The goals of the MFAS seminar are the following:
Arrange a regular meet-up for IMath students, faculty, and other affiliates to serve as a venue for presentations and discussion regarding topics on mathematical finance, stochastic calculus, and mathematical modelling of financial instruments and entities.
Aid students and faculty in producing and refining academic output thru collaborations and by gathering informed opinions and suggestions from researchers working in the same field of study.
Target members/audience: IMath students, faculty, and other affiliates
Regular schedule and venue: Mondays, 10am to 11am MB 116
Timeline: August 14 - Dec 4, 2017
If you are interested in attending, giving a talk, or if you want to receive announcements regarding the schedule, email usolon[at]math.upd.edu.ph.
Past Talks
Introduction to Credit Risk Management (From the Basel Accords to IRB)
Jose Maria Escaner IV (University of the Philippines, Diliman)
jlescaner[at]gmail.com
27 November 2017 | 10:00 AM - 11:00 AM | MB 116
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This introductory lecture is based on the e-Course program on Credit Risk Management of Prof. Pasquale Cirillo of TU Delft. In this lecture, we define what credit risk is and how it can be measured it as guided by Basel II Accords.
Reference: Introduction to Credit Risk Management, EdX
Utility and Ruin Probability Relationship Through Premium Equivalence
Jonathan Mamplata (University of the Philippines, Los Banos)
jbmamplata[at]up.edu.ph
20 November 2017 | 10:00 AM - 11:00 AM | MB 116
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Utility is a satisfaction received from consuming a commodity or service. Ruin prob- ability is the probability that an insurer’s surplus will fall below zero for a certain period of time. Many researches focused on the pricing insurance premium using the concept of utility functions. Also some focused on finding the probability that the company or insurer will be insolvent at any given point in time via the ruin probability. This discussion focused mainly on the calculation of the insurance premium by setting a tolerance level for the ruin probability. After the premium was solved, the corresponding utility will be determined.
It was assumed that the loss or claims amount follows an exponential distribution or geometric distribution. A closed form expression for the finite-time ruin probability will be determined. This closed form expression will be used to generate algorithm that calculated the corresponding ruin probability.
An Overview of Copulas
Lu Kevin S. Ong (University of the Philippines, Diliman)
lkong[at]math.upd.edu.ph
6 November 2017 | 10:00 AM - 11:00 AM | MB 116
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Copulas are functions that join multivariate distribution functions to their one-dimensional margins. The study of copulas plays a major role in the field of probability and statistics.
For this talk, we will discuss the basic properties of copulas and some of their pri- mary applications. One of these applications is the construction of families of bivariate distributions. We will also introduce the use of copulas in the theory of Markov processes.
On the Martingale Approach in deriving the Black-Scholes Partial Differential Equation
Daryl Allen Saddi (University of the Philippines, Diliman)
dasaddi[at]math.upd.edu.ph
23 October 2017 | 10:00 AM - 11:00 AM | MB 116
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The celebrated Black-Scholes partial differential equation describes the price evolution of a derivative asset under the assumptions of a complete market and that the instantaneous log returns of the underlying asset follows an Arithmetic Brownian Motion. In this talk, we shall introduce the notion of the conditional expectation and the idea of martingales, and then discuss certain theorems by Ito, Girsanov, among others. We shall then use such results to give a rigorous derivation of the Black-Scholes PDE.
Numerical Methods for Pricing Options and Some Applications in Ruin Theory
Aaron Ramos (University of the Philippines, Diliman)
aaronjramos13[at]gmail.com
9 October 2017 | 10:00 AM - 11:00 AM | MB 116
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In this talk, we show methods to simulate some stochastic processes and demonstrate how to use them in pricing options and computing the probability of ruin. First, we discuss generating random numbers, and then use it to numerically solve stochastic differential equations. We then use these approximate solutions to price a European call option and some other exotic options. We also show how to simulate the surplus process and approximate quantities related to ruin, particularly its probability.
Solution to the Three-Tower Problem with Variable Bet Size Using Recursions Based on Multigraphs
Ramon Marfil (University of the Philippines, Diliman)
rimarfil[at]math.upd.edu.ph
2 October 2017 | 10:00 AM - 11:00 AM | MB 116
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The Three-Tower Problem is a 3-player gambler's ruin model where two players are involved in an even-money bet during each round. The Three-Tower Problem is extended to a No Limit scenario, that is, more than one chip may be transferred from one player to another. Weighted directed multigraphs were constructed to model the transitions between chip states. Linear systems are constructed based on the connections between nodes in these graphs. Solutions for the placing probabilities of each player are obtained from these linear systems. Expected time until ruin is solved by modeling the game as a Markov process. A numerical algorithm is developed to solve the No Limit 3-Tower Problem for any positive integer chip total. The solution leads to exact values, and results show that the equities in the this model depend on the number, not just proportion, of chips each player holds.
Option Pricing and the Black-Scholes Model
Gian Paolo Samson (University of the Philippines, Diliman)
giansamson[at]gmail.com
25 September 2017 | 10:00 AM - 11:00 AM | MB 116
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In this presentation, we will define and characterize financial options and then derive an option's value through a one-period binomial pricing model. We will then discuss the concept of risk-neutral pricing and generalize the binomial model for multiple periods. The Black-Scholes Model for pricing European call options will be introduced and related to the binomial model. Finally, we will also introduce the Monte Carlo method for pricing and illustrate its relationship to the valuation of Black-Scholes.
Stochastic Differential Equations
Ulyses Solon (University of the Philippines, Diliman)
usolon[at]math.upd.edu.ph
18 September 2017 | 10:00 AM - 11:00 AM | MB 116
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In this last talk of a three-part presentation, we apply our theoretical background of probability spaces, stochastic processes, and Ito integrals to make sense of stochastic differential equations (SDEs), that is, equations of the form \[ dX_t = a(t, X_t) ~dt + b(t, X_t) ~dB_t, ~~X_0(\omega) = Y(\omega)\]
where $\{X_t\}$ is the stochastic process solution to the SDE, $\{B_t\}$ is a Brownian motion, and $Y$ is a random variable set as the initial condition for $X_t$.\\
We recall a few properties of ordinary differential equations, and then discuss those properties in the context of SDEs. We make quick mentions of the strong and weak solutions to SDEs and conditions for the existence of a unique strong solution.\\
We then end with a gallery of some well-known SDEs and the properties of these stochastic processes.
Brownian Motion, Ito Integrals
Ulyses Solon (University of the Philippines, Diliman)
usolon[at]math.upd.edu.ph
11 September 2017 | 10:00 AM - 11:00 AM | MB 116
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In this text-based talk, we recall the concept of a stochastic process and present a continuous-time stochastic process called Brownian motion (or Wiener process). We discuss some of its important properties. We then recall the notion of the Riemann and Riemann-Stieltjes integrals, and show the difficulties that arise when one defines an integral \[ \ds \int_0^t f(s) dB_s \] where ${B_s}$ is a Brownian motion, in terms of the Riemann-Stieltjes integral. We then end with definitions leading to the Ito integral, a treatment which solves this problem and makes it possible for us to talk about stochastic differential equations.
Surveying the underpinnings of quantitative finance and some recent non-traditional financial innovations
Rogemar Mamon (University of Western Ontario)
4 September 2017 | 3:00 PM - 4:00 PM | MB 116
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This presentation will highlight the pertinent quantitative methods employed in dealing with the core financial/actuarial problems such as pricing, risk management, and asset allocation. The technical aspects will centre on the probabilistic approach, yielding analytic or simulation- based solutions, and its intimate connection with the partial differential approach. The relevant inverse problem in finance will also be underscored. Examples will be given on contract valuation and measurement of risks dependent on stochastic variables related to some of the DOST-research priorities. More specifically, the demonstrative examples will go beyond the traditional trading of financial assets. They will focus on the modelling of prices, risk factors or indices and the development of their derivative products covering energy (electricity in particular), commodity, weather, disaster severity, and mortality, amongst others. If time permits, the valuation of patents and R&D via a real-options approach will be briefly discussed.
Probability Spaces, Random Variables, and Stochastic Processes
Ulyses Solon (University of the Philippines, Diliman)
usolon[at]math.upd.edu.ph
14 August 2017 | 10:00 AM - 11:00 AM | MB 116
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For our first meeting, we create the research group's goals, finalize the seminar schedule and timeline, and acquaint the seminar's attendees with a quick meet-and-greet.
In the talk-proper, we lay the groundwork for the definition of stochastic processes by recalling the notion of probability spaces and random variables. We discuss sigma algebras and probability measures, as well as probability distributions and expected values of random variables. We end with the definition of a stochastic process and some examples.