Download File

From Continuous to Discrete:

Studies on Continuity Corrections and Monte Carlo Simulation

with Applications to Barrier Options and American Options

Menghui Cao

Submitted in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy in the Graduate School of Arts and Sciences

COLUMBIA UNIVERSITY

2014 c 2013

Menghui Cao

From Continuous to Discrete: Studies on Continuity Corrections and Monte

Carlo Simulation with Applications to Barrier Options and American Options

Menghui Cao

This dissertation 1) shows continuity corrections for ﬁrst passage probabilities of Brownian bridge and barrier joint probabilities, which are applied to the pricing of two-dimensional barrier and partial barrier options, and 2) introduces new variance reduction techniques and computational improvements to Monte Carlo methods for pricing American options.

The joint distribution of Brownian motion and its first passage time has found applications in many areas, including sequential analysis, pricing of barrier options, and credit risk modeling. There are, however, no simple closed-form solutions for these joint probabilities in a discrete-time setting. Chapter 2 shows that, discrete two-dimensional barrier and partial barrier joint probabilities can be approximated by their continuous-time probabilities with remarkable accuracy after shifting the barrier away from the underlying by a factor of √ βσ ∆t. We achieve this through a uniform continuity correction theorem on the first passage probabilities for Brownian bridge, extending relevant results in Siegmund (1985a). The continuity corrections are applied to the pricing of two-dimensional barrier and partial barrier options, extending the results in Broadie, Glasserman & Kou (1997) on one-dimensional barrier options. One interesting aspect is that for type B partial barrier options, the barrier correction cannot be applied throughout one pricing formula, but only to some barrier values and leaving the other unchanged, the direction of correction may also vary within one formula. In Chapter 3 we introduce new variance reduction techniques and computational improvements to Monte Carlo methods for pricing American-style options. For simulation algorithms that compute lower bounds of American option values, we apply martingale control variates and introduce the local policy enhancement, which adopts a local simulation to improve the exercise policy. For duality-based upper bound methods, specifically the primal-dual simulation algorithm (Andersen and Broadie 2004), we have developed two improvements. One is sub-optimality checking, which saves unnecessary computation when it is sub-optimal to exercise the option along the sample path; the second is boundary distance grouping, which reduces computational time by skipping computation on selected sample paths based on the distance to the exercise boundary. Numerical results are given for single asset Bermudan options, moving window Asian options and Bermudan max options. In some examples the computational time is reduced by a factor of several hundred, while the confidence interval of the true option value is considerably tighter than before the improvements. Contents

List of Figures v

List of Tables vi

1 Introduction 1

1.1 Overview ...... 1

1.2 First Passage Times and Barrier Options ...... 2

1.3 Monte Carlo Simulation for Pricing American Options ...... 5

2 Continuity Corrections for Brownian Bridge with Applications to Barrier

Options 9

2.1 Overview ...... 9

2.2 Notations ...... 11

2.3 A Uniform Continuity Correction Theorem for Brownian Bridge ...... 16

2.4 Continuity Corrections for Barrier Joint Probabilities ...... 18

2.4.1 One-dimensional Barrier Joint Probabilities ...... 19

2.4.2 Two-dimensional Barrier and Type A Partial Barrier Joint Probabilities 20

2.4.3 Type B Partial Barrier Joint Probabilities ...... 25

i 2.4.4 Extensions to Other Basic Types of Barrier Joint Probabilities . . . . 29

2.5 Continuity Corrections for Two-dimensional Barrier and Type A Partial Bar-

rier Options ...... 33

2.5.1 Two-dimensional Barrier Options ...... 33

2.5.2 Type A Partial Barrier Options ...... 36

2.6 Continuity Corrections for Type B Partial Barrier Options ...... 39

2.6.1 Type B Partial Barrier Options ...... 39

2.6.2 An Example of Type B Partial Barrier Correction ...... 42

3 Improved Lower and Upper Bound Algorithms for Pricing American Op-

tions by Simulation 45

3.1 Overview ...... 45

3.2 Problem formulation ...... 47

3.3 Improvements to lower bound algorithms ...... 48

3.3.1 A brief review of the lower bound algorithm ...... 48

3.3.2 Distance to the exercise boundary ...... 50

3.3.3 Local policy enhancement ...... 51

3.3.4 Use of control variates ...... 52

3.4 Improvements to upper bound algorithms ...... 53

3.4.1 Duality-based upper bound algorithms ...... 53

3.4.2 Primal-dual simulation algorithm ...... 54

3.4.3 Sub-optimality checking ...... 57

ii 3.4.4 Boundary distance grouping ...... 58

3.5 Numerical results ...... 60

3.5.1 Single asset Bermudan options ...... 61

3.5.2 Moving window Asian options ...... 62

3.5.3 Symmetric Bermudan max options ...... 65

3.5.4 Asymmetric Bermudan max options ...... 65

3.5.5 Summary and other computational issues ...... 67

3.6 Conclusion ...... 70

4 Bibliography 72

A Appendix for Chapter 2 79

A.1 Proof of Theorem 1 ...... 79

A.2 Proof of Lemma 2, 3 and 4 ...... 84

A.2.1 Proof of Lemma 2 ...... 84

A.2.2 Proof of Lemma 3 ...... 85

A.2.3 Proof of Lemma 4 ...... 86

B Appendix for Chapter 3 89

B.1 Regression related issues ...... 89

B.1.1 ‘Regression now’ vs. ‘regression later’ ...... 89

B.1.2 Choice of basis functions ...... 90

B.2 Proof of Proposition 1 and 2 ...... 91

B.3 Numerical details in boundary distance grouping ...... 94 iii B.3.1 Estimators of the duality gap ...... 94

B.3.2 Eﬀective saving factor ...... 96

B.3.3 Two ways to estimate Var[D˜] ...... 98

iv List of Figures

2.1 An Illustration of Two-dimensional Barrier Joint Probability P2d(↓ b, > a) . . 12

2.2 An Illustration of Type A Partial Barrier Joint Probability PA(↓ b, > a) . . . 14

2.3 An Illustration of Type B Partial Barrier Joint Probability PB(> c, ↓ b, > a) 15

v List of Tables

2.1 Results for two-dimensional barrier and type A partial barrier joint probabilities* 24

2.2 Results for type B partial barrier joint probabilities* ...... 29

2.3 Results for two-dimensional barrier options* ...... 34

2.4 Results for two-dimensional barrier options with varying parameters (m, σ,

K, ρ, T )* ...... 37

2.5 Results for type A partial barrier options* ...... 39

2.6 Results for type B partial barrier options* ...... 42

3.1 Summary results for single asset Bermudan call options ...... 63

3.2 Summary results for moving window Asian call options ...... 64

3.3 Summary results for 5-asset symmetric Bermudan max call options . . . . . 66

3.4 Summary results for 5-asset asymmetric Bermudan max call options . . . . . 67

3.5 Impact of basis functions on 5-asset asymmetric Bermudan max call options 68

3.6 Lower bound improvements by local policy enhancement (5-asset Bermudan

max call) ...... 68

3.7 Eﬀective saving factor by sub-optimality checking and boundary distance

grouping ...... 69

vi 3.8 Computational time vs. number of exercise opportunities (moving window

Asian call) ...... 70

B.1 Regression now vs. regression later (single asset Bermudan call) ...... 90

B.2 Regression now vs. regression later (5-asset asymmetric Bermudan max call) 90

vii ACKNOWLEDGEMENTS

I am indebted to my advisor Professor Steven Kou and co-advisor Professor Mark Broadie, for their constant guidance, support, and inspiration throughout the time. Their enthusiasm and knowledge of the ﬁeld have had a profound inﬂuence upon my education, as well as my life and career. Without their continuous encouragement and guidance, this dissertation could not have been accomplished.

I would like to sincerely thank Professor Garud Iyengar, Paul Glasserman, and Martin

Haugh for serving on my defense committee and providing valuable comments and advices.

My work in this dissertation is closely related to many of their works, especially by Professor

Glasserman.

Thanks are also due to the brilliant faculty members of IEOR department at Columbia

University, in particular I would like to express my gratitude to Professor Donald Goldfarb for granting me the opportunity to study at IEOR and Professor Guillermo Gallego for the generous help throughout my study.

My stay at Columbia University has truly been memorable. I have built great friendships, enjoyed inspiring discussions and learned valuable knowledge from my peer students. To name a few, Zheng Wang, Giovanni Petrella, Bin Yu, Jingyi Li, Jan Cosyn, Anton Riabov,

Ge Zhang and Zhi Huang.

Finally, I am truly grateful to my family, my friends, and my colleagues for their trust, care, support and encouragement throughout this process.

viii Chapter 1

Introduction

1.1 Overview

An option is a type of financial derivatives contract that gives one party (the option holder) the right, but not the obligation, to perform a specified transaction with another party (the option writer) according to specified terms. The option can be of American style, which allows the option holder an exercise of the right at any time before the expiry date, or

European style, where the exercise is only allowed on the option expiry date. In between

American and European styles, there are Bermudan options that can be exercised on spe- ciﬁc dates up to the expiry date, essentially American-style options that can be exercised in discrete time. Stock options are usually of American style, stock index options and currency options are typically of European style, while most exotic interest rate options are of Bermu- dian style. Standard European or American options are usually considered vanilla options, in contrast to exotic options which have more complex structure, such as barrier options, lookback options, Asian options, and basket options.

Option pricing is often complicated because of the number of variables to consider and the complexity of underlying dynamics. It has evolved to be an important ﬁeld in quantitative 1 2

ﬁnance, thanks to the tremendous growth of derivatives market since the introduction of

Black-Scholes (Merton) model in the 1970s. Option pricing traces its roots to Bachelier

(1900) who invented Brownian motion to model options on French government bonds, which anticipated Einstein’s independent use of Brownian motion in physics by ﬁve years. It was not until the publication of the Black-Scholes (1973) option pricing formula that a theoretically consistent framework for pricing options became available. Scholes and Merton won the 1997

Nobel Prize in Economics for this milestone work.

Pricing options often reduces to computing the expectations of random variables (repre- senting option payoffs). Most options can be priced in closed-form under a continuous-time framework (such as Black-Scholes), American options and exotic options with complicated dynamics or discrete-time features often do not have closed-form solutions. Numerical methods, such as binomial and trinomial lattices, finite difference methods, and Monte Carlo simulation, are widely used for those cases.

This dissertation studies some problems arising in discrete time, from ﬁrst passage times and optimal stopping times for Gaussian random walks to the pricing of discrete barrier options and Bermudan options, in contrast to Brownian motions, continuous barrier options and American options in continuous time.

1.2 First Passage Times and Barrier Options

The joint distribution of Brownian motion and its ﬁrst passage time has found applications in many areas, including sequential analysis (see, for example, Siegmund 1985a), pricing of barrier options (see, for example, Merton 1973), and the modeling of credit risk (see, for 3 example, Black and Cox 1976). These joint probabilities usually have closed-form solutions under standard continuous-time assumptions (see, for example, Chuang 1994). However, closed-form solutions are not available for discrete-time probabilities on Gaussian random walks, other than quasi closed-form solutions using multivariate normal distributions.

Siegmund (1979, 1985a, 1985b) and Siegmund & Yuh (1982) show diﬀusion approximations to the discrete-time probabilities of random walks. Glasserman & Liu (1997) extends the corrected diﬀusion approximations to perturbed random walks. Higher order expansions for various expectations and probabilities are given in Blanchet & Glynn (2006), Chang &

Peres (1997), Howison (2007), Howison & Steinberg (2007), Keener (2013).

In this dissertation we are interested in four styles of barrier joint probabilities: (1) a

Brownian motion joint with its first passage time (one-dimensional barrier); (2) one Brow- nian motion joint with the first passage time of another correlated Brownian motion (two- dimensional barrier); (3) a Brownian motion joint with its first passage time before a fixed time t1 (type A partial barrier); (4) a Brownian motion joint with its first passage time after a fixed time t1 (type B partial barrier). There are eight basic types for each style of barrier joint probabilities, based on the direction of the barrier crossing (up or down), the triggered event (in or out), and the terminal value condition (above or below).

A barrier option is knocked in (activated) or out (extinguished) when the underlying asset price, index, or rate reaches a certain level during a specified period of time. The barrier monitoring can be in continuous or discrete time, the exercise style can be American or European, some barrier options pay a rebate upon the barrier crossing while others do not. In contrast to one-dimensional barrier options whose barrier crossing and payoff depend 4 on one underlying, there are two-dimensional barrier options for which the barrier crossing depends upon one underlying and the payoff on another, and partial barrier options whose barrier crossing is monitored within part of the option’s lifetime. Similar to the barrier joint probabilities, there are eight basic types for each style of barrier options. For further discussions on two-dimensional barrier and partial barrier options, reader is referred to Heynen &

Kat (1994a, 1994b) and Carr (1994).

If the barrier crossing is monitored in continuous time, closed-form solutions for barrier option prices are available under standard Black-Scholes models (Merton 1973). The real contracts, however, usually specify discrete monitoring, such as daily or weekly. While the price differences between continuous and discrete barrier options are often substantial, there are no easily-computed analytical pricing formulae for discrete barrier options. Nu- merical methods, such as lattice, finite difference, and Monte Carlo simulation, are usually straightforward and widely used, but can be very time-consuming and face difficulties in incorporating discrete monitoring (Broadie, Glasserman & Kou 1999).

There have been several approaches to the practical pricing of discrete barrier options.

Broadie, Glasserman & Kou (1997) show that discrete one-dimensional barrier option prices can be approximated with remarkable accuracy using the closed-form solutions for continuous barrier options, with a barrier shift as the continuity correction. Broadie, Glasserman & Kou

(1999) develop continuity corrections for other path-dependent options such as lookback and hindsight options. Howison (2007) derives an approximation based on partial diﬀerential equations. Petrella & Kou (2005) and Feng & Linetsky (2006) use Laplace and Hilbert transforms to price discrete barrier options. There is no simple closed-form pricing of discrete 5 two-dimensional barrier and partial barrier options in the literature to our best knowledge.

1.3 Monte Carlo Simulation for Pricing American Op- tions

American options, unlike European options that can be priced in closed-form in standard

Black-Scholes settings, usually do not have closed-form pricing formulas (except for a few cases). Lattice and ﬁnite diﬀerence methods are widely used, however, their computation time and memory requirement can increase polynomially with the number of underlying variables (curse of dimensionality), which makes them impractical in multi-dimensional and path-dependent cases.

Monte Carlo simulation has been used for a wide array of financial engineering problems, since Boyle (1977) first applied Monte Carlo simulation for pricing European contingent claims. It has many nice features, including the straightforward implementation, flexible framework, as well as the linear relationship of complexity and memory requirement with dimensionality of the problem. However, it was long considered that simulation with the forward nature could not be used to solve the optimal stopping problem of American options, for which the backward induction is needed. In recent years many simulation-based algorithms have been proposed for pricing American options (effectively Bermudan options that have American style but can be exercised in discrete time), most using a hybrid approach of simulation and dynamic programming to determine an exercise policy. Because these algorithms produce an exercise policy which is inferior to the optimal policy, they provide low-biased estimators of the true option values. We use the term lower bound algorithm to 6 refer to any method that produces a low-biased estimate of an American option value with a sub-optimal exercise strategy.1

In an earliest eﬀort, Bossaerts (1989) parameterizes the exercise strategy then solve for parameters that maximize the value of the option from simulation. Tilley (1993) proposes a bundling algorithm where the paths are bundled into groups according to the state variables.

Barraquand and Martineau (1995) introduces a stratiﬁcation method in which they partition the paths into groups by the intrinsic value instead of the state variables. Raymar and

Zwecher (1997) extends it to a two-dimensional stratiﬁcation to get an improved exercise strategy. Andersen (2000) proposes a method that parameterizes the exercise policy and then optimizes the parameters over a set of simulated paths.

Regression-based methods for pricing American options are proposed by Carriere (1996),

Tsitsiklis and Van Roy (1999) and Longstaff and Schwartz (2001). The least-squares method by Longstaff and Schwartz projects the conditional discounted payoffs onto basis functions of the state variables. The projected value is then used as the approximate continuation value, which is compared with the intrinsic value for determining the exercise strategy. Low-biased estimates of the option values can be obtained by generating a new, i.e., independent, set of simulation paths, and exercising according to the sub-optimal exercise strategy. Clément,

Lamberton and Protter (2002) analyze the convergence of the least-squares method. Glasser- man and Yu (2004) study the tradeoﬀ between the number of basis functions and the number of paths. Broadie, Glasserman and Ha (2000) propose a weighted Monte Carlo method in which the continuation value of the American option is expressed as a weighted sum of future

1The terms exercise strategy, stopping time and exercise policy will be used interchangeably in this chapter. 7 values and the weights are selected to optimize a convex objective function subject to known conditional expectations. Glasserman and Yu (2002) analyze this ‘regression later’ approach, compared to the ‘regression now’ approach implied in other regression-based methods.

One difficulty associated with lower bound algorithms is that of determining how well they estimate the true option value. If a high-biased estimator is obtained in addition to the low-biased estimator, a confidence interval can be constructed for the true option value, and the width of the confidence interval may be used as an accuracy measure for the algorithms. Broadie and Glasserman (1997, 2004) propose two convergent methods that generate both lower and upper bounds of the true option values, one based on simulated trees and the other a stochastic mesh method. Haugh and Kogan (2004) and Rogers (2002) independently develop dual formulations of the American option pricing problem based on martingale, which can be used to construct upper bounds of the option values. Andersen and Broadie (2004) show how duality-based upper bounds can be computed directly from any given exercise policy through a primal-dual simulation algorithm, leading to significant improvements in their practical implementation. We call any algorithm that produces a high-biased estimate of an American option value an upper bound algorithm.

Jamshidian (2004) introduces an alternative ’multiplicative’ duality approach. Chen and

Glasserman (2007) study the connections between multiplicative duality and additive martingale duality approaches. Belomestny et. al (2009) develop a variation of the martingale duality approach that does not require nested simulation. Rogers (2010) proposes a pure dual algorithm for pricing and hedging Bermudan options. Desai et. al (2012) introduce a pathwise optimization method based on the martingale duality and a convex optimization. 8

The duality-based upper bound estimator can often be represented as a lower bound estimator plus a penalty term. The penalty term, which may be viewed as the value of a non-standard lookback option, is a non-negative quantity that penalizes potentially incorrect exercise decisions made by the sub-optimal policy. Estimation of this penalty term requires nested simulations which is computationally demanding. Our work addresses this major shortcoming of the duality-based upper bound algorithm by introducing improvements that may signiﬁcantly reduce its computational time and variance. We also propose enhancements to lower bound algorithms which improve exercise policies and reduce the variance.

The rest of this dissertation is organized as follows: Chapter 2 shows continuity corrections for ﬁrst passage probabilities of Brownian bridge, as well as two-dimensional barrier and partial barrier joint probabilities, with direct applications to two-dimensional barrier and partial barrier options. Chapter 3 introduces new variance reduction techniques and computational improvements to both the lower and upper bound Monte Carlo methods for pricing American-style options. Numerical results are presented as well as theoretical deriva- tions. Some detailed proofs for Chapter 2 are placed in Appendix A, some implementation details and proofs for Chapter 3 can be found in Appendix B. Chapter 2

Continuity Corrections for Brownian Bridge with Applications to Barrier Options

2.1 Overview

In this chapter we show the continuity corrections for ﬁrst passage probabilities of Brownian bridge, as well as two-dimensional barrier and partial barrier joint probabilities, with direct applications to two-dimensional and partial barrier options. The contribution of this chapter is fourfold,

(i) We prove a uniform continuity correction theorem on the ﬁrst passage probabilities of Brownian bridge (Theorem 1), extending relevant results in Siegmund (1985a).

(ii) We show the continuity corrections for two-dimensional barrier, type A and type B partial barrier joint probabilities (Theorems 2, 3 and 4), based on the uniform continuity correction theorem.

(iii) We derive the continuity corrections for the pricing of two-dimensional barrier, type

A and type B partial barrier options (Corollaries 2, 3 and 4), based on the probability results.

(iv) We demonstrate that for type B partial barrier options, the barrier correction cannot 9 10 be simply applied throughout one pricing formula (like for most barrier options), but only to some barrier values and leaving the other unchanged, the direction of correction may also vary within one formula–based on the direction of barrier crossing it corresponds to.

The numerical results, for barrier joint probabilities (in Tables 2.1 and 2.2) and barrier option prices (in Tables 2.3 to 2.6), are calculated on a Pentium 4 1.8G Hz PC using continuous-time formulae–both ‘before’ and ‘after’ continuity corrections, and compared to

‘true’ values obtained via Monte Carlo simulation with 108 sample paths and appropriate control variates (for variance reduction). The continuous-time formulae takes 10−5 ∼ 10−4 second per option to evaluate, compared to 5 ∼ 25 minutes per option using Monte Carlo simulation. Selective sample paths are charted (in Figures 2.1 to 2.3) to help illustrate the three style of barrier joint probabilities studied.

The rest of this chapter is organized as follows: we introduce the notations in Section

2.2; in Section 2.3 we show a uniform continuity correction theorem on the ﬁrst passage probabilities of Brownian bridge; the continuity corrections for two-dimensional, type A partial, and type B partial barrier joint probabilities are shown in Section 2.4; in Section

2.5 we apply the probability results to obtain the continuity corrections for the pricing of two-dimensional barrier and type A partial barrier options; the continuity corrections for type B partial barrier options are derived in Section 2.6, with an example illustrating details on the pricing formulae and barrier corrections for type B partial barrier options. Some detailed proofs are given in the appendices. 11 2.2 Notations

Consider two correlated Brownian motions with drifts,

X(t) := X(0) + µ1t + σ1B1(t),Y (t) := Y (0) + µ2t + σ2B2(t),

where B1(t) and B2(t) are standard Brownian motions with correlation ρ, i.e., dB1(t)dB2(t) =

ρdt. The discrete-time version of the Brownian motions are two correlated Gaussian random

walks (Xn,Yn) with time interval ∆t = T/m,

n √ X X Xn := X0 + µ1n∆t + σ1 ∆tZi , n = 1, ..., m, i=1 n √ X Y Yn := Y0 + µ2n∆t + σ2 ∆tZi , n = 1, ..., m, i=1

X Y where (Zi ,Zi ) are independent pairs of standard bivariate Gaussian random variables with correlation ρ. Extremal values of the processes within a time interval can be deﬁned as

v v j j M|u := max X(t), m|u := min X(t); M|i := max Xk, m|i := min Xk. u≤t≤v u≤t≤v i≤k≤j i≤k≤j

The ﬁrst passage times of X(t) and Xn are

τ↑(X, x) := inf{t ≥ 0 : X(t) ≥ x}, τ↓(X, x) := inf{t ≥ 0 : X(t) ≤ x};

0 0 τ↑(X, x) := inf{i ≥ 0 : Xi ≥ x}, τ↓(X, x) := inf{i ≥ 0 : Xi ≤ x}.

Note that we assume X and Y to start from 0, i.e., X0 = 0 = Y0, by default in this chapter

(unless otherwise speciﬁed, for example, X0 = x in Px,y(·)).

The one-dimensional barrier joint probabilities that X down-crosses the barrier b (b < 0) prior to time T and exceeds a at T are

0 0 P1d(↓ b, > a) := P (τ↓(X, b) ≤ T,X(T ) > a),P1d(↓ b, > a) := P (τ↓(X, b) ≤ m, Xm > a). 12

Figure 2.1: An Illustration of Two-dimensional Barrier Joint Probability P2d(↓ b, > a)

0.20

Y(t) 0.10

a = 0.0 0.00 t

0.0 0.1 T = 0.2 X(t) -0.10 b = -0.1

-0.20

2d Path X 2d Path Y

The two-dimensional barrier joint probabilities that X down-crosses the barrier b (b < 0) prior to time T and Y exceeds a at T are

0 0 P2d(↓ b, > a) := P (τ↓(X, b) ≤ T,Y (T ) > a),P2d(↓ b, > a) := P (τ↓(X, b) ≤ m, Ym > a).

The corresponding two-dimensional barrier joint probabilities involving X(T ) are

P2d(↓ b, > c, > a) := P (τ↓(X, b) ≤ T,X(T ) > c, Y (T ) > a),

0 0 P2d(↓ b, > c, > a) := P (τ↓(X, b) ≤ m, Xm > c, Ym > a).

Figure 2.1 shows a sample path for P2d(↓ b, > a), illustrating the case of (↓ b, > a). 13

The type A partial barrier joint probabilities that X down-crosses the barrier b (b < 0) prior to time t1 (∆t = t1/m) and exceeds a at time T are

PA(↓ b, > a) := P (τ↓(X, b) ≤ t1,X(T ) > a),

0 0 PA(↓ b, > a) := P (τ↓(X, b) ≤ m, X(T ) > a).

The corresponding type A partial barrier joint probabilities involving X(t1) are

PA(↓ b, > c, > a) := P (τ↓(X, b) ≤ t1,X(t1) > c, X(T ) > a),

0 0 PA(↓ b, > c, > a) := P (τ↓(X, b) ≤ m, Xm > c, X(T ) > a).

˜ Figure 2.2 shows three sample paths for PA(↓ b, > a), illustrating the cases of (↓b, < a),

(↓ b, < a), and (↓ b, > a) respectively.

The type B partial barrier joint probabilities that X starts above c (c ≥ b) at time t1

(∆t = (T − t1)/m), down-crosses the barrier b between t1 and T , and exceeds a at T , are

PB(> c, ↓ b, > a) := P (X(t1) > c, min X(t) ≤ b, X(T ) > a) t1≤t≤T

t1 t1 t1 = P (X (0) > c, τ↓(X , b) ≤ T − t1,X (T − t1) > a),

0 PB(> c, ↓ b, > a) := P (X(t1) > c, min X(t1 + i∆t) ≤ b, X(T ) > a) 0≤i≤m

t1 0 t1 t1 = P (X0 > c, τ↓(X , b) ≤ m, Xm > a),

where Xs is the time-shifted process starting from X(s),

n √ s s X X X (t) := X(s) + µ1t + σ1B1(t),Xn := X(s) + (µ1∆t + σ1 ∆tZi ), i=1

and the extremal values of Xs are

s v s s v s s j s s j s M |u := max X (t), m |u := min X (t); M |i := max Xk, m |i := min Xk. u≤t≤v u≤t≤v i≤k≤j i≤k≤j 14

Figure 2.2: An Illustration of Type A Partial Barrier Joint Probability PA(↓ b, > a)

0.20

0.10

X(t) a = 0.0 0.00 t 0.0 t1 = 0.1 T = 0.2

-0.10

b = -0.1

-0.20

tA Path 1 tA Path 2 tA Path 3

The corresponding type B partial barrier joint probabilities not involving X(T ) are

t1 PB(> c, ↓ b) := P (X(t1) > c, τ↓(X , b) ≤ T − t1),

0 0 t1 PB(> c, ↓ b) := P (X(t1) > c, τ↓(X , b) ≤ m).

Figure 2.3 shows three sample paths for PB(> c, ↓ b, > a) with c = b, illustrating the cases of (> c, ↓˜b, > a), (> c, ↓ b, < a), and (> c, ↓ b, > a) respectively.

We deﬁne the conditional probabilities given initial values (and terminal values) as

0 Px(·) := P (·|X(0) = x),Px(·) := P (·|X0 = x),

0 Px,y(·) := P (·|X(0) = x, X(T ) = y),Px,y(·) := P (·|X0 = x, Xm = y),

where Px,y(·) is w.r.t. a Brownian bridge. We also deﬁne the non-crossing barrier joint 15

Figure 2.3: An Illustration of Type B Partial Barrier Joint Probability PB(> c, ↓ b, > a)

0.20

0.10

X(t) a = 0.0 0.00 t 0.0 t1 = 0.1 T = 0.2

-0.10 b = -0.1

-0.20

tB Path 1 tB Path 2 tB Path 3

probabilities as

˜ P1d(↓b, > a) := P (τ↓(X, b) > T, X(T ) > a),

˜ P2d(↓b, > a) := P (τ↓(X, b) > T, Y (T ) > a),

˜ PA(↓b, > a) := P (τ↓(X, b) > t1,X(T ) > a),

˜ t1 PB(> c, ↓b, > a) := P (X(t1) > c, τ↓(X , b) > T − t1,X(T ) > a), where ‘↓˜’ indicates the complement of ‘↓’ (i.e., the non-crossing event).

For the convenience of proof, we deﬁne

√ √ √ dm := σ1 T ln m, b−∆t = b − βσ1 ∆t, b+∆t = b + βσ1 ∆t,

C : a generic positive constant that satisﬁes an equality or inequality. 16

Remarks. In this chapter we deﬁne the probabilities with initial value and terminal value conditions exclusive of the thresholds a and c, e.g., X(t1) > c instead of X(t1) ≥ c. They are in fact invariant with the threshold inclusion or exclusion, e.g., P2d(↓ b, > c, > a) ≡ P2d(↓ b, ≥ c, > a), because the probability that a Brownian motion or a Gaussian random walk takes a speciﬁc value at any time is zero, e.g., P2d(↓ b, = c, > a) = 0.

2.3 A Uniform Continuity Correction Theorem for Brow- nian Bridge

Theorem 1 The following continuity corrections hold for ﬁrst passage probabilities of Brow- nian bridge, for each ε > 0.

(i) For an up-crossing ﬁrst passage probability (b > 0),

√ 1 P 0 (τ 0 (X, b) ≤ m) − P (τ (X, b + βσ ∆t) ≤ T ) = o(√ ), (2.3.1) δ,ζ ↑ δ,ζ ↑ 1 m

uniformly in −dm ≤ δ, ζ ≤ b − ε, i.e., δ and ζ are not arbitrarily close to b.

(ii) For an up-crossing ﬁrst passage probability (b > 0),

√ 1 P 0 (τ 0 (X, b) ≤ m) − P (τ (X, b + βσ ∆t) ≤ T ) = (ζ − b) · O(√ ), (2.3.2) δ,ζ ↑ δ,ζ ↑ 1 m

uniformly in −dm ≤ ζ ≤ b − ε ≤ δ < b, i.e., δ can be arbitrarily close to b while ζ is not;

√ 1 P 0 (τ 0 (X, b) ≤ m) − P (τ (X, b + βσ ∆t) ≤ T ) = (δ − b) · O(√ ), (2.3.3) δ,ζ ↑ δ,ζ ↑ 1 m

uniformly in −dm ≤ δ ≤ b − ε ≤ ζ < b, i.e., ζ can be arbitrarily close to b while δ is not.

(iii) For a down-crossing ﬁrst passage probability (b < 0),

√ 1 P 0 (τ 0 (X, b) ≤ m) − P (τ (X, b − βσ ∆t) ≤ T ) = o(√ ), (2.3.4) δ,ζ ↓ δ,ζ ↓ 1 m 17

uniformly in b + ε ≤ δ, ζ ≤ dm, i.e., δ and ζ are not arbitrarily close to b.

(iv) For a down-crossing ﬁrst passage probability (b < 0),

√ 1 P 0 (τ 0 (X, b) ≤ m) − P (τ (X, b − βσ ∆t) ≤ T ) = (ζ − b) · O(√ ), (2.3.5) δ,ζ ↓ δ,ζ ↓ 1 m

uniformly in b < δ ≤ b + ε ≤ ζ ≤ dm, i.e., δ can be arbitrarily close to b while ζ is not;

√ 1 P 0 (τ 0 (X, b) ≤ m) − P (τ (X, b − βσ ∆t) ≤ T ) = (δ − b) · O(√ ), (2.3.6) δ,ζ ↓ δ,ζ ↓ 1 m

uniformly in b < ζ ≤ b + ε ≤ δ ≤ dm, i.e., ζ can be arbitrarily close to b while δ is not.

Remarks. Theorem 1 extends some uniform convergence results in Siegmund (1985a, p.

224), from conditioning on either the initial or terminal value to be conditioning on both

(see Corollary 1 below for more details), also determines the order of convergence when δ or

ζ can be arbitrarily close to b. These results are critical for the proof of Theorem 4.

Corollary 1 The following uniform convergence results for conditional ﬁrst passage proba-

bilities in Siegmund (1985a, p. 224) are special cases of Theorem 1,

(m) −1/2 (i) “ Pξ {τ < m} = exp{−2(b + ρ+)(b + ρ+ − ξ)/m} + o(m ) ... is uniform in − log m ≤

ξ0 ≤ ζ − ε for each ε > 0”.

(m) −1/2 −1/2 (ii) “ Pλ,0 {τ < m} = exp{−2ζ(ζ − λ0)} + O(m ) uniformly in ζ − ε ≤ m λ < ζ for

each ε > 0”.

Proof. Part (i) is simply (2.3.1) with δ = 0, part (ii) follows directly from (2.3.2) with ζ = 0.

Note that we generalize the unit-variance and unit-horizon settings in Siegmund (1985a) to 18

2 σ1 and T in this chapter, and establish the following mapping of variables:

√ Sn → mXn, ρ+ → β, √ √ λ0 → δ, λ = mλ0 → mδ, √ √ ξ0 → ζ, ξ = mξ0 → mζ, √ √ ζ → b, b = mζ → mb,

0 √ √ 2ζ − ξ0 → ζb, ξ = m(2ζ − ξ0) → mζb.

For the proof of Theorem 1, we ﬁrst change the measure from P 0 to P 0 , then give a δ,ζ δ,ζb detailed uniform expansion involving changing of drifts for a discrete random walk. A key step is to recognize the uniform boundedness of the integrand in (A.2.2). As a comparison, the proof in Siegmund (1985a, p. 224) focuses on the expansion of the conditional probability

0 Pδ,ζ directly. Some techniques used in Siegmund (1985a), such as the time-reversal argument and the choice of Am, are also used in our proof. The detailed proof of Theorem 1 is shown in Appendix A.1.

2.4 Continuity Corrections for Barrier Joint Probabil- ities

We ﬁrst review some results on the continuity corrections for one-dimensional barrier joint probabilities, then show the continuity corrections for two-dimensional barrier and partial barrier joint probabilities. For simplicity we will show one basic type for each style of barrier joint probabilities, and provide the extensions to other basic types at the end of this section. 19

2.4.1 One-dimensional Barrier Joint Probabilities

Lemma 1 The following continuity corrections hold for one-dimensional barrier joint probabilities in ∀a ∈ R and ∀b < 0,

√ 1 P (τ 0 (X, b) ≤ m) = P (τ (X, b − βσ ∆t) ≤ T ) + o(√ ), (2.4.1) ↓ ↓ 1 m √ 1 P 0 (↓ b, > a) = P (↓ b − βσ ∆t, > a) + o(√ ), (2.4.2) 1d 1d 1 m √ 1 P (↓˜b − βσ ∆t, ≤ b) = o(√ ), (2.4.3) 1d 1 m √ 1 where β = −ζ( 2 )/ 2π ≈ 0.5826 and ζ(·) is the Riemann zeta function.

Proof. (2.4.1) and the a ≥ b case of (2.4.2) follow from Theorem 10.41 in Siegmund (1985a)

(also see Corollary 3.2 in Kou 2003). For the a < b case of (2.4.2),

0 0 0 0 P1d(↓ b, > a) = P (τ↓(X, b) ≤ m, Xm > a) = P (τ↓(X, b) ≤ m) − P1d(↓ b, ≤ a) √ 1 = P (τ (X, b − βσ ∆t) ≤ T ) + o(√ ) − P (X(T ) ≤ a) ↓ 1 m 1 = P (τ (X, b ) ≤ T ) + o(√ ) − P (↓ b , ≤ a) ↓ −∆t m 1d −∆t 1 = P (↓ b , > a) + o(√ ), 1d −∆t m

where the third equality results from (2.4.1) and the last equality holds because a < b−∆t for a suﬃciently large m. 20

(2.4.3) is a special case of Corollary 3.2 in Kou (2003). Alternatively we can show it as follows,

˜ P1d(↓b−∆t, ≤ b) = P (X(T ) ≤ b) − P1d(↓ b−∆t, ≤ b)

= P (X(T ) ≤ b) − P (τ↓(X, b−∆t) ≤ T ) + P1d(↓ b−∆t, > b) 1 = P (X ≤ b) − P (τ 0 (X, b) ≤ m) + P 0 (↓ b, > b) + o(√ ) m ↓ 1d m 1 = P (X ≤ b) − P 0 (↓ b, ≤ b) + o(√ ) m 1d m 1 1 = P (τ 0 (X, b) > m, X ≤ b) + o(√ ) = o(√ ), ↓ m m m

where the third equality results from (2.4.1) and (2.4.2), and the last equality holds because

0 P (τ↓(X, b) > m, Xm ≤ b) = 0. 2

2.4.2 Two-dimensional Barrier and Type A Partial Barrier Joint Probabilities

Theorem 2 The following continuity corrections hold for two-dimensional barrier joint

probabilities in ∀a, c ∈ R and ∀b < 0,

√ 1 P 0 (↓ b, > a) = P (↓ b − βσ ∆t, > a) + o(√ ), (2.4.4) 2d 2d 1 m √ 1 P 0 (↓ b, > c, > a) = P (↓ b − βσ ∆t, > c, > a) + o(√ ). (2.4.5) 2d 2d 1 m

Proof. Consider the continuous two-dimensional barrier joint probability,

P2d(↓ b, > c, > a) = P (τ↓(X, b) ≤ T,X(T ) > c, Y (T ) > a) Z ∞ = P0,ζ (τ↓(X, b) ≤ T )P (Y (T ) > a|X(T ) = ζ)fT (ζ)dζ, c

where ft(x) is the normal probability density of X(t),

2 1 (x − µ1t) ft(x) := f(X(t) = x) = exp − . p 2 2σ2t 2πσ1t 1 21

The discrete probability is

0 0 P2d(↓ b, > c, > a) = P (τ↓(X, b) ≤ m, Xm > c, Ym > a) Z ∞ 0 = P0,ζ (τ↓(X, b) ≤ m)P (Ym > a|Xm = ζ)fT (ζ)dζ, c and the continuity correction diﬀerence is

0 ∆P2d(↓ b, > c, > a) := P2d(↓ b, > c, > a) − P2d(↓ b−∆t, > c, > a) Z ∞ 0 = ∆P0,ζ (τ↓(X, b) ≤ m)P (Ym > a|Xm = ζ)fT (ζ)dζ, c where

0 0 ∆Px,y(τ↓(X, b) ≤ m) := Px,y(τ↓(X, b) ≤ m) − Px,y(τ↓(X, b−∆t) ≤ T ).

Similarly,

0 ∆P2d(↓ b, > a) := P2d(↓ b, > a) − P2d(↓ b−∆t, > a) Z ∞ 0 = ∆P0,ζ (τ↓(X, b) ≤ m)P (Ym > a|Xm = ζ)fT (ζ)dζ. −∞

Below we show (2.4.5) in three cases, with respect to c relative to the barrier b.

(i) c > b

Note that ∀a ∈ R and ∀b < 0,

0 |∆P2d(↓ b, > dm, > a)| ≤ max[P2d(↓ b, > dm, > a),P2d(↓ b−∆t, > dm, > a)]

≤ P (Xm > dm), while

1 − 1 (ln m)2 1 P (X > d ) = Φ(− ln m) ≤ C · e 2 = o(√ ). (2.4.6) m m (ln m)2 m 22

0 √1 By (2.3.4), ∆P0,ζ (τ↓(X, b) ≤ m) = o( m ) holds uniformly in b + ε ≤ ζ ≤ dm, therefore

|∆P2d(↓ b, > c, > a)| = |∆P2d(↓ b, ∈ (c, dm], > a) + ∆P2d(↓ b, > dm, > a)|

Z dm 0 ≤ ∆P0,ζ (τ↓(X, b) ≤ m|Xm = ζ)P (Ym > a|Xm = ζ)fT (ζ)dζ + P (Xm > dm) c

0 1 1 ≤ sup ∆P0,ζ (τ↓(X, b) ≤ m) · P (c ≤ Xm ≤ dm,Ym > a) + o(√ ) = o(√ ). c≤ζ≤dm m m

(ii) c = b

0 √1 By (2.3.6), ∆P0,ζ (τ↓(X, b) ≤ m) = b · O( m ) holds uniformly in b < ζ ≤ b + ε, therefore

Z b+ε 0 |∆P2d(↓ b, ∈ (b, b + ε], > a)| = ∆P0,ζ (τ↓(X, b) ≤ m)P (Ym > a|Xm = ζ)fT (ζ)dζ b

0 ≤ sup ∆P0,ζ (τ↓(X, b) ≤ m)P (Ym > a|Xm = ζ)fT (ζ) · ε b≤ζ≤b+ε 1 1 ≤ b · O(√ ) · C · ε = ε · O(√ ), m m

where the last inequality holds because the normal density fT (ζ) is bounded by a constant.

Applying case (i) here,

which holds for each ε > 0, and the result follows by letting ε → 0. 23

(iii) c < b

|∆P2d(↓ b, > c, > a)|

= |∆P2d(↓ b, > b, > a) + ∆P2d(↓ b, ∈ (b−∆t, b], > a) + ∆P2d(↓ b, ∈ (c, b−∆t], > a)| 1 ≤ o(√ ) + |P 0 (↓ b, ∈ (b , b], > a) − P (↓ b , ∈ (b , b], > a)| m 2d −∆t 2d −∆t −∆t 1 = |P (X(T ) ∈ (b , b],Y (T ) > a) − P (↓ b , ∈ (b , b], > a)| + o(√ ) −∆t 2d −∆t −∆t m 1 = P (↓˜b , ∈ (b , b], > a) + o(√ ) 2d −∆t −∆t m 1 1 ≤ P (↓˜b , ≤ b) + o(√ ) = o(√ ), 1d −∆t m m where the last equality follows from (2.4.3). This completes the proof of (2.4.5).

Finally, (2.4.4) follows case (iii) of (2.4.5) above, with any c < b (and ∀b < 0),

∆P2d(↓ b, > a) = ∆P2d(↓ b, > c, > a) + ∆P2d(↓ b, < c, > a) 1 = o(√ ) + P 0 (↓ b, < c, > a) − P (↓ b , < c, > a) m 2d 2d −∆t 1 1 = o(√ ) + P (X(T ) < c, Y (T ) > a) − P (X(T ) < c, Y (T ) > a) = o(√ ), m m

where the third equality holds because c < b−∆t for a large m. 2

Theorem 3 The following continuity corrections hold for type A partial barrier joint probabilities in ∀a, c ∈ R and ∀b < 0,

√ 1 P 0 (↓ b, > a) = P (↓ b − βσ ∆t, > a) + o(√ ), (2.4.7) A A 1 m √ 1 P 0 (↓ b, > c, > a) = P (↓ b − βσ ∆t, > c, > a) + o(√ ). (2.4.8) A A 1 m 24

Table 2.1: Results for two-dimensional barrier and type A partial barrier joint probabilities*

Prob. Barrier Continuous Corrected True Rel. Err. Rel. Err. Type (b < 0) (Before) (After) (s.e.) (Before) (After) Two- -0.02 0.684 0.617 0.617 (0.0001) 10.9% 0.0% dim -0.10 0.283 0.245 0.245 (0.0001) 15.5% -0.3% Barrier -0.18 0.084 0.069 0.070 (0.0001) 20.8% -0.7% Type A -0.02 0.634 0.541 0.542 (0.0001) 17.0% -0.0% Partial -0.10 0.143 0.109 0.110 (0.0001) 30.6% -0.9% Barrier -0.18 0.012 0.008 0.008 (0.0000) 49.4% -2.4%

* P2d(↓ b, > a) as in (2.4.4) and PA(↓ b, > a) as in (2.4.7). The parameters are a = −0.1, µ1 = µ2 = 0.1, σ1 = σ2 = 30%, ρ = 50%, t1 = 0.1, T = 0.2 and m = 25.

Proof. Theorem 3 directly follows from Theorem 2, since the type A partial barrier joint q t1 probability is a special case of two-dimensional barrier joint probability with ρ = T ,

PA(↓ b, > c, > a) = P (τ↓(X, b) ≤ t1,X(t1) > c, X(T ) > a) Z ∞ = P (τ↓(X, b) ≤ t1,X(T ) > a|X(t1) = δ)f(X(t1) = δ)dδ c Z ∞

= P (τ↓(X, b) ≤ t1|X(t1) = δ)P (X(T ) > a|X(t1) = δ)ft1 (δ)dδ. c

The rest of proof follows the same argument as for Theorem 2. 2

Table 2.1 shows that continuous two-dimensional barrier and type A partial barrier joint

probabilities can have substantially diﬀerent values from the discrete probabilities, with

relative errors (‘rel. err.’) ranging 10% − 50% compared to ‘true’ values generated by Monte

Carlo simulation (with 108 paths and appropriate control variates). The approximation

errors are dramatically reduced, to below 1% (except one case with a remote barrier), ‘after’

the continuity corrections. The computational speed of evaluating closed-form solutions, on

the other hand, is many orders of magnitude faster than the Monte Carlo simulation. 25

2.4.3 Type B Partial Barrier Joint Probabilities

Theorem 4 The following continuity corrections hold for type B partial barrier joint probabilities in ∀a, b ∈ R, c ≥ b.

√ 1 P 0 (> c, ↓ b) = P (> c, ↓ b − βσ ∆t) + o(√ ), (2.4.9) B B 1 m √ 1 P 0 (> c, ↓ b, > a) = P (> c, ↓ b − βσ ∆t, > a) + o(√ ). (2.4.10) B B 1 m

Remarks. The proof of continuity corrections for type B partial barrier joint probabilities is more complicated than that of two-dimensional and type A partial barrier joint probabilities, mainly because it requires the integral and uniform convergence over both the initial value

X(t1) and terminal value X(T ). The condition X(t1) > c (c ≥ b), which ties the relevant

√1 type B partial barrier joint probability to a down-crossing event, is necessary for the o( m ) convergence.

Proof. Consider the continuous type B partial barrier joint probability,

t1 PB(> c, ↓ b, > a) = P (X(t1) > c, τ↓(X , b) ≤ T − t1,X(T ) > a) Z ∞ Z ∞ t1 = Pδ,ζ (τ↓(X , b) ≤ T − t1)ft1,T (δ, ζ)dζdδ, c a q t1 where ft1,T (x, y) is the bivariate normal density of (X(t1),X(T )) with correlation T ,

ft1,T (x, y) := f(X(t1) = x, X(T ) = y) 1 (x − µ t )2T + (y − µ T )2t − 2(x − µ t )(y − µ T )t = exp − 1 1 1 1 1 1 1 1 . 2p 2 2πσ1 t1(T − t1) 2σ1t1(T − t1)

The discrete probability is

0 t1 0 t1 t1 PB(> c, ↓ b, > a) = P (X0 > c, τ↓(X , b) ≤ m, Xm > a) Z ∞ Z ∞ 0 t1 = Pδ,ζ (τ↓(X , b) ≤ m)ft1,T (δ, ζ)dζdδ, c a 26

and the continuity correction diﬀerence is

0 ∆PB(> c, ↓ b, > a) := PB(> c, ↓ b, > a) − PB(> c, ↓ b−∆t, > a) Z ∞ Z ∞ 0 t1 = ∆Pδ,ζ (τ↓(X , b) ≤ m)ft1,T (δ, ζ)dζdδ, c a

Similarly,

0 ∆PB(> c, ↓ b) := PB(> c, ↓ b) − PB(> c, ↓ b−∆t) Z ∞ Z ∞ 0 t1 = ∆Pδ,ζ (τ↓(X , b) ≤ m)ft1,T (δ, ζ)dζdδ. c −∞

Below we show (2.4.10) in ﬁve cases, with respect to a and c relative to the barrier b.

(i) a > b, c > b

0 In this case, both δ and ζ are not arbitrarily close to b. By (2.3.4), ∆Pδ,ζ (τ↓(X, b) ≤

√1 m) = o( m ) holds uniformly in b + ε ≤ δ, ζ ≤ dm for each ε > 0, therefore

|∆PB(> c, ↓ b, > a)|

= |∆PB(∈ (c, dm], ↓ b, ∈ (a, dm]) + ∆PB(> dm, ↓ b, ∈ (a, dm]) + ∆PB(> c, ↓ b, > dm)|

Z dm Z dm 0 t1 t1 t1 ≤ ∆Pδ,ζ (τ↓(X , b) ≤ m)ft1,T (δ, ζ)dζdδ + P (X0 > dm) + P (Xm > dm) c a 1 0 t1 ≤ sup ∆Pδ,ζ (τ↓(X , b) ≤ m) P (X(t1) ∈ (c, dm],X(T ) ∈ (a, dm]) + o(√ ) c≤δ≤dm,a≤ζ≤dm m 1 = o(√ ), m where the last inequality holds by applying (2.4.6) twice.

(ii) a > b, c = b 27

0 In this case, δ can be arbitrarily close to b (given c = b). By (2.3.5), ∆Pδ,ζ (τ↓(X, b) ≤

√1 m) = (ζ − b) · O( m ) holds uniformly in b < δ ≤ b + ε ≤ ζ ≤ dm for each ε > 0, therefore

|∆PB(> b, ↓ b, > a)|

= ∆PB(∈ (b, b + ε], ↓ b, ∈ (a, dm]) + ∆PB(∈ (b, b + ε], ↓ b, > dm]) + ∆PB(> b + ε, ↓ b, > a)

Z b+ε Z dm 0 t1 ≤ ∆Pδ,ζ (τ↓(X , b) ≤ m)ft1,T (δ, ζ)dζdδ + P (X(T ) > dm) + ∆PB(> b + ε, ↓ b, > a) δ=b ζ=a Z b+ε Z dm 1 1 1 = (ζ − b)ft1,T (δ, ζ)dζdδ · O(√ ) + o(√ ) + o(√ ), δ=b ζ=a m m m

where the last equality follows from (2.3.5), (2.4.6), and case (i) above. Since ft1,T (·, ·) is

∂ft1,T (δ,ζ) ζ−δ−µ1(T −t1) bounded by a constant and = − 2 ft1,T (δ, ζ), we have ∂ζ σ1 (T −t1)

|∆PB(> b, ↓ b, > a)|

Z b+ε Z dm 2 ∂ft1,T (δ, ζ) 1 = −σ1(T − t1) + (δ − b + µ1(T − t1))ft1,T (δ, ζ) dζdδ · O(√ ) δ=b ζ=a ∂ζ m Z b+ε 2 ≤ σ1(T − t1)|ft1,T (δ, a) − ft1,T (δ, dm)| + sup |ft1,T (δ, ζ)|(δ − b + µ1(T − t1)) dδ δ=b a≤ζ≤dm 1 ·O(√ ) m Z b+ε 1 1 ≤ C dδ · O(√ ) = ε · O(√ ), δ=b m m

which holds for each ε > 0, and the result follows by letting ε → 0.

(iii) a = b, c > b

0 In this case, ζ can be arbitrarily close to b (given a = b). By (2.3.6), ∆Pδ,ζ (τ↓(X, b) ≤

√1 m) = (δ − b) · O( m ) holds uniformly in b < ζ ≤ b + ε ≤ δ ≤ dm for each ε > 0, therefore by 28

a similar argument as for case (ii),

∆PB(> c, ↓ b, > b)

= ∆PB(∈ (c, dm], ↓ b, (b, b + ε]) + ∆PB(> dm, ↓ b, (b, b + ε]) + ∆PB(> c, ↓ b, > b + ε) 1 1 1 1 = ε · O(√ ) + o(√ ) + o(√ ) = ε · O(√ ), m m m m

which holds for each ε > 0, and the result follows by letting ε → 0.

(iv) a < b, c ≥ b

This case follows a similar argument as for case (iii) of (2.4.5),

|∆PB(> c, ↓ b, > a)|

= |∆PB(> c, ↓ b, ∈ (a, b−∆t]) + ∆PB(> c, ↓ b, ∈ [b−∆t, b]) + ∆PB(> c, ↓ b, > b)| 1 ≤ 0 + |P 0 (> c, ↓ b, ∈ [b , b]) − P (> c, ↓ b , ∈ [b , b])| + o(√ ) B −∆t B −∆t −∆t m 1 = P (X(t ) > c, X(T ) ∈ [b , b]) − P (> c, ↓ b , ∈ [b , b]) + o(√ ) 1 −∆t B −∆t −∆t m 1 1 = P (> c, ↓˜b , ∈ [b , b]) + o(√ ) ≤ P (m|T > b ,X(T ) ≤ b) + o(√ ) B −∆t −∆t m t1 −∆t m 1 1 ≤ P (↓˜b , ≤ b) + o(√ ) = o(√ ), 1d −∆t m m

where the last equality follows from (2.4.3).

(2.4.9) follows case (iv) of (2.4.10) above, with any a < b (and c ≥ b),

∆PB(> c, ↓ b) = ∆PB(> c, ↓ b, > a) + ∆PB(> c, ↓ b, ≤ a) 1 = o(√ ) + P 0 (> c, ↓ b, ≤ a) − P (> c, ↓ b , ≤ a) m B B −∆t 1 1 = o(√ ) + P (X(t ) > c, X(T ) ≤ a) − P (X(t ) > c, X(T ) ≤ a) = o(√ ), m 1 1 m

where the third equality holds because a < b−∆t for a large m. 29

Table 2.2: Results for type B partial barrier joint probabilities*

Barrier Continuous Corrected True Rel. Err. Rel. Err. (b < 0) (Before) (After) (s.e.) (Before) (After) -0.02 0.229 0.193 0.192 (0.0000) 19.0% 0.5% -0.10 0.122 0.088 0.088 (0.0000) 38.9% 0.7% -0.18 0.016 0.010 0.010 (0.0000) 62.7% -0.2%

* PB(> b, ↓ b, > a) as in (2.4.10) with c = b. The parameters are a = −0.1, µ1 = 0.1, σ1 = 30%, t1 = 0.1, T = 0.2 and m = 25.

(v) a = b, c = b

This case follows an application of (2.4.9) above, with c = b,

|∆PB(> b, ↓ b, > b)| = |∆PB(> b, ↓ b) − ∆PB(> b, ↓ b, ≤ b)|

1 1 0 = o(√ ) − ∆PB(> b, ↓ b, ≤ b) = o(√ ) − [P (> b, ↓ b, ≤ b) − PB(> b, ↓ b−∆t, ≤ b)] m m B

1 = o(√ ) − [P (X(t1) > b, X(T ) ≤ b) − PB(> b, ↓ b−∆t, ≤ b)] m 1 1 ≤ o(√ ) + P (> b, ↓˜b , ≤ b) ≤ P (m|T > b ,X(T ) ≤ b) + o(√ ) m B −∆t t1 −∆t m 1 1 ≤ P (↓˜b , ≤ b) + o(√ ) = o(√ ), 1d −∆t m m

where the second equality holds because of (2.4.9), the third equality is by deﬁnition of

∆PB(> c, ↓ b, ≤ a), and the last equality follows from (2.4.3). 2

Table 2.2 shows that the continuity corrections also work well for type B partial barrier

joint probabilities.

2.4.4 Extensions to Other Basic Types of Barrier Joint Probabil- ities

Theorems 2, 3 and 4 each address one basic type of the corresponding barrier joint prob-

abilities, continuity corrections for other basic types can be derived through the following 30

extensions.

(i) Extensions of Theorem 2

By (2.4.4), for a two-dimensional down-crossing barrier joint probability (b < 0) we have

1 P 0 (↓ b, > a) = P (↓ b , > a) + o(√ ). 2d 2d −∆t m

For the down-crossing probability with Y (T ) < a,

0 0 0 P2d(↓ b, < a) = P (τ↓(X, b) ≤ m) − P2d(↓ b, > a) 1 1 = P (τ (X, b ) ≤ T ) − P (↓ b , > a) + o(√ ) = P (↓ b , < a) + o(√ ). ↓ −∆t 2d −∆t m 2d −∆t m

For an up-crossing probability (b > 0),

1 P 0 (↑ b, < a) = P (τ 0 (−X, −b) ≤ m, Y < a) = P (τ (−X, −b ) ≤ T,Y (T ) < a) + o(√ ) 2d ↓ m ↓ −∆t m 1 = P (↑ b , < a) + o(√ ), 2d +∆t m

where the third equality follows by applying the down-crossing results above on −X, and

1 P 0 (↑ b, > a) = P (τ 0 (−X, −b) ≤ m, Y > a) = P (↑ b , > a) + o(√ ). 2d ↓ m 2d +∆t m

The extensions to non-crossing barrier joint probabilities can be derived using the probability

parity,

0 0 ˜ ˜ P2d(↓ b, > a) + P2d(↓b, > a) = P (Y (T ) > a) = P2d(↓ b, > a) + P2d(↓b, > a). 31

Thus we have

1 P 0 (↓˜b, > a) := P (τ 0 (X, b) > m, Y > a) = P (↓˜b , > a) + o(√ ) where b < 0, 2d ↓ m 2d −∆t m 1 P 0 (↓˜b, < a) := P (τ 0 (X, b) > m, Y < a) = P (↓˜b , < a) + o(√ ) where b < 0, 2d ↓ m 2d −∆t m 1 P 0 (↑˜b, > a) := P (τ 0 (X, b) > m, Y > a) = P (↑˜b , > a) + o(√ ) where b > 0, 2d ↑ m 2d +∆t m 1 P 0 (↑˜b, < a) := P (τ 0 (X, b) > m, Y < a) = P (↑˜b , < a) + o(√ ) where b > 0. 2d ↑ m 2d +∆t m

0 The extensions of (2.4.5), on P2d(↓ b, > c, > a), can be derived similarly.

(ii) Extensions of Theorem 3

By (2.4.7), for a type A partial barrier joint probability (b < 0) we have

1 P 0 (↓ b, > a) = P (↓ b , > a) + o(√ ). A A −∆t m

Similar to the extensions of (2.4.4), we can extend (2.4.7) as follows,

1 P 0 (↓ b, < a) = P (↓ b , < a) + o(√ ) where b < 0, A A −∆t m 1 P 0 (↑ b, < a) = P (↑ b , < a) + o(√ ) where b > 0, A A +∆t m 1 P 0 (↑ b, > a) = P (↑ b , > a) + o(√ ) where b > 0, A A +∆t m 1 P 0 (↓˜b, > a) = P (↓˜b , > a) + o(√ ) where b < 0, A A −∆t m 1 P 0 (↓˜b, < a) = P (↓˜b , < a) + o(√ ) where b < 0, A A −∆t m 1 P 0 (↑˜b, > a) = P (↑˜b , > a) + o(√ ) where b > 0, A A +∆t m 1 P 0 (↑˜b, < a) = P (↑˜b , < a) + o(√ ) where b > 0. A A +∆t m

0 The extensions of (2.4.8), on PA(↓ b, > c, > a), can be derived similarly.

(iii) Extensions of Theorem 4 32

By (2.4.10), for a type B partial barrier down-crossing probability (c ≥ b) we have

1 P 0 (> c, ↓ b, > a) = P (> c, ↓ b , > a) + o(√ ). B B −∆t m

For the down-crossing probability with X(T ) < a,

0 0 0 PB(> c, ↓ b, < a) = PB(> c, ↓ b) − PB(> c, ↓ b, > a) 1 1 = P 0 (> c, ↓ b ) − P (> c, ↓ b , > a) + o(√ ) = P (> c, ↓ b , < a) + o(√ ), B −∆t B −∆t m B −∆t m

which follows directly from (2.4.10) and (2.4.9). For the up-crossing probability (c ≤ b),

1 P 0 (< c, ↑ b, > a) = P 0 (> −c, ↓ −b, < −a| − X) = P (> −c, ↓ −b , < −a| − X) + o(√ ) B B B −∆t m 1 = P (< c, ↑ b , > a) + o(√ ), B +∆t m

where the third equality follows by applying the down-crossing results above on −X. Similar to the extensions of (2.4.4) and (2.4.7), we have

1 P 0 (< c, ↑ b, < a) = P (< c, ↑ b , < a) + o(√ ) where c ≤ b, B B +∆t m 1 P 0 (> c, ↓˜b, > a) = P (> c, ↓˜b , > a) + o(√ ) where c ≥ b, B B −∆t m 1 P 0 (< c, ↑˜b, > a) = P (< c, ↑˜b , > a) + o(√ ) where c ≤ b, B B +∆t m 1 P 0 (> c, ↓˜b, < a) = P (> c, ↓˜b , < a) + o(√ ) where c ≥ b, B B −∆t m 1 P 0 (< c, ↑˜b, < a) = P (< c, ↑˜b , < a) + o(√ ) where c ≤ b. B B +∆t m

0 The extensions of (2.4.9), on PB(> c, ↓ b), can be derived similarly. 33 2.5 Continuity Corrections for Two-dimensional Bar- rier and Type A Partial Barrier Options

In this section we show how to apply the continuity corrections on the barrier joint prob-

abilities to the pricing of two-dimensional barrier and type A partial barrier options. We

assume a complete ﬁnancial market and the underlying assets follow geometric Brownian

motion processes in a standard ﬁltered probability space (Ω, F , P ). Risk-free interest rate

is assumed to be constant throughout time. By the fundamental theorem of asset pricing,

there exists a unique equivalent martingale measure P ∗ (risk-neutral probability) and the price of any asset can be obtained by taking the expectation of discounted future payoﬀ under P ∗.

For a continuous barrier option, the barrier crossing is monitored in continuous time.

While for a discrete barrier option, the barrier crossing can only occur in discrete time, such as daily or monthly. For simplicity, we assume there is no rebate when the barrier is crossed.

2.5.1 Two-dimensional Barrier Options

Consider two asset price processes S(t) and R(t), a two-dimensional barrier option is knocked in or knocked out if R(t) crosses barrier H by maturity T . The payoﬀ, (S(T ) − K)+ for

a call or (K − S(T ))+ for a put, is received at T if the option has been knocked in or has

not been knocked out during its lifetime. Assume S(t) and R(t) follow geometric Brownian

motions with correlation ρ,

X(t) Y (t) R(t) = R0e ,S(t) = S0e ,

where X(t) and Y (t) are correlated Brownian motions as deﬁned in Section 2.2. 34

Table 2.3: Results for two-dimensional barrier options*

Option Continuous Corrected True Rel. Err. Rel. Err. Type Barrier (Before) (After) (s.e.) (Before) (After) Down- 91 1.573 1.309 1.311 (0.0004) 20.0% -0.2% and-in 95 3.022 2.579 2.581 (0.0005) 17.1% -0.1% Call 99 5.179 4.519 4.482 (0.0005) 15.6% 0.8% Up- 101 0.518 1.051 1.081 (0.0004) -52.1% -2.8% and-out 105 2.188 2.564 2.562 (0.0004) -14.6% 0.1% Put 109 3.299 3.546 3.543 (0.0004) - 6.9% 0.1%

* The parameters are S0 = R0 = 100, K = 100, σ1 = σ2 = 30%, r = 5%, ρ = 0.5, T = 0.2 and m = 50.

For a discrete barrier option, the barrier is monitored only at time i∆t, i = 0, 1, ..., m

(where ∆t = T/m). The discrete asset processes are

Xn Yn Rn = R0e ,Sn = S0e ,

where Xn and Yn are correlated Gaussian random walks as deﬁned in Section 2.2.

Corollary 2 Let V2d(H) be the value of a continuously monitored two-dimensional barrier

0 option with barrier H, and V2d(H) be the value of its discrete counterpart,

√ 1 V 0 (H) = V (He±βσ1 ∆t) + o(√ ), 2d 2d m

where ‘+’ applies if H > R0,‘−’ applies if H < R0.

Proof. We use the two-dimensional down-and-in call option as an example to show Corollary

2. A two-dimensional down-and-in call is knocked in if R(t) crosses the barrier H (where

+ H < R0) by maturity T , the payoﬀ (S(T ) − K) is received at T . 35

The value of a continuous two-dimensional down-and-in call is

−rT ∗ + DIC2d(H) = e E [(S(T ) − K) 1{τ↓(R,H) ≤ T }]

−rT ∗ = e E [(S(T ) − K)1{τ↓(R,H) ≤ T,S(T ) > K}]

−rT ∗ e S(T ) −rT ∗ = S0E 1{τ↓(X, b) ≤ T,Y (T ) > a} − e KE [1{τ↓(X, b) ≤ T,Y (T ) > a}] S0 ˆ −rT ∗ = S0P (τ↓(X, b) ≤ T,Y (T ) > a) − e KP (τ↓(X, b) ≤ T,Y (T ) > a)

ˆ −rT ∗ = S0P2d(↓ b, > a) − e KP2d(↓ b, > a),

where b = ln H < 0 and a = ln K , P ∗ is the risk-neutral measure and Pˆ is the equivalent R0 S0

measure deﬁned by,

−rT 2 dPˆ e S(T ) σ1 − 2 T +σ1B1(T ) ∗ = = e . dP S0

By the continuous-time Girsanov theorem,

2 ˆ ˆ X(t) = (µ1 + σ1)t + σ1B1(t),Y (t) = (µ2 + ρσ1σ2)t + σ2B2(t),

ˆ ˆ ˆ where B1(t) and B2(t) are standard Brownian motions with correlation ρ under P . The

discrete option value is

0 −rT ∗ 0 DIC2d(H) = e E [(S(T ) − K)1{τ↓(R,H) ≤ m, Sm > K}]

−rT ∗ e Sm 0 −rT ∗ 0 = S0E 1{τ↓(X, b) ≤ m, Ym > a} − e KE [1{τ↓(X, b) ≤ m, Ym > a}] S0 ˆ0 −rT ∗0 = S0P2d(↓ b, > a) − e KP2d(↓ b, > a), where the equivalent measure Pˆ is deﬁned by

ˆ −rT ( 2 m ) dP e Sm σ √ X = = exp − 1 m∆t + σ ∆t ZY . dP ∗ S 2 1 i 0 i=1 36

By the discrete-time Girsanov theorem,

n √ n √ 2 X ˆX X ˆY Xn = (µ1 + σ1)n∆t + σ1 ∆tZi ,Yn = (µ2 + ρσ1σ2)n∆t + σ2 ∆tZi , i=1 i=1

ˆX ˆY where (Zi , Zi ) are independent pairs of standard bivariate normal random variables with correlation ρ under Pˆ. By (2.4.4),

1 P ∗0(↓ b, > a) = P ∗ (↓ b , > a) + o(√ ), 2d 2d −∆t m 1 Pˆ0 (↓ b, > a) = Pˆ (↓ b , > a) + o(√ ), 2d 2d −∆t m

which leads to √ 1 DIC0 (H) = DIC (He−βσ1 ∆t) + o(√ ).2 2d 2d m

Table 2.3 shows that the continuity corrections are as eﬀective for two-dimensional barrier

options (as for barrier joint probabilities), with both down-and-in calls and up-and-out puts.

The results are robust with varying parameters (m, σ, K, ρ), generating approximation

errors below 1% (for all cases except when m = 10) as shown in Table 2.4. Continuity

corrections for other basic types of two-dimensional barrier options can be derived similarly

using the extensions of Theorem 2 in Section 2.4.

2.5.2 Type A Partial Barrier Options

For a partial barrier option, the barrier crossing is monitored only within part of the option’s

lifetime, more speciﬁcally [0, t1] for type A partial barrier options and [t1,T ] for type B partial

barrier options, where t1 is a pre-speciﬁed time between 0 and T . A discrete type A partial

barrier option has its barrier monitored at i∆t, i = 0, 1, ..., m (where ∆t = t1/m). 37

Table 2.4: Results for two-dimensional barrier options with varying parameters (m, σ, K, ρ, T )*

Varying Continuous Corrected True Rel. Err. Rel. Err. Parameters Barrier (Before) (After) (s.e.) (Before) (After) m = 10 91 1.573 1.034 1.049 (0.0004) 50.0% -1.4% 95 3.022 2.101 2.114 (0.0005) 43.0% -0.6% 99 5.179 3.787 3.617 (0.0005) 43.2% 4.7% m = 50 91 1.573 1.309 1.311 (0.0004) 20.0% 0.2% 95 3.022 2.579 2.581 (0.0005) 17.1% -0.1% 99 5.179 4.519 4.482 (0.0005) 15.6% 0.8% m = 50 91 0.452 0.364 0.365 (0.0002) 23.8% -0.3% σ1 = σ2 = 20% 95 1.399 1.171 1.173 (0.0003) 19.3% -0.2% 99 3.375 2.924 2.915 (0.0003) 15.8% 0.3% σ1 = σ2 = 20% 91 0.167 0.131 0.132 (0.0001) 26.5% -0.8% K = 105 95 0.591 0.484 0.484 (0.0002) 22.1% 0.0% 99 1.610 1.365 1.360 (0.0002) 18.4% 0.4% K = 105 91 1.080 0.991 0.995 (0.0002) 8.5% -0.4% ρ = −50% 95 1.585 1.504 1.506 (0.0002) 5.2% -0.1% 99 1.940 1.890 1.890 (0.0001) 2.6% 0.0% ρ = −50% 91 6.367 6.011 6.019 (0.0006) 5.8% -0.1% T = 1 95 7.216 6.907 6.912 (0.0005) 4.4% -0.1% 99 7.884 7.637 7.617 (0.0003) 3.5% 0.3%

* The parameters are S0 = R0 = 100, K = 100, r = 5%, σ1 = σ2 = 30%, ρ = 50%, T = 0.2 and m = 50 (except the varying ones as noted). 38

Corollary 3 Let VA(H) be the value of a continuously monitored type A partial barrier

0 option with barrier H, and VA(H) be the value of its discrete counterpart,

√ 1 V 0 (H) = V (He±βσ1 ∆t) + o(√ ), A A m

where ‘+’ applies if H > S0 and ‘−’ applies if H < S0.

Proof. The proof of Corollary 3 is straightforward given the similarity of type A partial

barrier options to the two-dimensional barrier options. We again use the down-and-in call as

an example. A type A partial down-and-in call is knocked in if S(t) ever crosses H (where

+ H < S0) between 0 and t1, and the payoﬀ is (S(T ) − K) received at T .

The value of a continuous type A partial down-and-in call is

−rT ∗ + ˆ −rT ∗ DICA(H) = e E [(S(T ) − K) 1{ min S(t) ≤ H}] = S0PA(↓ b, > a) − e KPA(↓ b, > a), 0≤t≤t1

where b = ln H < 0 and a = ln K . The discrete option value is S0 S0

0 −rT ∗ + ˆ0 −rT ∗0 DICA(H) = e E [(S(T ) − K) 1{ min Si ≤ H}] = S0PA(↓ b, > a) − e KPA (↓ b, > a). 0≤i≤m

By (2.4.7),

1 P ∗0(↓ b, > a) = P ∗(↓ b , > a) + o(√ ), A A −∆t m 1 Pˆ0 (↓ b, > a) = Pˆ (↓ b , > a) + o(√ ), A A −∆t m

which leads to √ 1 DIC0 (H) = DIC (He−βσ1 ∆t) + o(√ ).2 A A m

Table 2.5 shows the numerical results on type A partial barrier options, suggesting com-

parable level of accuracy as for two-dimensional barrier options. Continuity corrections for 39

Table 2.5: Results for type A partial barrier options*

Option Continuous Corrected True Rel. Err. Rel. Err. Type Barrier (Before) (After) (s.e.) (Before) (After) Down- 91 0.452 0.306 0.308 (0.0002) 46.8% -0.6% and-in 95 1.713 1.251 1.255 (0.0004) 36.5% -0.3% Call 99 4.722 3.684 3.625 (0.0005) 30.3% 1.6% Up- 101 0.882 1.725 1.773 (0.0004) -50.3% -2.7% and-out 105 3.261 3.674 3.671 (0.0003) -11.2% 0.1% Put 109 4.316 4.478 4.476 (0.0002) - 3.6% 0.0%

* The parameters are S0 = 100, K = 100, r = 5%, σ1 = 30%, T = 0.2, t1 = 0.1 and m = 25. other basic types of type A partial barrier options can be derived similarly by applying the extensions of Theorem 3 in Section 2.4.

2.6 Continuity Corrections for Type B Partial Barrier Options

2.6.1 Type B Partial Barrier Options

In contrast to type A partial barrier options for which the barrier is monitored between 0 and t1 (0 < t1 < T ), there are type B partial barrier options whose barrier monitoring is between t1 and maturity T . They can be further divided into type B1 and type B2 partial barrier options, depending on the trigger for barrier crossing: type B1 based on an actual crossing and type B2 based on the relative location of the underlying to the barrier.

The value of a type B1 partial knock-in call (activated if S(t) crosses H within [t1,T ]) is

−rT ∗ + ICB1(H) = e E [(S(T )−K) 1{S(t1) > H, min S(t) ≤ H or S(t1) < H, max S(t) ≥ H}]. t1≤t≤T t1≤t≤T

The value of a type B2 partial down-and-in call (activated if S(t) reaches below H in [t1,T ]) 40

−rT ∗ + DICB2(H) = e E [(S(T ) − K) 1{S(t1) > H, min S(t) ≤ H or S(t1) < H}]. t1≤t≤T

The discrete barrier is monitored only at t1 + i∆t, i = 0, 1, ..., m (where ∆t = (T − t1)/m).

Corollary 4 Let VB(H↑,H↓,H) be the value of a continuously monitored type B partial barrier option with barrier H, where H↑ corresponds to an up-crossing, H↓ corresponds to a down-crossing and H is not directly associated with a barrier crossing in the formula, and

0 VB(H) be the value of its discrete counterpart,

√ √ 1 V 0 (H) = V (Heβσ1 ∆t, He−βσ1 ∆t,H) + o(√ ). B B m

Remarks. The continuity corrections for type B partial barrier options are more compli-

cated than those for other barrier options. One cannot simply shift the barrier H throughout

the continuous-time formulae (as in the case of other barrier options), but only shift some

H, to the direction of the barrier crossing each H corresponds to, and leave the other H unchanged if it is not directly associated with a barrier crossing.

Proof. Below we prove Corollary 4 for type B1 and type B2 partial barrier options, using the type B1 partial knock-in call and type B2 partial down-and-in call as examples. Readers are referred to the example at the end of this section for details on the pricing formulae and how to shift the barrier for type B partial barrier options. (i) Type B1 41

First consider the case K < H, the value of a continuous type B1 partial knock-in call is

ICB1(H↑,H↓,H)|K

−rT ∗ + = e E [(S(T ) − K) 1{S(t1) > H, min S(t) ≤ H↓ or S(t1) < H, max S(t) ≥ H↑}] t1≤t≤T t1≤t≤T ˆ ˆ = S0[PB(> b, ↓ b↓, > a) + PB(< b, ↑ b↑, > a)]

−rT ∗ ∗ −e K[PB(> b, ↓ b↓, > a) + PB(< b, ↑ b↑, > a)], where b = ln H and a = ln K . The discrete option value is S0 S0

0 ICB1(H)|K

−rT ∗ + t1 t1 = e E [(S(T ) − K) 1{S(t1) > H, min Si ≤ H or S(t1) < H, max Si ≥ H}] 0≤i≤m 0≤i≤m ˆ0 ˆ0 −rT ∗0 ∗0 = S0[PB(> b, ↓ b, > a) + PB(< b, ↑ b, > a)] − e K[PB (> b, ↓ b, > a) + PB (< b, ↑ b, > a)].

Corollary 4 in this case can be easily derived by applying Theorem 4 and its extensions,

0 ˆ ˆ ICB1(H)|K b, ↓ b−∆t, > a) + PB(< b, ↑ b+∆t, > a)] 1 −e−rT K[P ∗ (> b, ↓ b , > a) + P ∗ (< b, ↑ b , > a)] + o(√ ) B −∆t B +∆t m √ √ 1 = IC (Heβσ1 ∆t, He−βσ1 ∆t,H)| + o(√ ). B1 K

ICB1(H↑,H↓,H)|K≥H

−rT ∗ = e E [(S(T ) − K)(1{S(t1) > H, min S(t) ≤ H↓,S(T ) > K} + 1{S(t1) ≤ H,S(T ) > K})] t1≤t≤T ˆ ˆ = S0[PB(> b, ↓ b↓, > a) + P (X(t1) ≤ b, X(T ) > a)]

−rT ∗ ∗ −e K[PB(> b, ↓ b↓, > a) + P (X(t1) ≤ b, X(T ) > a)],

and the continuity correction follows directly from (2.4.10) in the case of a ≥ b and c = b. (ii) Type B2 In the case of K ≥ H, the value of a continuous type B2 partial down-and-in call is the

same as the value of its type B1 counterpart,

−rT ∗ + DICB2(H↑,H↓,H)|K≥H = e E [(S(T ) − K) 1{ min S(t) ≤ H}] t1≤t≤T −rT ∗ = e E [(S(T ) − K)(1{S(t1) > H, min S(t) ≤ H↓,S(T ) > K} + 1{S(t1) ≤ H,S(T ) > K})], t1≤t≤T 42

Table 2.6: Results for type B partial barrier options*

Option Continuous Corrected True Rel. Err. Rel. Err. Type Barrier (Before) (After) (s.e.) (Before) (After) Type B1/B2 91 0.179 0.128 0.128 (0.0001) 39.8% 0.2% Knock-In 95 0.590 0.458 0.456 (0.0002) 29.4% 0.4% Call 99 1.390 1.156 1.150 (0.0003) 20.8% 0.5% (K ≥ H) 100 1.647 1.390 1.384 (0.0004) 19.0% 0.4% Type B1 101 1.919 1.644 1.636 (0.0004) 17.2% 0.4% (K < H) 105 3.021 2.694 2.681 (0.0005) 12.7% 0.5% 109 3.803 3.446 3.429 (0.0005) 10.9% 0.5% Type B2 101 1.919 1.645 1.637 (0.0004) 17.2% 0.5% (K < H) 105 3.069 2.777 2.764 (0.0005) 11.0% 0.5% 109 4.097 3.853 3.840 (0.0005) 6.7% 0.3%

* The parameters are S0 = 100, K = 100, r = 5%, σ = 30%, T = 0.2, t1 = 0.1 and m = 25. therefore the proof of type B1 applies.

In the case of K < H, the continuous option value is

−rT ∗ + DICB2(H↑,H↓,H)|K K} − 1{S(t1) > H, min S(t) ≥ H↓,S(T ) > H} t1≤t≤T ˆ ˆ ˜ −rT ∗ ∗ ˜ = S0[P (X(T ) > a) − PB(> b, ↓b↓, > b)] − e K[P (X(T ) > a) − PB(> b, ↓b↓, > b)], for which the continuity correction follows from (2.4.10) in the case of a = b and c = b. 2 Table 2.6 shows that the continuity corrections are even more eﬀective for type B partial barrier options, despite the complexity in barrier shifting. Continuity corrections for other type B partial barrier options can be derived similarly by applying the extensions of Theorem 4.

2.6.2 An Example of Type B Partial Barrier Correction

The barrier correction for type B partial barrier options is more complicated than that for one-dimensional, two-dimensional and type A partial barrier options. Here we illustrate the details using the type B1 knock-in call as an example, the complete pricing formulae are shown in K < H and K ≥ H cases, with H↑, H↓ and H identiﬁed respectively. The barrier 43

corrections for other type B partial barrier options can be derived similarly (see Heynen & Kat 1994b for more pricing details for type B partial barrier options).

First deﬁne the following coeﬃcients (noting a = K , b = H ), S0 S0

K H − ln + µ1T √ − ln + µ1t1 √ S0 S0 d2 = √ , d1 = d2 + σ1 T ; e2 = √ , e1 = e2 + σ1 t1; σ1 T σ1 t1 H H H| − ln + µ1T √ ln − 2 ln + µ1t1 √ S0 S0 S0 g2 = √ , g1 = g2 + σ1 T ; e2| = √ , e1| = e2| + σ1 t1; σ1 T σ1 t1 H H ln | √ ln | √ ˆ S0 ˆ ˆ S0 d2| = d2 + 2 √ , d1| = d2| + σ1 T ;e ˆ2| = e2 + 2 √ , eˆ1| =e ˆ2| + σ1 t1; σ1 T σ1 t1 ln H| √ S0 2 gˆ2| = g2 + 2 √ , gˆ2| =g ˆ2| + σ1 T ;µ ˆ1 = µ1 + σ1; σ1 T

where ‘|’ can be ‘↓’ (for a down-crossing) or ‘↑’ (for an up-crossing). Some coeﬃcients contain

H| which needs to be shifted in a continuity correction and the direction varies based on the relevant crossing, some contain H which corresponds to a threshold and does not need to be

shifted, the remaining contain both H| and H in which case some barrier values need to be shifted and others do not (within the same coeﬃcient). (i) In the case of K < H, the value of a continuous type B1 knock-in call is

−rT ∗ + T T ICB1|K b, m|t1 ≤ b})]

−rT ∗ T T T = e E [(S(T ) − K)(1{X(T ) > a} − 1{M|t1 < b} + 1{M|t1 < b, X(T ) < a} − 1{m|t1 > b})] h i ˆ ˆ T ˆ T ˆ T = S0 P (X(T ) > a) − P (M|t1 < b) + P (M|t1 < b, X(T ) < a) − P (m|t1 > b)

−rT ∗ ∗ T ∗ T ∗ T −e K P (X(T ) > a) − P (M|t1 < b) + P (M|t1 < b, X(T ) < a) − P (m|t1 > b) .

For the third probability item,

∗ T ∗ ∗ P (M|t1 < b, X(T ) < a) = P (X(t1) < b, X(T ) < a) − PB(< b, ↑ b↑, < a) 2µ1b↑ σ2 ˆ = Φ2(−e2↑, −d2, ρ) − e 1 Φ2(−eˆ2↑, −d2↑, −ρ),

q t1 where ρ = T and Φ2(·, ·, ρ) is the cumulative distribution function for standard bivariate normal with correlation ρ (and Φ(·) is the cumulative distribution function for standard 44

normal). Similarly,

∗ T ∗ ∗ P (M|t1 < b) = P (X(t1) < b, X(T ) < b) − PB(< b, ↑ b↑, < b) 2µ1b↑ σ2 = Φ2(−e2↑, −g2, ρ) − e 1 Φ2(ˆe2↑, −gˆ2↑, −ρ),

∗ T ∗ ∗ P (m|t1 > b) = P (X(t1) > b, X(T ) > b) − PB(> b, ↓ b↓, > b) 2µ1b↓ σ2 = Φ2(e2↓, g2, ρ) − e 1 Φ2(−eˆ2↓, gˆ2↓, −ρ).

2 ˆ By Girsanov Theorem, X(t) has a drift ofµ ˆ1 = µ1 + σ1 under the equivalent measure P

−rT deﬁned by dPˆ = e S(T ) , thus the continuous option value is dP ∗ S0

2ˆµ1b↑ σ2 ICB1(H↑,H↓,H)|K

2ˆµ1b↑ 2ˆµ1b↓ σ2 ˆ σ2 +Φ2(−e1↑, −d1, ρ) − e 1 Φ2(ˆe1↑, −d1↑, −ρ) − Φ2(e1↓, g1, ρ) + e 1 Φ2(−eˆ1↓, gˆ1↓, −ρ)]

2µ1b↑ −rT σ2 −e K[Φ(d2) − Φ2(−e2↑, −g2, ρ) + e 1 Φ2(ˆe2↑ − gˆ2↑, −ρ) + Φ2(−e2↑, −d2, ρ)

2µ1b↑ 2µ1b↓ σ2 ˆ σ2 −e 1 Φ2(ˆe2↑, −d2↑, −ρ) − Φ2(e2↓, g2, ρ) + e 1 Φ2(−eˆ2↓, gˆ2↓, −ρ)].

(ii) In the case of K ≥ H, the continuous option value is

ICB1(H↑,H↓,H)|K≥H ˆ ˆ = S0[PB(> b, ↓ b↓, > a) + P (X(t1) ≤ b, X(T ) > a)]

−rT ∗ ∗ −e K[PB(> b, ↓ b↓, > a) + P (X(t1) ≤ b, X(T ) > a)] 2ˆµ1b↓ σ2 ˆ = S0 e 1 Φ2(−eˆ1↓, d1↓, −ρ) + Φ2 (−e1↓, d1, −ρ)

2µ1b↓ −rT σ2 ˆ −e K e 1 Φ2(−eˆ2↓, d2↓, −ρ) + Φ2 (−e2↓, d2, −ρ) .

Finally, the continuity correction for the type B1 knock-in call is simply

√ √ 1 IC0 (H) = IC (Heβσ1 ∆t, He−βσ1 ∆t,H) + o(√ ), B1 B1 m √ √ βσ1 ∆t −βσ1 ∆t that is, replacing H↑ with He , H↓ with He and keeping the rest of H unchanged in the continuous-time pricing formulae above. Chapter 3

Improved Lower and Upper Bound Algorithms for Pricing American Options by Simulation

3.1 Overview

The improvements developed and tested in this chapter include martingale control variates and local policy enhancement for lower bound algorithms, and sub-optimality checking and boundary distance grouping enhancements for upper bound algorithms. The least-squares

Monte Carlo method introduced by Longstaﬀ and Schwartz (2001) is used as the lower bound algorithm and the primal-dual simulation algorithm by Andersen and Broadie (2004) is used as the upper bound method, although the improvements can be applied to other lower bound

and duality-based upper bound algorithms. Many lower bound algorithms approximate the option’s continuation value and compare it with the option’s intrinsic value to form a sub-optimal exercise policy. If the approximation

of the continuation value is inaccurate, it often leads to a poor exercise policy. To improve the exercise policy, we propose a local policy enhancement which employs sub-simulation to gain a better estimate of the continuation value in circumstances where the sub-optimal policy is likely to generate incorrect decisions. Then the sub-simulation estimate is compared with the intrinsic value to potentially override the original policy’s decision to exercise or continue. 45 46

In many upper bound algorithms, a time-consuming sub-simulation is carried out to estimate the option’s continuation value at every exercise time. We show in Section 3.4 that sub-simulation is not needed when the option is sub-optimal to exercise, that is, when the intrinsic value is lower than the continuation value. Based on this idea, sub-optimality checking is a simple technique to save computational work and improve the upper bound estimator. It states that we can skip the sub-simulations when the option’s intrinsic value is lower than an easily derived lower bound of the continuation value along the sample path. Despite being simple, this approach often leads to dramatic computational improvements in the upper bound algorithms, especially for out-of-the-money (OTM) options. Boundary distance grouping is another method to enhance the efficiency of duality-based upper bound algorithms. For many simulation paths, the penalty term that contributes to the upper bound estimator is zero. Thus it would be more efficient if we could identify in advance the paths with non-zero penalties. The goal of boundary distance grouping is to separate the sample paths into two groups, one group deemed more likely to produce zero penalties, the ‘zero’ group, and its complement, the ‘non-zero’ group. A sampling method is used to derive the upper bound estimator with much less computational effort, through the saving of sub-simulation, on the sample paths in the ‘zero’ group. The fewer paths there are in the ‘non-zero’ group, the greater will be the computational saving achieved by this method. While the saving is most significant for deep OTM options, the technique is useful for in-the-money (ITM) and at-the-money (ATM) options as well.

This chapter provides numerical results on single asset Bermudan options, moving window Asian options and Bermudan basket options, the latter two of which are difficult to price using lattice or finite difference methods. The techniques introduced in this chapter are general enough to be used for other types of Bermudan options, such as Bermudan interest rate swaptions. The research work for this chapter has been published (see Broadie and Cao 2008).

The rest of this chapter is organized as follows. In Section 3.2, the Bermudan option 47 pricing problem is formulated. Section 3.3 addresses the martingale control variates and local policy enhancement for the lower bound algorithms. Section 3.4 introduces sub-optimality checking and boundary distance grouping for the upper bound algorithms. Numerical results are shown in Section 3.5. In Section 3.6, we conclude and suggest directions for further research. Some numerical details, including a comparison between ‘regression now’ and

‘regression later,’ the choice of basis functions, the proofs of propositions, and the variance estimation for boundary distance grouping, are given in the appendices.

3.2 Problem formulation

We consider a complete ﬁnancial market where the assets are driven by Markov processes in a standard ﬁltered probability space (Ω, F, P). Let Bt denote the value at time t of $1 R t rsds invested in a risk-free money market account at time 0, Bt = e 0 , where rs denotes the

d instantaneous risk-free interest rate at time s. Let St be an R -valued Markov process with the initial state S0, which denotes the process of underlying asset prices or state variables of the model. There exists an equivalent probability measure Q, also known as the risk-neutral measure, under which discounted asset prices are martingales. Pricing of any contingent claim on the assets can be obtained by taking the expectation of discounted cash ﬂows with respect to the Q measure. Let Et[·] denote the conditional expectation under the Q

Q measure given the information up to time t, i.e., Et[·] = E [·|Ft]. We consider here discretely- exercisable American options, also known as Bermudan options, which may be exercised only at a ﬁnite number of time steps Γ = {t0, t1, ..., tn} where 0 = t0 < t1 < t2 < ... < tn ≤ T . τ is the stopping time which can take values in Γ. The intrinsic value ht is the option payoﬀ

+ upon exercise at time t, for example ht = (St − K) for a single asset call option with strike price K, where x+ := max(x, 0). The pricing of Bermudan options can be formulated as a primal-dual problem. The primal problem is to maximize the expected discounted option payoﬀ over all possible stopping 48

times, hτ Primal: V0 = sup E0 . (3.2.1) τ∈Γ Bτ

More generally, the discounted Bermudan option value at time ti < T is Vti hτ hti Vti+1 = sup Eti = max ,Eti , (3.2.2) Bti τ≥ti Bτ Bti Bti+1

where Vt/Bt is the discounted value process and the smallest super-martingale that dominates

ht/Bt on t ∈ Γ (see Lamberton and Lapeyre 1996). The stopping time which achieves the largest option value is denoted τ ∗.1 Haugh and Kogan (2004) and Rogers (2002) independently propose the dual formulation

of the problem. For an arbitrary adapted super-martingale process πt we have hτ hτ V0 = sup E0 = sup E0 + πτ − πτ τ∈Γ Bτ τ∈Γ Bτ hτ ≤ π0 + sup E0 − πτ τ∈Γ Bτ ht ≤ π0 + E0 max − πt , (3.2.3) t∈Γ Bt which gives an upper bound of V0. Based on this, the dual problem is to minimize the upper bound with respect to all adapted super-martingale processes, ht Dual: U0 = inf π0 + E0 max − πt , (3.2.4) π∈Π t∈Γ Bt where Π is the set of all adapted super-martingale processes. Haugh and Kogan (2004) show that the optimal values of the primal and the dual problems are equal, i.e., V0 = U0, and

∗ the optimal solution of the dual problem is achieved with πt being the discounted optimal value process.

3.3 Improvements to lower bound algorithms

3.3.1 A brief review of the lower bound algorithm

Most algorithms for pricing American options are lower bound algorithms, which produce low-biased estimates of American option values. They usually involve generating an exercise 1In general we use ‘*’ to indicate a variable or process associated with the optimal stopping time. 49

strategy and then valuing the option by following the exercise strategy. Let Lt be the lower bound value process associated with τ, which is deﬁned as Lt hτt = Et , (3.3.1) Bt Bτt where τt = inf{u ∈ Γ ∩ [t, T ]: 1u = 1} and 1t is the adapted exercise indicator process, which equals 1 if the sub-optimal strategy indicates exercise and 0 otherwise. Clearly, the sub-optimal exercise strategy is always dominated by the optimal strategy, hτ L0 = E0 ≤ V0, Bτ

in other words, L0 is a lower bound of the Bermudan option value V0.

τ We denote Qt, or Qt , as the option’s continuation value at time t under the sub-optimal strategy τ, Bti Qti = Eti Lti+1 , (3.3.2) Bti+1 ˜ ˜ and Qt as the approximation of the continuation value. In regression-based algorithms, Qt is the projected continuation value through a linear combination of the basis functions

b ˜ X ˆ Qti = βkfk(S1,ti , ..., Sd,ti ), (3.3.3) k=1 ˆ where βk is the regression coeﬃcient and fk(·) is the corresponding basis function. Low bias of sub-optimal policy is introduced when the decision from the sub-optimal

policy diﬀers from the optimal decision. Broadie and Glasserman (2004) propose policy ﬁxing to prevent some of these incorrect decisions: the option is considered for exercise only

if the exercise payoﬀ exceeds a lower limit of the continuation value, Qt. A straightforward choice for this exercise lower limit is the value of the corresponding European option if it

can be valued analytically. More generally it can be the value of any option dominated by the Bermudan option or the maximum among the values of all dominated options (e.g., the maximum among the values of European options that mature at each exercise time of

the Bermudan option). We apply policy ﬁxing for all lower bound computations in this 50

chapter. Note that we only use the values of single European options and not the maximum among multiple option values, because the latter invalidates the condition for Proposition 1

(refer to the proof of Proposition 1 in Appendix B.2 for more detail). Denote the adjusted ˜ ˜ approximate continuation value as Qt := max(Qt,Qt). The sub-optimal strategy with policy ﬁxing can be deﬁned as

˜ τt = inf{u ∈ Γ ∩ [t, T ]: hu > Qu}. (3.3.4)

If it is optimal to exercise the option and yet the sub-optimal exercise strategy indicates otherwise, i.e.,

∗ ˜ ∗ Qt < ht ≤ Qt, 1t = 1 and 1t = 0, (3.3.5) it is an incorrect continuation. Likewise when it is optimal to continue but the sub-optimal exercise strategy indicates exercise, i.e.,

∗ ˜ ∗ Qt ≥ ht > Qt, 1t = 0 and 1t = 1, (3.3.6) it is an incorrect exercise.

3.3.2 Distance to the exercise boundary

In this section we discuss an approach to quantify the distance of an option to the exercise boundary. The exercise boundary is the surface in the state space where the option holder, based on the exercise policy, is indifferent between holding and exercising the option. Ac- cordingly the sub-optimal exercise boundary can be defined as the set of states at which the ˜ adjusted approximate continuation value equals the exercise payoff, i.e., {ωt : Qt = ht}. The ˜ exercise region is where Qt < ht and the sub-optimal policy indicates exercise, and vice versa for the continuation region.

Incorrect decisions are more likely to occur when the option is ‘close’ to the exercise boundary. To determine how ‘close’ the option is from the sub-optimal exercise boundary we introduce a boundary distance measure

˜ dt := |Qt − ht|. (3.3.7) 51

This function is measured in units of the payoﬀ as opposed to the underlying state variables. It does not strictly satisfy the axioms of a distance function, but it does have similar

characteristics. In particular, dt is zero only when the option is on the sub-optimal exercise ˜ boundary and it increases as Qt deviates from ht. We can use it as a measure of closeness between the sample path and the sub-optimal exercise boundary. Alternative boundary ˜ ˜ distance measures include |Qt − ht|/ht and |Qt − ht|/St.

3.3.3 Local policy enhancement

The idea of local policy enhancement is to employ a sub-simulation to estimate the contin- ˆ ˜ uation value Qt and use that, instead of the approximate continuation value Qt, to make the exercise decision. Since the sub-simulation estimate is generally more accurate than

the approximate continuation value, this may improve the exercise policy, at the expense of additional computational eﬀort. It is computationally demanding, however, to perform a sub-simulation at every time

step. To achieve a good tradeoff between accuracy and computational cost, we would like to launch a sub-simulation only when an incorrect decision is considered more likely to be made. Specifically, we launch a sub-simulation at time t if the sample path is sufficiently

close to the exercise boundary. The simulation procedure for the lower bound algorithm with local policy enhancement is as follows:

(i) Simulate the path of state variables until either the sub-optimal policy indicates exer-

cise or the option matures.

˜ (ii) At each exercise time, compute ht, Qt, Qt, and dt. Continue if ht ≤ Qt, otherwise

˜ a. If dt > , follow the original sub-optimal strategy, exercise if ht > Qt, continue otherwise. 52

ˆ ˆ b. If dt ≤ , launch a sub-simulation with N paths to estimate Qt, exercise if ht > Qt, continue otherwise.

ˆ (iii) Repeat steps (i)-(ii) for NL sample paths, obtain the lower bound estimator L0 by averaging the discounted payoﬀs.

Due to the computational cost for sub-simulations, local policy enhancement may prove to be too expensive to apply for some Bermudan options.

3.3.4 Use of control variates

The fast and accurate estimation of an option’s continuation values is essential to the pricing of American options in both the lower and upper bound computations. We use the control variate technique to improve the eﬃciency of continuation value estimates. The control variate method is a broadly used technique for variance reduction (see, for example, Boyle, Broadie and Glasserman 1997), which adjusts the simulation estimates by quantities with known expectations. Assume we know the expectation of X and want to estimate E[Y ]. The control variate adjusted estimator is Y¯ −β(X¯ −E[X]), where β is the adjustment coeﬃcient.

∗ σY The variance-minimizing adjustment coeﬃcient is β = ρXY , which can be estimated from σX the X and Y samples. Broadie and Glasserman (2004) use European option values as controls for pricing Bermudan options and apply them in two levels: inner controls are used for estimating continuation values and outer controls are used for the mesh estimates. Control variates contribute to tighter price bounds in two ways, by reducing both the standard errors of the lower bound estimators and the bias of the upper bound estimators. Typically control variates are valued at a ﬁxed time, such as the European option’s maturity. Rasmussen (2005) and Broadie and Glasserman (2004) use control variates that are valued at the exercise time of the Bermudan option rather than at maturity, which leads to larger variance reduction because the control is sampled at an exercise time and so has a higher correlation with the Bermudan option value. This approach requires the control variate to have the martingale property and thus can be called a martingale control variate. 53

We apply this technique in our examples, speciﬁcally by taking single asset European option values at the exercise time as controls for single asset Bermudan options, and the geometric

Asian option values at the exercise time as controls for moving window Asian options. For Bermudan max options, since there is no simple analytic formula for European max options on more than two assets, we use the average of single asset European option values as the

martingale control. As discussed in Glasserman (2003), bias may be introduced if the same samples are used to estimate the adjustment coefficient β and the control variate adjusted value. In order to avoid bias which may sometimes be significant, we can fix the adjustment coefficient at a constant value. In our examples we fix the coefficient at one when estimating the single asset Bermudan option’s continuation value with European option value as the control, and

ﬁnd it to be generally eﬀective.

3.4 Improvements to upper bound algorithms

The improvements shown in this section can be applied to duality-based upper bound algorithms. In particular we use the primal-dual simulation algorithm of Andersen and Broadie (2004).

3.4.1 Duality-based upper bound algorithms

The dual problem of pricing Bermudan options is ht U0 = inf π0 + E0 max − πt . π∈Π t∈Γ Bt

Since the discounted value process Vt/Bt is a super-martingale, we can use Doob-Meyer

∗ decomposition to decompose it into a martingale process πt and an adapted increasing

∗ process At

Vt ∗ ∗ = πt − At . (3.4.1) Bt 54

This gives

ht ∗ ht Vt ∗ − πt = − − At ≤ 0, ∀t ∈ Γ, Bt Bt Bt

since ht ≤ Vt and A∗ ≥ 0. Hence, Bt Bt t ht ∗ max − πt ≤ 0 t∈Γ Bt

∗ and using the deﬁnition of U0 above we get U0 ≤ π0 = V0. But also V0 ≤ U0, so U0 = V0, i.e., there is no duality gap. For martingales other than π∗ there will be a gap between the resulting upper and lower bounds, so the question is how to construct a martingale process that leads to a tight upper bound when the optimal policy is not available.

3.4.2 Primal-dual simulation algorithm

The primal-dual simulation algorithm is a duality-based upper bound algorithm that builds upon simulation and can be used together with any lower-bound algorithm to generate an upper bound of Bermudan option values. We can decompose Lt/Bt as

Lt = πt − At, (3.4.2) Bt

where πt is an adapted martingale process deﬁned as,

π0 := L0, πt1 := Lt1 /Bt1 , Lti+1 Lti Lti+1 Lti πti+1 := πti + − − 1ti Eti − for 1 ≤ i ≤ n − 1. (3.4.3) Bti+1 Bti Bti+1 Bti

h L i Qti Lti Qti ti+1 Since = when 1ti = 0 and = Eti when 1ti = 1, we have Bti Bti Bti Bti+1

Lti+1 Qti πti+1 = πti + − . (3.4.4) Bti+1 Bti Deﬁne the upper bound increment D as ht D := max − πt , (3.4.5) t∈Γ Bt which can be viewed as the payoﬀ from a non-standard lookback call option, with the

discounted Bermudan option payoﬀ being the state variable and the adapted martingale 55 process being the ﬂoating strike. The duality gap D0 := E0[D] can be estimated by

¯ 1 PNH ˆ ˆ ˆ D := Di, and H0 = L0 +D0 will be the upper bound estimator from the primal-dual NH i=1 simulation algorithm. The sample variance of the upper bound estimator can be approximated as the sum of sample variances from the lower bound estimator and the duality gap estimator,

sˆ2 sˆ2 sˆ2 H ≈ L + D , (3.4.6) NH NL NH because the two estimators are uncorrelated when estimated independently. The simulation procedure for the primal-dual simulation algorithm is as follows:

(i) Simulate the path of state variables until the option matures.

(ii) At each exercise time, launch a sub-simulation with NS paths to estimate Qt/Bt and

update πt using equation (3.4.4).

(iii) Calculate the upper bound increment D for the current path.

ˆ (iv) Repeat steps (i)–(iii) for NH sample paths, estimate the duality gap D0 and combine ˆ ˆ it with L0 to obtain the upper bound estimator H0.

Implementation details are given in Anderson and Broadie (2004). Note that At is not necessarily an increasing process since Lt/Bt is not a super-martingale. In fact,

( h L i 0, 1t = 0, ti+1 Lti i A − A = −1 E − = ht Qt ti+1 ti ti ti Bt Bt i i i+1 i − , 1ti = 1, Bti Bti which decreases when an incorrect exercise decision is made. The ensuing Propositions 1 and 2 illustrate some properties of the primal-dual simulation algorithm. Proofs are provided in Appendix B.2.

Proposition 1

Ltk htk (i) If hti ≤ Qti for 1 ≤ i ≤ k, then πtk = and − πtk ≤ 0. Btk Btk 56

Q Q tl−1 Ltk htk tl−1 (ii) If hti ≤ Qti for l ≤ i ≤ k, then πtk = πtl−1 − + and − πtk ≤ − πtl−1 . Btl−1 Btk Btk Btl−1

Proposition 1(i) states that the martingale process πt is equal to the discounted lower bound value process and there is no contribution to the upper bound increment before the option enters the exercise region, and 1(ii) means that the computation of πt does not depend on the path during the period that option stays in the continuation region. It follows from 1(ii) that the sub-simulation is not needed when the option is sub-optimal to exercise. In independent work, Joshi (2007) derives results very akin to those shown in Proposition 1 by using a hedging portfolio argument. He shows that the upper bound increment is zero in the continuation region, simply by changing the payoﬀ function to negative inﬁnity when the option is sub-optimal to exercise.

If an option stays in the continuation region throughout its life, the upper bound increment D for the path is zero. The result holds even if the option stays in the continuation region until the ﬁnal step. Furthermore, if it is sub-optimal to exercise the option except in the last two exercise dates and the optimal exercise policy is available at the last step before maturity (for example, if the corresponding European option can be valued analytically),

Ltn−1 Vtn−1 htn−1 the upper bound increment D is also zero, because πtn−1 = = ≥ . Btn−1 Btn−1 Btn−1 Proposition 2 For a given sample path,

˜ (i) If ∃δ > 0 such that |Qt − Qt| < δ, and dt ≥ δ or ht ≤ Qt holds ∀t ∈ Γ, then At is an increasing process and D = 0 for the path.

˜ ∗ ∗ (ii) If ∃δ > 0 such that |Qt − Qt | < δ, and dt ≥ δ or ht ≤ Qt holds ∀t ∈ Γ, then 1t ≡ 1t .

The implication of Proposition 2 is that, given a uniformly good approximation of the ˜ sub-optimal continuation value (|Qt − Qt| is bounded above by a constant δ), the upper bound increment will be zero for a sample path if it never gets close to the sub-optimal exercise boundary. And if the approximation is uniformly good relative to the optimal

˜ ∗ continuation value (|Qt − Qt | is bounded above by a constant δ), the sub-optimal exercise 57 strategy will always coincide with the optimal strategy for the path never close to the sub- optimal boundary.

3.4.3 Sub-optimality checking

The primal-dual simulation algorithm launches a sub-simulation to estimate continuation values at every exercise time along the sample path. The continuation values are then used to determine the martingale process and eventually an upper bound increment. These sub-simulations are computationally demanding, however, many of them are not necessary.

Sub-optimality checking is an effective way to address this issue. It is based on the idea of Proposition 1, and can be easily implemented by comparing the option exercise payoff with the exercise lower limit Qt. The sub-simulations will be skipped when the exercise payoff is lower than the exercise lower limit, in other words, when it is sub-optimal to exercise the option. Despite being simple, sub-optimality checking may bring dramatic computational improvement, especially for deep OTM options. Efficiency of the simulation may be measured by the product of sample variance and simulation time, and we can define an effective saving factor (ESF) as the ratio of the efficiency before and after improvement. Since the sub-optimality checking reduces computational time without affecting variance, its ESF is simply the ratio of computational time before and after the improvement. The simulation procedure for the primal-dual algorithm with sub-optimality checking is as follows:

(i) Simulate the path of underlying variables until the option matures.

(ii) At each exercise time, if ht > Qt, launch a sub-simulation with NS paths to esti-

mate Qt/Bt and update πt using Proposition 1; otherwise skip the sub-simulation and proceed to next time step.

(iii) Calculate the upper bound increment D for the current path. 58

(iv) Repeat (i)-(iii) for NH sample paths, estimate the duality gap D0 and combine it with ˆ ˆ L0 to obtain the upper bound estimator H0.

3.4.4 Boundary distance grouping

By Proposition 2, when the sub-optimal strategy is close to optimal, many of the simulation

paths will have zero upper bound increments D. The algorithm, however, may spend a substantial amount of time to compute these zero values. We can eliminate much of this work by characterizing the paths that are more likely to produce non-zero upper bound

increments than others. We do so by identifying paths that, for at least once during their life, are ‘close’ to the sub-optimal exercise boundary. In boundary distance grouping, we separate the sample paths into two groups according

to the distance of each path to the sub-optimal exercise boundary. Paths that are ever within a certain distance to the boundary during the option’s life are placed into the ‘non- zero’ group, because it is suspected that the upper bound increment is non-zero. All other

paths, the ones that never get close to the sub-optimal exercise boundary, are placed into the ‘zero’ group. A sampling method is used to eliminate part of the simulation work for the ‘zero’ group when estimating upper bound increments. If the fraction of paths in the

‘non-zero’ group is small, the computational saving from doing this can be substantial. The two groups are deﬁned as follows:

Z := {ω : ∀t ∈ Γ, dt(ω) ≥ δ or ht ≤ Qt}, (3.4.7) ¯ Z := {ω : ∃t ∈ Γ, dt(ω) < δ and ht > Qt}. (3.4.8)

˜ If there exists a small constant δ0 > 0 such that P ({ω : maxt∈Γ |Qt(ω)−Qt(ω)| < δ0}) = 1,

the distance threshold δ could simply be chosen as δ0 so that by Proposition 2, D is zero for all the sample paths that belong to Z. In general δ0 is not known, and the appropriate choice of δ still remains, as we will address below.

2 Assume the D estimator has mean µD and variance σD. Without loss of generality, we ¯ assume nZ¯ out of the NH paths belong to group Z and are numbered from 1 to nZ¯, i.e., 59

¯ ¯ ω1, ..., ωnZ¯ ∈ Z, and ωnZ¯ +1, ..., ωNH ∈ Z. Let pZ be the probability that a sample path belongs to group Z¯,

¯ pZ¯ = P (ω ∈ Z) = P ({ω : ∃t ∈ Γ, dt(ω) < δ and ht > Qt}). (3.4.9)

The conditional means and variances for upper bound increments in the two groups are

2 2 µZ¯, σZ¯, µZ and σZ . In addition to the standard estimator which is the simple average, an alternative estimator of the duality gap can be constructed by estimating Dis from a selected ¯ set of paths, more specifically the nZ¯ paths in group Z and lZ paths randomly chosen from group Z (lZ NH − nZ¯). For simplicity, we pick the first lZ paths from group Z. The new estimator is   nZ¯ nZ¯ +lZ ˜ 1 X NH − nZ¯ X D :=  Di + Di , (3.4.10) NH lZ i=1 i=nZ¯ +1 which may be easily shown to be unbiased. Although the variance of D˜ is higher than the variance of D¯, the difference is usually small (see Appendix B.3). As shown in Appendix B.3, under certain conditions the effective saving factor of boundary distance grouping is simply the saving of computational time by only estimating Dis from group Z¯ paths instead of all paths, i.e.,

Var[D¯] · T ¯ T ESF = D ≈ 1 + DZ , (3.4.11) ˜ p ¯T Var[D] · TD˜ Z DZ¯

which goes to inﬁnity as pZ¯ → 0, TD¯ and TD˜ are the expected time to obtain the standard

estimator and the alternative estimator, TDZ¯ and TDZ are respectively the expected time to estimate upper bound increment D from a group Z¯ path and from a group Z path. Notice that after the grouping, we cannot directly estimate Var[D˜] by calculating the

sample variance from Dis because they are no longer identically distributed. Appendix B.3 gives two indirect methods for estimating the sample variance. The simulation procedure for primal-dual algorithm with boundary distance grouping is as follows:

(i) Generate np pilot paths as in the standard primal-dual algorithm. For each δ among a 60

0 set of values, estimate the parameters pZ¯, µZ¯, σZ¯, TP , TI , etc., and calculate lZ , then choose the δ0 that optimizes the eﬃcient measure.

(ii) Simulate the path of underlying variables until the option matures.

0 (iii) Estimate the boundary distance dt along the path, if ∃t ∈ Γ such that dt < δ and ¯ ht > Qt, assign the path to group Z, otherwise assign it to Z.

¯ 0 (iv) If the current path belongs to group Z or is among the ﬁrst lZ paths in group Z, estimate the upper bound increment D as in the regular primal-dual algorithm, otherwise skip it.

(v) Repeat steps (ii)-(iv) for NH sample paths, estimate the duality gap using the al- ˜ ˆ ternative estimator D and combine it with L0 to obtain the upper bound estimator ˆ H0.

3.5 Numerical results

Numerical results for single asset Bermudan options, moving window Asian options and Bermudan max options are presented in this section. The underlying assets are assumed to follow the standard single and multi-asset Black-Scholes model. In the results below, ˆ L0 is the lower bound estimator obtained through the least-squares method (Longstaﬀ and ˆ Schwartz 2001), tL is the computational time associated with it, H0 is the upper bound estimator obtained through the primal-dual simulation algorithm (Andersen and Broadie

2004), tH is the associated computational time, and tT = tL + tH is the total computational time. The point estimator is obtained by taking the average of lower bound and upper bound estimators. All computations are done on a Pentium 4 2.0GHz computer and computation

time is measured in minutes. In the four summary tables below (Tables 3.1-3.4), we show the improvements from methods introduced in this chapter, through measures including the low and high estimators,

the standard errors and the computational time. Each table is split into three panels: the top 61 panel contains results before improvement, the middle panel shows the reduction of upper bound computational time through sub-optimality checking and boundary distance grouping, and the bottom panel shows the additional variance reduction and estimator improvement through local policy enhancement and the martingale control variate. Note that the local policy enhancement is only used for moving window Asian options (Table 3.2), for which we

find the method effective without significantly increasing the computational cost. For all regression-based algorithms, which basis functions to use is often critical but not obvious. We summarize the choice of basis functions for our numerical examples, as well as the comparison between ‘regression later’ and ‘regression now,’ in Appendix B.1.

3.5.1 Single asset Bermudan options

The single asset Bermudan option is the most standard and simplest Bermudan-type option.

We assume the asset price follows the geometric Brownian motion process

dSt = (r − q)dt + σdWt, (3.5.1) St where Wt is standard Brownian motion. The payoﬀ upon exercise for a single asset Bermudan

+ call option at time t is (St − K) . The option and model parameters are defined as follows: σ is the annualized volatility, r is the continuously compounded risk-free interest rate, q is the continuously compounded dividend rate, K is the strike price, T is the maturity in years, and there are n + 1 exercise opportunities, equally spaced at time ti = iT/n, i = 0, 1, ..., n. In the implementation of lower bound algorithms, paths are usually simulated from the initial state for which the option value is desired, to determine the sub-optimal exercise policy. However, the optimal exercise policy is independent of this initial state. To approximate the optimal policy more efficiently, we disperse the initial state for regression, an idea independently proposed in Rasmussen (2005). The paths of state variables are generated from a distribution of initial states, more specifically by simulating the state variables from strike

K at time −T/2 instead of from S0 at time 0. This dispersion method can be particularly helpful when pricing deep OTM and deep ITM options, given that simulating paths from 62

the initial states of these options is likely to contribute little to ﬁnding the optimal exercise strategy, since most of the paths will be distant from the exercise boundary. The regression

only needs to be performed once for pricing options with same strike and different initial states, in which case the total computational time is significantly reduced. In terms of regression basis functions, using powers of European option values proves to be more efficient

than using powers of the underlying asset prices. Table 3.1 shows the improvements in pricing single asset Bermudan call options using techniques introduced in this chapter. It demonstrates that the simulation algorithm may

work remarkably well, even compared to the binomial method. In each of the seven cases, a tight conﬁdence interval containing the true value can be produced in a time comparable to, or less than, the binomial method. The widths of 95% conﬁdence intervals are all within

0.4% of the true option values.

3.5.2 Moving window Asian options

A moving window Asian option is a Bermudan-type option that can be exercised at any time ti before T (i ≥ m), with the payoﬀ dependent on the average of the asset prices during a period of ﬁxed length. Consider the asset price St following the geometric Brownian motion

process deﬁned in equation (3.5.1), and let Ati be the arithmetic average of St over the m periods up to time ti, i.e.,

i 1 X A = S . (3.5.2) ti m tk k=i−m+1

+ The moving window Asian option can be exercised at any time ti with payoﬀ (Ati − K)

+ for a call and (K − Ati ) for a put. Notice that it becomes a standard Asian option when m = n, and a single asset Bermudan option when m = 1. The European version of this option is a forward starting Asian option or Asian tail option. The early exercise feature, along with the payoff’s dependence on the historic average, makes the moving window Asian option difficult to value by lattice or finite difference methods. Monte Carlo simulation appears to be a good alternative to price these options. Poly- 63

Table 3.1: Summary results for single asset Bermudan call options ˆ ˆ S0 L0(s.e.) tL H0(s.e.) tH 95% C.I. tT Point est. True 70 0.1281(.0036) 0.03 0.1287(.0036) 4.26 0.1210, 0.1358 4.29 0.1284 0.1252 80 0.7075(.0090) 0.04 0.7130(.0091) 4.73 0.6898, 0.7309 4.77 0.7103 0.6934 90 2.3916(.0170) 0.05 2.4148(.0172) 6.12 2.3584, 2.4484 6.16 2.4032 2.3828 100 5.9078(.0253) 0.07 5.9728(.0257) 7.95 5.8583, 6.0231 8.02 5.9403 5.9152 110 11.7143(.0296) 0.08 11.8529(.0303) 8.33 11.6562, 11.9123 8.41 11.7836 11.7478 120 20.0000(.0000) 0.03 20.1899(.0076) 5.83 20.0000, 20.2049 5.87 20.0950 20.0063 130 30.0000(.0000) 0.00 30.0523(.0043) 3.55 30.0000, 30.0608 3.55 30.0261 30.0000 70 0.1281(.0036) 0.03 0.1289(.0037) 0.00 0.1210, 0.1361 0.03 0.1285 0.1252 80 0.7075(.0090) 0.04 0.7113(.0091) 0.01 0.6898, 0.7291 0.05 0.7094 0.6934 90 2.3916(.0170) 0.05 2.4154(.0172) 0.12 2.3584, 2.4490 0.16 2.4035 2.3828 100 5.9078(.0253) 0.07 5.9839(.0258) 0.61 5.8583, 6.0343 0.68 5.9458 5.9152 110 11.7143(.0296) 0.08 11.8614(.0303) 1.98 11.6562, 11.9208 2.06 11.7878 11.7478 120 20.0000(.0000) 0.03 20.2012(.0075) 2.09 20.0000, 20.2159 2.12 20.1006 20.0063 130 30.0000(.0000) 0.00 30.0494(.0040) 1.75 30.0000, 30.0572 1.75 30.0247 30.0000 70 0.1251(.0001) 0.03 0.1251(.0001) 0.00 0.1249, 0.1254 0.04 0.1251 0.1252 80 0.6931(.0003) 0.04 0.6932(.0003) 0.01 0.6925, 0.6939 0.05 0.6932 0.6934 90 2.3836(.0007) 0.05 2.3838(.0007) 0.12 2.3821, 2.3852 0.16 2.3837 2.3828 100 5.9167(.0013) 0.07 5.9172(.0013) 0.61 5.9141, 5.9198 0.68 5.9170 5.9152 110 11.7477(.0019) 0.08 11.7488(.0019) 1.98 11.7441, 11.7524 2.07 11.7482 11.7478 120 20.0032(.0015) 0.04 20.0109(.0016) 2.19 20.0003, 20.0139 2.22 20.0070 20.0063 130 30.0000(.0000) 0.00 30.0007(.0004) 0.71 30.0000, 30.0015 0.71 30.0004 30.0000

Note: Option parameters are σ = 20%, r = 5%, q = 10%, K = 100, T = 1, n = 50, b = 3, NR = 100, 000, NL = 100, 000, NH = 1000, NS = 500. The three panels respectively contain results before improvement (top), after the improvement of sub-optimality checking and boundary distance grouping (middle), and additionally with martingale control variate (bottom)–European call option value sampled at the exercise time in this case. The true value is obtained through a binomial lattice with 36,000 time steps, which takes approximately two minutes per option. 64

Table 3.2: Summary results for moving window Asian call options ˆ ˆ S0 L0(s.e.) tL H0(s.e.) tH 95% C.I. tT Point est. 70 0.345(.007) 0.04 0.345(.007) 3.08 0.331, 0.359 3.12 0.345 80 1.715(.017) 0.05 1.721(.017) 3.67 1.682, 1.754 3.72 1.718 90 5.203(.030) 0.05 5.226(.030) 5.13 5.144, 5.285 5.18 5.214 100 11.378(.043) 0.08 11.427(.044) 6.97 11.293, 11.512 7.05 11.403 110 19.918(.053) 0.10 19.992(.053) 8.24 19.814, 20.097 8.34 19.955 120 29.899(.059) 0.10 29.992(.060) 8.58 29.782, 30.109 8.68 29.945 130 40.389(.064) 0.10 40.490(.064) 8.61 40.264, 40.616 8.71 40.440 70 0.345(.007) 0.04 0.345(.007) 0.00 0.331, 0.358 0.04 0.345 80 1.715(.017) 0.05 1.721(.017) 0.01 1.682, 1.754 0.06 1.718 90 5.203(.030) 0.05 5.227(.030) 0.10 5.144, 5.286 0.15 5.215 100 11.378(.043) 0.08 11.419(.044) 0.25 11.294, 11.504 0.33 11.399 110 19.918(.053) 0.10 19.990(.054) 0.55 19.814, 20.095 0.65 19.954 120 29.899(.059) 0.10 29.995(.060) 1.26 29.782, 30.112 1.36 29.947 130 40.389(.064) 0.11 40.478(.064) 1.67 40.264, 40.604 1.78 40.433 70 0.338(.001) 0.08 0.338(.001) 0.00 0.336, 0.340 0.08 0.338 80 1.699(.003) 0.30 1.702(.003) 0.01 1.694, 1.708 0.31 1.701 90 5.199(.005) 0.91 5.206(.006) 0.11 5.189, 5.217 1.02 5.203 100 11.406(.007) 2.01 11.417(.008) 0.25 11.391, 11.433 2.26 11.411 110 19.967(.009) 3.36 19.987(.010) 0.55 19.949, 20.007 3.92 19.977 120 29.961(.010) 4.24 29.972(.011) 1.26 29.942, 30.993 5.50 29.967 130 40.443(.010) 4.22 40.453(.011) 1.68 40.423, 40.475 5.89 40.448

Note: Option parameters are σ = 20%, r = 5%, q = 0%, K = 100, T = 1, n = 50, m = 10, b = 6, NR = 100, 000, NL = 100, 000, NH = 1000, NS = 500. The three panels respectively contain results before improvement (top), after the improvement of sub-optimality checking and boundary distance grouping (middle), and additionally with local policy enhancement and martingale control variate (bottom)–geometric Asian option value sampled at the exercise time in this case. For the local policy enhancement = 0.5 and N = 100. nomials of underlying asset price and arithmetic average are used as the regression basis functions. As shown in Table 3.2, the moving window Asian call options can be priced with high precision using Monte Carlo methods along with the improvements in this chapter. For all seven cases, the 95% conﬁdence interval widths lie within 1% of the true option values, compared to 2-7 times that amount before improvements. The lower bound computing time is longer after the improvements due to the sub-simulations in local policy enhancement, but the total computational time is reduced in every case. 65

3.5.3 Symmetric Bermudan max options

A Bermudan max option is a discretely-exercisable option on multiple underlying assets

whose payoﬀ depends on the maximum among all asset prices. We assume the asset prices follow correlated geometric Brownian motion processes, i.e.,

dSj,t = (r − qj)dt + σjdWj,t, (3.5.3) Sj,t where Wj,t, j = 1, ..., d, are standard Brownian motions and the instantaneous correlation between Wj,t and Wk,t is ρjk. The payoﬀ of a 5-asset Bermudan max call option is

+ (max1≤j≤5 Sj,t − K) .

For simplicity, we assume qj = q, σj = σ and ρjk = ρ, for all j, k = 1, ..., d and j 6= k. We call this the symmetric case because the common parameter values mean the future asset

returns do not depend on the index of speciﬁc asset. Under these assumptions the assets are numerically indistinguishable, which facilitates simpliﬁcation in the choice of regression basis functions. In particular, the polynomials of sorted asset prices can be used as the

(non-distinguishing) basis functions, without referencing to a speciﬁc asset index. Table 3.3 provides pricing results for 5-asset Bermudan max call options before and after the improvements in this chapter. Considerably tighter price bounds and reduced

computational time are obtained, in magnitudes similar to that observed for the single asset Bermudan option and moving window Asian option. Next we consider the more general case, in which the assets have asymmetric parameters

and are thus distinguishable.

3.5.4 Asymmetric Bermudan max options

We use the 5-asset max call option with asymmetric volatilities (ranging from 8% to 40%) as an example. Table 3.4 shows that the magnitude of the improvements from the techniques in this chapter are comparable to their symmetric counterpart. The lower bound estimator

in the asymmetric case may be signiﬁcantly improved by including basis functions that 66

Table 3.3: Summary results for 5-asset symmetric Bermudan max call options ˆ ˆ S0 L0(s.e.) tL H0(s.e.) tH 95% C.I. tT Point est. 70 3.892(.006) 0.72 3.904(.006) 2.74 3.880, 3.916 3.46 3.898 80 9.002(.009) 0.81 9.015(.009) 3.01 8.984, 9.033 3.82 9.009 90 16.622(.012) 0.97 16.655(.012) 3.39 16.599, 16.679 4.36 16.638 100 26.120(.014) 1.12 26.176(.015) 3.72 26.093, 26.205 4.83 26.148 110 36.711(.016) 1.18 36.805(.017) 3.90 36.681, 36.838 5.08 36.758 120 47.849(.017) 1.18 47.985(.019) 3.92 47.816, 48.023 5.10 47.917 130 59.235(.018) 1.16 59.403(.021) 3.86 59.199, 59.445 5.02 59.319 70 3.892(.006) 0.72 3.901(.006) 0.05 3.880, 3.913 0.77 3.897 80 9.002(.009) 0.80 9.015(.009) 0.10 8.984, 9.033 0.90 9.008 90 16.622(.012) 0.97 16.662(.012) 0.45 16.599, 16.686 1.42 16.642 100 26.120(.014) 1.12 26.165(.015) 0.91 26.093, 26.194 2.03 26.142 110 36.711(.016) 1.19 36.786(.017) 1.33 36.681, 36.819 2.52 36.749 120 47.849(.017) 1.19 47.994(.020) 1.62 47.816, 48.033 2.81 47.921 130 59.235(.018) 1.16 59.395(.021) 2.14 59.199, 59.437 3.30 59.315 70 3.898(.001) 0.70 3.903(.001) 0.06 3.896, 3.906 0.76 3.901 80 9.008(.002) 0.79 9.014(.002) 0.10 9.004, 9.019 0.90 9.011 90 16.627(.004) 0.95 16.644(.004) 0.46 16.620, 16.653 1.41 16.636 100 26.125(.005) 1.09 26.152(.006) 0.91 26.115, 26.164 2.00 26.139 110 36.722(.006) 1.15 36.781(.009) 1.34 36.710, 36.798 2.49 36.752 120 47.862(.008) 1.16 47.988(.012) 1.62 47.847, 48.011 2.78 47.925 130 59.250(.009) 1.45 59.396(.013) 2.14 59.233, 59.423 3.59 59.323

Note: Option parameters are σ = 20%, q = 10%, r = 5%, K = 100, T = 3, ρ = 0, n = 9, b = 18, NR = 200, 000, NL = 2, 000, 000, NH = 1500 and NS = 1000. The three panels respectively contain results before improvement (top), after the improvement of sub-optimality checking and boundary distance grouping (middle), and additionally with martingale control variate (bottom)–average of European option values sampled at the exercise time in this case. 67

Table 3.4: Summary results for 5-asset asymmetric Bermudan max call options ˆ ˆ S0 L0(s.e.) tL H0(s.e.) tH 95% C.I. tT Point est. 70 11.756(.016) 0.74 11.850(.019) 2.75 11.723, 11.888 3.49 11.803 80 18.721(.020) 0.96 18.875(.024) 3.43 18.680, 18.921 4.39 18.798 90 27.455(.024) 1.25 27.664(.028) 4.26 27.407, 27.719 5.52 27.559 100 37.730(.028) 1.57 38.042(.033) 5.13 37.676, 38.107 6.70 37.886 110 49.162(.031) 1.75 49.555(.037) 5.73 49.101, 49.627 7.48 49.358 120 61.277(.034) 1.82 61.768(.040) 5.99 61.211, 61.848 7.81 61.523 130 73.709(.037) 1.83 74.263(.044) 6.07 73.638, 74.349 7.89 73.986 70 11.756(.016) 0.74 11.850(.019) 0.19 11.723, 11.883 0.93 11.801 80 18.721(.020) 0.96 18.875(.024) 0.38 18.680, 18.933 1.34 18.803 90 27.455(.024) 1.25 27.664(.028) 0.62 27.407, 27.741 1.87 27.570 100 37.730(.028) 1.57 38.042(.033) 1.47 37.676, 38.106 3.04 37.886 110 49.162(.031) 1.75 49.555(.037) 2.58 49.101, 49.626 4.33 49.357 120 61.277(.034) 1.82 61.768(.040) 3.17 61.211, 61.830 4.99 61.514 130 73.709(.037) 1.83 74.263(.044) 4.11 73.638, 74.351 5.94 73.986 70 11.778(.003) 0.75 11.842(.007) 0.19 11.772, 11.856 0.95 11.810 80 18.744(.004) 0.98 18.866(.011) 0.39 18.736, 18.887 1.38 18.805 90 27.480(.006) 1.29 27.659(.014) 0.62 27.468, 27.686 1.90 27.570 100 37.746(.008) 1.62 37.988(.016) 1.48 37.730, 38.020 3.10 37.867 110 49.175(.010) 1.79 49.492(.020) 2.58 49.155, 49.531 4.37 49.334 120 61.294(.015) 1.86 61.686(.023) 3.17 61.269, 61.730 5.04 61.490 130 73.723(.015) 1.88 74.184(.026) 4.12 73.694, 74.234 6.00 73.953

Note: Option parameters are the same as in Table 3.3, except that σi = 8%, 16%, 24%, 32% and 40% respectively for i = 1, 2, 3, 4, 5. The three panels respectively contain results before improvement (top), after the improvement of sub-optimality checking and boundary distance grouping (middle), and additionally with martingale control variate (bottom)–average of European option values sampled at the exercise time in this case. distinguish the assets (see Table 3.5). Nonetheless, for a reasonably symmetric or a large basket of assets, it is often more eﬃcient to use the non-distinguishing basis functions, because of the impracticality to include the large number of asset-speciﬁc basis functions.

3.5.5 Summary and other computational issues

Table 3.6 illustrates, using symmetric ﬁve-asset Bermudan max call option as an example, that the local policy enhancement can eﬀectively improve the lower bound estimator, especially when the original exercise policy is far from optimal. The lower bound estimator using 12 basis functions with local policy enhancement consistently outperforms the estimator us- 68

Table 3.5: Impact of basis functions on 5-asset asymmetric Bermudan max call options S0 = 90 S0 = 100 S0 = 110 ˆS Â ˆ ˆS Â ˆ ˆS Â ˆ b L0 L0 ∆L0 L0 L0 ∆L0 L0 L0 ∆L0 18 27.049 27.517 +0.468 37.089 37.807 +0.718 48.408 49.254 +0.846 12 27.325 27.480 +0.155 37.529 37.746 +0.217 48.910 49.175 +0.265

ˆS Note: Option parameters are the same as in Table 3.4. L0 represents the lower bound estimator ˆA using symmetric (non-distinguishing) basis functions, L0 is the estimator using asymmetric (distinguishing) basis functions. The 95% upper bounds for S0 = 90, 100, 110 are respectively 27.686, 38.020, and 49.531.

Table 3.6: Lower bound improvements by local policy enhancement (5-asset Bermudan max call) S0 = 90 S0 = 100 S0 = 110 ˆ Ê ˆ ˆ Ê ˆ ˆ Ê ˆ b L0 L0 ∆L0 L0 L0 ∆L0 L0 L0 ∆L0 6 16.563 16.613 +0.050 26.040 26.108 +0.068 36.632 36.713 +0.081 12 16.606 16.629 +0.023 26.106 26.138 +0.032 36.715 36.756 +0.041 18 16.627 16.634 +0.007 26.125 26.139 +0.014 36.722 36.750 +0.028 19 16.618 16.630 +0.012 26.113 26.134 +0.021 36.705 36.739 +0.034

Note: Option parameters are the same as in Table 3.3 except that the local policy enhancement is ˆ Ê used here with = 0.05 and N = 500. L0 is the regular lower bound estimator, L0 is the estimator with local policy enhancement. The 95% upper bounds for S0 = 90, 100, 110 are, respectively, 16.652, 26.170, and 36.804. ing 18 basis functions without local policy enhancement. This indicates that the local policy enhancement can help reduce the number of basis functions needed for regression in order to achieve the same level of accuracy. Table 3.7 shows the effective saving factor by sub-optimality checking and boundary distance grouping, which is calculated as the ratio of the product of computational time and variance of estimator, with and without improvements. Both methods are most effective on deep OTM options (for example, the S0 = 70 case for a moving window Asian call shows an effective saving factor of more than 1000), and show considerable improvements for ATM and ITM options. The sub-optimality checking shows greater improvements in most cases, while the boundary distance grouping works better for deep ITM single asset and max Bermudan options, for which many sample paths are far above the exercise boundary thus will be 69

Table 3.7: Eﬀective saving factor by sub-optimality checking and boundary distance grouping S S∗ B B S,B S,B Option type S0 tH tH ESF tH ESF tH ESF Single 70 4.22 0.01 739.7 0.03 196.8 0.00 N.A.∗∗ asset 90 5.87 0.24 32.3 1.13 6.9 0.10 92.3 Bermudan 110 7.90 3.26 2.3 3.71 2.6 1.95 6.0 call 130 3.58 4.21 1.2 1.18 4.5 1.08 5.4 Moving 70 3.05 0.01 1163.1 0.06 69.4 0.00 1172.1 window 90 5.00 0.15 43.3 2.30 2.7 0.11 62.0 Asian 110 7.97 1.04 9.1 4.88 1.9 0.55 16.5 call 130 8.37 2.33 4.4 6.33 1.3 1.68 5.7 Bermudan 70 2.72 0.10 26.5 0.32 8.2 0.06 47.2 max 90 3.49 1.01 3.4 1.16 2.4 0.46 7.4 call 110 4.04 3.93 1.0 2.70 1.6 1.34 1.6 130 4.08 4.30 1.0 2.30 1.6 2.14 1.6

∗: Effective saving factor (ESF) is defined in Section 3.4. ∗∗: Due to zero upper bound increment after improvements being the denominator. Note: Option parameters are the same as in Table 3.1, 3.2 and 3.3 respectively for three types S B S,B of options. tH , tH , tH , tH are the computational time for the primal-dual simulation estimates, respectively with no improvement, with sub-optimal checking, boundary distance grouping and two methods combined. placed in the ‘zero’ group and save the computation. Also notice that the improvements of two methods are not orthogonal, especially for OTM options, since the time saving for both methods comes mainly from the sample paths that never go beyond the exercise lower limit. More specifically, the expected time for estimating upper bound increment from a group Z

path is shorter than that from a group Z¯ path after applying the sub-optimality checking, which limits the additional saving through boundary distance grouping. Table 3.8 demonstrates how the computational time increases with the number of exercise

opportunities n, using the moving window Asian call option as an example. For the least-

squares lower bound estimator, the computational time tL increases linearly with n. After applying the local policy enhancement, dependence becomes between linear and quadratic,

because sub-simulations are performed, but only when the path is close to the sub-optimal exercise boundary.

The computational time for primal-dual simulation algorithm, tH , has a quadratic de- 70

Table 3.8: Computational time vs. number of exercise opportunities (moving window Asian call) S0 70 100 130 E S,B E S,B E S,B n tL tL tH tH tL tL tH tH tL tL tH tH 10 0.01 0.01 0.20 0.00 0.02 0.14 0.35 0.00 0.02 0.55 0.46 0.05 25 0.02 0.03 1.13 0.00 0.04 0.73 2.08 0.05 0.05 1.76 2.54 0.31 50 0.04 0.08 4.31 0.00 0.08 2.01 8.00 0.25 0.10 4.22 9.59 1.68 100 0.07 0.21 16.83 0.01 0.15 5.51 30.89 1.30 0.21 11.36 37.23 8.64

Note: Option parameters are the same as in Table 3.2 except the number of exercise opportunities n. The window size is set as m = n/5 to ensure the consistent window length. pendence on the number of exercise steps, as a sub-simulation is needed at every step of the sample path. By combining the sub-optimality checking and boundary distance grouping, the dependence becomes more than quadratic, while the computational time is actually reduced–that is because the boundary distance grouping is more eﬀective for options with fewer exercise opportunities, in which case there are fewer sample paths in the ‘non-zero’ group.

3.6 Conclusion

In this chapter we introduce new variance reduction techniques and computational improvements for both lower and upper bound Monte Carlo methods for pricing American-style options. Local policy enhancement may significantly improve the lower bound estimator of Bermu- dan option values, especially when the original exercise policy is far from optimal. Sub- optimality checking and boundary distance grouping are two methods that may reduce the computational time in duality-based upper bound algorithms by up to several hundred times. They both work best on out-of-the-money options. Sub-optimality checking is easy to implement and more effective in general, while boundary distance grouping performs better for options that are deep in-the-money. They can be combined to achieve a more significant reduction. 71

Tight lower and upper bounds for high-dimensional and path-dependent Bermudan options can be computed in the matter of seconds or a few minutes using the methods proposed here. Together they produce narrower conﬁdence intervals using less computational time, by improving the exercise policy, reducing the variance of the estimators and saving unnecessary computations. For all the numerical examples tested, widths of 95% conﬁdence intervals are within 1% of the option values, compared to 5% ∼ 10% before the improvements. And it takes up to 6 minutes to price each option, instead of several hours before the improvements. These improvements greatly enhance the practical use of Monte Carlo methods for pricing

complicated American options. Chapter 4

Bibliography

[1] Andersen, L. (2000). A Simple Approach to the Pricing of Bermudan Swaptions in the Multi-Factor LIBOR Market Model. Journal of Computational Finance, 3(1), 5-32.

[2] Andersen, L., and M. Broadie (2004). A Primal-Dual Simulation Algorithm for Pricing

Multi-Dimensional American Options. Management Science, 50(9), 1222-1234.

[3] Asmussen, S., and H. Albrecher (2010). Ruin Probabilities. Second Edition. World Sci- entiﬁc, Singapore.

[4] Barraquand, J., and D. Martineau (1995). Numerical Valuation of High Dimensional Multivariate American Securities. Journal of Financial and Quantitative Analysis,

30(3), 383-405.

[5] Beghin, L., and E. Orsingher (1999). On the Maximum of the Generalized Brownian Bridge. Lithuanian Mathematical Journal, 39, 157-167.

[6] Belomestny D, Bender C, Schoenmakers J (2009). True upper bounds for Bermudan products via non-nested Monte Carlo. Math. Finance, 19(1), 53-71.

[7] Black, F., and J. C. Cox (1976). Valuing Corporate Securities: Some Eﬀects of Bond

Indenture Provisions. Journal of Finance, 31, 351-367.

[8] Black, F., and M. Scholes (1973). The Pricing of Options and Corporate Liabilities.

Journal of Political Economy, May 1973. 72 73

[9] Blanchet J., and P. Glynn (2006). Complete Corrected Diﬀusion Approximations for the Maximum of a Random Walk. The Annals of Applied Probability, 16(2), 951-983.

[10] Bossaerts, P. (1989). Simulation Estimators of Optimal Early Exercise. Working paper,

Carnegie Mellon University.

[11] Boyle, P. (1977). Options: A Monte Carlo Approach. Journal of Financial Economics, 4, 323-338.

[12] Boyle, P., M. Broadie, and P. Glasserman (1997). Monte Carlo Methods for Security Pricing. Journal of Economic Dynamics and Control, 21, 1267-1321.

[13] Broadie, M., and M. Cao (2008). Improved Lower and Upper Bound Algorithms for

Pricing American Options by Simulation. Quantitative Finance, 8, 845-861.

[14] Broadie, M., and J. Detemple (1997). The Valuation of American Options on Multiple Assets. Mathematical Finance, 7, 241-286.

[15] Broadie, M., and P. Glasserman (1997). Pricing American-Style Securities Using Simu- lation. Journal of Economic Dynamics and Control, 21(8-9), 1323-1352.

[16] Broadie, M., and P. Glasserman (2004). A Stochastic Mesh Method for Pricing High-

Dimensional American Option. Journal of Computational Finance, 7(4), 35-72.

[17] Broadie, M., P. Glasserman, and Z. Ha (2000). Pricing American Options by Simulation Using a Stochastic Mesh with Optimized Weights. Probabilistic Constrained Optimiza- tion: Methodology and Applications, 32-50, S. Uryasev, ed., Kluwer, Norwell, Mass.

[18] Broadie, M., P. Glasserman, and S. G. Kou (1997). A continuity correction for discrete

barrier options. Math. Finance, 7, 325-349.

[19] Broadie, M., P. Glasserman, and S. G. Kou (1999). Connecting discrete and continuous path-dependent options. Fin. Stochastics, 3, 55-82. 74

[20] Carr, P. (1995). Two extensions to barrier option valuation. Appl. Math. Finance, 9, 173-209.

[21] Carriere, J. F. (1996). Valuation of the Early-Exercise Price for Derivative Securities using Simulations and Splines. Insurance: Mathematics and Economics 19, 19-30.

[22] Chang J., and Y. Peres (1997). Ladder heights, Gaussian random walks and the Riemann

zeta function. Annals of Probability, 25, 787-802.

[23] Chen N., P. Glasserman (2007). Additive and multiplicative duals for American option

pricing. Finance Stochastics, 11(2), 153-179.

[24] Chuang, C. S. (1996). Joint distributions of Brownian motion and its maximum, with a

generalization to correlated BM and applications to barrier options. Stat. Prob. Letters, 28, 81-90.

[25] Cl´ement, E., D. Lamberton, and P. Protter (2002). An Analysis of a Least Squares Regression Algorithm for American Option Pricing. Finance and Stochastics 17, 448-

471.

[26] Cox, J., S. Ross, and M. Rubinstein (1979). Option Pricing: A Simpliﬁed Approach.

Journal of Financial Economics, 7, 229-263.

[27] Davis, M., and I. Karatzas (1994). A Deterministic Approach to Optimal Stopping, with Applications. Probability, Statistics and Optimization: A Tribute to Peter Whittle, ed. F.P. Kelly, 455-466, John Wiley & Sons, New York and Chichester.

[28] Desai, V., V. Farias, C. Moallemi (2012). Pathwise Optimization for Optimal Stopping Problems. Management Science, Articles in Advance, 1-17.

[29] Feng, L., and V. Linetsky (2008). Pricing discretely monitored barrier options and de-

faultable bonds in Lvy process models: A fast Hilbert transform approach. Mathematical Finance, 18(3), 337-384. 75

[30] Glasserman, P. (2003). Monte Carlo Methods in Financial Engineering. Springer.

[31] Glasserman, P., and T. Liu (1997). Corrected diﬀusion approximations for a multistage production-inventory system. Mathematics of Operations Research, 22(1), 185-311.

[32] Glasserman, P., and B. Yu (2002). Simulation for American Options: Regression Now or

Regression Later? Monte Carlo and Quasi-Monte Carlo Methods 2002 (H. Niederreiter, ed.), 213-226. Springer, Berlin.

[33] Glasserman, P., and B. Yu (2004). Number of Paths Versus Number of Basis Functions in American Option Pricing. Annals of Applied Probability, 14(4), 2090-2119.

[34] Haugh, M., and L. Kogan (2004). Pricing American Options: A Duality Approach.

Operations Research, 52(2), 258-270.

[35] Heynen, R. C., and H. M. Kat (1994a). Crossing barriers. RISK, 7, 46-49. Correction, 1995), RISK, 8, 18.

[36] Heynen, R. C., and H. M. Kat (1994b). Partial barrier options. J. Financial Engineering, 3, 253-274.

[37] Howison, S. (2007). A matched asymptotic expansions approach to continuity correc-

tions for discretely sampled options. Part 2: Bermudan options. Appl. Math. Finance, 14, 91-104.

[38] Howison, S., and M. Steinberg (2007). A matched asymptotic expansions approach to continuity corrections for discretely sampled options. Part 1: barrier options. Appl.

Math. Finance, 14, 6389.

[39] Hull, J. (2003). Options, Futures and Other Derivatives. 5th ed. Prentice Hall.

[40] Jamshidian F. (2004). Numeraire-invariant option pricing & American, Bermudan, and trigger stream rollover. Technical report, NIB Capitial, University of Twente. 76

[41] Jensen, M., and M. Svenstrup (2002). Eﬃcient Control Variates and Strategies for Bermudan Swaptions in a LIBOR Market Model. Working paper, Aarhus School of

Business.

[42] Joshi, M. S. (2007). A Simple Derivation of and Improvements to Jamshidian’s and Rogers’ Upper Bound Methods for Bermudan Options. Applied Mathematical Finance, 14(3), 197-205.

[43] Karatzas, I. (1988). On the Pricing of American Options. Applied Mathematics and

Optimization, 17, 37-60.

[44] Karatzas, I., and S. Shreve (1991). Brownian Motion and Stochastic Calculus. Springer- Verlag, New York.

[45] Keener, R. (2013). Improving Brownian approximations for boundary crossing problems. Bernoulli, 19(1), 137153.

[46] Kou, S. G. (2003). On pricing of discrete barrier options. Statistica Sinica, 13, 955-964.

[47] Lamberton, D., and B. Lapeyre (1996). Introduction to Stochastic Calculus Applied to

Finance. Chapman & Hall/CRC, Boca Raton.

[48] Longstaﬀ, F., and E. Schwartz (2001). Valuing American Options By Simulation: A Simple Least-Squares Approach. Review of Financial Studies, 14, 113-148.

[49] Lorden, G. (1970). On excess over the boundary. Ann. of Math. Stat., 41, 520-527.

[50] Merton, R. C. (1973). Theory of rational option pricing. Bell J. Econ. Mgmt. Sci., 4, 141-183.

[51] Petrella, G., and S. G. Kou (2004). Numerical pricing of discrete barrier and lookback

options via Laplace transforms. Journal of Computational Finance, 8, 1-37. 77

[52] Rasmussen, N. (2005). Control Variates for Monte Carlo Valuation of American Options. Journal of Computational Finance, 9(1), 84-102.

[53] Raymar, S., and M. Zwecher (1997). A Monte Carlo Valuation of American Call Options

On the Maximum of Several Stocks. Journal of Derivatives, 5(1), 7-23.

[54] Rogers, L.C.G. (2002). Monte Carlo Valuation of American Options. Mathematical Fi- nance, 12(3), 271-286.

[55] Rogers, L.C.G. (2010). Dual valuation and hedging of Bermudan options. J. Financial Math., 1(1), 604-608.

[56] Rubinstein, M., and E. Reiner (1991). Breaking down the barriers. RISK, 4, 28-35.

[57] Siegmund, D. (1979). Corrected diﬀusion approximations in certain random walk prob-

lems. Adv. Appl. Prob., 11, 701-719.

[58] Siegmund, D. (1985a). Sequential Analysis: Tests and Conﬁdence Intervals. Springer- Verlag, New York.

[59] Siegmund, D. (1985b). Corrected diﬀusion approximations and their applications. Proc. Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer. Wadsworth, Belmont.

[60] Siegmund, D., and Y.-S. Yuh (1982). Brownian approximations for ﬁrst passage proba-

bilities. Z. Wahrsch. Verw. Gebiete, 59, 239-248.

[61] Tilley, J. (1993). Valuing American Options in a Path Simulation Model. Transactions

of the Society of Actuaries, 45, 83-104.

[62] Tsitsiklis, J., and B. Van Roy (1999). Optimal Stopping of Markov Processes: Hilbert Space Theory, Approximation Algorithms, and an Application to Pricing High- Dimensional Financial Derivatives. IEEE Transactions on Automatic Control, 44, 1840-

1851. 78

[63] Tsitsiklis, J., and B. Van Roy (2001). Regression Methods for Pricing Complex American-Style Options. IEEE Transactions on Neural Networks, 12, 694-703. 79

Appendix A

Appendix for Chapter 2

A.1 Proof of Theorem 1

We only need to show the proof of (2.3.1), (2.3.2) and (2.3.3) here, since the proof of (2.3.4), (2.3.5) and (2.3.6) is essentially the same. We ﬁrst deﬁne some variables and introduce three

Lemmas to facilitate the proof.

−1/3 ζ = 2b − ζ, R = X 0 − b, = (ln m) , b m τ↑ m

−2 0 m1 = m[1 − (ln m) ],Am = {τ↑(X, b) ≤ m1,Rm ≤ m}.

Lemma 2 The ﬁrst passage time for Brownian bridge has the following properties.

(i) For x, y < b, the cumulative probability distribution of τ↑ is

 n −2(b−x)(b−y) o  exp σ2T Φ(g(x, y, t)) + (1 − Φ(h(x, y, t))), t < T, P (τ ≤ t) = 1 (A.1.1) x,y ↑ n −2(b−x)(b−y) o  exp 2 , t = T, σ1 T where 2(b − x) t − (y − x) t − (b − x) b − x − (y − x) t g(x, y, t) := T T , h(x, y, t) := T . q 2 t q 2 t σ1t(1 − T ) σ1t(1 − T )

(ii) For x, y < b, the probability density function of τ↑ is t 2 b − x [b − x − (y − x) T ] fx,y(τ↑ = t) = q exp − 2 t . (A.1.2) 2 3 t 2σ1t(1 − T ) 2πσ1t (1 − T ) (iii) For x < b < y, the following equality holds, 1 1 E = . (A.1.3) x,y 1 − τ /T b−x ↑ 1 − y−x 80

b−x Remarks. Since Ex,y[τ↑] = y−x T , a known property for Brownian bridge, (A.1.3) implies h 1 i 1 that Ex,y = , suggesting that the expectation and function of the ﬁrst 1−τ↑/T 1−Ex,y[τ↑]/T passage time for Brownian bridge are interchangeable in certain cases.

Lemma 3 The ﬁrst passage probability for Brownian bridge has the following properties.

(i) Uniformly in −dm ≤ δ, ζ < b, 1 P 0 (τ 0 ≤ m, R > ) = o(√ ). (A.1.4) δ,ζ ↑ m m m

(ii) Uniformly in −dm ≤ δ < b, −dm ≤ ζ ≤ b − ε for each ε > 0 (ζb = 2b − ζ), 1 P 0 (τ 0 ≤ m) − P 0 (τ 0 ≤ m ,R ≤ ) = o(√ ), (A.1.5) δ,ζ ↑ δ,ζ ↑ 1 m m m

0 0 0 0 1 P (τ ≤ m) − P (τ ≤ m1,Rm ≤ m) = o(√ ). (A.1.6) δ,ζb ↑ δ,ζb ↑ m

Lemma 4 The overshoot Rm for Brownian bridge has the following properties.

(i) Uniformly in −dm ≤ δ, ζ ≤ b − ε for each ε > 0, " # √ 0 Rm βσ1 ∆t 1 E ; Am = + o(√ ), (A.1.7) δ,ζb 1 − τ 0 /m 1 − b−δ m ↑ ζb−δ  !2  R 1 E0 m ; A = O( ). (A.1.8) δ,ζb  0 m 1 − τ↑/m m

(ii) Uniformly in −dm ≤ ζ ≤ b − ε ≤ δ < b or −dm ≤ δ ≤ b − ε ≤ ζ < b for each ε > 0, " # R 1 E0 m ; A = O(√ ). (A.1.9) δ,ζb 0 m 1 − τ↑/m m The proofs for these Lemmas will be shown in Appendix A.2, given them we can proceed with the proof of Theorem 1. Let fn(x) be the normal density function, ( 2 ) 1 (x − µ1n∆t) fn(x) = exp − . p 2 2nσ2∆t 2πnσ1∆t 1 0 The density function of Xn (1 ≤ n ≤ m − 1) is fn(x − δ) under Pδ and fn(x − δ)fm−n(ζ −

0 0 0 x)/fm(ζ − δ) under Pδ,ζ . Therefore, the likelihood ratio of Xn under Pδ,ζ relative to Pδ is

fm−n(ζ − x) l1(n, x) := fm(ζ − δ) ( 2 2 ) 1 (ζ − x − µ1(m − n)∆t) (ζ − δ − µ1m∆t) = p exp − 2 + 2 (A.1.10). 1 − n/m 2(m − n)σ1∆t 2mσ1∆t 81

Similarly, the likelihood ratio of X under P 0 relative to P 0 is n δ,ζb δ

fm−n(ζb − x) l2(n, x) := fm(ζb − δ) ( 2 2 ) 1 (ζb − x − µ1(m − n)∆t) (ζb − δ − µ1m∆t) = p exp − 2 + 2 (A.1.11). 1 − n/m 2(m − n)σ1∆t 2mσ1∆t Since P 0 (·) and P 0 (·) do not depend on µ , we can choose µ = 0, and the likelihood ratio δ,ζ δ,ζb 1 1 of X under P 0 relative to P 0 (1 ≤ n ≤ m − 1) is n δ,ζ δ,ζb −1 fm−n(ζ − x) fm−n(ζb − x) l3(n, x) := fm(ζ − δ) fm(ζb − δ) 2 2 2 2 (ζ − x) (ζb − x) (ζb − δ) (ζ − δ) = exp − 2 + 2 − 2 + 2 2(m − n)σ1∆t 2(m − n)σ1∆t 2mσ1∆t 2mσ1∆t 2(b − x)(b − ζ) 2(b − δ)(ζ − b) = exp 2 + 2 (m − n)σ1∆t mσ1∆t −2(b − ζ) x − b = Pδ,ζ (τ↑ ≤ T ) exp 2 , (A.1.12) σ1T 1 − n/m

n −2(b−δ)(b−ζ) o where Pδ,ζ (τ↑ ≤ T ) = exp 2 by (A.1.1). σ1 T 0 0 0 0 √1 (i) By (A.1.5), we have Pδ,ζ (τ↑ ≤ m) − Pδ,ζ (τ↑ ≤ m1,Rm ≤ m) = o( m ) uniformly in

−dm ≤ δ, ζ ≤ b − ε for each ε > 0, therefore 1 P 0 (τ 0 ≤ m) − P (τ ≤ T ) = P 0 (τ 0 ≤ m ,R ≤ ) + o(√ ) − P (τ ≤ T ) δ,ζ ↑ δ,ζ ↑ δ,ζ ↑ 1 m m m δ,ζ ↑ " ( ) # −2(b − ζ) R 1 = P (τ ≤ T )E0 exp m ; A − P (τ ≤ T ) + o(√ ) δ,ζ ↑ δ,ζb 2 0 m δ,ζ ↑ σ1T 1 − τ↑/m m ( " ( ) # ) −2(b − ζ) R 1 = P (τ ≤ T ) E0 exp m − 1; A − 1 − P 0 (A ) + o(√ ), δ,ζ ↑ δ,ζb 2 0 m δ,ζb m σ1T 1 − τ↑/m m where the second equality follows from (A.1.12). Applying the inequality −x ≤ e−x − 1 ≤

x2 −x + 2 (x ≥ 0) to the ﬁrst term above, we have " # " ( ) # 2(b − ζ) R 2(b − ζ) R − E0 m ; A ≤ E0 exp − m − 1; A 2 δ,ζb 0 m δ,ζb 2 0 m σ1T 1 − τ↑/m σ1T 1 − τ↑/m " #  !2  2(b − ζ) R 2(b − ζ)2 R ≤ − E0 m ; A + E0 m ; A 2 δ,ζb 0 m 4 2 δ,ζb  0 m σ1T 1 − τ↑/m σ1T 1 − τ↑/m √ √ 2(b − ζ) βσ ∆t 1 1 2βσ ∆t(ζ − δ) 1 = − 1 + o(√ ) + C · O( ) = − 1 b + o(√ ), σ2T 1 − b−δ m m σ2T m 1 ζb−δ 1 82

where the ﬁrst equality follows from (A.1.7) and (A.1.8). By (A.1.6) and noting P 0 (τ 0 ≤ δ,ζb ↑

m) = 1 given δ < b < ζb,

0 0 0 0 1 1 − P (Am) = P (τ ≤ m) − P (Am) = o(√ ). δ,ζb δ,ζb ↑ δ,ζb m

Combining the two terms above, we have √ 2βσ ∆t(ζ − δ) 1 P 0 (τ 0 ≤ m) = P (τ ≤ T ) − 1 b P (τ ≤ T ) + o(√ ) δ,ζ ↑ δ,ζ ↑ σ2T δ,ζ ↑ m 1 √ ( ) 2(b − δ)(b − ζ) 2βσ1 ∆t(ζb − δ) 1 = exp − 2 exp − 2 + o(√ ) σ1T σ1T m 2 2(b+∆t − δ)(b+∆t − ζ) 2β 1 = exp − 2 exp + o(√ ) σ1T m m 1 = P (τ (X, b ) ≤ T ) + o(√ ), δ,ζ ↑ +∆t m

where ex = 1 + x + o(x)(x → 0) is applied in the second and last equalities.

(ii) In −dm ≤ ζ ≤ b − ε ≤ δ < b, the above proof of (2.3.1) breaks down because (A.1.5) and (A.1.6) do not apply when δ can be arbitrarily close to b. Given the likelihood ratio (A.1.12), we have " ( ) # −2(b − ζ) R P 0 (τ 0 ≤ m ,R ≤ ) = P (τ ≤ T )E0 exp m ; A . δ,ζ ↑ 1 m m δ,ζ ↑ δ,ζb 2 0 m σ1T 1 − τ↑/m

Applying the inequality 1 ≥ e−x ≥ 1 − x (x ≥ 0) to the equation above, we have

0 0 Pδ,ζ (τ↑ ≤ T ) ≥ Pδ,ζ (τ↑ ≤ m1,Rm ≤ m) " #! 2(b − ζ) R ≥ P (τ ≤ T ) 1 − E0 m ; A δ,ζ ↑ 2 δ,ζb 0 m σ1T 1 − τ↑/m " # 2(b − ζ) R ≥ P (τ ≤ T ) − E0 m ; A δ,ζ ↑ 2 δ,ζb 0 m σ1T 1 − τ↑/m 1 = P (τ ≤ T ) + (ζ − b) · O(√ ), (A.1.13) δ,ζ ↑ m 83 where the last equality follows from (A.1.9). Finally, with an application of (A.1.13), (A.2.1) and (A.1.4),

0 0 0 0 0 0 0 0 Pδ,ζ (τ↑ ≤ m) = Pδ,ζ (τ↑ ≤ m1,Rm ≤ m) + Pδ,ζ (m1 < τ↑ ≤ m, Rm ≤ m) + Pδ,ζ (τ↑ ≤ m, Rm > m) 1 1 1 = P (τ ≤ T ) + (ζ − b) · O(√ ) + o(√ ) + o(√ ) δ,ζ ↑ m m m 1 = P (τ ≤ T ) + (ζ − b) · O(√ ) δ,ζ ↑ m 1 = P (τ (X, b ) ≤ T ) + (ζ − b) · O(√ ), δ,ζ ↑ +∆t m where the last equality holds with an expansion of Pδ,ζ (τ↑ ≤ T ).

(iii) In −dm ≤ δ ≤ b − ε ≤ ζ < b, i.e., ζ can be arbitrarily close to b, the result follows ←− the similar proof for case (ii) above by applying a time-reversal argument. Let X be the time-reversal process of X that starts from X(T ) at time T and reverses the path of X,

n ←− ←− X √ X (t) := X(T ) − µ1t − σ1(B(T ) − B(T − t)), X n := Xm + −µ1∆t − σ1 ∆tZm−i . i=1

←− 2 X (t) is eﬀectively a Brownian motion starting from T with (−µ1, σ1). Since δ, ζ < b, we have ←− Pδ,ζ (τ↑ ≤ T ) = Pζ,δ(τ↑(X , b) ≤ T ) = Pζ,δ(τ↑ ≤ T ), where the second equality holds because Pδ,ζ does not depend on µ1. Similarly,

0 0 0 0 ←− 0 0 Pδ,ζ (τ↑ ≤ m) = Pζ,δ(τ↑(X , b) ≤ m) = Pζ,δ(τ↑ ≤ m).

Therefore with an application of case (ii), in −dm ≤ δ ≤ b − ε ≤ ζ < b,

1 P 0 (τ 0 ≤ m) = P 0 (τ 0 ≤ m) = P (τ (X, b ) ≤ T ) + (δ − b) · O(√ ) δ,ζ ↑ ζ,δ ↑ ζ,δ ↑ +∆t m 1 = P (τ (X, b ) ≤ T ) + (δ − b) · O(√ ). δ,ζ ↑ +∆t m 84 A.2 Proof of Lemma 2, 3 and 4

A.2.1 Proof of Lemma 2

(i) By Theorem 2.1 in Beghin and Orsingher (1999), the cumulative distribution function of

τ↑ for a unit-variance Brownian bridge starting at 0 and ending at y is

2b t −y t −b Z √T T −y2/2 Z ∞ −y2/2 −2b(b − y) t(1− t ) e e T P0,y(τ↑ ≤ t) = exp √ dy + √ dy. b−y t T −∞ 2π √ T 2π t(1− t ) T 2 Therefore for a generic Brownian bridge starting at x with variance σ1, −2(b − x)(b − y) Z g(x,y,t) e−y2/2 Z ∞ e−y2/2 Px,y(τ↑ ≤ t) = exp 2 √ dy + √ dy σ1T −∞ 2π h(x,y,t) 2π −2(b − x)(b − y) = exp 2 Φ(g(x, y, t)) + (1 − Φ(h(x, y, t))). σ1T Since Φ(g(x, y, T )) = 1 = Φ(h(x, y, T )), we have

−2(b − x)(b − y) Px,y(τ↑ ≤ T ) = exp 2 . σ1T

(ii) The probability density function of τ↑ can be derived by taking the partial derivative of the cumulative probability above,

∂ f (τ = t) = P (τ ≤ t) x,y ↑ ∂t x,y ↑   1 1 −2(b − x)(b − y) 2(b − x) T − (y − x) T g(x, y, t) 2 2t = exp 2 φ (g(x, y, t))  q − 2 t σ1(1 − ) σ1T 2 t 2σ1t(1 − T ) T σ1t(1 − T )   1 (y − x) T h(x, y, t) 2 2t +φ(h(x, y, t)) q + 2 t σ1(1 − ) 2 t 2σ1t(1 − T ) T σ1t(1 − T )   1 2(b − x) T h(x, y, t) − g(x, y, t) 2t = φ(h(x, y, t)) q + t (1 − ) 2 t 2t(1 − T ) T σ1t(1 − T ) 2σ2(b − x)(1 − t ) b − x h(x, y, t)2 = φ(h(x, y, t)) 1 T = exp − , t 3 q 2 2 2 t 2 2(σ1t(1 − )) 3 T 2πσ1t (1 − T ) where the second equality holds because

t 2 t 2 φ(g(x, y, t)) [(2b − x − y) T − (b − x)] − [b − x − (y − x) T ] 2(b − x)(b − y) = exp − 2 t = exp 2 . φ(h(x, y, t)) 2σ1t(1 − T ) σ1T 85

(iii) For x < b < y, 1 Z T t Ex,y = fx,y(τ↑ = t) 1 + dt 1 − τ↑/T 0 T − t Z T b − x [b − x − (y − x)t/T ]2 t = 1 + exp − dt p 2 3 2 0 2πσ1t (T − t)/T 2σ1t(T − t)/T T − t b − x Z T y − b [y − b − (y − x)(T − t)/T ]2 = 1 + exp − d(T − t) p 2 3 2 y − b 0 2πσ1t(T − t) /T 2σ1t(T − t)/T b − x Z T 1 = 1 + f (τ = T − t)d(T − t) = , y − b y,x ↓ b−x 0 1 − y−x R T where 0 fy,x(τ↓ = T − t)d(T − t) = 1 because x < b < y and τ↓ ∈ (0,T ). 2

A.2.2 Proof of Lemma 3

(i) In −dm ≤ δ, ζ < b, the conditional joint probability in question is bounded by a sum of unconditional normal probability terms below, m 0 0 X 0 0 Pδ,ζ (τ↑ ≤ m, Rm > m) = Pδ,ζ (τ↑ = n, Rm > m) n=1 m √ X √ ≤ P (σ1 ∆tZ1 + µ1∆t > b − δ + m) + P (σ1 ∆tZn + µ1∆t > m) √ n=2 ≤ mP (σ1 ∆tZ1 + µ1∆t > m), where µ1 = (ζ − δ)/T is the drift of the Brownian bridge, the ﬁrst inequality follows the

Brownian bridge property and the second inequality holds because −dm ≤ δ, ζ < b. Plugging

−1/3 in the probability function of Z1 and noting m = (ln m) 1/m, for a large m we have 0 0 −m + µ1∆t −m Pδ,ζ (τ↑ ≤ m, Rm > m) ≤ mΦ √ ≤ mΦ √ σ1 ∆t 2σ1 ∆t 4σ2∆t 2 1 1 m 2/3 −2/3 √ ≤ m · C · 2 exp − 2 = C · (ln m) exp{−C · m(ln m) } = o( ), m 8σ1∆t m where the last inequality holds because Φ(−x) ≤ C · x−2e−x2/2 (x > 0). (ii) The probability diﬀerential in question is bounded by the sum of two probability terms,

0 0 0 0 0 ≤ Pδ,ζ (τ↑ ≤ m) − Pδ,ζ (τ↑ ≤ m1,Rm ≤ m)

0 0 0 0 = Pδ,ζ (m1 < τ↑ ≤ m, Rm ≤ m) + Pδ,ζ (τ↑ ≤ m, Rm > m)

0 0 0 0 ≤ Pδ,ζ (m1 < τ↑ ≤ m) + Pδ,ζ (τ↑ ≤ m, Rm > m). 86

In −dm ≤ δ < b, −dm ≤ ζ ≤ b − ε, the ﬁrst probability term above is

m 0 0 X 0 0 0 0 Pδ,ζ (m1 < τ↑ ≤ m) = Pδ,ζ (τ↑ = n) ≤ (m − m1) sup Pδ,ζ (τ↑ = n). m1

2 ! 0 0 −ε − µ1T/(ln m) P (τ = n) ≤ P (Xm − Xm < −ε) = Φ = Φ(−C · (ln m)), δ,ζ ↑ 1 p 2 2 σ1T/(ln m)

where the inequality follows the Brownian bridge property. Applying the inequality Φ(−x) ≤

−2 −x2/2 −2 C · x e (x > 0) again and noting m1 = m[1 − (ln m) ],

m m 1 P 0 (m < τ 0 ≤ m) ≤ Φ(−C · (ln m)) ≤ C · · exp{−(ln m)2} = o(√ (A.2.1)). δ,ζ 1 ↑ (ln m)2 (ln m)4 m

0 0 √1 By (A.1.4), Pδ,ζ (τ↑ ≤ m, Rm > m) = o( m ) uniformly in −dm ≤ δ, ζ < b, therefore (A.1.5) follows, 1 P 0 (τ 0 ≤ m) − P 0 (τ 0 ≤ m ,R ≤ ) = o(√ ). δ,ζ ↑ δ,ζ ↑ 1 m m m

(A.1.6) follows a similar proof as (A.1.5), and P 0 (m < τ 0 ≤ m) = o( √1 ) given δ,ζb 1 ↑ m

ζb > b + ε. 2

A.2.3 Proof of Lemma 4

(i) In −dm ≤ δ, ζ ≤ b − ε, we can choose µ1 = (ζb − δ)/T and the likelihood ratio of Xn under P 0 relative to P 0 in (A.1.11) (τ 0 = n) is δ,ζb δ ↑

( 0 2 ) fm−τ 0 (ζb − Xτ 0 ) 0 ↑ ↑ 1 ζb − b − Rm − (ζb − δ)(1 − τ↑/m) l2(τ ,X 0 ) = = exp − . ↑ τ↑ q 2 0 fm(ζb − δ) 0 2σ1T (1 − τ↑/m) 1 − τ↑/m

Thus, we have

" # " # " 0 # l2(τ ,Xτ 0 ) 0 Rm 0 0 Rm 0 ↑ ↑ E ; A = E l (τ ,X 0 ) ; A = E R ; A . δ,ζb 0 m δ 2 ↑ τ↑ 0 m δ 0 m m 1 − τ↑/m 1 − τ↑/m 1 − τ↑/m 87

To show the uniform integrability, a key step is to recognize the uniform boundedness of 0 l2(τ↑,Xτ0 ) ↑ 0 in ζ ≤ b − ε, (1−τ↑/m)

0 ( 2 ) l (τ ,X 0 ) 0 2 ↑ τ↑ 1 ζb − b − Rm − (ζb − δ)(1 − τ↑/m) 0 = 3 exp − 2 0 ≤ C, (A.2.2) 1 − τ /m 0 2 2σ T (1 − τ /m) ↑ (1 − τ↑/m) 1 ↑

which holds because ζb − b − Rm = b − ζ − Rm ≥ ε − m ≥ ε/2 (for a large m) and

0 l (τ ,X 0 ) 2 ↑ τ↑ 0 ≤ sup e(x, y) ≤ C (0 ≤ x ≤ 1), 1 − τ↑/m y≥ε/2

where

2 1 [y − (ζb − δ)(1 − x)] e(x, y) := 3 exp − 2 , (1 − x) 2 2σ1T (1 − x) 2 [y − (ζb − δ)] lim e(x, y) = exp − 2 , lim e(x, y) = 0. x→0 2σ1T x→1

Lemma 10.11 in Siegmund (1985a) shows the convergence of all moments of Rm under

0 Pδ, √ 0 √ 0 2 Eδ mRm = βσ1 T + o(1),Eδ mRm = O(1). (A.2.3)

Lorden (1970) shows that all moments of Rm are uniformly bounded,

0 √ 0 2 sup Eδ mRm ≤ C1, sup Eδ mRm ≤ C2. (A.2.4) δ

1 Since Rm and 0 are asymptotically independent (similar to the proof of Proposition 4.4 1−τ↑/m in Asmussen & Albrecher 2010), we have " # √ R 1 1 βσ ∆t 1 E0 m ; A = E0 [R ]E + o(√ ) = 1 + o(√ ), δ,ζb 0 m δ m δ,ζb b−δ 1 − τ /m 1 − τ↑/T m 1 − m ↑ ζb−δ uniformly in −dm ≤ δ, ζ ≤ b − ε for each ε > 0, where the second equality follows from (A.2.3) and (A.1.3). (A.1.8) follows directly from the uniform boundedness in (A.2.2),

 !2  " 0 # R l2(τ↑,Xτ 0 ) 1 E0 m ; A = E0 ↑ R2 ; A ≤ C · E0 [R2 ] = O( ), δ,ζb  0 m δ 0 2 m m δ m 1 − τ↑/m (1 − τ↑/m) m 88

0 l2(τ↑,Xτ0 ) ↑ where the inequality holds because of the uniform boundedness of 0 2 (by a similar (1−τ↑/m) argument as for (A.2.2)) and the last equality results from (A.2.4).

(ii) In −dm ≤ ζ ≤ b − ε ≤ δ < b, since the conditional density does not depend on µ1, we can again let µ1 = (ζb − δ)/T .

" # " 0 # R l2(τ↑,Xτ 0 ) 1 E0 m ; A = E0 ↑ R ; A ≤ C · E0 [R ] = O(√ ), δ,ζb 0 m δ 0 m m δ m 1 − τ↑/m 1 − τ↑/m m

where the inequality follows (A.2.2) and the last equality results from (A.2.4). In −dm ≤ δ ≤ b − ε ≤ ζ < b, the result follows the same proof as above with a time-reversal argument. 2 89

Appendix B

Appendix for Chapter 3

B.1 Regression related issues

B.1.1 ‘Regression now’ vs. ‘regression later’

As a comparison with the standard least-squares estimator, we implement the ‘regression later’ technique (Broadie, Glasserman and Ha 2000, Glasserman and Yu 2002) on single asset Bermudan options and Bermudan max options. The ‘regression later’ approach requires the

2 use of martingale basis functions, thus we choose St, St and Et (European option value at time t) as the basis functions for single asset Bermudan options and the ﬁrst two powers of

2 each asset price {Si,t,Si,t} for Bermudan max options. The corresponding martingale basis

2 −(r−q)t −2(r−q)t−σ2t 2 −rt functions for St, St and Et are e St, e St and e Et. As shown in Tables B.1 and B.2, ‘regression later’ approach generates more accurate estimates than ‘regression now’ in most cases, especially when fewer paths are used for regression and the exercise policy is far from optimal. However, when the number of regression paths are suﬃciently large, ‘regression later’ does not lead to an observable improvement because the regression estimates approach their convergence limit and there is no room for additional improvement by ‘regression later’. Note also that, the requirement of martingale basis functions limits the use of this algorithm. 90

Table B.1: Regression now vs. regression later (single asset Bermudan call)

S0 = 90 S0 = 100 S0 = 110 ˆN ˆL ˆ ˆN ˆL ˆ ˆN ˆL ˆ NR L0 L0 ∆L0 L0 L0 ∆L0 L0 L0 ∆L0 10000 2.3759 2.3786 0.0027 5.8938 5.9017 0.0079 11.7141 11.7263 0.0122 100000 2.3832 2.3835 0.0003 5.9149 5.9154 0.0005 11.7442 11.7447 0.0005

Note: Option parameters are the same as in Table 3.1 except the number of basis functions b = 4 here.

Table B.2: Regression now vs. regression later (5-asset asymmetric Bermudan max call) S0 = 90 S0 = 100 S0 = 110 ˆN ˆL ˆ ˆN ˆL ˆ ˆN ˆL ˆ NR L0 L0 ∆L0 L0 L0 ∆L0 L0 L0 ∆L0 2000 27.268 27.310 0.042 37.473 37.489 0.016 48.817 48.858 0.041 200000 27.315 27.318 0.003 37.494 37.494 0.000 48.846 48.833 -0.013

Note: Option parameters are the same as in Table 3.4 except the number of basis functions b = 11 here.

B.1.2 Choice of basis functions

The choice of basis functions is critical for the regression-based algorithms. We list here the basis functions we use for our examples. Note that the constant c is counted as one of the basis functions.

2 For the single asset Bermudan options, we use 3 basis functions {c, Et,Et } where Et is the value of the European option at time t. The 6 basis functions used for the moving window Asian options include the polynomials

2 2 of St and At up to the second order {c, St, St , At, At , StAt}. Polynomials of sorted asset prices are used for the symmetric Bermudan max options.

0 Let Si,t be the i-th highest asset price at t. The 18 basis functions include the polynomials

0 02 0 02 0 0 03 03 0 02 0 0 0 0 04 05 02 0 up to the ﬁfth order: {c, S1, S1 , S2, S2 , S1S2, S1 , S2 , S3, S3 , S1S3, S2S3, S1 , S1 , S1 S2,

0 02 0 0 0 0 0 S1S2 , S1S2S3, S4S5}; 12-case includes the polynomials of three highest asset prices, i.e., the ﬁrst twelve in the list above; 6-case includes polynomials of two highest asset prices,

i.e., the ﬁrst six in the list above; 19-case is the same as in Longstaﬀ and Schwartz (2001),

0 02 03 04 05 0 02 0 02 0 02 0 02 0 0 0 0 0 0 0 0 which are {c, S1, S1 , S1 , S1 , S1 , S2, S2 , S3, S3 , S4, S4 , S5, S5 , S1S2, S2S3, S3S4, S4S5, 91

0 0 0 0 0 S1S2S3S4S5}. The 18 asset-distinguishing basis functions for the asymmetric Bermudan max options

0 02 0 02 0 0 03 2 2 2 2 2 03 are {c, S1, S1 , S2, S2 , S1S2, S1, S2, S3, S4, S5, S1 , S1 , S2 , S3 , S4 , S5 , S2 }; 12-case includes the ﬁrst twelve in the list above. For the comparisons between ‘regression now’ and ‘regression later,’ 4 basis functions

2 {c, St,St ,Et} are used for the single asset Bermudan options, 11 basis functions {c, S1,t,

2 2 2 2 2 S2,t, S3,t, S4,t, S5,t, S1,t, S2,t, S3,t, S4,t, S5,t} are used for the asymmetric Bermudan max options.

B.2 Proof of Proposition 1 and 2

In this section we give proofs of Propositions 1 and 2 in Section 3.4.

Proposition 1

Ltk htk (i) If hti ≤ Qti for 1 ≤ i ≤ k, then πtk = and − πtk ≤ 0. Btk Btk

Q Q tl−1 Ltk htk tl−1 (ii) If hti ≤ Qti for l ≤ i ≤ k, then πtk = πtl−1 − + and − πtk ≤ − πtl−1 . Btl−1 Btk Btk Btl−1

Proof.

(i) If hti ≤ Qti for 1 ≤ i ≤ k, then 1ti = 0, and

Ltk Ltk−1 πtk = πtk−1 + − Btk Btk−1

Ltk−1 Ltk−2 Ltk Ltk−1 = πtk−2 + − + − Btk−1 Btk−2 Btk Btk−1 k X Ltj Ltj−1 Lt = π + − = k , 0 B B B j=1 tj tj−1 tk

therefore,

Qt htk k Ltk − πtk ≤ − ≤ 0. Btk Btk Btk 92

Notice that the last inequality holds for instance if Qt is a Q-sub-martingale, which is valid with the choice of one European option value but invalid with the choice of

maximum among multiple option values.

(ii) If hti ≤ Qti for l ≤ i ≤ k

Ltk Ltk−1 πtk = πtk−1 + − Btk Btk−1

Ltk−1 Ltk−2 Ltk Ltk−1 = πtk−2 + − + − Btk−1 Btk−2 Btk Btk−1

Qtl−1 Ltk = πtl−1 − + . Btl−1 Btk

and

htk htk Qtl−1 Ltk Qtl−1 − πtk = − πtl−1 + − ≤ − πtl−1 .2 Btk Btk Btl−1 Btk Btl−1

Proposition 2 For a given sample path,

˜ (i) If ∃δ > 0 such that |Qt − Qt| < δ, and dt ≥ δ or ht ≤ Qt holds for ∀t ∈ Γ, then At is an increasing process and D = 0 for the path.

˜ ∗ (ii) If ∃δ > 0 such that |Qt − Qt | < δ, and dt ≥ δ or ht ≤ Qt holds for ∀t ∈ Γ, then

∗ 1t ≡ 1t .

Proof.

(i) For ∀ti ∈ Γ, if hti ≤ Qti , 1ti = 0 and Ati+1 − Ati = 0; otherwise if dti ≥ δ, ˜ (a) 1ti = 0, Ati+1 − Ati = 0 and hti − Qti < −δ,

˜ ˜ hti − Lti = hti − Qti = (hti − Qti ) + (Qti − Qti ) < −δ + δ = 0; 93

˜ (b) 1ti = 1, hti = Lti and hti − Qti > δ, hti Lti+1 Ati+1 − Ati = − Eti Bti Bti+1 1 = (hti − Qti ) Bti 1 ˜ ˜ = [(hti − Qti ) + (Qti − Qti )] Bti 1 > (δ − δ) = 0. Bti Finally, ht ht Lt D = max − πt = max − − At = 0. t Bt t Bt Bt

∗ ˜ ∗ (ii) For ∀t ∈ Γ, if ht ≤ Qt, 1t = 0 = 1t ; otherwise if dt ≥ δ, since |Qt − Qt | < δ, ∗ ˜ ∗ (a) 1t = 0, Qt > Qt − δ ≥ ht + δ − δ = ht and thus 1t = 0; ∗ ˜ ∗ (b) 1t = 1, Qt < Qt + δ ≤ ht − δ + δ = ht and thus 1t = 1.2

As we noted in the introduction, D may be interpreted as a penalty term for incorrect decisions. An incorrect continuation decision at t will be penalized and cause ht − π to be Bt t positive if the path never enters the exercise region before t. To see this, assume an incorrect continuation decision is made at ti for a sample path that has not been in the exercise region L ti ∗ ˜ before ti. By Proposition 1, πti = and Lti ≤ Qt < hti < Qti , Bti i Q∗ hti hti Lti hti ti − πti = − ≥ − > 0. Bti Bti Bti Bti Bti On the other hand, an incorrect exercise decision at t will only get penalized (i.e., D > 0 for the path) either when the option leaves and re-enters the exercise region or the option never comes back into the exercise region and matures OTM. For example, suppose an

incorrect exercise decision is made at tk, and tk is the ﬁrst time that the sample path enters the exercise region. Assume the next time that the sample path enters the exercise region is

at time tl, i.e., l = infi>k{i : 1ti = 1} ∧ n. By Proposition 1,

Qtk htl Ltk Qtk htl πtl = πtk − + = − + , Btk Btl Btk Btk Btl 94

thus Q∗ htl Qtk Ltk tk Ltk htk Ltk − πtl = − ≈ − > − = 0, Btl Btk Btk Btk Btk Btk Btk assuming the sub-optimal continuation value is a good approximate of the true continuation value.

B.3 Numerical details in boundary distance grouping

B.3.1 Estimators of the duality gap

N P H D ¯ i=1 i The standard estimator of the duality gap D0 is D = , and the alternative estimator NH using boundary distance grouping is

PnZ¯ NH −nZ¯ PnZ¯ +lZ i=1 Di + i=n +1 Di D˜ = lZ Z¯ . NH Assume ¯ ¯ 2 E[Di|Z] = µZ¯, Var[Di|Z] = σZ¯,

2 E[Di|Z] = µZ , Var[Di|Z] = σZ . ¯ Since each path belongs to group Z with probability pZ¯ and group Z with probability 1−pZ¯, nZ¯ is a binomial random variable with parameters (NH , pZ¯). We have E[nZ¯] = NH pZ¯, ˜ Var[nZ¯] = NH pZ¯(1 − pZ¯). By Proposition 2, if the Qt almost surely lies within δ0 from the sub-optimal continuation value, we can choose δ = δ0 so that D = 0 for almost all paths in group Z, thus µZ and σZ are both close to zero. The variance of the standard estimator is,

¯ ¯ ¯ Var[D] = E[Var(D|nZ¯)] + Var(E[D|nZ¯]) 2 2 nZ¯σZ¯ + (NH − nZ¯)σZ nZ¯µZ¯ + (NH − nZ¯)µZ = E 2 + V ar NH NH 2 1 2 1 2 2 (µZ¯ − µZ ) V ar(nZ¯) = σZ + pZ¯(σZ¯ − σZ ) + 2 NH NH NH 1 2 2 2 2 = [σZ + pZ¯(σZ¯ − σZ ) + pZ¯(1 − pZ¯)(µZ¯ − µZ ) ] NH 1 2 2 2 = [(1 − pZ¯)σZ + pZ¯σZ¯ + pZ¯(1 − pZ¯)(µZ¯ − µZ ) ] NH σ2 = D . NH 95

It is easy to verify the alternative estimator is unbiased, ˜ ˜ nZ¯ NH − nZ¯ E[D] = E[E[D|nZ¯]] = E µZ¯ + µZ = pZ¯µZ¯ + (1 − pZ¯)µZ = µ, NH NH

and the variance is

˜ ˜ ˜ Var[D] = E[Var(D|nZ¯)] + Var(E[D|nZ¯])    nZ¯ 2 nZ¯ +lZ 1 X (NH − nZ¯) X nZ¯ NH − nZ¯ = E  2 Var(Di) + 2 2 V ar  Di + V ar µZ¯ + µZ NH lZ NH NH NH i=1 i=nZ¯ +1 2 2 1 2 (NH − nZ¯) 2 (µZ¯ − µZ ) = E 2 nZ¯σZ¯ + 2 2 lZ σZ + 2 Var(nZ¯) NH lZ NH NH 2 2 2 pZ¯σZ¯ σZ 2 2 (µZ¯ − µZ ) = + 2 E[NH − 2nZ¯NH + nZ¯] + pZ¯(1 − pZ¯) NH lZ NH NH 1 2 2 1 2 1 2 1 2 = (1 − pZ¯) σZ + pZ¯(1 − pZ¯)σZ + pZ¯σZ¯ + pZ¯(1 − pZ¯)(µZ¯ − µZ ) . lZ lZ NH NH NH

For any choice of lZ ≤ NH (1 − pZ¯), where NH (1 − pZ¯) is the expected number of paths in Z,

˜ 1 2 2 1 2 1 2 Var[D] > (1 − pZ¯) σZ + pZ¯σZ¯ + pZ¯(1 − pZ¯)(µZ¯ − µZ ) lZ NH NH 1 2 2 2 ≥ [(1 − pZ¯)σZ + pZ¯σZ¯ + pZ¯(1 − pZ¯)(µZ¯ − µZ ) ] NH σ2 = D = Var[D¯], NH

which means the variance of D˜ is always greater than that of D¯. The ‘=’ sign in the second ¯ ˜ inequality holds when lZ = NH (1 − pZ¯), in which case Var[D] and Var[D] are only diﬀerent

1 2 by a small term 2 pZ¯σZ . The diﬀerence is due to the randomness in the number of paths NH in group Z¯ and Z. Generally,

2 ˜ σZ 1 2 2 2 Var[D] < + [(1 − pZ¯)σZ + pZ¯σZ¯ + pZ¯(1 − pZ¯)(µZ¯ − µZ ) ] lZ NH σ2 σ2 = D + Z , NH lZ

˜ 2 2 2 ¯ ¯ 2 i.e., Var[D] falls into [σD/NH , σD/NH +σZ /lZ ], i.e., [Var[D], Var[D]+σZ /lZ ], which is a tight

2 2 interval if σZ lZ σD/NH . 96

B.3.2 Eﬀective saving factor

The effective saving factor for boundary distance grouping can be calculated as the ratio of the efficiency before and after improvement, where the efficiency of simulation is measured by the product of sample variance and computational time. As denoted in Section 3.4, TP is the expected time spent for generating one sample path, TI is the expected time to identify

which group the path belongs to, TDZ¯ and TDZ are the expected time to estimate upper bound increment D from a group Z¯ path and from a group Z path respectively, typically

TP ,TI TDZ¯ ,TDZ . The total expected time for estimating D¯ is,

¯ ¯ ¯ ¯ ¯ TD ≈ NH TP + NH pZ TDZ¯ + NH (1 − pZ )TDZ = NH [TP + pZ TDZ¯ + (1 − pZ )TDZ ], and for D˜,

¯ ¯ TD˜ ≈ NH TP + NH TI + pZ NH TDZ¯ + lZ TDZ = NH (TP + TI + pZ TDZ¯ ) + lZ TDZ .

2 2 For a fixed boundary distance threshold δ > 0, parameters pZ¯, µZ¯, µZ , σZ¯, σZ can be estimated from simulation. We may maximize the effective saving factor with respect to lZ (the number of paths selected from group Z to estimate D) for a fixed δ and find the optimal δ0 from a pre-selected set of δ choices,

0 ˜ δ := arg(min Var[D] · TD˜ ). (B.3.1) δ

The variance and eﬃciency measure for D˜ are

˜ 1 2 1 1 2 1 2 Var[D] = (1 − pZ¯)σZ [(1 − pZ¯) + pZ¯] + pZ¯σZ¯ + pZ¯(1 − pZ¯)(µZ¯ − µZ ) lZ NH NH NH 1 2 2 1 2 1 2 ≈ (1 − pZ¯) σZ + pZ¯σZ¯ + pZ¯(1 − pZ¯)(µZ¯ − µZ ) , lZ NH NH 1 2 2 ˜ ¯ ¯ Var[D] · TD˜ = NH (1 − pZ ) σZ (TP + TI + pZ TDZ¯ ) lZ 2 2 p ¯σ T Z Z¯ DZ pZ¯(1 − pZ¯)(µZ¯ − µZ ) TDZ +lZ + lZ + constant. NH NH 97

Take the partial derivative with respect to lZ , ˜ ∂(Var[D] · TD˜ ) | 0 lZ =lZ ∂lZ 2 2 p ¯σ T 1 2 2 Z Z¯ DZ pZ¯(1 − pZ¯)(µZ¯ − µZ ) TDZ = − N (1 − p ¯) σ (T + T + p ¯T ) + + , 02 H Z Z P I Z DZ¯ lZ NH NH

thus the product achieves its minimum at s 2 2 ¯ 0 (1 − pZ¯) σZ TP + TI + pZ TDZ¯ lZ = · 2 NH = γNH , ¯ ¯ ¯ 2 pZ TDZ σZ¯ + (1 − pZ )(µZ − µZ ) where γ denotes the portion of the sample paths that we should choose to estimate the group

Z average. As

2 ˜ ∂ (Var[D] · TD˜ ) 2 2 2 = N (1 − p ¯) σ (T + T + p ¯T ) > 0, 2 3 H Z Z P I Z DZ¯ ∂lZ lZ

0 the function is strictly convex and lZ is the unique minimum. The eﬀective saving factor of boundary distance grouping can be calculated as, ¯ Var[D] · TD¯ ESF = |l =l0 ˜ Z Z Var[D] · TD˜ 2 2 2 (1 − p ¯)σ + p ¯σ + p ¯(1 − p ¯)(µ ¯ − µ ) T + p ¯T + (1 − p ¯)T Z Z Z Z¯ Z Z Z Z P Z DZ¯ Z DZ = 2 · . (1−pZ¯ ) 2 2 2 TP + TI + pZ¯TD ¯ + γTD γ σZ + pZ¯σZ¯ + pZ¯(1 − pZ¯)(µZ¯ − µZ ) Z Z

¯ ¯ Assume pZ ≈ 0, µZ ≈ 0 and pZ TDZ¯ TP + TI , then

s 2 σZ TDZ¯ γ ≈ 2 2 . (σZ¯ + µZ¯)TDZ

l0 σ2 If in addition we have σ2 Z D which leads to Var[D¯] ≈ Var[D˜], the eﬀective saving is Z NH essentially the saving of time spent for estimating D,

p ¯T + (1 − p ¯)T ESF ≈ Z DZ¯ Z DZ . ¯ pZ TDZ¯ + γTDZ If the boundary distance can eﬀectively identify the paths with zero upper bound incre-

ment, group Z will have approximately zero mean and variance, thus γ ≈ 0 and

T ESF ≈ 1 + DZ . ¯ pZ TDZ¯ 98

B.3.3 Two ways to estimate Var[D˜]

˜ We can not directly estimate the variance of the alternative estimator D from Dis because they are not i.i.d. random variables after grouping, in this section we show two indirect ways

to estimate it. One is through the batching procedure, in which we estimate the batch mean D˜ (j) with

NH boundary distance grouping from k independent batches (j = 1, ..., k), each using b k c simulation paths. D˜ (j) are i.i.d. estimates of D˜ and the sample variance can be calculated by

Pk D˜ (j) Pk (D˜ (j) − j=1 )2 sˆ2 = j=1 k , D k − 1 which is an unbiased estimator of Var[D˜]. The other alternative is to use a modiﬁed sample variance to approximate it. Similar to the regular sample variance, a modiﬁed sample variance estimators ˆ can be constructed as below,   nZ¯ nZ¯ +lZ ˆ 1 X ˜ 2 NH − nZ¯ X ˜ 2 θ =  (Di − D) + (Di − D)  , NH (NH − 1) lZ i=1 i=nZ¯ +1

or equivalently,   nZ¯ nZ¯ +lZ ˆ 1 X 2 NH − nZ¯ X 2 ˜ 2 θ =  Di + Di − NH D  . NH (NH − 1) lZ i=1 i=nZ¯ +1 99

The expectation of θˆ is

N −n ¯ n ¯ +l "PnZ¯ 2 2 # " H Z P Z Z 2 ˜ ˜ 2 # (D − 2D D˜ + D˜ ) i=n +1(Di − 2DiD + D ) E[θˆ] = E i=1 i i + E lZ Z¯ NH (NH − 1) NH (NH − 1)

" 2 2 2 # " 2 2 2 # n ¯(µ + σ ) − 2N D˜ (N − n ¯)(µ + σ ) + N D˜ = E Z Z¯ Z¯ H + E H Z Z Z H NH (NH − 1) NH (NH − 1)

2 2 2 2 2 " 2 2 # n ¯(µ + σ ) + (N − n ¯)(µ + σ ) − N D¯ D˜ − D¯ = E Z Z¯ Z¯ H Z Z Z H − E NH (NH − 1) NH − 1 " # D˜ 2 − D¯ 2 = Var[D¯] − E NH − 1 " # N (D˜ 2 − D¯ 2) = Var[D˜] − E H NH − 1 2 σ (N + p ¯ − l )(1 − p ¯) = Var[D˜] − Z H Z Z Z lZ NH − 1 2 ˜ σZ ≥ Var[D] − (1 − pZ¯) lZ σ2 ≥ Var[D˜] − Z . lZ

Thus our new estimator can be constructed as

2 2 ˆ σˆZ sˆD˜ = θ + lZ PnZ¯ +lZ n n +l n +l i=n +1 Di P Z¯ ˜ 2 NH −nZ¯ P Z¯ Z ˜ 2 P Z¯ Z Z¯ 2 i=1(Di − D) + i=n +1(Di − D) i=n +1(Di − ) = lZ Z¯ + Z¯ lZ , NH (NH − 1) lZ (lZ − 1) whose expectation is

2 2 2 ˆ σZ ˜ ˜ σZ E[ˆsD˜ ] = E[θ] + ∈ Var[D], Var[D] + . lZ lZ

Although the modiﬁed sample variance is not an unbiased estimator of the true variance,

h σ2 σ2 σ2 i it closely bounds the true variance from above. Since Var[D˜] ∈ D , D + Z , we have NH NH lZ h σ2 σ2 σ2 i l σ2 E[ˆs2 ] ∈ D , D + 2 Z , which is a tight interval if σ2 Z D . D˜ NH NH lZ Z NH