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Financial Markets with Stochastic Volatilities

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,

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Financial Markets with Stochastic Volatilities

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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 • 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) Abstract Dynamical + Systems RE = 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 f(z(t))=V(t)f(z) 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 F=F(z,x) x=x(t,w) f(z(t,w))=V(t,w)f(z) Linear Stochastic Operator Equation dV(t,w)/dt=T(x(t,w))V(t.w) Nonlinear Ordinary Stochastic Differential Equation dz(t,w)/dt=F(z(t,w),x(t,w)))

  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)) <a , Z (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 forvariance and volatility swaps with stochastic volatility • He & Wang (RBC Financial Group) (2002)-variance, volatility, covariance, correlation swaps for deterministic volatility

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

  15. Volatility swaps are forward contracts on future realized stock volatility Variance swaps are forward contract on future realized stock variance Variance and Volatility Swaps 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); Kvar 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 VolatilitySwapWork?

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

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

  27. Standard Brownian Motion andGeometric 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 deterministic interest rate, follows Cox-Ingersoll-Ross (CIR) process

  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 forStochastic 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: We use second order Taylor expansion for square root function. To calculate volatility swap we need not only E{V} (as inthe case of variance swap), but also Var{V}.

  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 • 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 are used in practice to define discrete sampled variance and volatility Logarithmic Returns: 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 • An expiration time T>t; • A maximum number N of exercise times; • 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

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