TD - Stochastic differential equations and instantaneously risk-free portfolios

Stochastic Differential Equations and Instantaneously Risk-Free Portfolios

Problem Overview
This study material delves into the application of stochastic differential equations (SDEs) in modeling asset prices, particularly focusing on deriving Ito’s lemma and constructing an instantaneously risk-free portfolio. By examining a continuously dividend-paying asset and its corresponding option, we will explore the fundamental relationships governing option pricing in stochastic environments.

Key Topics Covered:

• Asset Price Dynamics: We assume that an asset $S$ evolves according to the stochastic differential equation:

  Ito's Lemma: A heuristic derivation of Ito’s lemma for a sufficiently differentiable function $V(S,t)$ that depends on the asset price $S$ and time $t$ is provided. This derivation is crucial for understanding how the stochastic nature of asset prices affects option valuation.

• Instantaneously Risk-Free Portfolio: We construct an instantaneously risk-free portfolio by combining the asset with its option, ensuring no arbitrage opportunities. By examining the cash flows from this portfolio, we show that the option value $V(S,t)$ satisfies the partial differential equation:

  Special Case: $K(S,t)=g(t)S$: We analyze the case where $K(S,t)$ takes the form $g(t)S$, with $g(t)$ as a known function of time. In this scenario, solutions can be found in the form $V=f(t)S$. By solving for $f(t)$, we derive the explicit form of the option value $V(S,t)$.

• Option Delta: Finally, we show that the delta (the rate of change of the option value with respect to the asset price) for such an option is given by:

  This delta formula is critical for risk management and hedging strategies.

Applications:

• Option Pricing: The results obtained in this material are essential for pricing options on dividend-paying assets in a stochastic financial model.
• Risk Management: Understanding the delta of an option helps in managing the risk associated with changes in the underlying asset price.

This material is suited for students and professionals in financial engineering, quantitative finance, and risk management, particularly those interested in stochastic processes and options pricing theory.

Why Choose This Material?

• Detailed derivation of Ito’s lemma and its application in option pricing.
• Step-by-step guide to constructing an instantaneously risk-free portfolio.
• Practical insights into option delta for risk management.