Problem 1 - A Course in Financial Calculus

Question

Suppose that an asset price \(S_{t}\) is such that \(d S_{t}=\mu S_{t} d t+\sigma S_{t} d W_{t}\), where \(\left\\{W_{t}\right\\}_{t \geq 0}\) is, as usual, standard \(\mathbb{P}\)-Brownian motion. Let \(r\) denote the risk-free interest rate. The price of a riskless asset then follows \(d B_{t}=r B_{t} d t\). We write \(\left\\{\psi_{t}, \phi_{t}\right\\}\) for the portfolio consisting of \(\psi_{t}\) units of the riskless asset \(B_{t}\) and \(\phi_{t}\) units of \(S_{t}\) at time \(t\). For each of the following choices of \(\phi_{t}\), find \(\psi_{t}\) so that the portfolio \(\left\\{\psi_{t}, \phi_{t}\right\\}\) is self-financing. (Recall that the value of the portfolio at time \(t\) is \(V_{t}=\psi_{t} B_{t}+\phi_{t} S_{t}\) and that the portfolio is self-financing if \(d V_{t}=\psi_{t} d B_{t}+\phi_{t} d S_{t} \cdot\) ) (a) \(\phi_{t}=1\), (b) \(\phi_{t}=\int_{0}^{t} S_{u} d u\), (c) \(\phi_{t}=S_{t}\).

Step-by-Step Solution

Answer

(a) \(\psi_{t} = \frac{S_{t}}{B_{t}}\), (b) \(\psi_{t} = 0\), (c) \(\psi_{t} = \frac{2 \mu S_{t}^2}{r B_{t}}\).

Step 1: Understanding the Self-Financing Portfolio

A portfolio is self-financing if it does not require any additional input of cash. The change in the value of the portfolio, \(d V_{t}\), should match the changes due to the riskless and risky assets, calculated as \(d V_{t} = \psi_{t} d B_{t} + \phi_{t} d S_{t}\). We'll utilize this to find \(\psi_{t}\) for each \(\phi_{t}\).

Step 1: Finding \(\psi_{t}\) for \(\phi_{t} = 1\)

For \(\phi_{t} = 1\), the portfolio value is \(V_{t} = \psi_{t} B_{t} + S_{t}\). Differentiating, we find \(d V_{t} = \psi_{t} d B_{t} + d S_{t}\). Since \(d S_{t} = \mu S_{t} dt + \sigma S_{t} d W_{t}\) and \(d B_{t} = r B_{t} dt\), substitute to get \(d V_{t} = \psi_{t} r B_{t} dt + \mu S_{t} dt + \sigma S_{t} d W_{t}\). For self-financing, this must equal \(d V_{t} = \psi_{t} r B_{t} dt + \phi_{t} d S_{t}\), which becomes \(\psi_{t} (r B_{t} - \mu S_{t}) dt = 0\). Therefore, \(\psi_{t} = \frac{S_{t}}{B_{t}}\).

Step 2: Finding \(\psi_{t}\) for \(\phi_{t} = \int_{0}^{t} S_{u} du\)

With \(\phi_{t} = \int_{0}^{t} S_{u} du\), the portfolio value is \(V_{t} = \psi_{t} B_{t} + \int_{0}^{t} S_{u} du \cdot S_{t}\). Differentiate to find \(d V_{t} = \psi_{t} d B_{t} + S_{t}^{2} dt\). For self-financing, this must match \(d B_{t} + \phi_{t} d S_{t}\). Plugging in \(d B_{t}\) and \(d S_{t}\), the equality \(\psi_{t} r B_{t} dt = 0\) is simplified to \(\psi_{t} = 0\) since \(S_{t}^{2} dt\) does not affect the dependency on \(\phi_{t}\) in the integrand.

Step 3: Finding \(\psi_{t}\) for \(\phi_{t} = S_{t}\)

For \(\phi_{t} = S_{t}\), the portfolio value is \(V_{t} = \psi_{t} B_{t} + S_{t}^2\). Differentiating gives \(d V_{t} = \psi_{t} d B_{t} + 2 S_{t} d S_{t} + (d S_{t})^2\). Using \(d S_{t} = \mu S_{t} dt + \sigma S_{t} d W_{t}\), and considering terms up to \(dt\) for simplicity and ignoring \((d W_{t})^2\), \(d V_{t} = \psi_{t} r B_{t} dt + 2 S_{t}^2 (\mu dt + \sigma d W_{t})\). Equating for self-financing \(d V_{t} = \psi_{t} r B_{t} + S_{t} d S_{t}\), solve \(\psi_{t} r B_{t} = 2 \mu S_{t}^2\) leading to \(\psi_{t} = \frac{2 \mu S_{t}^2}{r B_{t}}\).

Key Concepts

Self-Financing Portfolio

A self-financing portfolio is a fascinating concept in stochastic calculus, particularly in financial modeling. This type of portfolio implies that no external funds are added or withdrawn at different times. Therefore, any change in the portfolio's value is entirely due to gains or losses from the assets it contains.
To determine if a portfolio is self-financing, you look at the change in its value, denoted by \(d V_{t}\).
This change must be the sum of the changes in the value of the risk-free asset and the risky asset. Mathematically, this is captured in the equation:
\[d V_{t} = \psi_{t} d B_{t} + \phi_{t} d S_{t}.\]
Here, \(\psi_{t}\) and \(\phi_{t}\) represent the number of units of the risk-free and risky assets at time \(t\) respectively. The crucial aspect is that no extra money should be needed to maintain the portfolio's value. Therefore, understanding self-financing portfolios helps in modeling and predicting asset prices without additional capital injection.

Brownian Motion

Brownian motion, also known as a Wiener process, plays a key role in stochastic calculus and financial modeling. Named after the botanist Robert Brown, it was initially observed as the random movement of particles suspended in a fluid. In financial contexts, Brownian motion is used to model the unpredictable path of asset prices over continuous time.
One of the essential features of Brownian motion is that it has independent increments. This means that the motion's change over a given period is independent of its change over any non-overlapping period.
Standard Brownian motion, denoted \(\{W_{t}\}_{t \geq 0}\), starts at zero and has a continuous path. It also has the property that \(W_{t+s} - W_{t}\) is normally distributed with mean 0 and variance \(s\). In the context of asset price modeling, the random part driven by Brownian motion captures the inherent uncertainty and volatility of asset prices. This randomness is crucial in predicting how an asset might behave under different market conditions.

Risk-Free Interest Rate

The risk-free interest rate is an essential concept in finance and is necessary for understanding the valuation of various financial assets. It represents the return on an investment with no risk of financial loss over time.
While actual risk-free assets do not exist due to inherent uncertainties in every investment, government bonds of financially stable countries are often used as benchmarks.
In mathematical terms, if \(B_{t}\) is the price of a risk-free asset, it is modeled by the differential equation:
\[d B_{t} = r B_{t} dt,\]
where \(r\) is the risk-free interest rate. This simple equation signifies that the asset grows continuously at the rate \(r\). The concept of risk-free interest rate is critical for calculating the present value of cash flows, pricing derivatives, and portfolio optimization. It serves as a baseline against which the return on risky assets can be measured.

Asset Price Modeling

Asset price modeling is a central topic in finance, involving the use of mathematical tools to describe how the prices of financial assets change over time. One popular model is the geometric Brownian motion (GBM), which is often used to model stock prices. This model assumes that the price of an asset follows a stochastic differential equation:
\[d S_{t} = \mu S_{t} dt + \sigma S_{t} d W_{t},\]where:

• \(S_{t}\) represents the price of the asset at time \(t\),
• \(\mu\) is the drift term, representing the expected return of the asset,
• \(\sigma\) represents the volatility of the asset, and
• \(d W_{t}\) is the increment of a standard Brownian motion.

This equation captures two main influences on price: the predictable trend (drift) and the random fluctuations (volatility). Asset price modeling assists in pricing options and other derivatives, managing risk, and strategic decision-making. Understanding this equation enables investors to estimate how an asset might behave in response to market forces.