- MATHEMATICAL BIOLOGY BY D A - YEON M IN # SIR Epidemics

CONTENTS - What is Mathematical epidemic modelling? -Threshold for SIR Epidemic -Code for SIR Model

What is Mathematical epidemic modelling?

Kermack and McKendrick introduced in 1927 and published their theory in a set of three articles from 1927, 1932, and Mathematical modelling To know how infectious diseases spread through a population Substantial reason of theory

A SSUMPTION The duration of the epidemic is short copared to the lifetime of its hosts. We can neglect birth and disease-unrelated death. The population is closed of constant size N.

Population : Susceptible→Infective→Removed

Threshold For SIR Epidemic

function f = SIR_model(Parameter, Tdata) global S0 I0 R0 id1 id2 ad_num a; S(1) = S0; I(1) = I0; R(1) = R0; b = Parameter(1); g = Parameter(2); tmpNt = length(Tdata); Nt = tmpNt + (tmpNt - 1)*ad_num; dt = (Tdata(end) - Tdata(1))/(Nt-1); for i=1:Nt S(i+1) = S(i) - b*S(i)*I(i)*dt; I(i+1) = I(i) + (b*S(i)*I(i) - g*I(i))*dt; R(i+1) = R(i) + g*I(i)*dt; end f = [ S(1:ad_num+1:Nt) I(1:ad_num+1:Nt) R(1:ad_num+1:Nt)]; Code For SIR Model

F INANCIAL M ATHEMATICS Call Option

C ONTENTS – What is financial mathematics? – What is derivatives? – Call option – Black Scholes model – Finite Difference Method

What is Financial Mathematics? Financial Mathematics : a field of applied mathematics, concerned with financial markets. Derive and extend the mathematical or numerical models without necessarily establishing a link to financial theory, taking observed market prices as input.

A contract that derives its value from the performance of an entity. Purposes : insuring against price movements(hedging), increasing exposure to price movements for speculation, getting access to otherwise hard-to-trade assets or markets forwards, futures, options, swaps … What is Derivatives?

A financial contract between two parties : The buyer has the right(not the obligation) to buy a commodity or financial instrument from the seller at a certain time for a certain price(the strike price) The seller has the obligation to sell the commodity or financial instrument to the buyer if the buyer decides. The buyer pays a fee (a premium) for this right. K : Strike price Call Option

Fischer Black and Myron Scholes first published in their 1973 paper, “The Pricing of Options and Corporate Liabilities” Derived a partial differential equation to estimate the price of the option over time Black-Scholes equation KEY Idea : to hedge the option by buying and selling the underlying asset To eliminate risk … delta hedging – basis of more complicated hedging strategies How we can decide price of Call option? - By Option Pricing model by Black Scholes

A SSUMPTION on the assets : (riskless rate) The rate of return on the riskless asset is constant. (random walk) The instantaneous log returns of the stock price is an infinitesimal random walk with drift- a geometric Brownian Motion, and its drift and volatility are constant. The stock does not pay a dividend. on the market: There is no arbitrage opportunity. It is possible to borrow and lend any amount, even fractional, of cash at the riskless rate and to buy and sell any amount, even fractional, of the stock. The transactions do not incur any fees or costs.

Price of Call Option by Closed Form

P ARTIAL D IFFERENTIAL E QUATION BY B LACK -S CHOLES

Finite Difference Method

Explicit Method

Implicit Method

Crank-Nicolson Method

clear; clc; S = 100 ; E = 100; r = 0.05; T = 1.0; Vm = 20; vol = 1; tol = 1.0e-6; iter = 1; maxit = 1000; d1 = (log(S/E)+(r+vol^2/2)*T)/(vol*sqrt(T)); d2 = d1-vol*sqrt(T); price = S*normcdf(d1)-E*exp(-r*T)*normcdf(d2); vega = S*sqrt(T)*normpdf(d1); fprintf('n\tvol\t\tprice\n') fprintf('%d\t%f\t%f\n', iter, vol, price) while abs(price - Vm) > tol && iter < maxit vol = vol - (price - Vm) / vega; d1 = (log(S/E)+(r+vol^2/2)*T)/(vol*sqrt(T) d2 = d1-vol*sqrt(T); price = S*normcdf(d1)-E*exp(-r*T)*normcdf(d2); vega = S*sqrt(T)*normpdf(d1); iter = iter + 1; fprintf('%d\t%f\t%f\n', iter, vol, price) end Implied_Vol = vol S: price of underlying asset, E: exercise price, R : riskless interest rate, T : maturity (year), Vm : market option price, vol : initial volatility, tol : tolerance, iter : iteration, maxit : max iteration Initial price using initial vol Partial price / partial vol Code for call option - By Newton Method Newton method