[Paper Review] Introduction to Stochastic Differential Equations (SDEs) for Finance

Andrew C. Papanicolaou|arXiv (Cornell University)|Apr 21, 2015

Stochastic processes and financial applications14 references3 citations

TL;DR

This paper provides a comprehensive introduction to stochastic differential equations (SDEs) in finance, focusing on the Black-Scholes framework and risk-neutral pricing via replication and equivalent martingale measures. It derives the Black-Scholes option pricing formula using Itô calculus, Feynman-Kac, and Fourier transform methods, establishing the foundation for derivative valuation and hedging in complete markets.

ABSTRACT

These are course notes on the application of SDEs to options pricing. The author was partially supported by NSF grant DMS-0739195.

Motivation & Objective

• To establish the theoretical foundation for derivative pricing using stochastic calculus in financial markets.
• To demonstrate how contingent claims can be replicated using dynamic trading strategies in a complete market.
• To derive the Black-Scholes option pricing formula through the Feynman-Kac formula and risk-neutral measure.
• To introduce key concepts such as delta hedging, Greeks, and implied volatility in option pricing.
• To extend the framework to stochastic volatility and control problems using the Hamilton-Jacobi-Bellman equation.

Proposed method

• Uses discrete-time, two-state market models to illustrate the principle of no-arbitrage and replication for option pricing.
• Applies Itô calculus to model continuous-time asset dynamics, including the Itô integral and Itô’s lemma for Ito processes.
• Derives the Black-Scholes partial differential equation (PDE) by requiring self-financing portfolios to replicate contingent claims.
• Employs the Feynman-Kac formula to connect the PDE solution to risk-neutral expectations under the equivalent martingale measure.
• Utilizes Fourier transform techniques to solve the heat and Black-Scholes equations, enabling analytical and semi-analytical pricing.
• Applies Girsanov’s theorem to change probability measures and derive the risk-neutral dynamics of asset prices.

Experimental results

Research questions

• RQ1 How can a contingent claim be priced without arbitrage in a complete financial market?
• RQ2 What conditions ensure that a derivative can be replicated by a dynamic portfolio of underlying assets and cash?
• RQ3 How does the risk-neutral measure simplify the pricing of derivatives using expectation under a changed probability measure?
• RQ4 What is the analytical solution to the Black-Scholes PDE for a European call option, and how is it derived?
• RQ5 How do the Greeks (delta, gamma, vega, etc.) quantify the sensitivity of option prices to underlying variables?

Key findings

• The call option price in the discrete two-state model is uniquely determined as $ C_0 = 0.5 $ through replication using $ \alpha = 1/2 $ shares and $ \beta = -1/2 $ in the bank account.
• The Black-Scholes PDE is derived as $ \left(\frac{\partial}{\partial t} + \frac{\sigma^2}{2}\frac{\partial^2}{\partial x^2} + \left(r - \frac{\sigma^2}{2}\right)\frac{\partial}{\partial x} - r\right)V = 0 $, with terminal condition $ V(T,x) = \psi(x) $.
• The solution to the Black-Scholes PDE is given by the risk-neutral expectation: $ V(t,x) = e^{-r(T-t)}\mathbb{E}^Q[\psi(X_T) \mid X_t = x] $, where $ X_t = \log(S_t) $.
• The Black-Scholes call option formula is recovered via Fourier inversion, yielding $ C(S,t) = S N(d_1) - K e^{-r(T-t)} N(d_2) $, with standard normal cumulative distribution functions.
• The Heston stochastic volatility model is introduced with an explicit formula for option pricing using characteristic functions.
• The paper establishes that the Black-Scholes Greeks (delta, gamma, vega, etc.) are derived from the partial derivatives of the option price with respect to underlying variables.