1. Andrey Itkin, Math 612-02 Selected Topics in Applied Mathematics – Computational Finance Andrey Itkin http://www.chem.ucla.edu/~itkin Course web page http://www.chem.ucla.edu/~itkin/CompFinanceC ourse/rutgers_course.html My email: itkin@chem.ucla.edu

2. Andrey 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

3. Andrey Itkin, Math 612-02 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.

4. Andrey Itkin, Math 612-02 (Numerical technique) Engineering Mathematics (Basic stochastic calculus) Finance (Derivative pricing And hedging) This course Excerpt from Yuji Yamada’s course

5. Andrey Itkin, Math 612-02 Lecture 1 1.Short overview of stochastic calculus 2.All we have to know about Black-Scholes 3.Traditional approach – binomial trees

6. 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

7. Andrey Itkin, Math 612-02 Movements in Time  t Su Sd S p 1 – p

8. 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) = 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

9. Andrey Itkin, Math 612-02 Solution to equations 2 equations, 3 unknown. One free choice: Cox Ross Rubinstein (CRR)

10. Andrey Itkin, Math 612-02 Alternative Solution By Jarrow and Rudd

11. Andrey Itkin, Math 612-02 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

12. Andrey Itkin, Math 612-02 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

13. Andrey Itkin, Math 612-02 C0C0 Single period binomial model Compare with portfolio process Two equations for two unknowns Solve these equations for  and 

14. Andrey Itkin, Math 612-02 Comparison principle for each state

15. Andrey Itkin, Math 612-02 It is (notationally) convenient to regardandas probabilities : Risk neutral probability (real probability is irrelevant)

16. Andrey Itkin, Math 612-02 Multi-period binomial lattice model Stock priceCall price Finite number of one step models

17. 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

18. Andrey Itkin, Math 612-02 Stock priceCall price Apply one step pricing formula at each step, and solve backward until initial price is obtained.

19. Andrey Itkin, Math 612-02 Perfect replication is possible Real probability is irrelevant Market is complete Risk neutral probability dominates the pricing formula Multi-period binomial lattice model

20. Andrey Itkin, Math 612-02 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

21. 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:

22. Andrey Itkin, Math 612-02 Binomial Trees and Option Pricing (Summary)

23. Andrey Itkin, Math 612-02 American Put Option S 0 = 50; X = 50; r =10%;  = 40%; T = 5 months = 0.4167;  t = 1 month = 0.0833 The parameters imply u = 1.1224; d = 0.8909; a = 1.0084; p = 0.5076

24. Andrey Itkin, Math 612-02 Example (continued) Figure 18.3

25. Andrey Itkin, Math 612-02 Calculation of Delta Delta is calculated from the nodes at time  t

26. Andrey Itkin, Math 612-02 Calculation of Gamma Gamma is calculated from the nodes at time 2  t

27. Andrey Itkin, Math 612-02 Calculation of Theta Theta is calculated from the central nodes at times 0 and 2  t

28. 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

29. Andrey Itkin, Math 612-02 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

30. Andrey Itkin, Math 612-02 Alternative solutions: Combine two steps of CRR: