Lyuu, Yuh-Dauh - Department of Computer Science and Information Engineering, National Taiwan University

• Computation Information
• Decidability and Recursive Languages Let L ( -{ })
• Reductions and Completeness c 2009 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 205
• Theory of Computation Final Examination on June 19, 2008
• Reduction of reachability to circuit value Note that both problems are in P.
• c2010 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 537 When Professors Scholes and Merton and I
• Interest Rate Barrier Options Pricing
• Theory of Computation Mid-Term Examination on November 10, 2009
• American Call Pricing in One Period Have to consider immediate exercise.
• What Is a Proof? A proof convinces a party of a certain claim.
• Undecidability c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 103
• Theory of Computation Mid-Term Examination on November 11, 2008
• The Extended CIR Model In the extended CIR model the short rate follows
• A General Analytic Formula for Path-Dependent Options
• Pseudo-Polynomial-Time pseudo-polynomial-time
• NP-Complete nae-satis able.
• (concluded) computation
• (continued) (continued)
• Option Pricing Models c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 188
• max cut Revisited The NP-complete max cut seeks to partition the nodes
• Spot and Forward Rates under Continuous Compounding
• reachability requirement
• BPP's Circuit Complexity Theorem 74 (Adleman (1978)) All languages in BPP
• Theory of Computation Solutions to Homework 4
• Locking in the Forward Rate f(n, m) Buy one n-period zero-coupon bond for 1/(1 + S(n))n
• A Quantum Cryptosystem With Perfect Secrecy and Message
• Theory of Computation Solutions to Homework 1
• The finite-difference formula for gamma is P(S + ) -2 P(S) + P(S -)
• Maximum Satisfiability Given a set of clauses, maxsat seeks the truth
• Logarithmic reachability
• composite''; probability
• exponentially. polynomially
• Continuous-Time Financial Mathematics c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 458
• Numerical Examples Assume S0 = 100, y0 = ln S0 = 4.60517, r = 0,
• Theory of Computation Lecture Prof. Yuh-Dauh Lyuu
• tsp (d) Is NP-Complete Corollary 40 tsp (d) is NP-complete.
• Composition Proposition
• Computability: well-de ned
• Completeness transformation
• AUTHOR QUERIES Journal id: RAEL_A_260259
• Analytics for geometric average trigger reset options
• An efficient convergent lattice algorithm for European Asian options
• On Accurate and Provably Efficient GARCH Option Pricing Algorithms
• Convertible Group Undeniable Signatures Yuh-Dauh Lyuu1
• Decidability and Recursive Languages Let L ( -{ })
• Undecidability in Logic and Mathematics First-order logic is undecidable.a
• Complete Problems and Complexity Classes Proposition 26 Let C and C be two complexity classes
• Theory of Computation Solutions to Homework 2
• The naesat (for "not-all-equal" sat) is like 3sat. But there must be a satisfying truth assignment under
• Theory of Computation Mid-Term Examination on November 09, 2010
• Knowledge in Proofs Suppose I know a satisfying assignment to a satisfiable
• Theory of Computation Final Examination on January 11, 2011
• Bond Price Volatility c 2011 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 74
• Department of Computer Science and Information Engineering College of Electrical Engineering and Computer Science
• Department of Computer Science and Information Engineering College of Electrical Engineering & Computer Science
• Department of Finance College of Management
• Option Pricing with Stochastic Volatility
• Group-Oriented Encryption and Student: Ming-Luen Wu
• Interpolation and American options pricing
• Pricing Parisian-Type Options A Thesis Submitted to the Graduate Institute of
• Any Expression Can Be Converted into CNFs and DNFs = xj: This is trivially true.
• Randomized Complexity Classes; RP Let N be a polynomial-time precise NTM that runs in
• Prover and Verifier There are two parties to a proof.
• Computation That Counts c 2004 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 615
• We are given an undirected graph G and a goal K. node cover asks if there is a set C with K or fewer
• Theory of Computation Lecture Prof. Yuh-Dauh Lyuu
• Boolean Logic c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 73
• The lower bound is rather tight because an upper bound In the proof, we counted the number of circuits.
• NP-Complete Problems c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 248
• Graph Coloring k-coloring asks if the nodes of a graph can be colored
• c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 607 The density of language L
• coNP Hardness and NP Hardnessa Proposition 46 If a coNP-hard problem is in NP, then
• Despite the simplicity of a circuit, the previous discussions imply the following
• Approximability c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 609
• On P vs. NP c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 654
• The Proof: and The approximate and of crude circuits CC(X) and
• Function Problems Are Not Harder than Decision Problems If P = NP
• Theory of Computation Due: 2008/11/25
• Theory of Computation Final Examination on January 13, 2009
• Undecidability in Logic and Mathematics First-order logic is undecidable.a
• bin packing We are given N positive integers a1, a2, . . . , aN , an
• BPP's Circuit Complexity Theorem 77 (Adleman (1978)) All languages in BPP
• Theory of Computation Final Examination on January 12, 2010
• 00.050.10.150.20.25 Backwardinduction(0,k)
• Principles of Financial Computing Prof. Yuh-Dauh Lyuu
• Geometric Brownian Motion Consider the geometric Brownian motion process
• Variance Reduction The statistical efficiency of Monte Carlo simulation can
• Problems; the Smile Options written on the same underlying asset usually do
• Delta Hedge The delta (hedge ratio) of a derivative f is defined as
• Matrix Computation c 2007 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 627
• Principles of Financial Computing Prof. Yuh-Dauh Lyuu
• Fundamental Statistical Concepts c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 135
• States and Their Transitions The tuple
• Introduction to Mortgage-Backed Securities c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 969
• Spot and Forward Rates under Continuous Compounding
• Option Pricing Models c 2009 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 193
• Toward the Black-Scholes Formula The binomial model seems to suffer from two unrealistic
• Forwards, Futures, Futures Options, Swaps c 2009 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 367
• Problems with the Monte Carlo Method The error bound is only probabilistic.
• Numerical Examples Assume S0 = 100, y0 = ln S0 = 4.60517, r = 0,
• Equilibrium Term Structure Models c 2009 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 874
• Principles of Financial Computing Prof. Yuh-Dauh Lyuu
• Forward Price The payoff of a forward contract at maturity is
• Brownian Motiona Brownian motion is a stochastic process { X(t), t 0 }
• Problems with the Monte Carlo Method The error bound is only probabilistic.
• Fixed-Income Options Consider a two-year 99 European call on the three-year,
• Theory of Computation Solutions to Homework 5
• Zero-Knowledge Proof of 3 Colorabilitya 1: for i = 1, 2, . . . , | E |2 do
• Remarks on Asian Option Pricing Asian option pricing is an active research area.
• Other Examples Subordinated debts as bull call spreads.
• Completenessa As reducibility is transitive, problems can be ordered
• This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research
• Logarithmic Space c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 776
• Pricing European and American Options with Extrapolation
• The exponent of m (p) is the least k Z+ Every residue s (p) has an exponent.
• BOPM and Black-Scholes Model The Black-Scholes formula needs five parameters: S, X,
• Unapproximability of tspa Theorem 80 The approximation threshold of tsp is 1
• Theory of Computation Mid-Term Examination on April 17, 2008
• Back to maxsat In maxsat, the i's are clauses.
• Computation That Counts c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 620
• Theory of Computation Lecture Prof. Yuh-Dauh Lyuu
• On the Computational Power of Players in Two-Person Strategic Games
• Cryptography c 2004 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 489
• Programming generality,
• Hamiltonian?'' mathematics,
• An Approximate Formula for 1-Year Asian Callsa 3 [ -+ t(1 -t/2) ] dt
• On the Possibility of Basing Oblivious Transfer on Weakened
• polynomial-time nondeterministic
• Computation Information
• Decidability Palindromes
• Probabilistic Polynomial)
• Numerical Methods c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 579
• Cryptanalysis of and improvement on the HwangChen multi-proxy
• Principles of Financial Computing Prof. Yuh-Dauh Lyuu
• levelcoupon Terminology
• GARCH Option Pricinga Options can be priced when the underlying asset's
• Theory of Computation Solutions to Homework 4
• Graph Isomorphism V1 = V2 = {1, 2, . . . , n}.
• Nondeterministic Space Complexity Classes Let L be a language.
• (1806--1871). Conjunctive
• (completeness). (soundness).
• satisfiability PSPACEComplete
• Satisfiability NPcomplete
• Take the binomial model with two assets. In a period, asset one's price can go from S to S1 or
• approximate approximate
• Proposition NP-complete
• (1845--1918). mathematics,
• Isomorphism permutation
• interaction. arbitrarily
• Set-Related NP-complete.
• Zero-Coupon Bonds (Pure Discount Bonds) The price of a zero-coupon bond that pays F dollars in
• Randomized Computation c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 419
• Department of Finance College of Management
• Price Behavior (2) A level-coupon bond sells
• Department and Graduate Institute of Finance College of Management
• bisection width bisection width is like max bisection except that it
• Decidability Palindromes
• NP-complete improvement
• deterministic. single-string
• volatility. atthemoney
• "!$#%!'& !$#)( 0 132'( 4!5 687 9 @ #%!A6B CEDGFIHQPRFIHTS
• When interest rates are stochastic, forward and futures prices are no longer theoretically identical.
• Binomial Distribution Denote the binomial distribution with parameters n
• Pricing Asian Options with Fourier Convolution
• 2.557.51012.51517.5
• positives). negatives).
• compositeness simultaneous
• discussions computation.
• (Complexity) (complexity)
• Time Series Analysis c 2005 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 708
• Turing-Computable Functions Let f : ( -{ })
• A Second Corollary of Cantor's Theorem Corollary 8 The set of all functions on N is not countable.
• Computation That Counts c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 730
• Comments on Lower-Bound Proofs They are usually difficult.
• Trinomial Tree Set up a trinomial approximation to the geometric
• Proof of Theorem (continued) Bob verifies that A
• Interest Rate Derivative Securities c 2005 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 772
• Foundations of Term Structure Modeling c 2005 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 838
• max bisection max cut becomes max bisection if we require that
• On the Least-Squares Monte Carlo for Pricing American Barrier Options
• NP-Complete nae-satis able.
• Computation Information
• Theory of Computation Mid-Term Examination on April 17, 2008
• Efficient Algorithms for Geometric-Average-Trigger Reset
• We had proved that max cut is NP-complete for multigraphs.
• Some Boolean Functions Need Exponential Circuitsa Theorem 15 (Shannon (1949)) For any n 2, there is
• Stochastic Processes and Brownian Motion c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 396
• Decidability and Recursive Languages Let L ( -{ })
• immediately. immediately.
• configuration
• Multivariate Contingent Claims They depend on two or more underlying assets.
• contradiction. contradiction.
• Computation Information
• NP-complete. NP-complete.
• Computability: well-de ned
• construction limitations,
• Computation assignments
• On the Complexity of the Ritchken Sankarasubramanian
• Backward Induction on the RT Tree After the RT tree is constructed, it can be used to price
• Department of Finance College of Management
• NPComplete polynomially
• (Complexity) (complexity)
• NPcomplete NPcomplete
• Generalized expression.
• Intractable NP-complete
• Brownian Motiona Brownian motion is a stochastic process { X(t), t 0 }
• Underlying Pays a Continuous Dividend Yield of q The value of a forward contract at any time prior to T is
• Function Problems c 2004 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 219
• Forward Rates Spot Rates Yield Curve The FV of $1 at time n can be derived in two ways.
• Sensitivity Analysis of Options c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 264
• The Primality Problem An integer p is prime if p > 1 and all positive numbers
• Reductions and Completeness c 2004 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 106
• On Accurate Trinomial GARCH Option Pricing
• By the linear speedup theorem, any polynomial time bound can be represented by its leading term nk
• The Reachability Method The computation of a time-bounded TM can be
• 1/51/21/1 1/3 1/4 2/1 2/2 2/3 2/4
• Department of Computer Science and Information Engineering College of Electrical Engineering and Computer Science
• Matrix Computation c 2011 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 707
• Ito's Lemma (continued) Theorem 18 (Alternative Ito's Lemma) Let
• certi cates Proposition
• NPcomplete NPcomplete
• Residuacity (ajp)(bjp).
• cryptosystem probability
• composite''; probability
• Pricing CDOs with the Fourier Transform Method
• Theory of Computation Lecture Prof. Yuh-Dauh Lyuu
• The Secret-Key Agreement Problem Exchanging messages securely using a private-key
• On P vs. NP c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 672
• The Primality Problem An integer p is prime if p > 1 and all positive numbers
• Graph Coloring k-coloring asks if the nodes of a graph can be colored
• (continued) Furthermore,
• transformation restrictions
• Proposition undecidable
• SET COVERING SET PACKING
• Extrapolation It is a method to speed up numerical convergence.
• De Morgan'sa De Morgan's laws say that
• Another Variant of 3sat Proposition 32 3sat is NP-complete for expressions in
• Nondeterministic Space Complexity Classes Let L be a language.
• NP-Complete Problems c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 248
• Continuous-Time Derivatives Pricing c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 477
• Monte Carlo Algorithmsa The randomized bipartite perfect matching algorithm is
• Oblivious Polynomial Evaluation and Oblivious Neural Learning
• Backward Induction on the RT Tree After the RT tree is constructed, it can be used to price
• Correlated Trinomial Modela Two risky assets S1 and S2 follow
• Toward the Black-Scholes Formula The binomial model seems to suffer from two unrealistic
• Time Complexity The multistring TM is the basis of our notion of the
• Completenessa As reducibility is transitive, problems can be ordered
• The Reachability Method The computation of a time-bounded TM can be
• A Few Calculations Let p = 13.
• Coupon Effect on the Yield to Maturity Under a normal spot rate curve, a coupon bond has a
• Time Complexity under Nondeterminism Nondeterministic machine N decides L in time f(n),
• polynomial-time nondeterministic
• Completeness transformation
• NTU International Conference on Economics, Finance and Accounting
• Complexity of the Ritchken-Trevor-Cakici-Topyan GARCH Option Pricing Algorithm
• Primality Tests primes asks if a number N is a prime.
• Despite the simplicity of a circuit, the previous discussions imply the following
• Prepayment Vector The PSA tries to capture how prepayments vary with
• Legendre's Law of Quadratic Reciprocitya Let p and q be two odd primes.
• General Derivatives Pricing In general the underlying asset S may not be traded.
• Numerical Valuation of Discrete Barrier Options with
• More Undecidability {M : M halts on all inputs}.
• Reductions and Completeness c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 192
• Bond Price Volatility c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 71
• Sample Term Structure We shall construct interest rate trees consistent with the
• Toward the Black-Scholes Formula The binomial model seems to suffer from two unrealistic
• Department of Finance College of Management
• Spread of Nonbenchmark Bonds Model prices calculated by the calibrated tree as a rule
• Theory of Computation Solutions to Homework 1
• Polynomial Space c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 678
• Lower Bounds for Asian Options
• Fixed-Income Options Consider a two-year 99 European call on the three-year,
• Legendre's Law of Quadratic Reciprocitya Let p and q be two odd primes.
• The Circuit Complexity of P Proposition 69 All languages in P have polynomial
• Nondeterminism Nondeterministic
• NPcomplete improvements
• (single-string) con guration
• Theory of Computation Lecture Prof. Yuh-Dauh Lyuu
• Theory of Computation Solutions to Homework 2
• Extended Church's Thesis All "reasonably succinct encodings" of problems are
• A Fully Public-Key Traitor-Tracing Scheme YUH-DAUH LYUU
• Control Variates Use the analytic solution of a similar yet simpler
• Spread of Nonbenchmark Bonds Model prices calculated by the calibrated tree as a rule
• Implicit Methods Suppose we use t = tj+1 in Eq. (64) on p. 601 instead.
• (continued) probability.
• Fast Fourier Transform with Applications to Pricing Discrete European Barrier Options
• Consider the stochastic process Xi, n 1 },
• Lengths of Boolean Formulas for the Threshold Functiona Define the boolean function Tk(x1, . . . , xn) to be 1 if at
• Consider the stochastic process Xi, n 1 },
• "w = 1, x = 0, y = 0, z = 1" Adds 46 to Cycle Count
• The Traveling Salesman Problem We are given n cities 1, 2, . . . , n and integer distances dij
• Department of Computer Science and Information Engineering College of Electrical Engineering & Computer Science
• Group Undeniable Signatures YUH-DAUH LYUU
• Logic, Language, and Computation
• Two Algorithms for Maximum and Minimum Weighted
• Put-Call Parity (Castelli, 1877) C = P + S -PV(X). (19)
• Theory of Computation Lecture Prof. Yuh-Dauh Lyuu
• min cut and max cut A cut in an undirected graph G = (V, E) is a partition
• Theory of Computation Lecture Prof. Yuh-Dauh Lyuu
• Maximum Satisfiability Given a set of clauses, maxsat seeks the truth
• Graph Isomorphism V1 = V2 = {1, 2, . . . , n}.
• Algorithms Comparisona So which algorithm is better, binomial or trinomial?
• Backward Induction on the RT Tree After the RT tree is constructed, it can be used to price
• Equilibrium Term Structure Models c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 854
• Barrier Optionsa Their payoff depends on whether the underlying asset's
• Exponents and Primitive Roots From Fermat's "little" theorem, all exponents divide
• Complementing a TM's Halting States Let M decide L, and M be M after "yes" "no".
• Proof of Theorem (continued) Clearly, if A > 0, the protocol convinces Bob of this.
• coNP and Function Problems c 2004 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 331
• Pricing Discrete Barrier Options Barrier options whose barrier is monitored only at
• The IASTED International Conference on Financial Engineering and Applications
• Nondeterministic Satis ability
• NPComplete polynomially
• Exponential NP-Complete
• Undecidability contradiction.
• max bisection max cut becomes max bisection if we require that
• Delivery and Hedging Delivery ties the futures price to the spot price.
• Option Pricing Models c 2011 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 198
• Sensitivity Analysis of Options c 2005 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 257
• Theorem 7 The set of all subsets of N (2N ) is infinite and
• The Black-Karasinski Modela The BK model stipulates that the short rate follows
• Introduction In Taiwan'...nancial market, in order to adpat to "Globalization" and satisfy
• Principles of Financial Computing Prof. Yuh-Dauh Lyuu
• Daily Monitoring Almost all barrier options monitor the barrier only for
• What Is a Proof? A proof convinces a party of a certain claim.
• connectives: expressions
• Nondeterministic deterministic
• Theory of Computation Lecture Prof. Yuh-Dauh Lyuu
• Approximability, Unapproximability, and Between knapsack, node cover, maxsat, and max cut have
• Theory of Computation Solutions to Homework 3
• configuration x yes/noR(x)
• An Accurate and Provably Efficient GARCH Option Pricing Tree
• Random Walk Works for 2sat Theorem 60 Suppose the random walk algorithm with
• Recursive and Recursively Enumerable Languages Proposition 2 If L is recursive, then it is recursively
• The Hull-White Model: Calibration with Irregular Trinomial Trees
• Sequential Tranche Paydown In the sequential tranche paydown structure, for
• Day Count Conventions: Actual/Actual The first "actual" refers to the actual number of days in
• Pricing Asian Options on Lattices Tian-Shyr Dai
• Department of Computer Science and Information Engineering College of Electrical Engineering and Computer Science
• The Cox-Ingersoll-Ross Modela It is the following square-root short rate model
• Principles of Financial Computing Prof. Yuh-Dauh Lyuu
• Extensions of Options Theory c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 274
• Answers to the Final Examination on January 12, 2005 Problem 1 Answer
• Levin Reduction and Parsimonious Reductions The reduction R in Cook's theorem (p. 266) is such that
• k-sat, where k Z+ , is the special case of sat.
• The BDT Model: Continuous-Time Limit The continuous-time limit of the BDT model is
• Alexander Razborov (1963) c 2009 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 710
• Theory of Computation Solutions to Homework 5
• Principles of Financial Computing Prof. Yuh-Dauh Lyuu
• deterministic randomization
• polynomially polynomially
• Satisfiability A boolean expression is satisfiable if there is a truth
• Private Information Retrieval Does Not Imply One-way Permutations
• The Number of Witnesses to Compositeness Theorem 71 (Solovay and Strassen (1977)) If N is an
• The Reachability Method The computation of a time-bounded TM can be
• Ti -1,j -1 Ti -1,j + 1Ti -1,j
• Tracking Error Revisited Define the dollar gamma as S2
• Theory of Computation Due: 2010/10/26
• The term yield denotes the return of investment. Two widely used yields are the bond equivalent yield
• Reductions and Completeness c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 192
• Option Pricing Models c 2005 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 187
• Theory of Computation Lecture Prof. Yuh-Dauh Lyuu
• Acceptability and Recursively Enumerable Languages Let L ( -{ })
• Alternative (Burnside's
• Application: compositions
• Mathematics Information
• conservation (continued)
• Mathematics Information
• colorability computationally
• otherwise). otherwise).
• parenthesize Illustration
• (continued) (concluded)
• computationally Illustration
• contradiction cancellation
• Characterization constructive.
• Application: Convention
• (1883{1960). partitioned
• combinatorial nonnegative
• simultaneous (continued)
• Combinatorics mathematics
• (Surjective) Necessarily,
• 2-Colorings (1)(2)(3)(4)
• Compositions composition
• characteristic (concluded)
• unreasonable (1914{1984)
• Probabilistic nonconstructive
• capacities, applications.
• Multiplicative factorization
• Propositional Connectives
• Applications capacities,
• antisymmetric, transitive.
• (concluded) nonnegative
• (continued) generality,
• (associativity). (distributive
• Calculations substitution)
• Mathematics Information
• Combinatorial particular,
• Probabilistic nonconstructive
• (concluded) partition-of-integers
• (1765{1822). contradiction
• Innumerable applications
• Barrier Optionsa Their payoff depends on whether the underlying asset's
• 10 20 30 40 50
• 9 = 0 + 0 + 2 + 2 + 2 + 3 9 = 4 + 4 + 1 4444444444445
• Application: No Return to the Origin until End What is the number of ways a binomial random walk that is
• Discrete Mathematics Lecture Notes Prof. Yuh-Dauh Lyuu
• Orbit of 111 Stabilizer of 111
• The Binomial Theorema (x + y)(x + y) (x + y).
• Stiring number of the 2nd kind 1/51/21/1 1/3 1/4
• f(m)-f(m-1)
• Applications besides Exploiting Arbitrage Opportunitiesa
• Monotone Increasing Functions with f(x) x Theorem 44 There are
• Bond Price Volatility c2012 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 75
• Properties of Integers: Mathematical c2012 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 160
• Option Basics c2012 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 153
• ((ab)(cd)) (((ab)c)d) (a(b(cd))) ((a(bc))d) (a((bc)d))
• 50100150200 Logplotofn!
• Principles of Financial Computing Prof. Yuh-Dauh Lyuu