Deterministic and Stochastic Topics in Computational Finance

The following sections are included:

• Preface
• Chapters Diagram
• List of Notations and Symbols
• Contents

The following sections are included:

• Determinism and Semi-determinism
• How to Measure
• An Uncertainty Principle
• Market Completeness
• Market Efficiency
• Stopping Times
• Martingales and Submartingales
• Optional Stopping Theorem
• Can Random be Deterministic?
• Change of Time Scale

Regression is used when calibrating financial models to the market, i.e. finding the values of the model parameters for which the model best fits the market observations with respect to a certain goodness of fit measure. Regression can be either deterministic, or stochastic. The former looks for a best fit of the market data by a curve, while the latter approximates the data by a stochastic process, which depends on some parameters. Regression provides a procedure by which these parameters can be inferred such that the model approximates the true data, minimizing a certain error.

Elementary Calculus provides powerful methods for modeling phenomena from the real world. However, the real world is imperfect, due to multiple perturbation effects, and in order to study it properly, one needs to employ methods of Stochastic Calculus.

This chapter deals with the following three financial equivalent instruments: bonds, forward rates and yield curves. Knowing any one of these provides full information on the other two. This makes possible to approach the interest rate market from multiple perspectives. The second part of the chapter presents several bond prices computations in the case when the spot rate is stochastic.

The price of a stock can be modeled by a continuous stochastic process which is the sum between a predictable and an unpredictable part. However, this type of model does not take into account market crashes. If those are to be taken into consideration, the stock price needs to contain a third component that models unexpected jumps. We shall also discuss the time when a stock reaches a given barrier, correlation of two stocks, the running maximum distribution of a stock and stock averages such as arithmetic, geometric and harmonic.

In this chapter we shall present and use the method of risk-neutral valuation for a large number of European options from simple to complex. The main assumption is that the interest rate and volatility are constant and the stock is log-normally distributed.

The following sections are included:

• Martingale Measures
  • Is the stock price S_t a martingale?
  • Risk-neutral World and Martingale Measure
  • Finding the Risk-Neutral Measure
• Risk-neutral World Density Functions
• Self-financing Portfolios
• The Sharpe Ratio
• Risk-neutral Valuation for Derivatives

This chapter provides an alternative way to price derivative securities using tools of partial differential equations. The idea is to eliminate the stochastic news factor and to obtain a deterministic partial differential equation, which was considered for the first time by Black, Scholes and Merton in the early 1970s.

In this chapter we shall develop the Black-Scholes equation in the case of Asian derivatives and we shall discuss the particular cases of options and forward contracts on weighted averages. In the case of the latter contracts we obtain closed form solutions, while for the former we apply the reduction variable method to decrease the number of variables and discuss the solution.

American options are options that are allowed to be exercised at any time before maturity. Because of this extra feature, they tend to be more expensive than their European counterparts. Exact pricing formulas exist just for perpetuities, while they are missing for finitely lived American options. However, some approximative formulas can be worked out.

The following sections are included:

• Heston Equation
  • Dynamics of the Model
  • Setting up the Heston PDE
  • Simplifying the Heston PDE
• Boundary Conditions for European Call
  • PDEs for π₁ and π₂
  • Finding π₁ and π₂
  • Solving the Riccati Equation
  • Finding the Function C(τ,ω)
  • The Final Formula
  • Concluding Formula
• Delta of a Call
  • Interpretation of f₂
• American Options
  • Calls
  • Call Perpetuity
  • Analytic Approximation to a Call

This chapter develops a theory similar to the one done for the Heston model, with the difference that the underlying asset stochastic volatility follows this time a GARCH process.

We include the AR(1) stochastic volatility model for pedagogical reasons, so that the reader can notice the similarities between the Heston, GARCH and AR(1) models as well as their main differences. The AR(1) model can be solved in a similar way as the Heston model.

This chapter deals with option pricing in the case when the underlying asset has a stochastic rate of return. The reason for including this chapter is that the subject of this work was the topic of the 2013 Nobel Price in Economics, see Campbell [18] and Cochrane [22] for details. These conclusions were based on a long line of empirical research, which has documented the extent to which expected return on assets has predictive components. This research leads to an Ornstain-Uhlenbeck process for the expected return on stocks and bonds in continuous time. The reader is referred to Campbell et al. [19] for the explanation of how researchers estimate expected return models in continuous time.

Here we provide hints and solutions to selected exercises. The interested reader should be able to derive the solutions to the rest based on the experience gained from the chapter examples and other similar solved exercises.

The following sections are included:

• Appendix A: Useful Transforms
  • The Fourier Transform
  • The Laplace Transform
• Appendix B: Probability Concepts
  • Events and Probability
  • Stochastic Processes
  • Expectation
  • Conditional Expectations
  • Martingales
  • Submartingales and their Properties
  • Jensen’s Inequality
• Appendix C: Elements of Stochastic Calculus
  • The Poisson Process
  • The Brownian Motion
  • Exponential Process
  • Ito’s Lemma
  • Girsanov Theorem
  • Ito Integral
  • Brownian Motion with Drift
  • The Generator of an Ito Diffusion
• Appendix D: Series and Equations
  • Confluent Hypergeometric Equation
  • Duhamel’s Principle

The following sections are included:

• Bibliography
• Index