Kris Glover Analyzing PDEs

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THE ANALYSIS OF PDES ARISING IN

NONLINEAR AND NON-STANDARD

OPTION PRICING

A thesis submitted to the University of Manchester

for the degree of Doctor of Philosophy

in the Faculty of Engineering and Physical Sciences

2008

Kristoffer John Glover

School of Mathematics


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Contents

Abstract 11

Declaration 12

Copyright Statement 13

Acknowledgements 14

Dedication 15

1 Introduction 16

1.1 Evidence of increased interest in liquidity . . . . . . . . . . . . . . . . 17

1.2 A brief history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.3 Derivative pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.3.1 European options . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.3.2 Arbitrage pricing . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.3.3 The Feynman-Kac representation theorem . . . . . . . . . . . 24

1.3.4 From Feynman-Kac to Black-Scholes . . . . . . . . . . . . . . 25

1.3.5 American options . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.3.6 Optimal stopping problems . . . . . . . . . . . . . . . . . . . 29

1.3.7 Free-boundary problems . . . . . . . . . . . . . . . . . . . . . 30

1.4 Supply and demand economics . . . . . . . . . . . . . . . . . . . . . . 32

1.5 Liquidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

1.5.1 Defining liquidity . . . . . . . . . . . . . . . . . . . . . . . . . 35

1.5.2 Measuring liquidity . . . . . . . . . . . . . . . . . . . . . . . . 361.6 Price formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

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1.7 Option pricing in illiquid markets: a literature review . . . . . . . . . 40

1.8 Introduction to perturbation methods . . . . . . . . . . . . . . . . . . 45

1.9 Layout of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2 The Modelling Framework 48

2.1 Technical asides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.1.1 Markovian processes . . . . . . . . . . . . . . . . . . . . . . . 53

2.1.2 Applicability of Itos formula . . . . . . . . . . . . . . . . . . . 54

2.2 Alternative models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

2.2.1 Transaction-cost models . . . . . . . . . . . . . . . . . . . . . 56

2.2.2 Reaction-function (equilibrium) models . . . . . . . . . . . . . 57

2.2.3 Reduced-form SDE models . . . . . . . . . . . . . . . . . . . . 58

2.3 A unified framework . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

2.3.1 Cetin et al. (2004) . . . . . . . . . . . . . . . . . . . . . . . . 59

2.3.2 Platen and Schweizer (1998) . . . . . . . . . . . . . . . . . . . 59

2.3.3 Mancino and Ogawa (2003) . . . . . . . . . . . . . . . . . . . 60

2.3.4 Lyukov (2004) . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

2.3.5 Sircar and Papanicolaou (1998) . . . . . . . . . . . . . . . . . 61

3 First-order Feedback Model 64

3.1 Analysis close to expiry: European options . . . . . . . . . . . . . . . 67

3.2 Analysis close to expiry: American put options . . . . . . . . . . . . . 72

3.3 The vanishing of the denominator . . . . . . . . . . . . . . . . . . . . 77

4 Full-feedback Model 83

4.1 Put-call parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.2 A solution by inspection . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.3 Similarity solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.4 Perturbation expansions . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.5 Numerical solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.6 Analysis close to expiry . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.7 Numerical results - full problem . . . . . . . . . . . . . . . . . . . . . 97

4.7.1 A second solution regime . . . . . . . . . . . . . . . . . . . . . 99

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5 Smoothed Payoffs - Another Breakdown 102

5.1 Local analysis about the singularities . . . . . . . . . . . . . . . . . . 106

5.1.1 Asymptotic matching . . . . . . . . . . . . . . . . . . . . . . . 108

5.1.2 Properties of the inner solution . . . . . . . . . . . . . . . . . 110

5.1.3 Introduction to phase-plane analysis . . . . . . . . . . . . . . 111

5.1.4 Deriving an autonomous system . . . . . . . . . . . . . . . . . 114

5.1.5 Behaviour of the fixed points . . . . . . . . . . . . . . . . . . 116

5.1.6 Structure of the phase portrait . . . . . . . . . . . . . . . . . 120

5.1.7 Other fixed points . . . . . . . . . . . . . . . . . . . . . . . . . 122

6 Perpetual Options 127

6.1 Analytic solutions and perturbation methods . . . . . . . . . . . . . . 131

7 Other Models 137

7.1 Frey (1998, 2000) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

7.2 Frey and Patie (2002) . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

7.3 Sircar and Papanicolaou (1998) . . . . . . . . . . . . . . . . . . . . . 140

7.4 Bakstein and Howison (2003) . . . . . . . . . . . . . . . . . . . . . . 1417.4.1 Non-smooth solutions to the Bakstein and Howison (2003) model146

7.4.2 New non-smooth solutions to the Black-Scholes equation . . . 147

7.5 Liu and Yong (2005) . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

7.5.1 Vanishing of the denominator . . . . . . . . . . . . . . . . . . 150

7.6 Jonsson and Keppo (2002) . . . . . . . . . . . . . . . . . . . . . . . . 152

7.6.1 Connections with the other modelling frameworks . . . . . . . 154

8 Explaining the Stock Pinning Phenomenon 155

8.1 Linear price impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

8.2 Nonlinear price impact . . . . . . . . . . . . . . . . . . . . . . . . . . 159

8.3 A new nonlinear price impact model . . . . . . . . . . . . . . . . . . 161

9 The British Option 164

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

9.2 The no-arbitrage price . . . . . . . . . . . . . . . . . . . . . . . . . . 167

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9.2.1 The gain function . . . . . . . . . . . . . . . . . . . . . . . . . 170

9.3 Numerical treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

9.4 Free boundary analysis far from expiry . . . . . . . . . . . . . . . . . 175

9.5 Analysis close to expiry . . . . . . . . . . . . . . . . . . . . . . . . . . 180

9.6 Financial analysis of the British put option . . . . . . . . . . . . . . . 186

9.7 The British call option . . . . . . . . . . . . . . . . . . . . . . . . . . 193

9.7.1 Analysis far from expiry . . . . . . . . . . . . . . . . . . . . . 196

9.7.2 Analysis close to expiry . . . . . . . . . . . . . . . . . . . . . 198

9.8 Integral representations of the free boundary . . . . . . . . . . . . . . 198

9.8.1 The American put option . . . . . . . . . . . . . . . . . . . . 198

9.8.2 The British put option . . . . . . . . . . . . . . . . . . . . . . 201

10 Conclusions 204

A Maximum Principles 223

A.1 Nonlinear equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

A.2 Uniqueness of PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

A.2.1 The linear Black-Scholes equation . . . . . . . . . . . . . . . . 228A.2.2 The nonlinear (illiquid) Black-Scholes equation . . . . . . . . . 229

A.3 Monotonicity in . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

B Non-dimensionalisation of the British Put 232

C The Probability Density Function 233

Word count 69834

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List of Figures

3.1 Value of European call options with first-order feedback (T = 1, r =

0.04, = 0.2, K = 1) for = 0, 1, 2, 5, 10; the variation with

appears to be monotonic. . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.2 Value of European put options with first-order feedback (T = 1, r =

0.04, = 0.2, K = 1) for = 0, 1, 2, 5, 10; the variation with

appears to be monotonic. . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.3 Asymptotic Matching. . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.4 Inner solution minus the payoff for put and call options, r = 0.04,

= 0.2, K = 1 and for = 0.1, 0.15, 0.2, . . ., 0.4. . . . . . . . . . . . 72

3.5 Value of American put options, T = 1, r = 0.04, = 0.2, K = 1 and

for = 0, 1, 2, 5, 10; the variation with appears to be monotonic. . 73

3.6 First-order feedback put (with early exercise), location of free bound-

ary (as 0) with , K = 1, r = 0.04, = 0.2. . . . . . . . . . . . . 763.7 Location of the vanishing of the denominator of (2.9) with = 0.1,

K = 1, r = 0.04 and = 0.2. . . . . . . . . . . . . . . . . . . . . . . 78

3.8 The first derivative () of the Black-Scholes equation (3.2) (dotted

line) and the first order feedback PDE (2.9) (solid line) for = 0.01,

0.015, . . ., 0.05. Compare the location of the vanishing denominator 3.7. 79

3.9 The second derivative () of the Black-Scholes equation (3.2) (dotted

line) and the first order feedback PDE (2.9) (solid line) for = 0.01,

0.015, . . ., 0.05. Compare the location of the vanishing denominator 3.7. 79

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3.10 First-order feedback put option value for two different values of at

various times to expiry; = 0.0125, 0.0375, 0.075. For r = 0.04,

= 0.2, K = 1 and = 0.09 (solid line) and = 0.1 (dotted line).

Compare with figure 3.7. . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.1 The leading order correction term V1(S, ) to the Black-Scholes (i.e.

= 0) European put option for various time to expiry. K = 1,

r = 0.04, = 0.2, T = 1 and = 0.1, 0.2, . . . , 1. . . . . . . . . . . . . 92

4.2 Deltas for full-feedback (European) put, K = 1, r = 0.04, = 0.2,

= 0.1 and T = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.3 Local ( 0) solution of a full-feedback put, K = 1, = 0.1, r = 0.04and = 1, 0.95, . . ., 0.15. . . . . . . . . . . . . . . . . . . . . . . . . 96

4.4 Full feedback put, K = 1, r = 0.04, = 0.2 and = 0.1; modified

numerical scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.5 Full feedback put, K = 1, r = 0.04, = 0.2 and = 0.1; modified


4.6 Full feedback call, K = 1, r = 0.04, = 0.2 and = 0.1; modified


4.7 Full feedback put, smoothed payoff, K = 1, r = 0.04, = 0.1, = 0.1

and = 0.01. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.1 Phase portrait of the autonomous system (5.24). Note the fixed point

at u =24380

13 , v = 0 and the field direction lines. The dotted line rep-

resents an analytic envelope for the phase portrait close to the singular

line v =

5u

3 , cf. equation (5.31). . . . . . . . . . . . . . . . . . . . . . 120

6.1 Full feedback American put, K = 1, r = 0.04, = 0.2, = 0.25,

= 0.15 (smoothed payoff), = 0, 1, . . . , 10. Note that we are in the

regime < 2 and so we should expect no singular behaviour. . . . . 128

6.2 Perpetual full-feedback American put, K = 1, r = 0.04, = 0.2,

= 0, 0.1, 0.2, . . . , 1.1; free-boundary location as indicated. . . . . . . 130

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6.3 The first order correction to the Black-Scholes perpetual American put

option (solid line) compared to the difference of the fully numerical

option value with the Black-Scholes (dotted line). K = 1, r = 0.04,

= 0.2 and = 0.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

6.4 The first order correction to the Black-Scholes perpetual American put

option (solid line) compared to the difference of the fully numerical

option value with the Black-Scholes (dotted line) for various values of

. K = 1, r = 0.04, = 0.2 and = 0.1, 0.5, 1. . . . . . . . . . . . . 136

7.1 Location of the vanishing of the denominator of the Frey (1998, 2000)

(solid line) and Schonbucher and Wilmott (2000) (dotted line) model

with = 0.1, K = 1, r = 0.04 and = 0.2. . . . . . . . . . . . . . . 139

7.2 Local ( 0) call solution of the Sircar and Papanicolaou (1998)model K = 1, r = 0.04, = 0.2, and = 0, 0.05, . . . , 0.2. . . . . . . . 141

7.3 Local ( 0) put solution of the Sircar and Papanicolaou (1998)model K = 1, r = 0.04, = 0.2, and = 0 0.05, . . ., 0.3. . . . . . . . 142

7.4 Solution to equation (7.9) for a put option with = 0.01, 0.5, 1, . . .,

5, = 0.2, r = 0.04, K = 1, and = 1.5. . . . . . . . . . . . . . . . . 144

7.5 Local ( 0) put solution of the Bakstein and Howison (2003) modelK = 1, r = 0.04, = 0.2, = 1.5, and = 5, -4.75, . . ., 5. . . . . . 147

7.6 Local ( 0) call solution of the Bakstein and Howison (2003) modelK = 1, r = 0.04, = 0.2, = 1.5, and = 5, -4.75, . . ., 5. . . . . . 148

7.7 Non-smooth solution of the Black-Scholes equation. K = 1, r = 0.04,

= 0.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

7.8 Location of the vanishing of the denominator for the Liu and Yong

(2005) model for various value of . K = 1, r = 0.04, = 0.2,

= 0.1, and = 1 105, 2 105, . . ., 1 106. . . . . . . . . . . . . . 1517.9 Local ( 0) call solution of the Jonsson and Keppo (2002) model

K = 1, = 0.2, and a = 1, -0.9, . . ., 1. . . . . . . . . . . . . . . . . 1537.10 Local ( 0) put solution of the Jonsson and Keppo (2002) model

K = 1, = 0.2, and a =

1, -0.9, . . ., 1. . . . . . . . . . . . . . . . . 153

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8.1 The pinning probability (8.5) for values ofnE = 0.5, 1, . . ., 5. T t =0.1, K = 1, and = 0.2. . . . . . . . . . . . . . . . . . . . . . . . . . 158

8.2 Comparing the pinning probability associated with (8.6) (solid line)

with the model of Avellaneda and Lipkin (2003) (dotted line) for nE =

0.1, T t = 0.1, K = 1, = 0.2, and r + 12

2 = 0. . . . . . . . . . . . 159

8.3 Solution to (8.7) for p = 0.8, 0.9, . . . , 1.2, T t = 0.1, K = 1, = 0.2and r + 1

22 = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

8.4 Solution to equation (8.8) (solid line) compared to (8.5) (dotted line)

for T = 0.1, K = 1, = 0.2, and r + 12

2 = 0. . . . . . . . . . . . . . 163

9.1 The British put option free boundary for varying values of the contract

drift. T = 1, K = 1, = 0.4, r = 0.1, D = 0, and c = 0.11, 0.115,

0.12, . . ., 0.16. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

9.2 The British put option free boundary for varying volatilities. T = 1,

K = 1, c = 0.125, r = 0.1, D = 0, and = 0.05, 0.1, . . ., 0.5. . . . . 173

9.3 The zero of the H-function, i.e. Sh(t), for varying values of the contract

drift. c = 0.102, 0.104, . . . , 1. T = 50, K = 1, r = 0.1, D = 0, and

= 0.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

9.4 The asymptotic approximation for the British put option free bound-

ary close to expiry, i.e. (9.27) (dotted line) compared with fully nu-

merical value (solid line). T = 0.01, = 0.4, r = 0.1, c = 0.125, and

D = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

9.5 Location of the free boundary for the British (solid line) and American

(dotted line) put option under investigation in figures 9.6, 9.7 and 9.8.

T = 1, K = 1, = 0.4, r = 0.1, c = 0.125, and D = 0. . . . . . . . . 189

9.6 The difference in the percentage return of the British put option and

the American put option at every possible stopping location. The solid

lines denote contours at increments of 10% from -10% to 60%. The

dotted line represents the zero contour. S0 = 1, T = 1, K = 1, = 0.4,

r = 0.1, D = 0, and c = 0.125. . . . . . . . . . . . . . . . . . . . . . 190

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9.7 The difference in the percentage return of the British put option and

the European put option. Again the solid lines denote contours at

increments of 10% from 0% to 70%. The dotted line represents the

zero contour. S0 = 1, T = 1, K = 1, = 0.4, r = 0.1, D = 0, and

c = 0.125. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

9.8 The difference in the percentage return of the American put option and

the European put option. The solid lines denote contours at increments

of 10% from -70% to 30%. The dotted line represents the zero contour.

S0 = 1, T = 1, K = 1, = 0.4, r = 0.1, and D = 0. Note the change

of orientation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

9.9 Schematic representation of the regions in which at-the-money Eu-

ropean, American and British put option would provide the greatest

return on an investment. The dotted lines represent the free bound-

aries of the American and British put option for reference. T = 1,

K = 1, = 0.4, r = 0.1 and D = 0. . . . . . . . . . . . . . . . . . . . 192

9.10 Figures representing the region in which American put options would

provide a greater expected return that its British option counterpart,

for increasing moneyness. T = 1, K = 1, = 0.4, r = 0.1 and D = 0. 194

9.11 The British call option free boundary for varying values of the contract

drift. T = 1, K = 1, = 0.4, r = 0.1, D = 0, and c = 0.05, 0.055,

0.06, . . ., 0.09. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

9.12 The British call option free boundary for varying volatilities. T = 1,

K = 1, c = 0.08, r = 0.1, D = 0, and = 0.05, 0.1, . . ., 0.5. . . . . . 195

9.13 The asymptotic approximation for the British call option free bound-ary close to expiry, i.e. (9.32) (dotted line) compared with fully nu-

merical value (solid line). T = 0.01, K = 1, = 0.4, r = 0.1, c = 0.08

and D = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

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The University of ManchesterKristoffer John Glover

Doctor of Philosophy

The Analysis of PDEs Arising in Nonlinear and Non-standard OptionPricing

October 23, 2008

This thesis examines two distinct classes of problem in which nonlinearities arise inoption pricing theory. In the first class, we consider the effects of the inclusion of fi-nite liquidity into the Black-Scholes-Merton option pricing model, which for the mostpart result in highly nonlinear partial differential equations (PDEs). In particular,we investigate a model studied by Schonbucher and Wilmott (2000) and furthermore,

show how many of the proposed existing models in the literature can be placed intoa unified analytical framework. Detailed analysis reveals that the form of the nonlin-earities introduced can lead to serious solution difficulties for standard (put and call)payoff conditions. One is associated with the infinite gamma and in such regimesit is necessary to admit solutions with discontinuous deltas, and perhaps even moredisturbingly, negative option values. A second failure (applicable to smoothed payofffunctions) is caused by a singularity in the coefficient of the diffusion term in theoption-pricing equation. It is concluded in this case is that the model irretrievablybreaks down and there is insufficient financial modelling in the pricing equation.The repercussions for American options are also considered.

In the second class of problem, we investigate the properties of the recently intro-duced British option (Peskir and Samee, 2008a,b), a new non-standard class of earlyexercise option, which can help to mediate the effects of a finitely liquid market,since the contract does not require the holder to enter the market and hence incurliquidation costs. Here we choose to focus on the interesting nonlinear behaviour ofthe early-exercise boundary, specifically for times close to, and far from, expiry.

In both classes, detailed asymptotic analysis, coupled with advanced numerical tech-niques (informed by the asymptotics) are exploited to extract the relevant dynamics.

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Declaration

No portion of the work referred to in this thesis has been

submitted in support of an application for another degree

or qualification of this or any other university or other

institute of learning.

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Copyright Statement

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Acknowledgements

I am extremely grateful to my supervisors Professor Peter W. Duck and David P.

Newton for their expert guidance and continued support throughout the course of

this Ph.D. In particular, I thank Peter for his boundless knowledge, enthusiasm and

efficiency, and David for his caring supervision and his confidence in my abilities. In

addition, EPSRC funding is gratefully acknowledged.

I thank my parents for their love and unwavering support for which these mere

expressions of gratitude do not suffice. Thank you to my colleagues and friends

for their invaluable advice and numerous enlightening discussions. In particular, toGoran Peskir for his time and enthusiasm for the subject, and to Erik Ekstrom for

his insight and friendship.

To my close friends, both old and new, and in particular to Jonathan Causey, Helen

Burnip, Philip Haines, John Heap, Sebastian Law and Vicky Thompson, I thank you

for creating the good times and for being there through the bad. I hope, despite

the distances between us, our friendships can continue to flourish. Finally, I thank

Hannah for everything, I hope we both find what were looking for.

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Dedication

To Gran, in loving memory.

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Chapter 1

Introduction

Nowadays people know the price of everything and the value of nothing.

- Oscar Wilde (1854-1900)

The Picture of Dorian Gray (1891)

Mathematical finance is not a branch of the physical sciences. There are no laws of

nature just waiting to be discovered; one is not trying to model Mother Nature and

her laws, but the nature of man and his markets. However, this does not preclude

us from trying to quantify the financial markets and to utilise the powerful tools of

mathematics in order to better understand such markets.

Since the definitive papers of Black and Scholes (1973) and Merton (1973), much of

the work undertaken in mathematical finance has been aimed at relaxing a number

of the modelling assumptions. One of the more subtle was that the market in the

underlying asset1 was infinitely (or perfectly) elastic, such that trading had no impact

on the price of the underlying. If we relax this assumption, then we see some rather

interesting and possibly counterintuitive behaviours. As we shall show later, this is

partly due to the fact that any model incorporating such a feature will inevitably

lead to nonlinear behaviour (feedback). In particular, we shall be concerned for the

most part with nonlinear partial differential equations (PDEs) arising from the study1Termed underlying in the sequel.

16


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CHAPTER 1. INTRODUCTION 17

of finitely elastic markets. Work that has led to this class of PDEs in finance to

date includes Whalley and Wilmott (1993) in relation to transaction costs, which

was one of the first nonlinear PDEs to arise in the field of mathematical finance.

In addition, there is the so called Black-Scholes-Barrenblatt equation introduced by

Avellaneda et al. (1995) in the study of uncertain volatility models. These models

involve optimisation over all possible values of volatility, and as a result are also

highly nonlinear.

The aim of modelling the behaviour of the underlying is to capture the dynamics

of the observed market prices as faithfully as possible. One approach to incorporate

these dynamics is to find a stochastic process that fits most closely the distribution of

returns of the underlying. This is an exogenous strategy, and as such provides little

insight into which of the many factors affecting the price dynamics are actually the

most important. In addition, the exogenous processes required tend to be difficult to

handle mathematically, for example Levy processes. An alternative approach (and

that to be followed in this thesis) is to retain one the simplest stochastic process,

geometric Brownian motion, but to provide an endogenous mechanism by which

the dynamics differ from this standard geometric Brownian motion. This provides

much greater insight into how prices are actually formed in the market, and has the

advantage of being consistent with the bulk of the literature over the past thirty-five

years.

In this chapter we introduce the basic ideas and concepts and review the results of

the classical Black-Scholes-Merton option pricing theory used in later chapters. It

is by no means a complete treatment of the relevant theories, just enough for the

unfamiliar reader to understand the contributions of the following chapters.

1.1 Evidence of increased interest in liquidity

Recent worries about the health of the modern financial system have deterred people

from getting involved in the derivatives markets. This has resulted in trading volumes


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decreasing and hence increased liquidity problems. David Oakley of the Financial

Times2 warns that

...the sharp slowdown in these [derivative] markets is a serious warning

sign of the growing problems in the financial world as they are usually

highly liquid, turning over vast amounts of trade every day.

Further, Rachel Lomax, the Bank of Englands Deputy Governor goes on to describe

the recent financial turmoil in the wake of the American sub-prime mortgage prob-

lems3 as

...the largest ever peacetime liquidity crisis.

The current liquidity crisis can be traced back to the collapse of the US sub-prime

mortgage market. In August 2007 the Financial Times is quoted as saying that4

...as market turmoil rises financial problems are no longer simply confined

to a risky corner of the US mortgage market. This stems from another

key theme now haunting the markets: namely that liquidity is evaporating

from numerous corners of the financial world, as both investors in hedge

funds and the banks that lend to them try to cut and run from recent

losses.

Clearly, in times of crisis, liquidity becomes an ever important issue, motivating

further investigation into the effects of reduced liquidity on all aspects of the financial

markets. In a recent blog entry regarding the sub-prime induced liquidity crisis Paul

Wilmott states that5

...this should spur on the implementation of mathematical models for

feedback... which may in turn help banks and regulators to ensure that

2See Derivative liquidity crisis to continue, David Oakley, FT.com, November 23 2007.3Quoted in Bank deputy downbeat on economy, Chris Giles, FT.com, February 27 2008.4See Liquidity alarm bells sound, Paul J Davies, Gillian Tett, Joanna Chung and Stacy-Marie

Ishmael, FT.com, August 1 2007.5Quoted in Science in Finance IV: The feedback effect Paul Wilmott, blog entry at http://www.

wilmott.com/blogs/paul/, January 29 2008.


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the press that derivatives are currently getting is not as bad as it could

be.

1.2 A brief history

In 1828 Robert Brown (1773-1858), a Scottish botanist, observed the apparently ran-

dom motion of pollen particles suspended in water and subsequently during the 19th

century it became clear that the pollen particles were being bombarded by a multi-

tude of molecules of the surrounding water, whose aggregate effect was (apparently)

random. In addition, wherever we look we see a random world and therefore Brow-

nian motion (named in honour of Robert Brown) is an invaluable tool for describing

this randomness. In fact, the ubiquitous nature of Brownian motion can be seen as

the dynamic counterpart of the ubiquitous nature of the normal distribution, which

rests ultimately on the Central Limit Theorem.6

The origins of much of financial mathematics trace back to a dissertation (entitled

Theorie de la speculation7) published in 1900 by Louis Bachelier (1870-1946). In

it he proposed to model the movement of stock prices with a diffusion process or

Brownian motion. Note that this was five years before Einsteins seminal paper

outlining the theory of Brownian motion, and it was not until the 1920s that the

rigorous mathematical underpinnings of the theory of Brownian motion was provided

by Norbert Wiener (1894-1964).

Meanwhile, as quantum mechanics emerged in the 1920s it began to become clear

that the quantum picture is both inescapable at the subatomic level and intrinsically

probabilistic. The work of Richard P. Feynman (1918-1988) in the late 1940s on quan-

tum mechanics using path integrals, introduced the Wiener measure into the heart

of quantum theory. Feynmans work was made mathematically rigorous by Mark

Kac (1914-1984) and the so-called Feynman-Kac formula, which gives a stochastic

6See for example Jacod and Protter (2003).7For a translated version with commentary and a foreword by Paul Samuelson see Davis and

Etheridge (2006).


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representation for the solution to certain classes of PDEs, was introduced (see section

1.3.3).

In 1944 Kiyoshi Ito (1915-) went on to develop stochastic calculus, the machinery

needed in order to use Brownian motion to model stock prices successfully, and which

would later become an essential tool of modern finance. However, it was not until 1965

that economist Paul Samuelson (1915-) resurrected Bacheliers work and advocated

Itos geometric Brownian motion model as a suitable model for stock price movements.

After this it was not long until Black, Scholes and Merton wrote down their famous

equation for the price of a European call and put option in 1969, work for which the

surviving members (Scholes and Merton) received the Nobel Prize for economics in

1997.

A more comprehensive overview of the early years of mathematical finance can be

found in Jarrow and Protter (2004).

1.3 Derivative pricing

When we discretise a problem it becomes easier to define or understand but much

harder to solve without the use of continuous time calculus; this thesis deals solely

with continuous time models. In continuous-time modelling there are two main ap-

proaches to calculating the price of a given derivative security, the so-called martin-

gale approach and the PDE approach. In the former, a stochastic process for the

underlying is specified and an equivalent probability measure is found that turns the

discounted underlying into a martingale. The price of the derivative is then defined

as the conditional expectation of its discounted payoff under this new (risk-neutral)

measure. Alternatively, in the PDE approach, a stochastic process for the underlying

is likewise specified and then Itos formula for a function of the underlying stochastic

process is used to derive a PDE involving the coefficients of the underlying process.

The two approaches are deeply linked via the famous Feynman-Kac formula(outlined

in section 1.3.3) and it should be noted that both approaches can be used for complete


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and incomplete markets. In the latter case, arriving at a unique price for a derivative

requires additional assumptions. If one is using the martingale approach, then this

arbitrariness is reflected in the choice ofequivalent martingale measure, whereas using

the PDE approach the choice of martingale measure is analogous to specifying the

so-called market price of risk of the non-traded variable. Since the models introduced

in this thesis result in complete markets,8 i.e. all sources of risk are traded, both the

martingale approach and the PDE approach should arrive at the same price.

The fair price of a derivative security (and all other financial instruments) is de-

termined by the expected discounted value of some future payoff, which is itself

dependent on the future value of the underlying asset. Of course, the future value

of the underlying is not known a priori, and price processes are often modelled by

stochastic processes. Therefore, an understanding of the behaviour of such stochastic

processes is a valuable prerequisite for the study of derivative pricing; this section

attempts to provide such an understanding. The derivative securities studied in this

thesis, without exception, are options contracts. A brief overview of the types of

contracts referred to in the main body of the thesis will be considered next.

1.3.1 European options

European options are the simplest type of options contract and within this class the

most common are call options and put options. The holder of a call option written

on a certain underlying asset (usually a stock) has the right, but not the obligation,

to buy the underlying at some pre-determined date, denoted T, and at some pre-

determined price, denoted K. If the underlying at time t = T, ST, is worth more

then K then the (rational) holder would exercise the option and make a profit STK.Alternatively, if ST is less than K, then the holder would not exercise, resulting in

the option expiring worthless. Thus, the value of the call option at expiry (T) is

given by

VC(ST, T) = (ST K)+ := max{ST K, 0}. (1.1)8Under suitable restrictions, see chapter 2.


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Similarly, the holder of a put option has the right to sell the underlying for the

exercise price K, resulting in the value of the put option:

VP(ST, T) = (K ST)+ := max{K ST, 0}. (1.2)

The functions (1.1) and (1.2) are called payoff profiles and will be referred to as such

throughout this thesis. There are, of course, many different options contracts with

more general payoff profiles, h(ST) say. For an option to be described as European,

its contract must specify that exercise is only possible at a single maturity time, T.

Note that these contracts dependent only on the price of the underlying at expiry,

ST, and not on the path of the price prior to maturity; this results in tractability in

many situations. Options that allow exercise at times prior to expiry are said to have

an early-exercise feature. More specifically if the option allows exercise at any time

prior to expiry such an option is referred to as an American option. These options

are very popular in practise, and will play an important role in much of this thesis.

Indeed we shall return to them shortly in section 1.3.5.

1.3.2 Arbitrage pricing

An arbitrage opportunity corresponds to a risk-free profit. More formally, it is the

opportunity to construct a trading strategy (i.e. buying and selling financial instru-

ments) in such a way that the initial investment (at t = 0) is zero and the wealth at

time T is non-negative with a non-zero probability of a strictly positive wealth. In

an efficient market there should be no such arbitrage opportunities and indeed theseminal work by Black and Scholes (1973) and Merton (1973) used the no-arbitrage

principle to arrive at a unique price for the fair value of an option contract. To state

their results, we have a market consisting of a bank account which grows according

to the (deterministic) dynamics

dB = rBdt,

and one risky asset, with stochastic price dynamics

dSt = Stdt + StdWPt , (1.3)


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where r is the positive (constant) interest rate, the drift and the volatility of

the underlying price process. WPt denotes a standard Brownian motion under the

probability measure P. The fair value or price of a European option contract V(S, t)

with payoff profile h(ST) can be shown to be given by

V(S, t) = EQS,t

er(Tt)h(ST)

, (1.4)

in words, the expected discounted future payoff. The indices indicate that the pro-

cess for St is started at S at time t and also that the expectation is calculated under

the so-called risk-neutral probability measure, Q, as opposed to the real world mea-

sure, P, defined by the process (1.3).9 The risk-neutral measure is defined as the

unique measure equivalent to P under which the discounted price process is a mar-

tingale. Consequently, the stock price process (1.3) can then be described in terms

of a standard Q-Brownian motion WQt as

dSt = rStdt + StdWQt . (1.5)

Note that the dynamics of St under the risk-neutral measure Q are the same as

the dynamics under the real-world measure P, except that the drift of St under Q

is equal to the interest rate r instead of . Consequently the drift parameter

does not appear anywhere in the pricing formula for European options; this fact

undoubtedly contributed to the widespread application of the Black-Scholes-Merton

pricing methodology in the years subsequent to its publication, since in practise

the drift parameter is notoriously difficult to measure from past time series of the

underlying process.10

The model analysed above is an example of a complete market model. The simplest

definition of a complete market is one in which every derivative security can be repli-

cated by a self-financing trading strategy in the stock and bond. In this model, any

security whose payoff h(ST) is known at time T (where h(ST) is any FT-measurable9This subtlety was the main innovation of option pricing research in the 1970s. Prior to this,

expectations had been taken under the real world measure P.

10In fact, Liptser and Shiryaev (2001) show that the expected waiting time to obtain an estimateof the drift (via the naive approximation St/t) that is within of the true drift is proportional to2. For example if = 0.01 it would take 10, 000 years to obtain such an estimate.


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random variable with E [h2(ST)] < ) can be replicated by some unique self-financingtrading strategy. Finally, we note that in a complete market, a characterisation of

the arbitrage-free principle is that there exists a unique equivalent martingale mea-

sure Q, under which the discounted prices of traded securities are martingales. For

more on this characterisation see the original works of Harrison and Kreps (1979)

and Harrison and Pliska (1981, 1983).

Expected values of solutions to stochastic differential equations (SDEs), such as the

pricing equation (1.4), are linked to the solution of (linear) parabolic partial differen-

tial equations (PDEs) via the famous Feynman-Kac representation theorem. Thus,

the price of a European option can be studied using both stochastic methods and

parabolic PDE methods; this thesis focuses primarily on the latter. In the following

section we describe the Feynman-Kac representation theorem.

1.3.3 The Feynman-Kac representation theorem

Suppose we are given the PDE for the unknown function u(S, t)

u

t+

1

22(S, t)

2u

S2+ (S, t)

u

S= 0, (1.6)

subject to the final condition

u(S, T) = h(S),

where (S, t), (S, t) and h(S) are known functions and T a parameter. This equation

is sometimes called the Kolmogorov backward equation. The Feynman-Kac formula

tells us that the solution can be written as an expectation,

u(S, t) = EPS,t [h(ST)]

where St is a stochastic process given by the equation

dSt = (St, t)dt + (St, t)dWPt . (1.7)

The indices on the expectation indicates that the process St is started at S at time t

and in addition the superscript indicates that the expectation is taken under the prob-

ability measure P, corresponding to the stochastic process (1.7), with a P-Brownian


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motion WPt . This useful representation allows us to solve deterministic PDEs via

stochastic methods and, conversely, expectations of functions of stochastic processes

via deterministic PDEs.

Proof. The proof of the Feynman-Kac representation is fairly straightforward and so

we shall outline the basic idea here. Consider an unknown function u(S, t). Applying

Itos formula we have

du =

u

t+ (S, t)

u

S+

1

22(S, t)

2u

S2

dt + (S, t)

u

SdWPt .

Now, by assumption the O(dt) terms above are zero if u(S, t) is assumed to be the

solution of the PDE (1.6). Integrating the above equation we obtain

Tt

du = u(ST, T) u(St, t) =Tt

(S, t)u

SdWPt .

Next, taking expectations and reorganising a little we arrive at

u(S, t) = EPS,t [u(ST, T)] EPS,t T

t

(S, t)u

SdWPt .

Finally, it can be shown that the expectation of an Ito integral with respect to a

Brownian motion is zero (see, for example, prop. 4.4 of Bjork, 2004) resulting in the

required result

u(S, t) = EPS,t [u(ST, T)] = EPS,t [h(ST)] .

1.3.4 From Feynman-Kac to Black-Scholes

Having satisfied ourselves of the validity of the Feynman-Kac representation theorem,

we can now use it to represent the expectation given in (1.4), representing the price

of a European option, as the solution to a second-order linear parabolic PDE. The

first point to note is that (1.4) involves discounting and so it is useful to make the

transformationV(S, t) = er(Tt)u(S, t)


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in equation (1.6) to obtain the PDE

V

t+

1

22(S, t)

2V

S2+ (S, t)

V

S rV = 0,

which we have shown can be represented as the conditional expectation

V(S, t) = EPS,t

er(Tt)h(ST)

.

However, note that the expectation in (1.4) is taken under the risk-neutral measure

Q and so the corresponding PDE representation of (1.4) is given by

V

t +

1

2

2

S

22V

S2 + rS

V

S rV = 0, (1.8)with the following condition

V(S, T) = h(S), (1.9a)

V(0, t) = h(0)er(Tt), (1.9b)

V(S, t) h(S)er(Tt) as S , (1.9c)

where we have used the risk-neutral process (1.5). Note that in what follows this

shall be referred to as in the Black-Scholes equation (which should also be credited

to Merton). Moreover, if we assume a stochastic process of the much more general

form (1.7), then the corresponding (generalised) Black-Scholes equation obtained via

the Feynman-Kac formula is given by11

LBS(V) = Vt

+1

22(S, t)

2V

S2+ rS

V

S rV = 0, (1.10)

with the same boundary conditions as previously, i.e. (1.9).

However, it can be shown that standard Feynman-Kac type results only hold under

(quite restrictive) analytic conditions on the coefficients of the SDE and PDE, as-

sumptions that are often not satisfied by many models used in practise. Remarkably,

this problem is often glossed over or simply not mentioned in the literature. What

follows is a brief overview of the some of these analytic conditions. In some sense the11Again note the independence of the real-world drift (S, t).


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behaviour of the models presented in this thesis can be attributed to the failure of

the coefficients of the relevant equations to satisfy the conditions outline below.

In order for the conditional expectation (1.4) to be the unique classical solution to the

Black-Scholes equation (1.10) with the conditions (1.9) then the diffusion coefficient

(S, t) must be sufficiently regular. More precisely, it must be locally Lipschitz, i.e.

|(S1, t) (S2, t)| C|S1 S2|

for some C > 0, and also satisfy a linear growth condition in S, i.e.

|(S, t)

| D(1 +

|S|)

for some constant D > 0.12 Another condition is that the operator LBS must beuniformly elliptic, meaning (in this one-dimensional situation) that the coefficient

(S, t) must be strictly positive at every point in the solution domain (S, t) [0, T], where is the domain of the process St, for example = {S > 0} for geometricBrownian motion. In other words, we have the restriction that

2(S, t) > 0 (S, t),

i.e. the diffusion coefficient 2(S, t) cannot degenerate be zero. Note that even in the

simplest cases, such as geometric Brownian motion where (S, t) = S, the volatility

term degenerates in certain regions of state space. Specifically limS0 (S, t) = 0.

We can avoid this difficulty here (and also in many other more general situations)

by making the change of variable x = log S giving (x, t) = which is no longer

degenerate.

1.3.5 American options

Unlike European options discussed in section 1.3.1, American options have the extra

feature that they can be exercised at any time prior to expiry, T. The time at which

12The stochastic process derived in chapter 2 can be seen to exhibit singular behaviour and, assuch, these conditions are no longer satisfied. Hence, we are no longer in a regime where standard

results from SDE and PDE theory can be applied. In addition, here the non-Lipschitz nature of thecoefficients means that the solutions to the corresponding SDE need no longer remain continuous;jumps may be seen at the location where the diffusion coefficient becomes singular.


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the option is exercised is called the exercise time and because the market cannot be

anticipated, the holder of the option needs to decide whether to exercise at each

point in time based only on the information up to time t

T (i.e. the information

contained in the filtration Ft).

The terms European and American were first coined in Samuelson (1965) and the

story behind their naming is noteworthy. According to a private communication

with Robert C. Merton, Samuelson visited many practitioners on Wall Street prior to

writing his paper. One of his industry contacts explained to him that there were two

types of options available, one more complex (that could be exercised early) and one

much simpler (that could only be exercised at expiry). The practitioner commented

that only the more sophisticated European mind (as opposed to the American mind)

could understand the former. In response, when Samuelson (an American) wrote the

paper, he used the European and American prefixes but reversed the ordering.

If the payoff profile is given by h, and the holder of the American option decides to

exercise early then she receives the amount h(S) at time . Using the theory ofoptimal stopping (cf. Peskir and Shiryaev, 2006), the unique no-arbitrage price of an

American option can be shown to be given intuitively by

V(S, t) = suptT

EQS,t

er(t)h(S)

, (1.11)

i.e. the supremum of the expected value of the discounted payoff over all random times

that are stopping times with respect to the filtration generated by the Brownian

motion used to specify the dynamics of the underlying process for St. This is a rather

intuitive definition of the American option price.

Immediately from the definition (1.11) we have the inequality

V(S, t) h(S) (1.12)

since the stopping time = t is included in the supremum. This is a natural condition

since if V(S, t) < h(S) then there would be an obvious instant arbitrage at time t.


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In addition, choosing = T gives the further inequality

V(S, t) VE(S, t),

where VE(S, t) is the corresponding European option price. Again, this is intuitive,

since an American option gives its holder more rights than the corresponding Euro-

pean option with the same payoff function and expiration date.

Another point to note is that when pricing American options we cannot, without

loss of generality, set the interest rate to zero, which can be done for their European

counterparts. Pricing American derivatives is mathematically more involved thanthe European case and closed-form expressions for American option prices are rarely

obtained. However, it can be shown by no-arbitrage arguments that, for nonnegative

interest rates and no dividends, the price of an American call option is the same as

its corresponding European option (see, for example, prop. 7.14 of Bjork, 2004). In

other words, the supremum in expression (1.11) is attained for the stopping time

= T when considering the payoff function of a call option. Thus, the price of an

American call reduces to the price of a European call, which does have an explicit

formula, first derived by Black and Scholes (1973).

It can also be shown that the price of an American put option is, in general, strictly

higher than the price of the corresponding European put option. Indeed it can be

seen (directly from its well-known analytic expression) that the European put option

price crosses below the payoff function (1.2) for sufficiently small S, violating the

condition (1.12). Hence the value of the American contract cannot coincide with

that of its European counterpart. We therefore use a put option as our canonical

example of an American option throughout the remainder of this thesis.

1.3.6 Optimal stopping problems

The observant reader may have noticed already that there is a strong link betweenpricing American options and optimal stopping problems. When faced with an optimal


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stopping problem, there are two facets of the solution that we are most interested

in. The first is to determine the price of the option V (called the value function in

optimal stopping terminology) and the second to determine the optimal strategy for

the option holder, in other words to determine the stopping time that realises the

supremum in (1.11). Determining the value function will be discussed shortly, but

first we state a key result from the theory of optimal stopping. If the function h is

continuous, in addition to some other technical conditions,13 then the supremum is

attained for the stopping time

:= inf{

u

t : V(Su, u) = h(Su)}

,

i.e. the first time that the price of the American option drops down to the value of

its payoff. Alternatively, and more practically, the optimal stopping time can be

formulated as the first exit time from the continuation region defined by

C := {(S, t) : V(S, t) > h(S)},

i.e. as := inf{u t : (Su, u) / C}.

The continuation region is so named due to the fact that in this region it is not

optimal to exercise the option. Clearly, if the value V(S, t) at some time t is strictly

larger than the payoff profile h(S), then it is not optimal to exercise the option.

1.3.7 Free-boundary problems

Analogous to the Feynman-Kac representation theorem for European options (out-

lined in section 1.3.3), the price of American options can be shown to satisfy partial

differential inequalities. For a nonnegative payoff function h, the price of an American

13See, for example, Peskir and Shiryaev (2006)


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option as defined in (1.11) is given by the solution to the following linear complemen-

tarity problem:

V(S, t) h(S, t), (1.13a)LBS(V) = V

t+

1

22S2

2V

S2+ rS

V

S rV 0, (1.13b)

LBS(V).

h(S, t) V(S, t) = 0, (1.13c)to be solved in the entire domain {(S, t) : S > 0, 0 t T} with the final conditionV(S, T) = h(S).

Further to this, it can be shown that the Black-Scholes equation holds at all points inthe continuation region and that at the boundary of the continuation region, we must

apply the smooth pasting or smooth fit principle.14 This principle states that the value

function V(S, t) must be at least C1,1 differentiable,15 not only in the continuation

regions, but also over the boundary of the continuation regions, denoted by C. Italso transpires that for a standard American put option there is an increasing function

Sf(t), the free boundary, separating the continuation region from the stopping region,

compare Jacka (1991). As such the linear complementarity problem (1.13) can be

formulated as the free-boundary problem16

V

t+

1

22S2

2V

S2+ rS

V

S rV = 0, (1.14a)

V(Sf, t) = K Sf, (1.14b)

VS(Sf, t) = 1, (1.14c)

V(S, T) = (K S)+

, (1.14d)

V(S, t) 0 as S , (1.14e)

to be solved in the domain {(S, t) : 0 t T, S > S f(t)}, in other words theboundary of the domain is to be solved as part of the problem. This implies that for

S > Sf(t) the value V(S, t) must satisfy V(S, t) > (K S)+, and for S Sf(t) the14In fact the principle of smooth fit in probability, the principle of no arbitrage in finance and the

conservation of energy law in the physical sciences can be seen as different formulations of the same

principle. This is alluded to in Peskir (2005b).15At least for points at which the payoff profile h(S, t) is C1,1 differentiable.16See for example Karatzas and Shreve (1998)


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value satisfies V(S, t) = (K S)+. Furthermore, the existence and uniqueness of thefree-boundary problem (1.14) can be proved. In addition, for put options without

dividends, Chen et al. (2008) have recently proved the convexity of the resulting free

boundary.

Explicit solutions to parabolic free-boundary problems are rare, however it can be

shown (cf. Jacka, 1991) that the American put option free boundary Sf(t) is a mono-

tonically increasing function and that it approaches K as t approaches T. The asymp-

totic behaviour of Sf(t) for times close to expiry can also be determined and indeed

this shall be expounded upon in further detail in chapter 9.

1.4 Supply and demand economics

Many of the models presented in this thesis make assumptions about the structure

of the markets and the intentions of the participants of these idealised markets. This

motivates a brief discussion of how prices are actually formed in these markets, in

short a discussion of supply and demand, the backbone of a market economy.

Starting with the basics, a market is a place where buyers (providing demand) and

sellers (providing supply) meet. In a free market, prices are determined solely by

the interaction of demand and supply; nothing more, nothing less. In addition, all

being equal, there will be more demand for an asset at a lower price than at a

higher price and, hence, we should expect an inverse relationship between price and

quantity demanded. Conversely, an increase in price will usually lead to an increase

in the number of people wishing to sell at that price, hence we should expect a

positive relationship between price and supply. In the economics literature, these

relationships are often called the law of demand and the law of supply. In a market,

the price at which supply matches demand is often called the equilibrium price or

market clearing price, so called because it is at this price that all the surpluses are

cleared from the market and the forces of demand and supply are not acting to change

this equilibrium. If disequilibrium exists, then the forces of demand and supply will


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automatically adjust the market to equilibrium. With excess demand, prices will be

forced upwards due to the shortage that exists, and with excess supply, prices will be

forced downwards, due to the surplus that exists.

An important concept crucial to the models discussed in this thesis is that ofelasticity.

At its heart this concept is a purely mathematical one which aims to measure the

responsiveness of one variable to a change in another variable. More specifically given

any functional relationship y = f(x) the point elasticity, , is defined as

=dy/y

dx/x=

dy

dx

x

y=

d(log y)

d(log x),

i.e. the ratio of percentage changes. Similarly, given a function of more than one

variable y = f(x1, x2, . . . , xn) the partial point elasticities are given by

i =y

xi

xiy

=(log y)

(log xi).

Applied to the economics of supply and demand the price elasticity of demand (PED)

is defined as

PED = dq/qdp/p = dqdppq ,

where q is the quantity demanded of an asset and p is the price per unit of that asset.

The PED measures the responsiveness of the quantity demanded to the change in

price. PED > 1 implies that the good is price elastic, PED < 1 implies that the

good is price inelastic and when PED = 1 we have unit elasticity. The limiting cases

PED = 0 and PED = imply that the asset is perfectly price inelastic and elastic

respectively. The price elasticity of supply (PES) is defined similarly.

An important point to note at this stage is that elasticity and liquidity are not the

same, though there is a tendency to confuse the two. Elasticity defines a relationship

between price and the quantity demanded (as defined above), whereas liquidity is

concerned with the availability to trade the underlying asset at a given price. How-

ever (unlike elasticity) liquidity is not a well-defined concept, hence there is much

ambiguity in the connection between the two concepts. The next section explores the

concepts of liquidity in much more detail.


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1.5 Liquidity

Risk can be classified into the following categories17

Market risk,

Credit risk,

Model risk,

Operational risk,

Liquidity risk.

The standard models implicitly assume that the only risk experienced by a trader

is that due to the uncertain nature of the market. More relevant to this thesis,

these standard models assume that the trader will not experience any liquidity risk,

implicitly assuming a level of liquidity that is without limits. Liquidity risk arises

in situations where a party interested in trading an asset cannot do so because she

cannot find a willing counter-party to that trade. Liquidity risk becomes particularly

important to parties who are about to hold or currently hold an asset, since it affects

their ability to trade. In fact one of the most important attributes of financial markets

is to provide immediate liquidity to investors. Of course, some markets are more liquid

than others, and the liquidity of a given market varies over time and in addition can

dramatically dry up in times of crisis.

Recent crises in the financial markets have triggered studies on the subject of market

liquidity. For example, the stock market crises in October 1987 and 1989, the Asian

crisis in 1997 and the problems at Long-Term Capital Management Fund (LTCM)

led the Committee on the Global Financial System to conduct several studies dis-

cussing the importance of liquid financial markets, including Bank for International

Settlements (1999) and Bank for International Settlements (2001).

17See Protter (2006).


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1.5.1 Defining liquidity

Market liquidity is often associated with the ability to quickly buy or sell a particular

item without causing a significant movement in the price. However, the conceptof liquidity is multifaceted and ill-defined. Many researchers have attempted to do

so but the best that can be done is to classify its many dimensions. Kyle (1985)

describes market liquidity in terms of three attributes, namely the tightness, depth

and resilience of the market. Liu (2006) identifies four dimensions to liquidity, namely,

trading quantity, trading speed, trading cost, and price impact. Alternatively, Sarr

and Lybek (2002) state that liquid markets exhibit five characteristics: tightness, i.e.

having low transaction costs, such as a small bid-ask spread as well as other implicit

costs; immediacy, i.e. the speed with which orders can be executed, reflecting the

efficiency of the trading, clearing and settlement systems; depth, i.e. the existence of

abundant orders both above and below the price at which an asset currently trades;

breadth, i.e. orders are both numerous and large in volume with minimal impact on

prices; and finally resiliency, i.e. new orders flow quickly to correct order imbalances.

Clearly, liquidity is a tricky concept to define (let alone measure), and due to this

multidimensional nature comparing individual assets liquidities is also problematic,

since one asset could be more liquid along one dimension of liquidity while the other

is more liquid in a different dimension. One particular interpretation of liquidity in

the literature fits nicely with the philosophy of this thesis; Howison (2005) states

that market liquidity can manifest itself in three possible forms. First, there is a

difference between the prices for buying and selling the asset, the so-called bid-ask

spread. Second, the price paid for trading the asset depends on the quantity traded,

due to limited availability of a stock at the quoted price. In fact, even for a highly

liquid market, trading beyond the quoted depth of the market usually results in a

higher purchase price (or a lower selling price) for part, if not all, of the trade; this

is often termed the liquidation cost. Third, and most relevant to this thesis, is that

the action of a large trade may itself impact the price, independent of all the other

factors affecting the price dynamics; this is termed price impact.


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1.5.2 Measuring liquidity

Because there are many dimensions of liquidity, there is no single method for mea-

suring it. Measures which are often used in the empirical literature on liquidity and

asset pricing include the bid-ask spreads, various measures of the price impact of

order flow, and various measures of order flow. Measures of the price impact of or-

der flow include price changes regressed on signed volume, or absolute price changes

regressed on absolute volume, or daily changes regressed on daily volume. Measures

of volume include numbers of trades and daily volume measured in dollars. Of all

these measures, the price impact of order flow is perhaps the most widely used, the

advantage of this measure being that it is based on the actual observed price changes

associated with trades. However, despite the advantages of using the price impact of

order flow as a measure of liquidity, tricky econometric issues, such as measurement

error, selection bias and simultaneity bias are involved when using this measure.

Sarr and Lybek (2002) classify the existing liquidity measures into four categories.18

The first is transaction cost measures that capture the costs of trading financial

assets and trading frictions in secondary markets. One particularly intuitive measure

of transaction costs is the percentage bid-ask spread, defined as

BAS = 2

PA PBPA + PB

,

where the ask price PA and bid price PB can be calculated from the quotes on the

market or using a weighted average of actual executed trades over a period of time,

the latter being a better estimate of the actual transaction costs since trades may not

take place at the actual quoted prices, in this case the spread is called the realised

spread. In the second category are volume-based measures that attempt to distinguish

liquid markets by the volume of transactions compared to the price variability, this

is primarily used to measure the breadth and depth of the market. Trading volume

is traditionally used to measure the existence of numerous market participants and

transactions and is defined as

Vol =n

i=1

PiQi (1.15)

18See Sarr and Lybek (2002) for a good review of many examples of each class of liquidity measure.


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where Vol is the dollar volume traded, Pi and Qi are prices and quantities of the i-th

trade during a specified period. This can be given more meaning by relating it to

the outstanding volume of the asset. The resulting turnover rate gives an indication

of the number of times the outstanding volume of the asset changes hands. The

turnover can thus be defined as

TO =Vol

NP

where Vol is the trading volume defined in (1.15), N is the outstanding stock of the

asset and P is the average price of the n trades in (1.15). There are many other

volume-based measures. The third category of liquidity measures are equilibrium

price-based measures that try to capture orderly movements towards equilibrium

prices; in the main these attempt to measure resiliency of the market. The fourth

and final category, and the most relevant to the focus of this thesis, are market-impact

measures that attempt to differentiate between price movements due to the degree

of liquidity from other factors, such as general market conditions or arrival of new

information; these attempt to measure both elements of resiliency and speed of price

discovery.

However, clearly no single measure can manage to fully capture the multifaceted na-

ture of liquidity, and as such there is no universally accepted measure of liquidity.

Most of the existing literature attempting to measure liquidity has focused on the

different dimensions of liquidity individually. In fact this problem of no universal liq-

uidity measure has resulted in many unanswered questions in market microstructure

theory, which focuses on determining the processes by which information is incorpo-

rated into prices. One such question is whether liquidity is priced in asset returns.

For example Amihud and Mendelson (1986) (who simply use the bid-ask spread)

and Datar et al. (1998) (who instead use the turnover rate) argue that liquidity is

priced, whereas others, such as Chalmers and Kadlec (1998), Chen and Kan (1995)

and Eleswarapu and Reinganum (1993) suggest that it is not.

More recently Liu (2006) introduced a new measure of liquidity (called the standard-

ised turnover-adjusted number of zero trading volumes over the prior 12 months ) that


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aims to capture multiple dimensions of liquidity. Using this measure Liu (2006) out-

lines a two-factor, liquidity risk adjusted capital asset pricing model (CAPM) that

well explains the cross-section of stock returns, (possibly) answering the question

whether liquidity is priced. In addition, the new two-factor CAPM model is able to

account for the book-to-market effect, which the Fama and French (1996) three-factor

model fails to explain.

1.6 Price formation

We have alluded to the fact that the price of financial instruments may be considered

as entirely dependent on supply and demand. However knowledge about how these

prices are actually formed in the market are of great interest, since we wish to see ex-

actly whereabouts in the price formation process liquidity issues become important.

From a market microstructure perspective, price movements are caused primarily

through the arrival of information. The dynamics by which this information is incor-

porated into the current price is addressed in the market microstructure literature,

where many models of price formation have been proposed; for an overview of this

topic see OHara (1995). Such models are not referred to specifically in this thesis

and so it suffices to describe briefly the role of some of the more important market

participants.

One of the most important members of any financial market are the so-called market

makers. These are individuals or firms that will take both long and short positions

in a given security in order to facilitate trading, and thus add to the liquidity and

depth of the market. The market-maker accepts a certain level of risk in holding the

financial instrument or commodity but hopes to be compensated by making a profit

on the bid-ask spread.

In the United States, many markets have official market makers for each given se-

curity, known as specialists. Their main function being to provide the other side of

trades when there are short-term buy-and-sell-side imbalances in customers orders.


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In return, the specialist is granted various informational and trade execution advan-

tages. On the London Stock Exchange (LSE) there are official market makers for

many securities (except for the largest and most heavily traded companies, which

instead use an automated system called SETS). On the LSE one can always buy and

sell stock; each stock always has at least two market makers and they are obliged to

deal. This is in contrast with much smaller order driven markets in which it can be

extremely difficult to determine at what price one would be able to buy or sell any

of the many illiquid stocks.

In traditional exchange floor markets the burden of providing liquidity is given to

market makers or specialists. Nowadays, however, most financial markets have be-

come fully electronic and operate on what is called a matched bargain or order driven

basis. In these markets, when a buyers bid price meets a sellers offer price the stock

exchanges matching system will decide that a deal has been executed. In an order-

driven market there are numerous types of orders that can be placed, each catering to

the different needs of different market participants. The two main type of orders are

the market order, which is an order to buy or sell immediately at the best available

price, and as such gives no guarantee on the price but is guaranteed to be executed

immediately. Alternatively we have limit orders which are not to be executed unless

the specified price is met (or bettered) by current bids or asks. Here, we are not

guaranteed execution but we are guaranteed price. It should, however, be noted that

limit orders often incur higher commission fees. Further, in these order-driven mar-

kets liquidity now becomes self-organised, in the sense that any agent can choose, atany instant of time, either to provide or to consume liquidity; providing liquidity by

posting limit orders or consuming liquidity by issuing a market order.

The introduction of electronic markets has seen a sharp increase in another type of

market participant, the program trader. A program trader is one who uses a computer

to automate his trades. This may be to exploit arbitrage opportunities such as index

arbitrage (the misalignment of the price of an index and the sum of its constituent

stocks) or to perform portfolio insurance, the automated execution of a deterministic


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hedging strategy. Program traders are thought to have been a contributing factor

of the October 19, 1987 market crash19 and to be responsible for an increased stock

market volatility, since they quickly dump large orders on the market at critical times.

These large orders can contribute to the existing momentum of the market, thereby

increasing market volatility. This shall be seen in a more mathematical framework

in chapter 3.

1.7 Option pricing in illiquid markets: a literature

review

Authors such as Kreps (1979) and Bick (1987, 1990) have placed the classical Black-

Scholes-Merton formulation into the framework of a consistent model for market

equilibrium with interacting agents having very specific investment characteristics

(see section 1.6). Moreover Bick (1987, 1990) showed how geometric Brownian mo-

tion, one of the fundamental assumptions of the Black-Scholes-Merton model, can be

derived in a general equilibrium model with price-taking agents.

Furthermore Follmer and Schweizer (1993) were the first to use a microeconomic

approach to construct diffusion models for asset price movements. They define in-

formation traders who believe in a fundamental value of the asset, and noise traders

whose demands are from hedging requirements. They derived equilibrium diffusion

models for the asset price based on interaction between the two. Many of the models

discussed in this thesis such as Platen and Schweizer (1998), Sircar and Papanicolaou

(1998) and Schonbucher and Wilmott (2000) were inspired by the temporary equi-

librium approach of Follmer and Schweizer (1993). Starting from a microeconomic

equilibrium and deriving a diffusion model for stock prices which endogenously in-

corporates the demand due to hedgers and in particular delta hedgers.

19Jacklin et al. (1992) argue that one of the causes was actually information about the extentof portfolio insurance-motivated trading suddenly becoming known to the rest of the market. Thisprompted the realisation that assets had been overvalued because the information content of tradesinduced by hedging concerns had been misinterpreted. Consequently, general price levels fell sharply.


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The literature on liquidity falls broadly into two approaches. The first involves the

price impact due to a large trade. In such models the large trader can move the price

by his actions. Jarrow (1992, 1994) provided a discrete-time model which allows

the large trader to impact the market via some reaction function. He showed that

the price of a derivative in this framework must be equal to the hedge cost, but

this cost, and hence the price, is dependent on the large traders position in the

underlying and the derivative asset; leading to nonlinearity. However in markets that

allow large traders to impact the price of the asset there is the possibility of price

manipulation and so called market corners and market squeezes. A market corner

is a successful effort of a trader to manipulate the price of a futures contract by

gaining effective control over trading in the futures and the supply of the deliverable

goods. In a market squeeze, the trader achieves control by disruption in the supply of

the cash commodity. Although price manipulation violates the Commodity Exchange

Act, there have been many examples of such activities, especially in (less regulated)

developing markets. An example of a market corner is the Hunt silver manipulation

of 1979-1980, a detailed and readable account of which can be found in Williams

(1995). An example of a market squeeze is the (alleged) soybean manipulation of 1989

for which more details can be found in Pirrong (2004). However in the theoretical

framework proposed by Jarrow (1992, 1994) it was showen that to prevent any such

manipulation the price impact mechanism must not exhibit any delay. In addition a

sufficient condition to exclude profitable market manipulation (in discrete-time) was

given, i.e. that the price mechanism must be independent of the history of the trades,

and only dependent on the current position of the trades. Bank and Baum (2004)

later extended Jarrows results to continuous time.

Moreover, in the presence of price impact, it is not clear that an option is still perfectly

replicable; hence it is no longer straightforward how to derive option prices from

the prices of the underlying. Frey and Stremme (1997) studied the perturbation of

volatility induced by a delta hedging strategy for a European option whose price is

given by a classical Black-Scholes formula with constant volatility. They concluded


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that if a hedging strategy is used which does not take into account the feedback effect

(which we term first-order feedback), then it is not possible to replicate perfectly

an option, and hence there is still risk associated with hedging in illiquid markets.

They did show, however, that increasing heterogeneity of the distribution of hedged

contracts reduces both the level and price sensitivity of this un-hedged risk. Frey

(1998, 2000) then showed that if feedback is taken into account in a more general

hedging strategy (which we term full feedback), then it is possible to replicate an

option perfectly (provided certain conditions on market liquidity and the nonlinearity

of the payoff condition are satisfied). In the discrete-time framework of Jarrow (1994),

the question as to whether options could be perfectly replicated in a finitely elastic

market reduces to solving (recursively) a finite number of equations. In the continuous

time framework of Frey (1998), this can be characterised more succinctly as the

solution of a nonlinear PDE, for which Frey (1998) gave existence and uniqueness

results. These results, however, place a heavy restriction on the amount of market

illiquidity that the model allows and rely on the terminal payoff being sufficiently

smooth, both of which can be seen as undesirable restrictions.20 Frey and Patie

(2002) extended the work of Frey (2000) with an asset dependent liquidity parameter

which attempts to incorporate so called liquidity drops, whereby market liquidity

drops if the stock price drops, the aim being to reproduce, more effectively, the

volatility smile.

Other continuous time models similar to Frey (1998) include Schonbucher and Wilmott

(2000), who used a market microstructure equilibrium model to derive a modifiedstochastic process under the influence of price impact. The PDEs derived by these

latter authors correspond to those derived in chapter 2 of the present study. Sircar

and Papanicolaou (1998) derived a slightly different nonlinear PDE that depends on

the exogenous income process of the reference traders and the relative size of the

program traders. Platen and Schweizer (1998) proposed a model using an approach

that attempted to explain the volatility smile and its skewness endogenously and

20For further discussion on these restrictions see chapter 4.


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Mancino and Ogawa (2003) proposed a very similar model in the same vein. Lyukov

(2004) then extended the model of Platen and Schweizer (1998) with more realistic

assumptions about market equilibrium conditions (taking into account the presence

of a market maker) and also obtained a very similar nonlinear PDE to that derived

in chapter 2. Another tweak of these models was made by Liu and Yong (2005) who

attempted to regularise the PDE close to expiry. The majority of these models will

be considered in more detail in chapter 7.

The second approach to liquidity seen in the literature involves the price impact due

to the immediacy provisions of market makers. In these models, supply and demand

are equalised by the market maker in the short-term market. The approach is relevant

if an agent wishes to trade a large amount in a short time. These models have been

considered by Rogers and Singh (2006) and Cetin and Rogers (2007), amongst others,

who propose a series of independent auctions. The main difference with the first class

of models is that these are now local in time models, without long-term effects, i.e.

the actions of the traders do not influence the underlying stochastic process. These

models eliminate the feedback effects discussed above and, as such, they are concerned

more with the liquidation cost than permanent price impact. Bakstein and Howison

(2003) adopted a similar approach to Rogers and Singh (2006) but the former study

leads to feedback effects, which the latter study was trying to avoid. Another model

in this category is the work of Cetin et al. (2004), who modelled the liquidation cost

as dependent on the quadratic v

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Kris Glover Analyzing PDEs

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