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Selected Topics in Applied Mathematics – Computational Finance

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Andrey Itkin

http://www.chem.ucla.edu/~itkin

Course web page

http://www.chem.ucla.edu/~itkin/CompFinanceCourse/rutgers_course.html

My email: [email protected]

Selected Topics in Applied Mathematics – Computational FinanceAndrey Itkin, Math 612-02

What is computational finance?

- Why computational?
- New sophisticated models
- Performance issue
- Calibration
- Data issue and historical data
- Market demand for quant people
- Pre-requisites:
- Stochastic calculus and related math
- Financial models
- Numerical methods
- Programming

Result: CF - very complex subject

Andrey Itkin, Math 612-02

Course outline?

- Closed-form solutions (BS world, stochastic volatility and Heston world, interest rates and Vasichek and Hull-White world)
- Almost closed-form solutions – FFT, Laplace transform
- Traditional probabilistic solutions – binomial, trinomial and implied trees
- Modern solutions – finite-difference
- Last chance - world of Monte Carlo, stochastic integration
- Calibration – gradient optimizers, Levenberg-Marquardt
- Advanced optimization – pattern search
- Specificity of various financial instruments – exotics, variance products, complex payoffs.
- Programming issues: Design of financial software, Excel/VBA-C++ bridge, Matlab-C++ bridge
- Levy processes, VG, SSM.

Andrey Itkin, Math 612-02

Mathematics

(Basic stochastic

calculus)

Engineering

Finance

(Numerical

technique)

(Derivative pricing

And hedging)

Excerpt from Yuji Yamada’s course

This course

Andrey Itkin, Math 612-02

Lecture 1

- Short overview of stochastic calculus
- All we have to know about Black-Scholes
- Traditional approach – binomial trees

Andrey Itkin, Math 612-02

Binomial Trees

- Binomial trees are used to approximate the movements in the price of a stock or other asset
- In each small interval of time the stock price is assumed to move up by a proportional amount u or to move down by a proportional amount d

Andrey Itkin, Math 612-02

Equation of tree Parameters

- We choose the tree parameters p, u, and d so that the tree gives correct values for the mean & standard deviation of the stock price changes in a risk-neutral world

(from John Hull: ) – the expected value of the stock price

E(Q) = Serdt= pSu + (1– p ) Sd

Log-normal process: var = S2e2rdt(eσ2dt -1) = E(Q2) – [E(Q)]2

s2dt = pu2 + (1– p )d 2 – [pu + (1– p )d ]2

Andrey Itkin, Math 612-02

Solution to equations

2 equations, 3 unknown. One free choice:

Cox Ross Rubinstein

(CRR)

Andrey Itkin, Math 612-02

Pro and Contra

- CRR – it leads to negative probabilities when σ < |(r-q)√dt|.
- Jarrow and Rudd – not as easy to calculate gamma and rho.
- If many time steps are chosen – low performance

Andrey Itkin, Math 612-02

t=0

t=1

(1+r)W

uS

p

Stock

Bond

W

S

1-p

d<1+r

dS

(1+r)W

Portfolio

X0=DS+qW

An alternative exlanations. Single period binomial model

(Excerpt from Yuji Yamada’s course)

X1(uS)=DuS+q(1+r)W

X1(dS)=DdS+q(1+r)W

Andrey Itkin, Math 612-02

C0

- Compare with portfolio process

Two equations for

two unknowns

Single period binomial model

Solve these equations for D and q

Andrey Itkin, Math 612-02

It is (notationally) convenient to regard

and

as probabilities

: Risk neutral probability (real probability is irrelevant)

Andrey Itkin, Math 612-02

Stock price

Call price

Finite number

of one step

models

Multi-period binomial lattice model

Andrey Itkin, Math 612-02

Backwards Induction

- We know the value of the option at the final nodes
- We work back through the tree using risk-neutral valuation to calculate the value of the option at each node, testing for early exercise when appropriate

Andrey Itkin, Math 612-02

Stock price

Call price

Apply one step pricing formula at each step, and solve

backward until initial price is obtained.

Andrey Itkin, Math 612-02

Market is complete

Multi-period binomial lattice model

- Perfect replication is possible

- Real probability is irrelevant

- Risk neutral probability dominates the pricing formula

Andrey Itkin, Math 612-02

Binomial Trees and Option Pricing(Two Fundamental Theorem of Asset Pricing (FTAP))

- 1st: The no-arbitrage assumption implies there exists (at least) a probability measure Q called risk-neutral, or risk-adjusted, or equivalent martingale measure, under which the discounted prices are martingales
- 2nd: Assuming complete market and no-arbitrage: there exists a unique risk-adjusted probability measure Q; any contingent claim has a unique price that is the discounted Q-expectation of its final pay-off

Andrey Itkin, Math 612-02

Binomial Trees and Option Pricing(Cox-Ross-Rubinstein Formula)

- Cox-Ross-Rubinstein Formula:
- J is the set of integers between 0 and N:
- Risk-neutral probability:

Andrey Itkin, Math 612-02

Binomial Trees and Option Pricing(Summary)

Andrey Itkin, Math 612-02

American Put Option

S0 = 50; X = 50; r =10%; s = 40%;

T = 5 months = 0.4167;

dt = 1 month = 0.0833

The parameters imply

u = 1.1224; d = 0.8909;

a = 1.0084; p = 0.5076

Andrey Itkin, Math 612-02

Example (continued)Figure 18.3

Andrey Itkin, Math 612-02

Calculation of Theta

Theta is calculated from the central nodes at times 0 and 2dt

Andrey Itkin, Math 612-02

Calculation of Vega

- We can proceed as follows
- Construct a new tree with a volatility of 41% instead of 40%.
- Value of option is 4.62
- Vega is

Andrey Itkin, Math 612-02

Su

pu

pm

S

S

pd

Sd

Trinomial Tree (Hull P.409)Again we want to match the mean and standard deviation of price changes. Terms of higher order than dtare ignored

Equivalent to explicit FD of 1st order

Andrey Itkin, Math 612-02

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