Problem 13 - A Course in Financial Calculus

Question

Suppose that the US dollar/Japanese Yen exchange rate follows the stochastic differential equation $$ d S_{t}=\mu S_{t} d t+\sigma S_{t} d W_{t} $$ for some constants \(\mu\) and \(\sigma\). You are told that the expected \$/¥and \(¥ /\) exchange rates in one years time are both \(2 S_{0} .\) Is this possible?

Step-by-Step Solution

Answer

No, the conditions contradict each other; they are not possible simultaneously.

Step 1: Understanding the Problem

We are given a stochastic differential equation that describes the evolution of the exchange rate between the US dollar and Japanese Yen: \( d S_t = \mu S_t dt + \sigma S_t dW_t \). We need to determine if the expected exchange rates in one year's time could be twice the initial rate \( S_0 \).

Step 2: Using Ito's Lemma

Since \( S_t \) is a geometric Brownian motion, we can apply Ito's lemma to find the expected value of \( S_t \). The solution to the SDE is \( S_t = S_0 e^{(\mu - \frac{1}{2} \sigma^2)t + \sigma W_t} \).

Step 3: Calculating Expected Value

The expected value at time \( t \) is given by \( \mathbb{E}[S_t] = S_0 e^{\mu t} \). For \( t = 1 \) year, \( \mathbb{E}[S_1] = S_0 e^{\mu} \). According to the problem, this should equal \( 2S_0 \).

Step 4: Equating to Given Condition

Set \( \mathbb{E}[S_1] = 2S_0 \): \( S_0 e^{\mu} = 2 S_0 \). Solving this equation gives \( e^{\mu} = 2 \), thus \( \mu = \ln(2) \).

Step 5: Analyzing the Expected Reciprocal

The expectation of \( \frac{1}{S_1} \) is \( \mathbb{E}\left[ \frac{1}{S_1} \right] = e^{-\mu} \). Since the expected ¥/\$ rate is also \( 2/S_1 \), this provides additional constraints.

Step 6: Verification

Both \( \mathbb{E}[S_1] \) and \( \mathbb{E}\left[ \frac{1}{S_1} \right] \) reflect the same rate, implying \( \mathbb{E}[S_1] \cdot \mathbb{E}\left[ \frac{1}{S_1} \right] = 1 \), which generally doesn't match: except for \( S_1 \) being deterministic, this is generally contradictory.

Key Concepts

Geometric Brownian Motion

Geometric Brownian Motion (GBM) is a popular stochastic process used to model the dynamics of financial systems, including stock prices and exchange rates. It is characterized by a continuous path that evolves in a random yet predictable manner.

In the context of exchange rates, like the US dollar/Japanese Yen, GBM is used to describe how these rates change over time. The general form of a GBM is given by the stochastic differential equation (SDE):

• \(d S_t = \mu S_t dt + \sigma S_t d W_t\)

Here:

• \(S_t\) represents the exchange rate at time \(t\).
• \(\mu\) is the drift coefficient, reflecting the expected rate of return.
• \(\sigma\) is the volatility coefficient, indicating the randomness or variability in the exchange rate.
• \(dW_t\) represents a Wiener process or Brownian motion, capturing the random effects.

Thanks to its mathematical properties and ability to incorporate both trend and randomness, GBM is highly useful in finance for modeling scenarios where future value changes are log-normally distributed.

Ito's Lemma

Ito's Lemma is a fundamental tool in stochastic calculus, used to find the derivative of a function that depends on a stochastic process. It extends the rules of differentiation to functions affected by Brownian motion.

When we apply Ito's Lemma to the solution of a geometric Brownian motion, such as our exchange rate model, it allows us to express \( S_t \) in terms of its stochastic components:

• \(S_t = S_0 e^{(\mu - \frac{1}{2} \sigma^2)t + \sigma W_t}\)

This formula is crucial because it enables us to predict the expected future value of the exchange rate by considering both the deterministic and stochastic effects.

By leveraging Ito's Lemma, we can make complex calculations more tractable, which is essential in evaluating financial models where models like GBM are used.

Expected Value Calculation

Calculating the expected value over time is a core aspect of stochastic processes like Geometric Brownian Motion. The expected value reveals the average outcome that we anticipate based on the process's inherent trends and randomness.

In our exercise:

• The expected value of the exchange rate at time \(t\) is represented by \(\mathbb{E}[S_t] = S_0 e^{\mu t}\).

This expectation signifies how much, on average, the exchange rate is expected to increase or decrease over a given time, assuming constant drift and no unforeseen large shifts.

In practical terms, especially for exchange rate modeling, calculating the expected future value helps in making informed economic and financial decisions, ensuring strategies can account for likely movements in currency values.

Exchange Rate Modelling

Exchange rate modeling involves using statistical and mathematical methods to predict future currency value movements. A robust model helps businesses and investors manage risks associated with currency fluctuations.

In our context, the stochastic differential equation \(d S_t = \mu S_t dt + \sigma S_t d W_t\) plays a pivotal role in modeling the US dollar/Japanese Yen exchange rate. This approach provides insights into:

• The average rate change, modeled by the drift \(\mu\), reflecting overall economic trends.
• The randomness introduced by volatility \(\sigma\), capturing market uncertainty and external shocks.

Efficient exchange rate forecasting using such models allows for better hedging strategies, reliable pricing of products in international markets, and improved cash flow planning. By understanding these parameters and how they interact, financial planners and strategists can enhance their approaches to dealing with forex risks.