8.2 Stochastic interest rates In the same way we developed a model for the asset price as a lognormal walk, suppose that the interest rate r is governed by a stochastic differential equation dr = w(r, t)dX + u(r, t)dt. The functional form of w(r, t) and u(r, t) determines the behaviour of the spot 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 interest rate of 5%. Instead, we hedge with bonds of different maturity dates. We set up a portfolio comprising two bonds with different maturities T1 and T , namely V and V respectively. We hold one V bond and ∆ of V 2 1 2 1 − 2 bonds, and so Π= V ∆V . 1 − 2 Using the above stochastic differential equation for the interest rate, in conjunction with Ito’s Lemma, gives the change in this portfolio in a time dt: ∂V ∂V ∂2V dΠ= 1 dt + 1 dr + 1 w2 1 dt ∂t ∂r 2 ∂r2 ∂V ∂V ∂2V ∆( 2 dt + 2 dr + 1 w2 2 dt). − ∂t ∂r 2 ∂r2 From this we see that the choice ∂V ∂V ∆= 1 / 2 ∂r ∂r eliminates the random component of dΠ. We then have ∂V ∂2V ∂V ∂V ∂V ∂2V dΠ= 1 + 1 w2 1 1 / 2 ( 2 + 1 w2 2 ) dt ∂t 2 ∂r2 − ∂r ∂r ∂t 2 ∂r2   = rΠdt ∂V ∂V = r(V V 1 / 2 )dt, 1 − 2 ∂r ∂r where we have used arbitrage arguments to set the return on the portfolio to equal the risk-free (spot) rate. Gathering all the V1 terms on the left-hand-side and all the V2 terms on the right-hand-side yields ∂V ∂2V ∂V ∂V ∂2V ∂V ( 1 + 1 w2 1 rV )/ 1 = ( 2 + 1 w2 2 rV )/ 2 ∂t 2 ∂r2 − 1 ∂r ∂t 2 ∂r2 − 2 ∂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 the maturity date. Thus ∂V ∂2V ∂V ( + 1 w2 rV )/ = a(r, t) ∂t 2 ∂r2 − ∂r 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 ∂2V ∂V + 1 w2 + (u λw) rV = 0, ∂t 2 ∂r2 − ∂r − 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 timestep dt the bond V changes in value by

∂V ∂V ∂2V ∂V dV = w dX + ( + 1 w2 + u )dt. ∂r ∂t 2 ∂r2 ∂r From the PDE derived above for V we can rewrite the bracketed term, giving ∂V ∂V dV = w dX + (wλ + rV )dt, ∂r ∂r or ∂V dV rV dt = w (dX + λdt). − ∂r 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 level of risk. In return for taking the extra risk the portfolio profits by an extra λdt per unit of extra risk dX. The function λ is called the market price of risk.

8.5 The Vasicek model This takes the form

1 dr = (η γr)dt + β 2 dX − The model is tractable - explicit formulae exist. For a zero-coupon bond, value is A(t;T ) rB(t;T ) e − Substituting into the PDE, and considering the O(r0) and O(r) terms separately (see examples 8), yields

1 γ(T t) B = (1 e− − ) γ −

1 1 βB2 A = (B T + t)(ηγ λγβ 2 1 β) γ2 − − − 2 − 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

rB(t;T ) A(t; T )e−

51 Substituting into the PDE, and considering the O(r0) and O(r) terms separately, yields dA = ηA(r)B(t) dt dB = (γ + λ√αB + 1 αB2 1 dt 2 − with A(T ) = 1, B(T ) = 0 The solution is given by

2η/α 2ξe(ξ+ψ)(T t)/2 A(t)= − (ξ + ψ)(eξ(T t) 1) + 2ξ ( − − ) 2(eξ(T t) 1) B(t)= − − (ξ + ψ)(eξ(T t) 1) + 2ξ − − where ψ = γ + λ√α, ξ = ψ2 + 2α p

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 contract are 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 a value Su or higher. If the asset reaches this level (obviously from below) then it is ‘knocked out’, becoming worthless. As well as ‘out’ options, there are also ‘in’ options which only receive a payoff if a level is reached, otherwise they 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 want to pay for the upside potential, believing that the movement of the underlying 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 be quite readily incorporated into the PDE methodology - we only need to know whether or not the barrier has been triggered; we do not need any other information about the path. This is in contrast to other more exotic types of option, such as Asian options (where, for example, the payoff may depend on the average value of the underlying during the lifetime of the option contact). Consider the value of a barrier contract before the barrier has been trig- gered. The value still satisfies the Black-Scholes equation

∂V ∂2V ∂V + 1 σ2S2 + rS rV = 0. ∂t 2 ∂S2 ∂S −

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

V (Su,t)=0 for t

53 9.3 In barriers An ‘in’ barrier option only has a payoff if the barrier is triggered. If the barrier is not triggered, then the option expires worthless. The value in the option is the potential to hit the barrier. If the option is an up-and-in contract then on the upper barrier the contract must have the same value as a vanilla contract (say Vv(S,t)). We then have

V (Su,t)= Vv(Su,t) for 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 therefore solve for the vanilla option first, before solving for the value of the barrier option.

9.4 Down-and-out call options

Consider the down-and-out call option with barrier level Sd below the strike 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

1 2r/σ2 V = S − Vv(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

S 1 2r/σ2 2 V (S,t)= Vv(S,t) ( ) − Vv(Sd /S,t). − Sd We can confirm this is the solution (from the above we know this will satisfy the Black-Scholes equation). If we substitute S = Sd, we find V (Sd,t) = 0. 2 2 Since Sd /S < X for S>Sd, the value of Vv(Sd /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 not the barrier is triggered, we still obtain the vanilla payoff at expiry. Thus the value of a down-and-in call option is

S 1 2r/σ2 2 ( ) − Vv(Sd /S,t). Sd

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