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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,

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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,

University of Calgary, Calgary, Alberta

October 28 , 2004

  • Introduction
  • Research:

-Random Evolutions (REs);

-Applications of REs;


-Financial and Insurance Mathematics;

-Stochastic Models with Delay and Applications to Finance;

-Stochastic Models in Economics;

--Financial Mathematics: Option Pricing, Stability, Control, Swaps


--Swing Options

--Future Work

random evolutions re
Random Evolutions (RE)


Dynamical +


RE =

Random Media

Operator Evolution +




Random Process



applications of res
Applications of REs


Nonlinear Ordinary

Differential Equations


Linear Operator Equation







Linear Stochastic

Operator Equation


Nonlinear Ordinary

Stochastic Differential



another names for random evolutions
Another Names for Random Evolutions
  • Hidden Markov (or other) Models
  • Regime-Switching Models
applications of res biomathematics
Applications of REs (biomathematics)
  • Evolution of biological systems

Example: Logistic growth model

applications of res financial mathematics
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
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.

introduction to swaps
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

Bonds (bank accounts)


Forward contract

Swaps-agreements between two counterparts to exchange cash flows in the future to a prearrange formula


Security-a piece of paper representing a promise

Basic Securities

Derivative Securities

variance and volatility swaps
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)


types of volatilities
Types of Volatilities

Deterministic Volatility=

Deterministic Function of Time

Stochastic Volatility=

Deterministic Function of Time+Risk (“Noise”)

deterministic volatility
Deterministic Volatility
  • Realized (Observed) Variance and Volatility
  • Payoff for Variance and Volatility Swaps
  • Example
realized continuous deterministic variance and volatility
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
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
Volatility Swaps

A Volatility Swap is a forward contract on realized volatility.

Its payoff at expiration is equal to:

example payoff for volatility and variance swaps
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.

models of stock price
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 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 prices
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


Geometric Brownian Motion

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

standard brownian motion and geometric brownian motion
Standard Brownian Motion andGeometric Brownian Motion

Standard Brownian motion

Geometric Brownian motion

stochastic volatility models
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
Heston Model for Stock Price and Variance

Model for Stock Price (geometric Brownian motion):


deterministic interest rate,

follows Cox-Ingersoll-Ross (CIR) process

cox ingersoll ross cir model for stochastic volatility
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 .

valuing of variance swap for stochastic volatility
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},



valuing of volatility swap for stochastic volatility
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}.

calculation of var v
Calculation of Var[V]

Variance of V is equal to:

We need EV^2, because we have (EV)^2:

calculation of var v continuation
Calculation of Var[V] (continuation)

After calculations:

Finally we obtain:

numerical example s p60 canada index42
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

Logarithmic Returns:


realized discrete sampled variance and volatility
Realized Discrete Sampled Variance and Volatility

Realized Discrete Sampled Variance:

Realized Discrete Sampled Volatility:

swing options
Swing Options
  • Financial Instrument (derivative) consisting of
  • An expiration time T>t;
  • A maximum number N of exercise times;
  • The selection of exercise times


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
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

future work in financial mathematics
Future Work in Financial Mathematics
  • Swaps with Jumps
  • Swaps with Regime-Switching Components
  • Swing Options with Jumps
  • Swing Options with Regime-Switching Components