Change of Time Method: Application to Mathematical Finance. I. Anatoliy Swishchuk Math & Comp Finance Lab Dept of Math & Stat, U of C ‘Lunch at the Lab’ Talk October 18, 2005
Outline Change of Time Method (CTM) for Martingale (Wiener Process) CTM in General Setting CTM for SDEs Geometrical Brownian Motion and CTM: Solution Black-Scholes Formula by CTM Cox-Ingersoll-Ross Process and CTM: Solution Variance and Volatility Swaps by CTM
CTM for Martingales
CTM in General Setting. I.
CTM in General Setting. II.
CTM for SDEs. I.
CTM for SDEs. II.
Idea of Proof. I.
Idea of Proof. II.
Geometric Brownian Motion
Change of Time Method for GBM
Solution for GBM Equation Using Change of Time
Properties of the Process
Properties of the Solution of GBM Using Change of Time Method
Option Pricing
European Call Option Pricing (Pay-Off Function)
European Call Option Pricing
Black-Scholes Formula
Stock Price under Risk-Neutral Measure
Explicit Expression for
European Call Option Through
Derivation of Black - Scholes Formula I
Derivation of Black-Scholes Formula II (continuation)
Derivation of Black - Scholes Formula III (continuation)
Derivation of Black - Scholes Formula IV (continuation)
Heston Model (Stochastic Volatility Model)
Explicit Solution for CIR Process: CTM
Proof. I.
Proof. II.
Properties of
Heston Model
Variance Swap for Heston Model. I.
Variance Swap for Heston Model. II.
Pricing of Variance Swap in Heston Model. I.
Pricing of Variance Swap in Heston Model. II.
Proof
Volatility Swap for Heston Model. I.
Volatility Swap for Heston Model. II.
Pricing of Volatility Swap for Heston Model. I.
Pricing of Volatility Swap for Heston Model. II.
Proof. I.
Proof. II.
Proof. III.
Proof. IV.
Proof. V.
References. I.
References. II.
References. III.
References. IV.
References. V.
References. VI.
References. VII.
References. VIII.
References. IX.
References. X. Elliott, R., Chan, L. and T. K. Siu (2005) “Pricing Volatility Swaps Under Heston's Volatility Model with Regime Switching ”
The End Thank you for your Attention!