financial markets with stochastic volatilities - markov modelling
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Financial Markets with Stochastic Volatilities
Anatoliy SwishchukMathematical and Computational Finance Lab
Department of Mathematics & StatisticsUniversity of Calgary, Calgary, AB, Canada
Seminar TalkMathematical and Computational Finance Lab
Department of Mathematics and Statistics,
University of Calgary, Calgary, Alberta
October 28 , 2004
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
Random Evolutions (RE)
RE = Abstract Dynamical + Systems
Random Media
Operator Evolution +EquationsdV(t)/dt= T(x)V(t)
Random Process
x(t,w)
dV(t,w)/dt=T(x(t,w))V(t,w)
Applications of REs
Nonlinear Ordinary Differential Equations
dz/dt=F(z)
Linear Operator Equationdf(z(t))/dt=F(z(t))df(z(t))/dzdV(t)f/dt=TV(t)fT:=F(z)d/dz
Nonlinear Ordinary Stochastic Differential Equationdz(t,w)/dt=F(z(t,w),x(t,w)))
Linear Stochastic Operator EquationdV(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)
Another Names for Random Evolutions
• Hidden Markov (or other) Models
• Regime-Switching Models
Applications of REs (traffic process)
• Traffic Process
Applications of REs (Storage Processes)
• Storage Processes
Applications of REs (Risk Process)
Applications of REs (biomathematics)
• Evolution of biological systems
Example: Logistic growth model
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)
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 scalardiffusion factor, Z (t) is a Markov chain.
SDDE and Applications to Finance(Option Pricing and Continuous-Time
GARCH Model)
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
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
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
Types of Volatilities
Deterministic Volatility=Deterministic Function of Time
Stochastic Volatility=Deterministic Function of Time+Risk (“Noise”)
Deterministic Volatility
• Realized (Observed) Variance and Volatility
• Payoff for Variance and Volatility Swaps
• Example
Realized Continuous Deterministic Variance and Volatility
Realized (or Observed) Continuous Variance:
Realized Continuous Volatility:
where is a stock volatility, is expiration date or maturity.
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;
Volatility Swaps
A Volatility Swap is a forward contract on realized volatility.
Its payoff at expiration is equal to:
How does the Volatility Swap Work?
Example: Payoff for Volatility and Variance Swaps
Kvar = (18%)^2; N = $50,000/(one volatility point)^2.
Strike price Kvol =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%:
Models of Stock Price
• Bachelier Model (1900)-first model
• Samuelson Model (1965)- Geometric Brownian Motion-the most popular
Simulated Brownian Motion and Paths of Daily Stock Prices
Simulated Brownian motion
Paths of daily stock prices of 5 German companies for 3 years
Bachelier Model of Stock Prices1). L. Bachelier (1900) introduced the first model for stock price based on Brownian motion
Drawback of Bachelier model: negative value of stock price
2). P. Samuelson (1965) introduced geometric (or economic, or logarithmic) Brownian motion
Geometric Brownian Motion
Standard Brownian Motion andGeometric Brownian Motion
Standard Brownian motion
Geometric Brownian motion
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
Heston Model for Stock Price and Variance
Model for Stock Price (geometric Brownian motion):
or
follows Cox-Ingersoll-Ross (CIR) process
deterministic interest rate,
Heston Model: Variance follows CIR process
or
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 pointstowards the current mean value .
Key Result: Explicit Solution for CIR Equation
Solution:
Here
Properties of the Process
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
Calculation E[V]
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.
Calculation of Var[V]
Variance of V is equal to:
We need EV^2, because we have (EV)^2:
Calculation of Var[V] (continuation)After calculations:
Finally we obtain:
Covariance and Correlation Swaps
Pricing Covariance and Correlation Swaps
Numerical Example:S&P60 Canada Index
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)
Logarithmic Returns
Logarithmic Returns:
Logarithmic returns are used in practice to define discrete sampled variance and volatility
where
Realized Discrete Sampled Variance and Volatility
Realized Discrete Sampled Variance:
Realized Discrete Sampled Volatility:
Statistics on Log-Returns of S&P60 Canada Index for 5 years (1997-2002)
Histograms of Log. Returns for S&P60 Canada Index
Figure 1: Convexity Adjustment
Figure 2: S&P60 Canada Index Volatility Swap
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.
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
The Swing Option Value
If
then
Future Work in Financial Mathematics
• Swaps with Jumps
• Swaps with Regime-Switching Components
• Swing Options with Jumps
• Swing Options with Regime-Switching Components
Thank you for your attention !