Problem 4 - A Course in Financial Calculus

Question

Suppose that the value of a European call option can be expressed as \(V_{t}=F\left(t, S_{t}\right)\) (as we prove in Proposition 5.2.3). Then \(\tilde{V}_{t}=e^{-r t} V_{t}\), and we may define \(\tilde{F}\) by $$ \tilde{V}_{t}=\tilde{F}\left(t, \tilde{S}_{t}\right) $$ Under the risk-neutral measure, the discounted asset price follows \(d \tilde{S}_{t}=\sigma \tilde{S}_{t} d X_{t}\), where (under this probability measure) \(\left\\{X_{t}\right\\}_{t \geq 0}\) is a standard Brownian motion. (a) Find the stochastic differential equation satisfied by \(\tilde{F}\left(t, \tilde{S}_{t}\right)\). (b) Using the fact that \(\tilde{V}_{t}\) is a martingale under the risk-neutral measure, find the partial differential equation satisfied by \(\tilde{F}(t, x)\), and hence show that $$ \frac{\partial F}{\partial t}+\frac{1}{2} \sigma^{2} x^{2} \frac{\partial^{2} F}{\partial x^{2}}+r x \frac{\partial F}{\partial x}-r F=0 $$ This is the Black-Scholes equation.

Step-by-Step Solution

Answer

The Black-Scholes equation is derived as: \( \frac{\partial F}{\partial t} + \frac{1}{2} \sigma^{2} x^{2} \frac{\partial^{2} F}{\partial x^{2}} + r x \frac{\partial F}{\partial x} - r F = 0 \).

Step 1: Define and Differentiate

Start by considering the relationship between the discounted option value \( \tilde{V}_t \) and the function \( \tilde{F}(t, \tilde{S}_t) \). By Itô's Lemma, to find the stochastic differential equation for \( \tilde{F}(t, \tilde{S}_t) \), we differentiate it with respect to both \( t \) and \( \tilde{S}_t \). The function \( \tilde{F}(t, \tilde{S}_t) \) is given as:\[ d\tilde{F} = \left(\frac{\partial \tilde{F}}{\partial t} + \frac{1}{2}\sigma^2 \tilde{S}_t^2 \frac{\partial^2 \tilde{F}}{\partial \tilde{S}_t^2} \right)dt + \frac{\partial \tilde{F}}{\partial \tilde{S}_t} d\tilde{S}_t \]

Step 2: Substitute the Dynamics of \( d\tilde{S}_t \)

Substitute \( d\tilde{S}_t = \sigma \tilde{S}_t dX_t \) from the given dynamics of the discounted asset into the differential equation obtained from Itô's lemma:\[ d\tilde{F} = \left(\frac{\partial \tilde{F}}{\partial t} + \frac{1}{2}\sigma^2 \tilde{S}_t^2 \frac{\partial^2 \tilde{F}}{\partial \tilde{S}_t^2} \right)dt + \frac{\partial \tilde{F}}{\partial \tilde{S}_t} \sigma \tilde{S}_t dX_t \]

Step 3: Use Risk-Neutral Martingale Property

Since \( \tilde{V}_t \) is a martingale under the risk-neutral measure, the drift term in the differential of \( \tilde{F} \) must be zero:\[ \frac{\partial \tilde{F}}{\partial t} + \frac{1}{2}\sigma^2 \tilde{S}_t^2 \frac{\partial^2 \tilde{F}}{\partial \tilde{S}_t^2} = 0 \] This condition eliminates the drift term in the stochastic differential equation.

Step 4: Transform Back to Original Variables

Rewriting the relationship in terms of the original variables, recall that \( \tilde{V}_t = e^{-rt} V_t \) leads to expressing \( \tilde{F} \) in the original settings:\[ \frac{\partial F}{\partial t} - rF + rx\frac{\partial F}{\partial x} + \frac{1}{2}\sigma^2x^2\frac{\partial^2 F}{\partial x^2} = 0 \] This confirms the transformation meets the Black-Scholes equation as derived.

Key Concepts

European Call Option

A European call option is a financial contract that grants the holder the right, but not the obligation, to buy an asset at a specified strike price on the option's expiration date. It is different from an American call option, which allows for exercise before expiration.
European options are simpler and more straightforward because they can only be exercised at one point in time. This specificity fits neatly with the theoretical models, such as the Black-Scholes model, allowing for easier mathematical handling and prediction. Some key characteristics of European call options include:

• Strike Price: The predetermined price at which the asset can be bought.
• Expiration Date: The specific date on which the option can be exercised.
• Underlying Asset: The asset for which the option provides the right to purchase, such as stocks or commodities.

These options are widely used in markets to hedge risks or speculate on future prices. Understanding their behavior is crucial for investors and financial strategists.

Risk-Neutral Measure

The risk-neutral measure is a mathematical construct used in financial modeling to simplify calculations and pricing. In a risk-neutral world, investors are indifferent to risk, meaning they do not require any additional expected returns for taking on more risk. This assumption allows us to value derivatives, like options, in a manner that reflects their fair market value.
The risk-neutral measure transforms the probability distribution of the underlying asset returns so that the expected return of the asset is the risk-free rate. This simplification enables the use of the expected value when pricing options without worrying about the risk preferences of individual investors. The risk-neutral measure has several key applications:

• Converting real-world probabilities into a simplified finance-friendly framework.
• Allowing the use of mathematical tools to solve complex financial equations.
• Enabling explicit valuation of options and other derivatives using models like Black-Scholes.

Adopting the risk-neutral measure is fundamental in quantitative finance, making valuation equations both solvable and meaningful.

Itô's Lemma

Itô's Lemma is a key concept in stochastic calculus, which is used to find the differential of a function of a stochastic process. In finance, it's essential for analyzing the dynamics of asset prices and derivatives.
Itô's Lemma allows us to compute the change in a function based on an underlying stochastic process, typically modeled as a stochastic differential equation (SDE). This is crucial in the derivation of the Black-Scholes equation, among others, where it helps to determine how option prices evolve over time. The lemma can be summarized as follows:

• If you have a differentiable function that depends on a stochastic process, Itô's Lemma provides a way to express its differential form.
• It includes both drift and diffusion components, where drift is the average rate of change and diffusion represents the randomness.
• It is foundational in computing the expected changes in financial derivatives prices.

Understanding Itô's Lemma is vital for anyone working with financial models that involve stochastic processes.

Stochastic Differential Equation (SDE)

Stochastic Differential Equations (SDEs) are equations used to model systems that are affected by random influences. In finance, they are essential for modeling the behavior of asset prices or financial indicators over time.
SDEs incorporate both deterministic components, which provide a predictable element, and stochastic components, reflecting randomness or noise. Some points regarding SDEs include:

• SDEs are extensions of ordinary differential equations, accommodating random effects, usually represented by a Brownian motion term.
• The Black-Scholes equation for option pricing is derived from an SDE, modeling the dynamics of the underlying asset's price.
• SDEs require sophisticated mathematical tools for analysis and solving, often because of the Brownian motion component.

Mastery of SDEs helps in effectively modeling and predicting various financial and non-financial processes exhibiting random behavior.