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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
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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
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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)
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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)
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Another Names for Random Evolutions Hidden Markov (or other) Models Regime-Switching Models
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Applications of REs (traffic process) Traffic Process
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Applications of REs (Storage Processes) Storage Processes
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Applications of REs (Risk Process)
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Applications of REs (biomathematics) Evolution of biological systems Example: Logistic growth model
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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)
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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.
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SDDE and Applications to Finance (Option Pricing and Continuous-Time GARCH Model)
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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
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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
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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
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Types of Volatilities Deterministic Volatility= Deterministic Function of Time Stochastic Volatility= Deterministic Function of Time+Risk (“Noise”)
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Deterministic Volatility Realized (Observed) Variance and Volatility Payoff for Variance and Volatility Swaps Example
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Realized Continuous Deterministic Variance and Volatility Realized (or Observed) Continuous Variance: Realized Continuous Volatility: where is a stock volatility, is expiration date or maturity.
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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 ;
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Volatility Swaps A Volatility Swap is a forward contract on realized volatility. Its payoff at expiration is equal to :
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How does the Volatility Swap Work?
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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%:
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Models of Stock Price Bachelier Model (1900)-first model Samuelson Model (1965)- Geometric Brownian Motion-the most popular
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Simulated Brownian Motion and Paths of Daily Stock Prices Simulated Brownian motion Paths of daily stock prices of 5 German companies for 3 years
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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
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2). P. Samuelson (1965) introduced geometric (or economic, or logarithmic) Brownian motion Geometric Brownian Motion
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Standard Brownian Motion and Geometric Brownian Motion Standard Brownian motion Geometric Brownian motion
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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
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Heston Model for Stock Price and Variance Model for Stock Price (geometric Brownian motion): or follows Cox-Ingersoll-Ross (CIR) process deterministic interest rate,
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Heston Model: Variance follows CIR process or
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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.
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Key Result: Explicit Solution for CIR Equation Solution: Here
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Properties of the Process
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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
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Calculation E[V]
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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.
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Calculation of Var[V] Variance of V is equal to: We need EV^2, because we have (EV)^2:
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Calculation of Var[V] (continuation) After calculations: Finally we obtain:
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Covariance and Correlation Swaps
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Pricing Covariance and Correlation Swaps
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Numerical Example: S&P60 Canada Index
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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)
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Logarithmic Returns Logarithmic Returns: Logarithmic returns are used in practice to define discrete sampled variance and volatility where
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Realized Discrete Sampled Variance and Volatility Realized Discrete Sampled Variance: Realized Discrete Sampled Volatility:
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Statistics on Log-Returns of S&P60 Canada Index for 5 years (1997-2002)
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Histograms of Log. Returns for S&P60 Canada Index
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Figure 1: Convexity Adjustment
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Figure 2: S&P60 Canada Index Volatility Swap
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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.
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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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The Swing Option Value If then
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Future Work in Financial Mathematics Swaps with Jumps Swaps with Regime-Switching Components Swing Options with Jumps Swing Options with Regime-Switching Components
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Thank you for your attention !
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