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Selected Topics in Applied Mathematics – Computational Finance

Andrey Itkin http://www.chem.ucla.edu/~itkin Course web page http://www.chem.ucla.edu/~itkin/CompFinanceCourse/rutgers_course.html My email : itkin@chem.ucla.edu. Selected Topics in Applied Mathematics – Computational Finance. What is computational finance?. Why computational?

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Selected Topics in Applied Mathematics – Computational Finance

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

  2. 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

  3. 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

  4. 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

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

  6. 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

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

  8. 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

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

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

  11. 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

  12. t=0 t=1 (1+r)W uS p Stock Bond W S 1-p d<1+r<u 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

  13. 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

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

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

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

  17. 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

  18. 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

  19. 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

  20. 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

  21. 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

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

  23. 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

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

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

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

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

  28. 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

  29. 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

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

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