Change of Time Method: Application to Mathematical Finance. I.

1
Change of Time MethodApplication 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

2
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

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CTM for Martingales
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CTM in General Setting. I .
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CTM in General Setting. II.
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CTM for SDEs. I.
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CTM for SDEs. II.
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Idea of Proof. I.
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Idea of Proof. II.
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Geometric Brownian Motion
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Change of Time Method for GBM
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Solution for GBM EquationUsing Change of Time
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Properties of the Process
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Properties of the Solution of GBMUsing Change of
Time Method
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Option Pricing
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European Call Option Pricing(Pay-Off Function)
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European Call Option Pricing
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Black-Scholes Formula
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Stock Price under Risk-Neutral Measure
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Explicit Expression for
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European Call Option Through
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Derivation of Black - Scholes Formula I
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Derivation of Black-Scholes Formula II
(continuation)
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Derivation of Black - Scholes Formula III
(continuation)
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Derivation of Black - Scholes Formula IV
(continuation)
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Heston Model (Stochastic Volatility Model)
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Explicit Solution for CIR Process CTM
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Proof. I.
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Proof. II.
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Properties of
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Properties of
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Heston Model
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Variance Swap for Heston Model. I.
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Variance Swap for Heston Model. II.
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Pricing of Variance Swap in Heston Model. I.
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Pricing of Variance Swap in Heston Model. II.
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Proof
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Volatility Swap for Heston Model. I.
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Volatility Swap for Heston Model. II.
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Pricing of Volatility Swap for Heston Model. I.
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Pricing of Volatility Swap for Heston Model. II.
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Proof. I.
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Proof. II.
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Proof. III.
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Proof. IV.
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Proof. V.
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References. I.
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References. II.
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References. III.
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References. IV.
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References. V.
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References. VI.
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References. VII.
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References. VIII.
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References. IX.
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References. X.

• Elliott, R., Chan, L. and T. K. Siu (2005)
  Pricing Volatility Swaps Under Heston's
  Volatility Model with Regime Switching

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The End

• Thank you for your Attention!