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Some Applications of Log-Ergodic Processes: Ergodic Trading Model and Call Option Pricing Using the Irrational Rotation

Kiarash Firouzi, Mohammad Jelodari Mamaghani

Abstract

Due to the increasing popularity of futures trading among financial market participants, the risk management of these instruments is crucial. In this paper, we introduce a model for estimating the ideal time for leaving a trading position on a stock. Also, using ergodic theorems, we investigate the European call option pricing problem using a stochastic irrational rotation on the unit circle. Utilizing the properties of log-ergodic processes, we use the time average of the stochastic process of risky assets instead of expectations in our calculations.

Some Applications of Log-Ergodic Processes: Ergodic Trading Model and Call Option Pricing Using the Irrational Rotation

Abstract

Due to the increasing popularity of futures trading among financial market participants, the risk management of these instruments is crucial. In this paper, we introduce a model for estimating the ideal time for leaving a trading position on a stock. Also, using ergodic theorems, we investigate the European call option pricing problem using a stochastic irrational rotation on the unit circle. Utilizing the properties of log-ergodic processes, we use the time average of the stochastic process of risky assets instead of expectations in our calculations.

Paper Structure

This paper contains 15 sections, 18 theorems, 96 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Trading Model: Mean Reversion Timing for $Z_\delta$
  4. The Setup
  5. The Dynamics of Recurrence Times
  6. An Example
  7. Estimating $t_{M_i}$
  8. Irrational Rotation
  9. Stochastic Process $\theta_t$
  10. Properties of the Irrational Rotation
  11. European Call Option Pricing Using The Irrational Rotation
  12. Black–Scholes Ergodic Partial Differential Equation
  13. Solving the Heat Equation
  14. European Call Option Price: Solution of the Equation \ref{['BSP']}
  15. Conclusion

Key Result

Lemma 2.1

For all $c>0$ and any mean ergodic process $Y_t=D_t+R_t$, where $c$ is a constant.

Figures (3)

  • Figure 1: The plot of the stock price process $S_t$.
  • Figure 2: Corresponding mean ergodic $Z_\delta$ process of $S_t$.
  • Figure 3: The behavior of the process $R_{\theta_t}$ on the unit circle relative to the stock price.

Theorems & Definitions (25)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.1
  • Proposition 2.1
  • Remark 2.1
  • Theorem 2.2
  • Theorem 3.1
  • Theorem 3.2
  • Definition 4.1
  • ...and 15 more