Financial Markets with Stochastic Volatilities - markov modelling
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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), aka Markov models;
    • -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)) <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 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 Securities Derivative 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. 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: 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
    • 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
  • 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 !