Problem 10 - A Course in Financial Calculus

Question

Let \(C\left(t, S_{t}\right)\) and \(P\left(t, S_{t}\right)\) denote the values of a European call and put option with the same exercise price, \(K\), and expiry time, \(T\). Show that \(C(t, x)-P(t, x)\) also satisfies the Black-Scholes equation with the final data \(C(T, x)-P(T, x)=x-K .\) Deduce that \(x-K e^{-r(T-t)}\) is also a solution of the Black-Scholes equation. Interpret these results financially.

Step-by-Step Solution

Answer

The difference satisfies the Black-Scholes equation and represents "put-call parity." The solution essentially relates the call, put, and the underlying asset's present value.

Step 1: Understand the Black-Scholes Equation

The Black-Scholes equation for a European call option is given as: \[ \frac{\partial C}{\partial t} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 C}{\partial S^2} + rS \frac{\partial C}{\partial S} - rC = 0 \] and for a European put option, it is: \[ \frac{\partial P}{\partial t} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 P}{\partial S^2} + rS \frac{\partial P}{\partial S} - rP = 0 \] where \(r\) is the risk-free interest rate, \(\sigma\) is the volatility, and \(S\) is the stock price.

Step 2: Derive the Black-Scholes Equation for the Difference

Consider the difference \(C(t, S) - P(t, S)\). We need to show this satisfies the Black-Scholes equation. Using the derivatives: \[ \frac{\partial}{\partial t}(C - P) = \frac{\partial C}{\partial t} - \frac{\partial P}{\partial t} \] \[ \frac{\partial^2}{\partial S^2}(C - P) = \frac{\partial^2 C}{\partial S^2} - \frac{\partial^2 P}{\partial S^2} \] \[ \frac{\partial}{\partial S}(C - P) = \frac{\partial C}{\partial S} - \frac{\partial P}{\partial S} \] Substitute these into the Black-Scholes equation: \[ \frac{\partial}{\partial t}(C - P) + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2}{\partial S^2}(C - P) + rS \frac{\partial}{\partial S}(C - P) - r(C - P) = 0 \] which shows \(C(t, S) - P(t, S)\) satisfies the Black-Scholes equation.

Step 3: Final Condition Verification

At expiration \(T\), the payoff for a call is \(\max(S_T - K, 0)\) and for a put \(\max(K - S_T, 0)\). Therefore: \[ C(T, S) - P(T, S) = (S_T - K) \] So the final condition \(C(T, x) - P(T, x) = x - K \) is satisfied.

Step 4: Deduce Symmetry Solution

From previous steps, since \(C(t, S) - P(t, S)\) follows the Black-Scholes equation, the line of symmetry in the form \( x - Ke^{-r(T-t)} \) must also satisfy the Black-Scholes equation through temporal translation, as it adjusts for the present value of \(K\) using continuous compounding at rate \(r\).

Step 5: Financial Interpretation

The results indicate that the difference \(C(t, S) - P(t, S)\) represents the arbitrage-free pricing involving the underlying stock's present value, suggesting the symmetry solution \( x - Ke^{-r(T-t)}\) reflects this arbitrage-free condition where an underlying investment using stock and bonds would be at equivalent value.

Key Concepts

European Call and Put Options

European call and put options are financial instruments used to speculate on the future price movements of underlying assets like stocks. A European call option gives the holder the right, but not the obligation, to purchase an asset at a specified strike price, known as the exercise price, on a specific date called the expiration date. This can be beneficial if you anticipate that the asset's price will rise before expiration. If it does, you can buy the asset at the lower strike price, potentially selling it at a higher market price for a profit.

On the flip side, a European put option allows the holder to sell an asset at a predefined strike price on the same expiration date. This type of option can be advantageous if you predict that the asset's price will decline. In this situation, you can sell the asset at the higher strike price, even if the market price is lower, securing a potential profit. Unlike American options, which can be exercised at any time before expiration, European options can only be exercised at expiration, allowing for strategic planning and limited opportunities for exercise.

Risk-Free Interest Rate

The risk-free interest rate is central to the Black-Scholes model. It's an idealized rate of return on an investment with zero risks, often represented by government bond yields. In the context of options pricing, it’s crucial because it reflects the opportunity cost of investing capital elsewhere. When valuing options, the risk-free rate is used to discount the future payoff of the option to present value.

Essentially, it is the rate at which all arbitrage-free pricing is adjusted, ensuring that the value of the option reflects the time value of money. If you invest your money, you forgo potential interest, so discounting future payoffs by the risk-free rate accounts for this lost opportunity. In practical terms, the higher the risk-free rate, the lower the present value of future cash flows, affecting options pricing by reducing their theoretical value.

Volatility

Volatility in the financial markets measures the extent to which the price of an asset fluctuates over time, and it’s a key component in the valuation of options using the Black-Scholes model. Higher volatility indicates a greater range of possible future prices, potentially increasing an option's value due to the higher likelihood of hitting profitable levels.

In the Black-Scholes equation, volatility is represented by the Greek letter sigma (C3). It affects both call and put options, influencing the probability of the option finishing "in the money" at expiration. A higher C3 increases the chances of significant price changes, thus raising the premium of options as investors anticipate more considerable variability.

• For a call option, high volatility boosts potential profits if prices rise.
• For a put option, it enhances profits if prices fall.

Volatility is pivotal in financial markets and is one of the main factors traders analyze when pricing and hedging options.

Arbitrage-Free Pricing

Arbitrage-free pricing refers to the process of determining the value of financial instruments to ensure there are no opportunities for riskless profit through arbitrage. In the Black-Scholes equation, this principle implies that the price of options should be such that no arbitrage opportunities exist. In simpler terms, one should not be able to buy and sell combinations of financial instruments to make a guaranteed profit without risk.

The concepts indicated by the Black-Scholes model, such as the risk-free interest rate and option pricing, rely heavily on this principle. The equation ensures that the price of a call and put option reflects the no-arbitrage condition by using the logic of time value of money and market efficiency. If you combine a call option, a put option, and the underlying asset, the resulting portfolio should yield no extra profit or loss beyond what could be expected based on the risk-free rate, assuming perfect market conditions and no frictions.

This principle is essential for maintaining fair, transparent, and efficient markets, as it prevents traders from exploiting inefficiencies for undue gain, ensuring credibility in financial transactions and pricing.