1. Financial Markets with Stochastic Volatilities Anatoliy Swishchuk Mathematical and Computational Finance Lab Department of Mathematics & Statistics University of Calgary, Calgary, AB, Canada Seminar Talk Mathematical and Computational Finance Lab Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta October 28 , 2004

3. Random Evolutions (RE) RE = Abstract Dynamical + Systems Random Media Operator Evolution + Equations dV(t)/dt= T(x)V(t) Random Process x(t,w) dV(t,w)/dt=T(x(t,w))V(t,w)

4. Applications of REs Nonlinear Ordinary Differential Equations dz/dt=F(z ) Linear Operator Equation df(z(t))/dt=F(z(t))df(z(t))/dz dV(t)f/dt=TV(t)f T:=F(z)d/dz Nonlinear Ordinary Stochastic Differential Equation dz(t,w)/dt=F(z(t,w),x(t,w))) Linear Stochastic Operator Equation dV(t,w)/dt=T(x(t,w))V(t.w) F=F(z,x) x=x(t,w) f(z(t))=V(t)f(z) f(z(t,w))=V(t,w)f(z)

8. Applications of REs (Risk Process)

12. SDDE and Applications to Finance (Option Pricing and Continuous-Time GARCH Model)

16. Types of Volatilities Deterministic Volatility= Deterministic Function of Time Stochastic Volatility= Deterministic Function of Time+Risk (“Noise”)

18. Realized Continuous Deterministic Variance and Volatility Realized (or Observed) Continuous Variance: Realized Continuous Volatility: where is a stock volatility , is expiration date or maturity.

19. Variance Swaps A Variance Swap is a forward contract on realized variance. Its payoff at expiration is equal to N is a notional amount ($/variance); K var is a strike price ;

20. Volatility Swaps A Volatility Swap is a forward contract on realized volatility. Its payoff at expiration is equal to :

21. How does the Volatility Swap Work?

22. Example: Payoff for Volatility and Variance Swaps K var = (18%)^2; N = $50,000/( one volatility point )^2. Strike price K vol =18% ; Realized Volatility =21%; N =$50,000/( volatility point ). Payment(HF to D )=$50,000(21%-18%)=$150,000. For Volatility Swap : For Variance Swap : Payment(D to HF )=$50,000(18%-12%)=$300,000. b) volatility decreased to 12%: a) volatility increased to 21%:

24. Simulated Brownian Motion and Paths of Daily Stock Prices Simulated Brownian motion Paths of daily stock prices of 5 German companies for 3 years

25. Bachelier Model of Stock Prices 1). L. Bachelier (1900) introduced the first model for stock price based on Brownian motion Drawback of Bachelier model : negative value of stock price

26. 2). P. Samuelson (1965) introduced geometric (or economic, or logarithmic) Brownian motion Geometric Brownian Motion

27. Standard Brownian Motion and Geometric Brownian Motion Standard Brownian motion Geometric Brownian motion

29. Heston Model for Stock Price and Variance Model for Stock Price (geometric Brownian motion): or follows Cox-Ingersoll-Ross (CIR) process deterministic interest rate,

30. Heston Model: Variance follows CIR process or

31. Cox-Ingersoll-Ross (CIR) Model for Stochastic Volatility The model is a mean-reverting process, which pushes away from zero to keep it positive . The drift term is a restoring force which always points towards the current mean value .

32. Key Result: Explicit Solution for CIR Equation Solution: Here

33. Properties of the Process

34. Valuing of Variance Swap for Stochastic Volatility Value of Variance Swap (present value): where E is an expectation (or mean value), r is interest rate . To calculate variance swap we need only E{V}, where and

35. Calculation E[V]

36. Valuing of Volatility Swap for Stochastic Volatility Value of volatility swap: To calculate volatility swap we need not only E{V} ( as in the case of variance swap ), but also Var{V}. We use second order Taylor expansion for square root function .

37. Calculation of Var[V] Variance of V is equal to: We need EV^2 , because we have (EV)^2:

38. Calculation of Var[V] (continuation) After calculations: Finally we obtain:

39. Covariance and Correlation Swaps

40. Pricing Covariance and Correlation Swaps

41. Numerical Example: S&P60 Canada Index

43. Logarithmic Returns Logarithmic Returns: Logarithmic returns are used in practice to define discrete sampled variance and volatility where

44. Realized Discrete Sampled Variance and Volatility Realized Discrete Sampled Variance: Realized Discrete Sampled Volatility:

45. Statistics on Log-Returns of S&P60 Canada Index for 5 years (1997-2002)

46. Histograms of Log. Returns for S&P60 Canada Index

47. Figure 1: Convexity Adjustment

48. Figure 2: S&P60 Canada Index Volatility Swap

50. Pricing of Swing Options G(S) -payoff function (amount received per unit of the underlying commodity S if the option is exercised) b G (S)- reward, if b units of the swing are exercised

51. The Swing Option Value If then

53. Thank you for your attention !