Question

Consider the standard Black-Scholes model. An innovative company, F\& H INC, has produced the derivative "the Golden Logarithm", henceforth abbreviated as the GL. The holder of a GL with maturity time T, denoted as \(G L(T)\), will, at time \(T\), obtain the sum \(\ln S(T) .\) Note that if \(S(T)<1\) this means that the holder has to pay a positive amount to \(F \& H I N C\). Determine the arbitrage free price process for the \(G L(T)\).

Short answer

Answer: The arbitrage free price process for the Golden Logarithm (GL) in the standard Black-Scholes model is given by \(V(t, S) = \ln S + (r-\frac{1}{2}\sigma^2)(T-t)\), where \(t\) is the time, \(S\) is the underlying asset price, \(T\) is the maturity time, \(r\) is the interest rate, and \(\sigma\) is the volatility.

Step-by-step solution

Write down the Black-Scholes PDE

In the standard Black-Scholes model, the price process V(t, S) of a derivative satisfies the following partial differential equation: \begin{equation} \frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV = 0, \end{equation} where \(V(t, S)\) is the price process, \(\sigma\) is the volatility, \(S\) is the underlying asset price, \(t\) is the time, and \(r\) is the interest rate.

Set up the problem with the given conditions

We need to find the price process for the GL. The holder of a GL with maturity time T will receive the sum \(\ln S(T)\) at time T. We write the price process of the GL as \(V(t, S)\): The payoff at maturity is: \begin{equation} V(T, S) = \ln S(T), \end{equation}

Solve the Black-Scholes PDE using the given conditions

We need to solve the Black-Scholes PDE with the given boundary condition. The most suitable method is using the Feynman-Kac Theorem. Feynman-Kac Theorem states that the solution to the Black-Scholes PDE can be expressed as the expected value of the discounted payoff of the derivative under the risk-neutral probability measure \(Q\). The discounted payoff of the GL at time T is \(\frac{1}{B(T)} \ln S(T)\), where \(B(T)\) is the bond price at time T. Under the risk-neutral probability measure, the expected value of the discounted payoff is: \begin{equation} V(t, S) = e^{-r(T-t)} \mathbb{E}_Q \left[ \ln S(T) \mid S(t) = S \right], \end{equation}

Calculate the expected value under the risk-neutral probability measure

Because we are in the Black-Scholes framework, the risk-neutral dynamics of the stock price are given by the following stochastic differential equation: \begin{equation} \frac{dS}{S} = r dt + \sigma dW, \end{equation} where \(W\) is a standard Brownian motion under the risk-neutral probability measure. By integrating this equation and taking exponentials, we get: \begin{equation} S(T) = S(t) \exp \left( \left( r - \frac{1}{2} \sigma^2 \right) (T-t) + \sigma \sqrt{T-t} Z \right), \end{equation} where \(Z\) is a standard normal random variable. We can now plug this into the expected value of the discounted payoff: \begin{equation} V(t, S) = e^{-r(T-t)} \mathbb{E}_Q \left[ \ln \left( S(t) \exp \left( \left( r - \frac{1}{2} \sigma^2 \right) (T-t) + \sigma \sqrt{T-t} Z \right) \right) \mid S(t) = S \right]. \end{equation}

Compute the expectation and simplify the expression

We can now compute the expectation and simplify the expression: \begin{align} V(t, S) &= e^{-r(T-t)} \mathbb{E}_Q \left[ \ln S(t) + \left( r - \frac{1}{2} \sigma^2 \right) (T-t) + \sigma \sqrt{T-t} Z \mid S(t) = S \right] \\ &= e^{-r(T-t)} \left[ \ln S(t) + \left( r - \frac{1}{2} \sigma^2 \right) (T-t) + \sigma \sqrt{T-t} \mathbb{E}_Q \left[ Z \right] \right] \\ &= \ln S + (r-\frac{1}{2}\sigma^2)(T-t), \end{align} since \(\mathbb{E}_Q[Z] = 0\). The arbitrage free price process for the Golden Logarithm (GL) in the standard Black-Scholes model is: \begin{equation} V(t, S) = \ln S + (r-\frac{1}{2}\sigma^2)(T-t). \end{equation}

Key concepts

Arbitrage Free Pricing

Arbitrage free pricing means that the derivative price is calculated so there are no possible riskless profits. In financial markets, arbitrage opportunities are rare because they are quickly exploited, leading to price adjustments.
To achieve arbitrage-free pricing, one typically uses models like the Black-Scholes model, which aligns the price of a derivative with the future probabilities of the underlying asset’s prices.
In our context, the Black-Scholes model assists in ensuring that the pricing of the Golden Logarithm (GL) reflects its expected market behavior, thus avoiding any chance of gaining a riskless profit by trading GL at its calculated price.

Partial Differential Equations

Partial differential equations (PDEs) play a pivotal role in derivative pricing, especially under the Black-Scholes framework. A PDE articulates how the price of a derivative should change over time in response to changes in the underlying asset price. It incorporates factors such as volatility and interest rates.
For the GL, we use a Black-Scholes PDE of the form:

• \[ \frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV = 0,\]

This equation helps determine the fair price of the GL by integrating all necessary financial inputs through a mathematical foundation. The solution to this PDE, under given boundary conditions, gives us the derivative’s price path over time.

Risk-Neutral Measure

The risk-neutral measure is a method used in financial mathematics to make complex models analytically tractable. Under this measure, all investors are assumed to be indifferent to risk. Thus, derivative pricing simplifies to just discounting expected payoffs at the risk-free rate.
In the Black-Scholes model, the risk-neutral measure allows us to compute expected values as if the mean return of the underlying stock equals the risk-free rate. For the GL, the risk-neutral dynamics of the stock are described by:

• \[ \frac{dS}{S} = r dt + \sigma dW, \]

where \( W \) represents Brownian motion. This approach ensures that the derivative pricing is coherent with market goods and allows financial models to predict how prices behave without needing to account for investor risk preferences.

Derivative Pricing

Derivative pricing involves calculating a fair value for financial contracts that derive their value from underlying securities, like stocks. The Black-Scholes model is widely used for this purpose due to its mathematical rigor and ability to incorporate the complexities of the market, like volatility and interest rates.
With the Golden Logarithm, derivative pricing focuses on establishing the present value of its future payoffs, which under the Black-Scholes model aligns with:

• \[ V(t, S) = \ln S + (r-\frac{1}{2}\sigma^2)(T-t), \]

in a risk-neutral world. This expression accounts for both current stock valuation and anticipated market movements, allowing investors to make informed decisions about buying or selling the GL.