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

2. Outline Introduction Research: -Random Evolutions (REs); -Applications of REs; -Biomathematics; -Financial and Insurance Mathematics; -Stochastic Models with Delay and Applications to Finance; -Stochastic Models in Economics; --Financial Mathematics: Option Pricing, Stability, Control, Swaps --Swaps --Swing Options --Future Work

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)

5. Another Names for Random Evolutions Hidden Markov (or other) Models Regime-Switching Models

6. Applications of REs (traffic process) Traffic Process

7. Applications of REs (Storage Processes) Storage Processes

8. Applications of REs (Risk Process)

9. Applications of REs (biomathematics) Evolution of biological systems Example: Logistic growth model

10. Applications of REs (Financial Mathematics) Financial Mathematics ((B,S)-security market in random environment or regime- switching (B,S)-security market or hidden Markov (B,S)-security market)

11. Application of REs (Financial Mathematics) Pricing Electricity Calls (R. Elliott, G. Sick and M. Stein, September 28, 2000, working paper) The spot price S (t) of electricity S (t)=f (t) g (t) exp (X (t)), where f (t) is an annual periodic factor, g (t) is a daily periodic factor, X (t) is a scalar diffusion factor, Z (t) is a Markov chain.

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

13. Introduction to Swaps Bachelier (1900)-used Brownian motion to model stock price Samuelson (1965)-geometric Brownian motion Black-Scholes (1973)-first option pricing formula Merton (1973)-option pricing formula for jump model Cox, Ingersoll & Ross (1985), Hull & White (1987) - stochastic volatility models Heston (1993)-model of stock price with stochastic volatility Brockhaus & Long (2000)-formulae for variance and volatility swaps with stochastic volatility He & Wang (RBC Financial Group) (2002)-variance, volatility, covariance, correlation swaps for deterministic volatility

14. Swaps Stock Bonds ( bank accounts ) Option Forward contract Swaps-agreements between two counterparts to exchange cash flows in the future to a prearrange formula Basic SecuritiesDerivative Securities Security -a piece of paper representing a promise

15. Variance and Volatility Swaps Volatility swaps are forward contracts on future realized stock volatility Variance swaps are forward contract on future realized stock variance Forward contract-an agreement to buy or sell something at a future date for a set price (forward price) Variance is a measure of the uncertainty of a stock price. Volatility (standard deviation) is the square root of the variance (the amount of “noise”, risk or variability in stock price) Variance=(Volatility)^2

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

17. Deterministic Volatility Realized (Observed) Variance and Volatility Payoff for Variance and Volatility Swaps Example

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%:

23. Models of Stock Price Bachelier Model (1900)-first model Samuelson Model (1965)- Geometric Brownian Motion-the most popular

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

28. Stochastic Volatility Models Cox-Ingersol-Ross (CIR) Model for Stochastic Volatility Heston Model for Stock Price with Stochastic Volatility as CIR Model Key Result: Explicit Solution of CIR Equation! We Use New Approach-Change of Time-to Solve CIR Equation Valuing of Variance and Volatility Swaps for Stochastic Volatility

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

42. We apply the obtained analytical solutions to price a swap on the volatility of the S&P60 Canada Index for five years (January 1997- February 2002) These data were kindly presented to author by Raymond Theoret (University of Quebec, Montreal, Quebec,Canada) and Pierre Rostan (Bank of Montreal, Montreal, Quebec,Canada)

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

49. Swing Options Financial Instrument (derivative) consisting of 1)An expiration time T>t; 2)A maximum number N of exercise times; 3)The selection of exercise times t1<=t2<=…<=tN; 4) the selection of amounts x1,x2,…, xN, xi=>0, i=1,2,…,N, so that x1+x2+…+xN<=H; 5) A refraction time d such that t<=t1<t1+d<=t2<t2+d<=t3<=…<=tN<=T; 6) There is a bound M such that xi<=M, i=1,2,…,N.

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

52. Future Work in Financial Mathematics Swaps with Jumps Swaps with Regime-Switching Components Swing Options with Jumps Swing Options with Regime-Switching Components

53. Thank you for your attention !