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 !