OPTIMAL STOPPING PROBLEMS WITH APPLICATIONS TO MATHEMATICAL FINANCE A thesis submitted to the University of Manchester for the degree of Doctor of Philosophy in the Faculty of Engineering and Physical Sciences 2014 Yerkin Kitapbayev School of Mathematics Contents Abstract 9 Declaration 10 Copyright Statement 11 Acknowledgements 13 1 Introduction 14 2 The American lookback option with fixed strike 23 2.1 Introduction . 23 2.2 Formulation of the problem and its reduction . 24 2.3 The two-dimensional problem . 26 2.4 The arbitrage-free price and stopping region . 39 2.5 Conclusion . 41 3 The British lookback option with fixed strike 43 3.1 Introduction . 43 3.2 Basic motivation for the British lookback option with fixed strike . 45 3.3 The British lookback option with fixed strike: Definition and basic properties . 48 3.4 The arbitrage-free price and the rational exercise boundary . 58 3.5 The financial analysis . 64 2 4 The American swing put option 69 4.1 Introduction . 69 4.2 Formulation of the swing put option problem . 71 4.3 Free-boundary analysis of the swing option with n = 2 . 75 4.4 Solution of the swing option with n rights . 98 5 Shout put option 104 5.1 Introduction . 104 5.2 Formulation of the problem . 107 5.3 Free-boundary problem . 109 5.4 The arbitrage-free price of the shout option . 116 5.5 The financial analysis . 121 6 Smooth-fit principle for exponential L´evymodel 127 6.1 Introduction . 127 6.2 Model setting . 128 6.3 American put option on finite horizon . 130 6.4 Smooth-fit principle: review of existing results . 131 6.5 Case d = 0 and remarks for the case d+ < 0 < d ............ 132 Bibliography 137 Word count 21,440 3 List of Tables 3.1 Returns observed upon exercising the British lookback option with fixed strike compared with returns observed upon exercising the Amer- ican lookback option with fixed strike. The returns are expressed as a percentage of the original option price paid by the buyer (rounded to _ µc m K − the nearest integer), i.e.R(t; m; s)=100 = (sG (t; s ) K)=V (0; 1; 1) + and RA(t; m; s)=100 = (m − K) =VA(0; 1; 1). The parameter set is µc = 0:05, K = 1:2, T = 1, r = 0:1, σ = 0:4 and the initial stock price equals 1. 66 3.2 Returns observed upon exercising the British lookback option with fixed strike compared with returns observed upon selling the European lookback option with fixed strike. The returns are expressed as a percentage of the original option price paid by the buyer (rounded to _ µc m K − the nearest integer), i.e.R(t; m; s)=100 = (sG (t; s ) K)=V (0; 1; 1) and RE(t; m; s)=100 = VE(t; m; s)=VE(0; 1; 1). The parameter set is µc = 0:05, K = 1:2, T = 1, r = 0:1, σ = 0:4 and the initial stock price equals 1. 67 5.1 Returns observed upon shouting (average discounted payoff at T ) the shout put option R(t; x)=100 = e−r(T −t)G(t; x)=V (0;K), exercising the + American put option RA(t; x)=100 = (K − x) =VA(0;K) and exercis- B ing the British put option RB(t; x)=100 = G (t; x))=VB(0;K). The parameter set is K = 10, T = 1, r = 0:1, σ = 0:4, µc = 0:13. 122 4 5.2 Returns observed upon selling the shout put option R(t; x)=100 = −r(T −t) e V (t; x)=V (0;K), selling the American put option RA(t; x)=100 = VA(t; x)=VA(0;K), selling the European put option RE(t; x)=100 = VE(t; x)=VE(0;K) and selling the British put option RB(t; x)=100 = VB(t; x)=VB(0;K). The parameter set is K = 10, T = 1, r = 0:1, σ = 0:4, µc = 0:13. ............................ 125 5 List of Figures 2.1 A computer drawing of the optimal stopping boundary b for the prob- lem (2.13) in the case Ke = 1:5;T = 1; r = 0:1; σ = 0:4 with the boundary condition b(T ) = Kee−rT > 1: ................. 29 2.2 A computer drawing of the optimal stopping boundary s 7! g(t; s) for 1) t = 0 (upper) and 2) t = 0:3 (lower) in the case K = 1:2;T = 1; r = 0:1; σ = 0:4. The limit of g(t; · ) at zero is greater than K for every t. 41 3.1 A computer drawing of the optimal stopping boundary b for the prob- e K lem (3.22) in the case K = 1:2;S0 = 1; K = = 1:2;T = 1; µc = S0 ∗ ≈ 0:05 < µc 0:075; r = 0:1; σ = 0:4 with the boundary condition b(T ) = Kee−rT > 1 and the starting point x = K < h(0). 56 S0 3.2 A computer drawing showing how the optimal stopping boundary b for the problem (3.22) increases as one decreases the contract drift. There are four different cases: 1)µc = 0:074; 2)µc = 0:07; 3)µc = 0:05; 4)µc = −∞ (the latter corresponds to the American lookback option e K problem). The set of parameters: K = 1:2;S0 = 1; K = = 1:2;T = S0 1; r = 0:1; σ = 0:4 with the boundary condition b(T ) = Kee−rT > 1, the starting point x = K and the root of (3.9) µ∗ ≈ 0:075. 57 S0 c 3.3 A computer drawing of the rational exercise boundary s 7! g(t; s) for 1)t = 0 (top at s = 0); 2)t = 0:3; 3)t = 0:6 (bottom at s = 0) in e K ∗ the case K = 1:2;S0 = 1; K = = 1:2;T = 1; µc = 0:05 < µ ≈ S0 c 0:075; r = 0:1; σ = 0:4. The limit of g(t; · ) at zero is greater than K for every t.................................. 62 6 3.4 A computer drawing showing how the rational exercise boundary g for the problem (3.16) increases as one decreases the contract drift for fixed t = 0. There are four different cases: 1)µc = 0:074; 2)µc = 0:05; 3)µc = −0:05; 4)µc = −∞ (the latter corresponds to the American lookback option problem). All boundaries have the same limit at s = 0. The set ∗ ≈ of parameters: K = 1:2;S0 = 1;T = 1; µc 0:075; r = 0:1; σ = 0:4. The rational exercise boundary in the case µc = 0:074 is discontinuous at s∗ ≈ 1:14 < K.............................. 63 4.1 A computer drawing of the optimal exercise boundaries t 7! b(2)(t) and t 7! c(2)(t) for the problem (4.16) in the case K = 1, r = 0:1 (annual), σ = 0:4 (annual), T = 11 months, δ = 1 month. The decreasing (2) boundary c is finite on [0;Tδ] but it takes values much larger than those of b(2) on [0; 8] and therefore in order to present the structure of the continuation set in a clear way we only plot the vertical axis up to x =3. ................................... 94 4.2 A computer drawing of the lower optimal exercise boundary t 7! b(2)(t) of problem (4.16) and the optimal exercise boundary t 7! b(1)(t) of problem (4.4) (American put) in the case K = 1, r = 0:1 (annual), σ = 0:4 (annual), T = 11 months, δ = 1 month. 95 4.3 Structure of the optimal exercise boundaries t 7! b(n)(t) (lower bound- ary) and t 7! c(n)(t) (upper boundary) of problem (4.8) with n = 2; 3; 4 and t 7! b(1)(t) of problem (4.4) (American put) in the case K = 1, r = 0:1 (annual), σ = 0:4 (annual), T = 11 months, δ = 1 month. 102 5.1 A computer drawing of the optimal shouting boundary t 7! b(t) (up- per) for the shout put option (5.10) and the optimal exercise boundary t 7! bA(t) (lower) for the American put option in the case K = 10, r = 0:1, σ = 0:4 , T =1.......................... 117 7 5.2 A computer drawing showing the (dark grey) region S in which the shout put option outperforms the American put option, and the region A in which the American put option outperforms the shout put option. The parameter set is the same as in Figure 5.1 above (K = 10, r = 0:1, σ = 0:4 , T =1). ............................. 123 5.3 A computer drawing showing the (dark grey) region S in which the shout put option outperforms the British put option, and the surround- ing region B in which the British put option outperforms the shout put option. The parameter set is the same as in Figure 5.1 above (K = 10, r = 0:1, σ = 0:4 , T = 1). 124 8 The University of Manchester Yerkin Kitapbayev Doctor of Philosophy Optimal stopping problems with applications to mathematical finance December 14, 2014 The main contribution of the present thesis is a solution to finite horizon optimal stopping problems associated with pricing several exotic options, namely the Ameri- can lookback option with fixed strike, the British lookback option with fixed strike, American swing put option and shout put option. We assume the geometric Brown- ian motion model and under the Markovian setting we reduce the optimal stopping problems to free-boundary problems.