The document describes the Cox-Ross-Rubinstein binomial model for pricing options. It involves modeling the underlying asset price as following a binomial process over discrete time periods, with the price having a probability p of moving up by a factor u or a probability q of moving down by a factor d. The model values the option by working backwards through a binomial tree. An example is provided to demonstrate generating the binomial tree and calculating option values at each node.
The document summarizes a lesson on game theory and linear programming. It discusses using linear programming to find optimal strategies in zero-sum games represented by payoff matrices. It provides examples of solving for optimal strategies in Rock-Paper-Scissors and another sample game. The key steps of formulating the column player's problem as a linear program to minimize the maximum payoff for the row player are outlined.
The document is a questionnaire given to an audience to provide feedback on film promotion materials for a psychological thriller film. It contains 15 questions with both open-ended and closed-ended questions about the film trailer, poster, and magazine cover. Based on the responses, the materials were generally found to appropriately convey the genre and plot while intriguing the audience about what would happen in the film. However, some responses also provided areas for improvement, such as ensuring the film title is clear in the trailer or using a voiceover to further explain the plot.
Depth-first search (DFS) is an algorithm that explores all the vertices reachable from a starting vertex by traversing edges in a depth-first manner. DFS uses a stack data structure to keep track of vertices to visit. It colors vertices white, gray, and black to indicate their status. DFS runs in O(V+E) time and can be used for applications like topological sorting and finding strongly connected components. The edges discovered during DFS can be classified as tree, back, forward, or cross edges based on the order in which vertices are discovered.
The document discusses backward difference, which is a method used in interpolation and numerical integration of functions. It involves taking the difference between function values at equally spaced points, working backward from the last point. The first backward difference is between the last two points, the second is the difference between the first differences, and so on. An example calculates the backward differences for a set of data points and uses the formula to interpolate the function value at x=42.
This document presents an algorithm for solving maze problems using backtracking. It first defines a maze as a confusing network of paths and hedges designed as a puzzle. It then explains that mazes can be solved by computers by devising an algorithm for a given maze. Specifically, it proposes using a backtracking approach due to the predefined constraints of mazes. The document provides pseudocode for a recursive backtracking algorithm that uses a depth-first search to find a clear path from the starting to ending position in a maze, represented as a 2D array. It analyzes the time complexity of this algorithm and suggests ways to potentially improve it.
▪ Developed a mathematical puzzle solver using brute force algorithm and Constraint Satisfaction Problems in C++
▪ Employed backtracking methodology to ensure guaranteed solution to Sudoku Problem in order of milliseconds
▪ Implemented AI practices like Forwarding Checking, Most Constrained Heuristic & ARC-Consistency in an incremental manner
The document discusses shortest path problems and algorithms. It defines the shortest path problem as finding the minimum weight path between two vertices in a weighted graph. It presents the Bellman-Ford algorithm, which can handle graphs with negative edge weights but detects negative cycles. It also presents Dijkstra's algorithm, which only works for graphs without negative edge weights. Key steps of the algorithms include initialization, relaxation of edges to update distance estimates, and ensuring the shortest path property is satisfied.
The document discusses minimum spanning trees and two algorithms for finding them: Prim's algorithm and Kruskal's algorithm. Prim's algorithm works by growing a spanning tree from an initial node, always adding the edge with the lowest weight that connects to a node not yet in the tree. Kruskal's algorithm sorts the edges by weight and builds up a spanning tree by adding edges in order as long as they do not form cycles. Both algorithms run on undirected, weighted graphs and produce optimal minimum spanning trees.
The document summarizes a lesson on game theory and linear programming. It discusses using linear programming to find optimal strategies in zero-sum games represented by payoff matrices. It provides examples of solving for optimal strategies in Rock-Paper-Scissors and another sample game. The key steps of formulating the column player's problem as a linear program to minimize the maximum payoff for the row player are outlined.
The document is a questionnaire given to an audience to provide feedback on film promotion materials for a psychological thriller film. It contains 15 questions with both open-ended and closed-ended questions about the film trailer, poster, and magazine cover. Based on the responses, the materials were generally found to appropriately convey the genre and plot while intriguing the audience about what would happen in the film. However, some responses also provided areas for improvement, such as ensuring the film title is clear in the trailer or using a voiceover to further explain the plot.
Depth-first search (DFS) is an algorithm that explores all the vertices reachable from a starting vertex by traversing edges in a depth-first manner. DFS uses a stack data structure to keep track of vertices to visit. It colors vertices white, gray, and black to indicate their status. DFS runs in O(V+E) time and can be used for applications like topological sorting and finding strongly connected components. The edges discovered during DFS can be classified as tree, back, forward, or cross edges based on the order in which vertices are discovered.
The document discusses backward difference, which is a method used in interpolation and numerical integration of functions. It involves taking the difference between function values at equally spaced points, working backward from the last point. The first backward difference is between the last two points, the second is the difference between the first differences, and so on. An example calculates the backward differences for a set of data points and uses the formula to interpolate the function value at x=42.
This document presents an algorithm for solving maze problems using backtracking. It first defines a maze as a confusing network of paths and hedges designed as a puzzle. It then explains that mazes can be solved by computers by devising an algorithm for a given maze. Specifically, it proposes using a backtracking approach due to the predefined constraints of mazes. The document provides pseudocode for a recursive backtracking algorithm that uses a depth-first search to find a clear path from the starting to ending position in a maze, represented as a 2D array. It analyzes the time complexity of this algorithm and suggests ways to potentially improve it.
▪ Developed a mathematical puzzle solver using brute force algorithm and Constraint Satisfaction Problems in C++
▪ Employed backtracking methodology to ensure guaranteed solution to Sudoku Problem in order of milliseconds
▪ Implemented AI practices like Forwarding Checking, Most Constrained Heuristic & ARC-Consistency in an incremental manner
The document discusses shortest path problems and algorithms. It defines the shortest path problem as finding the minimum weight path between two vertices in a weighted graph. It presents the Bellman-Ford algorithm, which can handle graphs with negative edge weights but detects negative cycles. It also presents Dijkstra's algorithm, which only works for graphs without negative edge weights. Key steps of the algorithms include initialization, relaxation of edges to update distance estimates, and ensuring the shortest path property is satisfied.
The document discusses minimum spanning trees and two algorithms for finding them: Prim's algorithm and Kruskal's algorithm. Prim's algorithm works by growing a spanning tree from an initial node, always adding the edge with the lowest weight that connects to a node not yet in the tree. Kruskal's algorithm sorts the edges by weight and builds up a spanning tree by adding edges in order as long as they do not form cycles. Both algorithms run on undirected, weighted graphs and produce optimal minimum spanning trees.
The document provides information on potential equity investments for a 100 million peso fund, including sector outlooks, individual stock analyses, and optimization recommendations. Key sectors identified are consumer staples, utilities, financials, real estate, and tourism. Stock screening is done based on fundamentals like P/E, P/B, growth, and technical analyses. Optimization suggests allocations like 50% in SECB and BPI for banks, and 55% in MPI for holding firms. The fund needs to achieve a 10% annual return while meeting other constraints.
The Philippine economy is expected to continue strong growth in the coming years, driven by robust consumer spending, increased investment, and sustained government spending. Inflation will remain low and interest rates are expected to stay at current levels, supporting economic activity. The current account surplus and prudent fiscal management have improved the country's credit ratings and investment environment.
The document describes an options pricing tool that allows users to:
1) Select the type of option, underlying asset, and valuation model
2) Input variables like price, strike price, yield, and maturity
3) View the calculated option price either in a new worksheet or predefined range
Monte Carlo simulation is a numerical method used to model probabilistic outcomes in complex systems. It works by simulating random variables many times according to a probability distribution. This allows estimating statistics like expected values. In finance, it is commonly used to price exotic options by simulating the behavior of the underlying asset over time and calculating the option payoff. The method proceeds in stages: defining distributions, simulating variables, repeating to increase accuracy. It is flexible but computationally intensive.
Modern Portfolio Theory provides a framework for constructing investment portfolios to maximize expected return based on a given level of market risk. It assumes investors aim to reduce risk through diversification among assets with low correlations. Markowitz models show how to combine assets to obtain an efficient portfolio with the highest return for a given risk. Mean-variance optimization identifies the portfolio on the efficient frontier with the best risk-return tradeoff. However, the theory relies on historical data and assumptions that may not always hold in real markets.
Strategic asset allocation involves setting long-term target allocations for different asset classes based on an investor's risk tolerance, while tactical asset allocation periodically adjusts the asset mix in response to changing market conditions in order to potentially boost returns or reduce risk in the short-term. While strategic asset allocation focuses on systematic market risk and has historically been the main source of risk for portfolios, tactical asset allocation aims to generate excess returns over benchmarks through shorter-term trading ideas and thematic adjustments based on valuations and market sentiment.
The document provides information on potential equity investments for a 100 million peso fund, including sector outlooks, individual stock analyses, and optimization recommendations. Key sectors identified are consumer staples, utilities, financials, real estate, and tourism. Stock screening is done based on fundamentals like P/E, P/B, growth, and technical analyses. Optimization suggests allocations like 50% in SECB and BPI for banks, and 55% in MPI for holding firms. The fund needs to achieve a 10% annual return while meeting other constraints.
The Philippine economy is expected to continue strong growth in the coming years, driven by robust consumer spending, increased investment, and sustained government spending. Inflation will remain low and interest rates are expected to stay at current levels, supporting economic activity. The current account surplus and prudent fiscal management have improved the country's credit ratings and investment environment.
The document describes an options pricing tool that allows users to:
1) Select the type of option, underlying asset, and valuation model
2) Input variables like price, strike price, yield, and maturity
3) View the calculated option price either in a new worksheet or predefined range
Monte Carlo simulation is a numerical method used to model probabilistic outcomes in complex systems. It works by simulating random variables many times according to a probability distribution. This allows estimating statistics like expected values. In finance, it is commonly used to price exotic options by simulating the behavior of the underlying asset over time and calculating the option payoff. The method proceeds in stages: defining distributions, simulating variables, repeating to increase accuracy. It is flexible but computationally intensive.
Modern Portfolio Theory provides a framework for constructing investment portfolios to maximize expected return based on a given level of market risk. It assumes investors aim to reduce risk through diversification among assets with low correlations. Markowitz models show how to combine assets to obtain an efficient portfolio with the highest return for a given risk. Mean-variance optimization identifies the portfolio on the efficient frontier with the best risk-return tradeoff. However, the theory relies on historical data and assumptions that may not always hold in real markets.
Strategic asset allocation involves setting long-term target allocations for different asset classes based on an investor's risk tolerance, while tactical asset allocation periodically adjusts the asset mix in response to changing market conditions in order to potentially boost returns or reduce risk in the short-term. While strategic asset allocation focuses on systematic market risk and has historically been the main source of risk for portfolios, tactical asset allocation aims to generate excess returns over benchmarks through shorter-term trading ideas and thematic adjustments based on valuations and market sentiment.
1. Cox, Ross and Rubinstein
Binomial Trees
Acedo Fabia Reyes Sorbito Vidamo
2. Report Outline
1
• Overview
2
• General Assumptions
3
• Steps and Formulas
4
• Example
5
• Summary
3. Overview
• A type of binomial asset pricing model first proposed by John
C. Cox, Stephen A. Ross and Mark Rubinstein (1979).
• “Simple and efficient numerical procedure for valuing
options for which premature exercise may be optional”
• “All corporate securities can be interpreted as portfolios of
puts and calls on the asset of the firm.”
• Uses discrete time model of varying price over time of the
underlying financial instrument
• Uses binomial tree of possible price of the underlying asset ;
each nodes valuation is performed iteratively
4. Assumptions
uS with probability p
S
dS with probability q = p ‒ 1
• Underlying asset price S follows a multiplicative binomial
process over discrete period.
• Rate of return on the stock over each period can have two
possible values.
• u and d parameters are constant over the whole tree.
5. Assumptions
• u and d are chosen so that u = 1/d .
• Interest rates are assumed constant, d < Rf < u. It means that
there is no arbitrage opportunity.
• No taxes, transaction cost, or margin requirements
• The underlying doesn't pay dividends over the life of the
option.
6. Steps and Formulas
Step 1. Compute for the Risk free Return
r is the one period rate of return
r = EXP(i*(t/n)) t is term in years
p = (r-d)/(u-d) n is the number of periods
q=1-p p is the risk-neutral probability up move
q is the risk-neutral probability down move
Step 2. Generate the price of the tree
uxS S is the price of underlying asset,
S u is the up move factor with probability p,
dxS d is the down move factor with probability q
7. Steps and Formulas
Step 3. Calculation of option value at each final node
(Backward Induction)
Sn is the computed
At Final Node n:
underlying asset price
If it is a Call Option, then use MAX(0,Sn-K)
at node n
If it is a Put Option, then use MAX(K-Sn,0)
K is the strike price
Step 4. Sequential calculation of the option value at each
preceding node
Cu is the older upper
At other Nodes 0 to n-1 option price
other nodes = [p * Cu + q * Cd] / r Cd is the older lower
option price
8. Example:
Step 1. Compute for the Risk free Return
Stock price [S] $ 60.00 Given
Interest rate [i] 5.00% Given
Strike price [K] $55.00 Given
Term in years [t] 1 Given
Number of periods - quarterly [n] 4 Given
Up move factor [u] 1.05 Given
Down move factor [d] 0.9524 d = 1/u
One period rate of return [r] 1.0126 r = EXP(i*(t/n))
Risk-neutral probability - up move [p] 61.67% p = (r-d)/(u-d)
Risk-neutral probability - down move [q] 38.33% q=1-p
Notes: The price of LDI stock is $60/share and the one-year interest rate is
0.05. We wish to price one-year call option with a strike price of $55. Using a
four-step tree (quarterly) with assumed stock price factor increase of 1.05, we
will compute for the price of the underlying asset and the call option.
9. Example:
Step 2. Generate the price of the tree
Formula: CRR Tree:
0 1 … n 0 1 2 3 4
Suuuu 72.93
Suuu 69.46
Suu Suuud 66.15 66.15
Su Suud 63.00 63.00
S Sud Suudd 60.00 60.00 60.00
Sd Sudd 57.14 57.14
Sdd Suddd 54.42 54.42
Sddd 51.83
Sdddd 49.36
S is the price of underlying asset, S = $ 60
u is the up move factor u = 1.05
d is the down move factor d = 0.9524
n is the number of periods n=4
10. Example:
Step 3. Calculation of option value at each final node
CRR Tree: Binomial Tree for Pricing a $55 Call Option
0 1 2 3 4 0 1 2 3 4
72.93 17.93
69.46
66.15 66.15 11.15
63.00 63.00
60.00 60.00 60.00 5.00
57.14 57.14
54.42 54.42 -
51.83
49.36 -
Given: K = $ 55
At Final Node n:
Sample Computation:
If it is a Call Option, then use MAX(0,Sn-K)
MAX(0, 72.93-55) = 17.93
If it is a Put Option, then use MAX(K-Sn,0)
MAX(0, 66.15-55) = 11.15
11. Example:
Step 4. Calculation of the option value at each preceding node
Binomial Tree for Pricing a $55 Call Option
At other Nodes 0 to n-1
other nodes = [p * Cu + q * Cd] / r 0 1 2 3 4
where
Cu is the older upper option price 17.93
Cd is the older lower option price 15.14
12.51 11.15
Given: p = 1.05, q = 0.9524, r = 1.0126 10.06 8.68
7.87 6.44 5.00
Sample Computation: 4.62 3.04
1.85 -
O31 = [1.05*17.93+0.9524*11.15]/1.0126 -
= 15.14 -
O32 = [1.05*11.15+0.9524*5.00]/1.0126
= 8.68
O21 = [1.05*15.14+0.9524*8.68]/1.0126
= 12.51
12. Summary and Conclusions
• Cox-Ross-Rubinstein Model is one of many available
binomial options pricing models. It is a simplified alternative
numerical method that can be used for practical
computations of complex option values. It assumes a
constant interest rate (risk free return), absence of arbitrage
opportunities and constant probability of underlying assets
upward (u) and downward (d) movement.
• Options priced derived from Cox-Ross-Rubinstein binomial
tree can be used in formulating strategy that will
generate/ lock in pure arbitrage profits if the market price of
an option differs from the value given by the model.
13. References:
• Cox, J.C., Ross S.A, Rubinstein, M., Option Pricing : A Simplified
Approach. (1979). Published in Journal of Finance and
Economics
• Watsham, Terry J., and Parramore, Keith. Quantitative
Methods in Finance. (1997)
• http://investexcel.net/736/binomial-option-pricing-excel/
• http://www.sitmo.com/article/binomial-and-trinomial-trees/
• http://en.wikipedia.org/wiki/Binomial_options_pricing_mode
l
• http://sfb649.wiwi.hu-
berlin.de/fedc_homepage/xplore/tutorials/xlghtmlnode63.ht
ml#bin-fig2
• http://www.terry.uga.edu/~mayhew/Old/chapter9.pdf
Step 1. Binomial model acts similarly to the asset that exists in a risk neutral world.pu+qd = exp(i*∆t) = r, where ∆t = t/nt = term of the optionn= number of periodsIts variance: pu^2 + qd^2 – (exp(i*∆t))^2 =𝜎^2∆tStep 1. Binomial model acts similarly to the asset that exists in a risk neutral world.pu+qd = exp(i*∆t) = r, where ∆t = t/nt = term of the optionn= number of periodsIts variance: pu^2 + qd^2 – (exp(i*∆t))^2 =𝜎^2∆t
Notice that the lattice is symmetrical, that is due to the assumption that d=1/u (ud=1).