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- 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 <ul><li>Introduction </li></ul><ul><li>Research: </li></ul><ul><li>-Random Evolutions (REs), aka Markov models; </li></ul><ul><li>-Applications of REs; </li></ul><ul><li>-Biomathematics; </li></ul><ul><li>-Financial and Insurance Mathematics; </li></ul><ul><li>-Stochastic Models with Delay and Applications to Finance; </li></ul><ul><li>-Stochastic Models in Economics; </li></ul><ul><li>--Financial Mathematics: Option Pricing, Stability, Control, Swaps </li></ul><ul><li>--Swaps </li></ul><ul><li>--Swing Options </li></ul><ul><li>--Future Work </li></ul>
- 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 <ul><li>Hidden Markov (or other) Models </li></ul><ul><li>Regime-Switching Models </li></ul>
- 6. Applications of REs (traffic process) <ul><li>Traffic Process </li></ul>
- 7. Applications of REs (Storage Processes) <ul><li>Storage Processes </li></ul>
- 8. Applications of REs (Risk Process)
- 9. Applications of REs (biomathematics) <ul><li>Evolution of biological systems </li></ul>Example: Logistic growth model
- 10. Applications of REs (Financial Mathematics) <ul><li>Financial Mathematics ((B,S)-security market in random environment or regime-switching (B,S)-security market or hidden Markov (B,S)-security market) </li></ul>
- 11. Application of REs (Financial Mathematics) <ul><li>Pricing Electricity Calls (R. Elliott, G. Sick and M. Stein, September 28, 2000, working paper) </li></ul><ul><li>The spot price S (t) of electricity </li></ul><ul><li>S (t)=f (t) g (t) exp (X (t)) <a , Z (t))>, </li></ul><ul><li>where f (t) is an annual periodic factor, g (t) </li></ul><ul><li>is a daily periodic factor, X (t) is a scalar </li></ul><ul><li>diffusion factor, Z (t) is a Markov chain. </li></ul>
- 12. SDDE and Applications to Finance (Option Pricing and Continuous-Time GARCH Model)
- 13. Introduction to Swaps <ul><li>Bachelier (1900)- used Brownian motion to model stock price </li></ul><ul><li>Samuelson (1965)- geometric Brownian motion </li></ul><ul><li>Black-Scholes (1973)- first option pricing formula </li></ul><ul><li>Merton (1973)- option pricing formula for jump model </li></ul><ul><li>Cox, Ingersoll & Ross (1985), Hull & White (1987) - stochastic volatility models </li></ul><ul><li>Heston (1993)- model of stock price with stochastic volatility </li></ul><ul><li>Brockhaus & Long (2000)- formulae for variance and volatility swaps with stochastic volatility </li></ul><ul><li>He & Wang (RBC Financial Group) (2002)- variance, volatility, covariance, correlation swaps for deterministic volatility </li></ul>
- 14. Swaps <ul><li>Stock </li></ul><ul><li>Bonds ( bank accounts ) </li></ul><ul><li>Option </li></ul><ul><li>Forward contract </li></ul><ul><li>Swaps - agreements between two counterparts to exchange cash flows in the future to a prearrange formula </li></ul>Basic Securities Derivative Securities Security - a piece of paper representing a promise
- 15. Variance and Volatility Swaps <ul><li>Volatility swaps are forward contracts on future realized stock volatility </li></ul><ul><li>Variance swaps are forward contract on future realized stock variance </li></ul>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 <ul><li>Realized (Observed) Variance and Volatility </li></ul><ul><li>Payoff for Variance and Volatility Swaps </li></ul><ul><li>Example </li></ul>
- 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 <ul><li>Bachelier Model (1900)-first model </li></ul><ul><li>Samuelson Model (1965)- Geometric Brownian Motion-the most popular </li></ul>
- 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 <ul><li>Cox-Ingersol-Ross (CIR) Model for Stochastic Volatility </li></ul><ul><li>Heston Model for Stock Price with Stochastic Volatility as CIR Model </li></ul><ul><li>Key Result: Explicit Solution of CIR Equation! </li></ul><ul><li>We Use New Approach-Change of Time-to Solve CIR Equation </li></ul><ul><li>Valuing of Variance and Volatility Swaps for Stochastic Volatility </li></ul>
- 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. Numerical Example: S&P60 Canada Index <ul><li>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) </li></ul><ul><li>These data were kindly presented to author by </li></ul><ul><li>Raymond Theoret (University of Quebec, </li></ul><ul><li>Montreal, Quebec,Canada) and Pierre Rostan </li></ul><ul><li>(Bank of Montreal, Montreal, Quebec,Canada) </li></ul>
- 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 <ul><li>Financial Instrument (derivative) consisting of </li></ul><ul><li>An expiration time T>t; </li></ul><ul><li>A maximum number N of exercise times; </li></ul><ul><li>The selection of exercise times </li></ul><ul><li>t1<=t2<=…<=tN; </li></ul><ul><li>4) the selection of amounts x1,x2,…, xN, xi=>0, i=1,2,…,N, so that x1+x2+…+xN<=H; </li></ul><ul><li>5) A refraction time d such that t<=t1<t1+d<=t2<t2+d<=t3<=…<=tN<=T; </li></ul><ul><li>6) There is a bound M such that xi<=M, i=1,2,…,N. </li></ul>
- 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 <ul><li>Swaps with Jumps </li></ul><ul><li>Swaps with Regime-Switching Components </li></ul><ul><li>Swing Options with Jumps </li></ul><ul><li>Swing Options with Regime-Switching Components </li></ul>
- 53. Thank you for your attention !

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