1. The new results in mathematical finance

2. Mathematical finance Basic facts

3. The main goal of mathematical finance: mathematical modelling of the equity market Problems: The modelling of the primary assets prices The modelling of the derivatives prices (options, futures contracts) The management of the investment portfolio

4. Mathematical tool Probability theory Theory of random processes Optimization techniques Partial differential equations Mathematical statistics Computational mathematics

5. The elementary (binomial) market model In the market are available: -asset without risk B, which price varies in the determined way: -risk asset S, which price varies in the stochastic way. It is multiplied on u or d and is described by random walk on a binomial tree:

6. Option prices in the binomial model The Call option gives to its owner the right to buy the asset at the moment of time N under in advance stipulated price K There is the unique “fair” price, which doesn`t give possibility neither seller nor buyer to receive profit without risk:

7. The Black-Scholes model The price of asset B is The price of asset S is W – standard Brownian motion (continuous random walk) Trajectories of Brownian motion are continuous but nondifferentiable

8. Option prices in the model based on the Brownian motion Stochastic analysis studies such models Fair price of the Call option is determined by the Black-Scholes formula:

9. The rate of exchange: €/$, march 2009 Working out of the models considering unpredictable jumps is necessary

10. Portfolio choice: profitability and risk For the analysis of the stochastic price of a portfolio containing risk assets such characteristics as average profitability (mathematical expectation) and risk (dispersion) are used For the fixed value of average profitability there is a portfolio with the least dispersion According to the Markowitz theory the rational investor chooses the portfolio on the so-called effective front

11. Hedging with the set probability The hedge plays a role of the protective tool (insurance of transactions) on the equity market The top hedge: the seller of the financial commitment interests the start capital, which will allow him to pay off with probability 1 The quantile hedging: the minimal start capital is determined, which will allow to pay off with probability α:

12. A little history 1900 – Bachelier L. the theory of speculation: stochastic processes with contunious time, Brownian motion, option prices 1952 – The Markowitz theory: the idea of risk diversification 1973 – Transition to floating exchange rates, the beginning of active trade in options 1973 – Black, Scholes, Merton: Dynamic trading strategies, use of stochastic calculation, modern theory of option prices Probability theory, stochastic calculation: From mathematics to business Leading magazines: Finance and Stochastics, Mathematical Finance

13. The achievements of our department in mathematical finance

14. The calculation of the optimal portfolio in the «model with boundaries» The model description The problem definition

15. Let,then The calculation of the optimal portfolio capital

16. «The model with boundaries» Let us determine the sequence of “boundaries” So, in the intervals between the “boundaries” the parameters of the model take on the constant values.

17. «The model with three boudnaries»

18. The Monte Carlo method To partition the section [0,T] on parts and mark To discretize the Wiener process: To discretize the process S:

19. The Monte Carlo method To obtain the sequence of stopping moments the number of simulations of the process S To calculate the capital of the optimal portfolio:

20. The method of discrete approximation

21. The following recurrent formulas for the calculation of the optimal portfolio on the binary tree are true:

22. The parallel algorithm

23. For the following formula is true: the financial commitment and the banking account at the moment of time on the path number i of the binary tree The function displayBits(i) translates the binary number i to the decimal representation and writes to the massiv

24. The parallel algorithm Algorithm 1. 1. 2. 3. 4.

25. The parallel algorithm 5. If,then 4. 6. 7. 8. If,then 2. 9. 10. Return

26. The parallel algorithm Algorithm 2. 1. 2. 3.

27. The parallel algorithm 4. If,then 3. 5. 6. 7. Return Let us suppose that we have processors and

28. The parallel algorithm Algorithm 3. ……….

29. Calculation of the optimal portfolio The number of the risk assets

30. Fast pricing of American and barrier options under Levy processes Historical background. The problem of options pricing under Levy processes has attracted much attention in the recent years. Methods: drawbacks. Monte Carlo (MC) methods: slow Galerkin methods: implementation is complicated Finite difference schemes (FDS): application entails a detailed analysis of the underlying Levy process Wiener-Hopf factorization (FWH) method: no efficient numerical realization of the method was suggested in general case

31. FWH-method The main goals: To suggest universal, fast, simple and efficient method for pricing American and barrier options under a wide class of Levy processes To demonstrate the accuracy and fast convergence of the method comparing with the results obtained by other methods

32. FWH-method: theoretical background Carrs randomization reduces the pricing problem to a sequence of stationary problems Each problem in the sequence can be solved explicitly using FWH method Explicit formulas in terms of expected present value (EPV) operators are available However, no efficient numerical realization of the action of EPV-operators was suggested

33. FWH-method: main ideas FWH-method FWH-method is based on an efficient approximation of the Wiener-Hopf factors in the exact formula for the solution and real fast Fourier transform (FFT) and inverse fast Fourier transform (iFFT). Key ideas Symbol of EPV-operator in the exact formula for the solution – characteristic function of infinitely divisible distribution (i.d.d.) Factorization of the characteristic function into the product of the leading function and the subordinate function The leading function is the characteristic function of Variance Gamma (VG) distribution without drift The subordinate function can be factorized by using Poisson type approximation Real FFT and iFFT can be applied

34. The generalized Black-Scholes equation for the down-and-out put option S.I.Boyarchenko and S.Z.Levendorskii (2000,2002): the option price is the solution of the boundary problem

35. The algorithm of problem solving The reduction to the family of one-dimensional problems on the axle for the integro-differential (pseudodifferential(PDO)) equation by the instrumentality of the following methods: -the method of horizontal lines with time discretization -the method of one-sided Laplase transformation regarding to the temporal variable The solution of the family problems by the instrumentality of the approximate Wiener-Hopf factorization -to apply the Wiener-Hopf factorization method in the operating form. To write the solution in terms of operators and to apply the FWH-method. -the FDS reduces the problem to the solution of the system with Teplith matrix. Then apply the approximate Wiener-Hoph factorization and write the approximate solution in terms of Lorenth matrices The detection of the approximate solution of the problem on the basis of the solutions of the family of one-dimensional problems

36. Analytic method of lines or Carr randomization Carr randomization reduces the pricing problem to a sequence of stationary problems.

37. FDS Construction of any FDS involves: -discretization in space and time -truncation of large jumps -approximation of small jumps The result is a linear system that needs to be solved at each time step.

38. FWH-method The main contribution of FWH-method is an efficient numerical realization of EPV-operators. Pricing barrier options For s=N-1, N-2, …, 0 define Then

39. FWH-method Wiener-Hopf factorization formula Let q>0, T~Exp(q) and are the supremum and infimum processes EPV-operators PDO The operator form of FWH:

40. Construction of approximation of factors in the Wiener-Hopf factorization formula Setup - order of Levy process - characterictic exponent

41. Explicit formulas for approximations of For small positive d and large M= set

42. DFT and RDFT The discrete Fourier transform (DFT) DFT for real-valued functions In our case, the data consist of a real-valued array The resulting transform satisfies: It has the same “degrees of freedom” as the original real data set. To distinguish DFT of real functions we will use notation RDFT.

43. Approximation of using FFT Approximation of Fix the space step d>0 and number of the space points The partition of the normalized log-price domain: The partition of the frequency domain:

44. Approximation of using FFT Approximation of Set Approximation of

45. Summary: FWH-method FWH method is more accurate than FDS method, because it deals with the approximation of the inverse operators The approximation does not imply that the option value is of class Approximate formulas in FWH-method for are needed at the first and last steps At all intermediate steps the exact analytic expression is used The results obtained by FWH, MC and FDS methods are in extremely good agreement

46. Future work: FWH-method Areas of potential research: Efficient computation of the Greeks Regime switching Levy models The Heston model The Bates model

47. Tempered stable Levy processes (TSL) We compare the results obtained by MC, FDS and FWH methods. FDS-method is described in S.Levendorskii, O.Kudryavtsev, V.Zherder (2005). For the MC calculations we used 500000 paths with time step=0.00005. For simulating trajectories of the tempered stable (KoBoL process) we implemented the code of J.Poirot and P.Tankov (www.math.jussieu.fr/~tankov/). Thewww.math.jussieu.fr/~tankov/ program uses the algorithm in Madan and Yor (2005), see also Poirot and Tankov (2006). We consider the down-and-out put option with strike K, barrier H and time to expiry T. PC-characteristics: Intel Core (TM) 2 Due CPU, 1.8 GHz, RAM 1024 Mb, under Windows Vista.

48. TSL as example: MC, FDS and FWH methods. Down-and-out put prices is TSL model, KoBoL parameters: Option parameters: Algorithm parameters: d – space step, N – number of time steps, S – spot price.

49. The large market model Kabanov, Kramkov (1994) The earlier model: APT Ross (1976), Huberman (1982) The sequence of “small” markets on the filtered probability spaces On the market number n the discount assets prices are described by vector semimartingale The investor capital integrable process

50. The asymptotic arbitrage Let - the set of capital-processes on the “small” market number n On the large market there is no asymptotic arbitrage (NAA) if On the large market there is strong asymptotic arbitrage (SAA) if

51. The diffusion models - independent standard Wiener processes The discount prices of risk assets The etalon portfolio

52. The diffusion models NAA SAA The similar results: Kabanov, Kramkov (1998) The assumptions weaker: the etalon portfolios may exist without eqvivalent local martingale measures on the “small” markets

53. The discrete time market model with infinite horizon and one log-normal stock -independent stochastic values with standard normal distribution One risk asset sequences are not stochastic Lets consider the sequence of small markets on the time intervals [0,…,n].

54. The discrete time market model with infinite horizon and one log-normal stock Shiryaev (1998): Let

55. The discrete time market model with infinite horizon and one log-normal stock Theorem.

56. Thank you!