Andrey Itkin, Math Selected Topics in Applied Mathematics – Computational Finance Andrey Itkin Course web page ourse/rutgers_course.html My

Andrey Itkin, Math 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 Course outline? 1.Closed-form solutions (BS world, stochastic volatility and Heston world, interest rates and Vasichek and Hull-White world) 2.Almost closed-form solutions – FFT, Laplace transform 3.Traditional probabilistic solutions – binomial, trinomial and implied trees 4.Modern solutions – finite-difference 5.Last chance - world of Monte Carlo, stochastic integration 6.Calibration – gradient optimizers, Levenberg-Marquardt 7.Advanced optimization – pattern search 8.Specificity of various financial instruments – exotics, variance products, complex payoffs. 9.Programming issues: Design of financial software, Excel/VBA-C++ bridge, Matlab-C++ bridge 10.Levy processes, VG, SSM.

Andrey Itkin, Math (Numerical technique) Engineering Mathematics (Basic stochastic calculus) Finance (Derivative pricing And hedging) This course Excerpt from Yuji Yamada’s course

Andrey Itkin, Math Lecture 1 1.Short overview of stochastic calculus 2.All we have to know about Black-Scholes 3.Traditional approach – binomial trees

Andrey Itkin, Math 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 Movements in Time  t Su Sd S p 1 – p

Andrey Itkin, Math 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) = Se r  t = pSu + (1– p ) Sd Log-normal process: var = S 2 e 2r  t (e σ2  t -1 ) = E(Q 2 ) – [E(Q)] 2  2  t = pu 2 + (1– p )d 2 – [pu + (1– p )d ] 2

Andrey Itkin, Math Solution to equations 2 equations, 3 unknown. One free choice: Cox Ross Rubinstein (CRR)

Andrey Itkin, Math Alternative Solution By Jarrow and Rudd

Andrey Itkin, Math Pro and Contra CRR – it leads to negative probabilities when σ < |(r-q)√  t|. Jarrow and Rudd – not as easy to calculate gamma and rho. If many time steps are chosen – low performance

Andrey Itkin, Math An alternative exlanations. Single period binomial model ( Excerpt from Yuji Yamada’s course ) X 1 (uS)=  uS+  r)  X 1 (dS)=  dS+  r)  t=1 uS dS p 1-p t=0 S Stock (1+r)  d<1+r<u  Bond X 0 =  S+  Portfolio

Andrey Itkin, Math C0C0 Single period binomial model Compare with portfolio process Two equations for two unknowns Solve these equations for  and 

Andrey Itkin, Math Comparison principle for each state

Andrey Itkin, Math It is (notationally) convenient to regardandas probabilities : Risk neutral probability (real probability is irrelevant)

Andrey Itkin, Math Multi-period binomial lattice model Stock priceCall price Finite number of one step models

Andrey Itkin, Math 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 Stock priceCall price Apply one step pricing formula at each step, and solve backward until initial price is obtained.

Andrey Itkin, Math Perfect replication is possible Real probability is irrelevant Market is complete Risk neutral probability dominates the pricing formula Multi-period binomial lattice model

Andrey Itkin, Math Binomial Trees and Option Pricing (Two Fundamental Theorem of Asset Pricing (FTAP)) 1 st : T he 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 2 nd : 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 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 Binomial Trees and Option Pricing (Summary)

Andrey Itkin, Math American Put Option S 0 = 50; X = 50; r =10%;  = 40%; T = 5 months = ;  t = 1 month = The parameters imply u = ; d = ; a = ; p =

Andrey Itkin, Math Example (continued) Figure 18.3

Andrey Itkin, Math Calculation of Delta Delta is calculated from the nodes at time  t

Andrey Itkin, Math Calculation of Gamma Gamma is calculated from the nodes at time 2  t

Andrey Itkin, Math Calculation of Theta Theta is calculated from the central nodes at times 0 and 2  t

Andrey Itkin, Math 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 Trinomial Tree (Hull P.409) Again we want to match the mean and standard deviation of price changes. Terms of higher order than  t are ignored SS Sd Su pupu pmpm pdpd Equivalent to explicit FD of 1 st order

Andrey Itkin, Math Alternative solutions: Combine two steps of CRR: