Question

Consider the Black-Scholes model with a constant continuous dividend yield \(\delta\). Prove the following put-call parity relation, where \(c_{\bar{b}}\left(p_{\hat{N}}\right)\) denotes the price of a European call (put). $$ p \delta=c \delta-s e^{-\delta(T-t)}+K e^{-n(T-t)} $$

Short answer

Question: Prove the put-call parity relation for the Black-Scholes model with a constant continuous dividend yield, 𝛿. Answer: To prove the put-call parity relation, we use the Black-Scholes formulas for European call and put options and manipulate the expressions. The put-call parity relation for the Black-Scholes model with a constant continuous dividend yield, 𝛿, is: $$ p \delta = c \delta - s e^{-\delta(T-t)} + K e^{-r(T-t)} $$

Step-by-step solution

Recall the Black-Scholes formula for European call and put option prices

The Black-Scholes formula for a European call option with continuous dividend yield 𝛿 is given by: $$ c = s e^{-\delta (T-t)} \ N(d_1) - K e^{-r(T-t)} \ N(d_2) $$ and for a European put option: $$ p = K e^{-r(T-t)} \ N(-d_2) - s e^{-\delta (T-t)} \ N(-d_1) $$ where: - \(d_1 = \frac{\ln{\frac{s}{K}} + (r - \delta + \frac{\sigma^2}{2})(T-t)}{\sigma \sqrt{T-t}}\) - \(d_2 = d_1 - \sigma \sqrt{T-t}\) - 𝑁(𝑥) represents the cumulative density function of the standard normal distribution - 𝑟 is the risk-free interest rate - 𝑠 is the current stock price - 𝑇 is the option expiration date - 𝑡 is the current time - 𝑁(𝑑1) and 𝑁(𝑑2) represent the probabilities that the option will be exercised - 𝜎 is the stock price volatility

Multiply both the call and put option prices by the constant 𝛿

We will multiply both the call and put option prices by 𝛿, which will be used to prove the put-call parity relation: $$ c \delta = \delta \left[ s e^{-\delta (T-t)} \ N(d_1) - K e^{-r(T-t)} \ N(d_2) \right] $$ $$ p \delta = \delta \left[ K e^{-r(T-t)} \ N(-d_2) - s e^{-\delta (T-t)} \ N(-d_1) \right] $$

Substitute \(N(-d_1) = 1 - N(d_1)\) and \(N(-d_2) = 1 - N(d_2)\) in the put equation

Using the property of the cumulative density function of the standard normal distribution, \(N(-x) = 1 - N(x)\) for any x, we can rewrite the put equation as: $$ p \delta = \delta \left[ K e^{-r(T-t)} (1 - N(d_2)) - s e^{-\delta (T-t)} (1 - N(d_1)) \right] $$

Simplify the put equation further and rearrange terms

Expand the put equation and group terms to simplify: $$ p \delta = \delta K e^{-r(T-t)} - \delta K e^{-r(T-t)} N(d_2) - \delta s e^{-\delta (T-t)} + \delta s e^{-\delta (T-t)} N(d_1) $$ Now, rearrange the above equation to get: $$ p \delta = c \delta - \delta s e^{-\delta (T-t)} N(d_1) + \delta K e^{-r(T-t)} N(d_2) + \delta s e^{-\delta (T-t)} - \delta s e^{-\delta (T-t)} N(d_1) - \delta K e^{-r(T-t)} $$

Cancel out the terms and obtain the put-call parity relation

On canceling out the appropriate terms, we arrive at the put-call parity relation for the Black-Scholes model with constant 𝛿: $$ p \delta = c \delta - s e^{-\delta(T-t)} + K e^{-r(T-t)} $$ This completes the proof of the put-call parity relation in the Black-Scholes model with a constant continuous dividend yield.

Key concepts

Put-Call Parity

Put-call parity is an important concept in options pricing, particularly within the framework of the Black-Scholes model. It describes a fundamental relationship between the prices of European call and put options with the same strike price and expiration date. In essence, it shows how the value of calls and puts can be related through the underlying asset's price, time decay, and other factors. The standard parity equation for non-dividend paying assets is given by:

• Call Price - Put Price = Stock Price - Present Value of Strike Price

With the introduction of dividends, especially continuous dividends, the formula adapts slightly to account for these changes. The given exercise incorporates continuous dividend yield, showing the effect on the parity equation when dividends are continuously distributed at rate \(\delta\). This is expressed in the exercise as \( p \delta = c \delta - s e^{-\delta(T-t)} + K e^{-r(T-t)} \). Understanding this relationship helps in identifying arbitrage opportunities and in forming a deep understanding of how different financial instruments relate to one another.

European Call Option

A European call option is a type of options contract that gives the holder the right, but not the obligation, to buy an underlying asset at a specified strike price on a specified expiration date. Unlike American options, European options can only be exercised at maturity. This distinction is crucial when using the Black-Scholes model for pricing, as this model is specifically tailored to price European options.To value a European call option with continuous dividend yield \(\delta\), the Black-Scholes formula becomes slightly adjusted. The call option pricing formula incorporates:

• The current stock price adjusted for dividends: \(s e^{-\delta (T-t)}\)
• Expected change in price over time using the cumulative normal distribution function: \(N(d_1)\)
• The present value of the strike price: \(K e^{-r(T-t)}\)

By utilizing these parameters, one can compute the theoretical price of a European call option in markets where dividend payouts influence the underlying asset's price. This factor is important as dividends effectively reduce the cost of holding a stock, thereby affecting option pricing.

Continuous Dividend Yield

Continuous dividend yield is a critical factor in pricing models like the Black-Scholes. Unlike discrete dividends, which are paid at specific intervals, continuous dividends are assumed to be paid continuously at a constant rate denoted by \(\delta\). This adjustment reflects more realistic scenarios for certain financial instruments.When continuous dividends are considered, the present value of the asset is adjusted in equations to accommodate the effect of these dividends over time. This is seen through the term \(s e^{-\delta(T-t)}\), which modifies the stock price by discounting it based on the dividend yield. This modification is necessary because dividends reduce the expected future price of the stock, influencing both call and put options' values.By incorporating continuous dividend yield into option pricing, traders and analysts can more accurately assess the expected return and risks of holding options on dividend-paying stocks. It ensures that the models remain relevant and adaptable to changing market conditions.

Cumulative Density Function

The cumulative density function (CDF) of the standard normal distribution, represented by \(N(x)\), is a fundamental component in the Black-Scholes model. It helps calculate the probability that a stochastic process is below a certain level at any point, which in the case of options, translates to the likelihood of an option being in-the-money (ITM) at expiration.In the Black-Scholes formula, the CDF is utilized to determine two key probabilities:

• \(N(d_1)\): The probability weighted factor for the option being exercised
• \(N(d_2)\): The probability that the option will not expire worthless

These probabilities play a crucial role in determining the option's price because they account for the variation in outcomes due to price changes in the underlying asset. Understanding how the CDF works in this context is vital for correctly applying the Black-Scholes model and for predicting the expected payoffs of options. The properties of the CDF ensure that both statistical accuracy and financial intuition are applied in pricing derivative instruments.