1. Numerical Analysis
Yu Jieun

2. 01 02 03 Index SIR model Finite Difference Method, FDE
Implicit
Explicit
Crank-Nicolson
MATLAB Code 03
Monte Carlo Simulation
MATLAB Code

3. 01. SIR model
Mathematical modeling of spread of disease

4. 01 Variables ① S = S(t) : the number of susceptible individuals
Independent variable : time t
Dependent variables
① S = S(t) : the number of susceptible individuals
② I = I(t) : the number of infected individuals
③ R = R(t) : the number of removed individuals

5. 01
Assumptions
We can ignore birth and disease-unrelated death, like immigration, since the rate of infection and recovery is much faster than the time scale of births and deaths.
The population is therefore closed, of constant size N,
𝑁=𝑆 𝑟 +𝐼 𝑟 +𝑅 𝑟

6. 01
Assumptions
Each infected individual has a fixed number 𝛽 of contacts per day that are sufficient to spread the disease and a fixed fraction 𝛾 of the infected group will recover during any given day.
𝑑𝑆 𝑑𝑟 =−𝛽𝐼𝑆, 𝑑𝐼 𝑑𝑟 =𝛽𝐼𝑆−𝛾𝐼, 𝑑𝑅 𝑑𝑟 =𝛾𝐼

7. 01
Assumptions
These processes which occur simultaneously are referred to as ‘the Law of Mass Action’, that the rate of contact between two groups in a population is proportional to the size of each of the groups concerned.

8. 01
Suspectible
Infective S I R
Removed
𝛽𝐼𝑆 𝛾𝐼

9. 01
MATLAB Code

10. 01
General graph of SIR model

11. 02. Finite Difference Model

12. 02 Finite Difference Method
Black & Scholes modelled stochastic differential equation about the price which follow Geometric Brownian Motion.
where V is the price of the option as a function of stock price S and time t, r is the risk-free interest rate, 𝜎 is the volatility of the stock.

13. 02 Finite Difference Method
FDM is a method of calculating a numerical approximation to the partial differential equations to differential equations.
It refers to the differential equation which is approximated to the derivative of a continuous function in the following forms.
Forward Difference : 𝑢 𝑥 𝑥,𝑡 = 𝑢 𝑥+ℎ,𝑡 −𝑢(𝑥,𝑡) ℎ +𝑂 ℎ
Backward Difference : 𝑢 𝑥 𝑥,𝑡 = 𝑢 𝑥,𝑡 −𝑢(𝑥−ℎ,𝑡) ℎ +𝑂 ℎ
Central Difference : 𝑢 𝑥 𝑥,𝑡 = 𝑢 𝑥+ℎ,𝑡 −𝑢(𝑥−ℎ,𝑡) 2ℎ +𝑂 ℎ 2 .

14. 02 Finite Difference Method
For this, we constitutes a two-dimensional grid consisting of a variable of a partial differential equation.

15. 02
1. Explicit FDM
If we use the forward difference at time and a second-order
central difference for the space derivative at position 𝑥 𝑗 .
we get the recurrence equation:
                 
                                                            
We can obtain     from the other values this way :
                                                                                   
where                       

16. 02
2. Implicit FDM
If we use the backward difference at time and a second-order
central difference for the space derivative at position 𝑥 𝑗 .
we get the recurrence equation:
We can obtain from solving a system of linear equations:

17. 02 3. Crank-Nicolson method
Finally if we use the central difference at time
and a second-order central difference for the space derivative at position
we get the recurrence equation:
                                                                                                                                                    
We can obtain    from solving a system of linear equations:
                                                                 

18. 02
MATLAB Code – Implicit FDM , call option

19. 02
MATLAB Code

20. 03. Monte Carlo Simulation

21. 03 Monte Carlo Simulation
Monte Carlo Simulation is a problem solving technique used to approximate the probability of certain outcomes by running multiple trial runs, called simulations, using random variables.

22. 03
MATLAB Code
The following MATLAB scripts performs the Monte-carlo simulation.

23. 03
MATLAB Code

24. THANK YOU