- 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