Question

Put-call parity: Denote by \(C_{t}\) and \(P_{t}\) respectively the prices at time \(t\) of a European call and a European put option, each with maturity \(T\) and strike \(K\). Assume that the risk-free rate of interest is constant, \(r\), and that there is no arbitrage in the market. Show that for each \(t \leq T\), $$ C_{t}-P_{t}=S_{t}-K e^{-r(T-t)} $$.

Short answer

The put-call parity is given by \( C_{t} - P_{t} = S_{t} - K e^{-r(T-t)} \), showing the relationship between call and put option prices and the current underlying asset price.

Step-by-step solution

Understanding Put-Call Parity

Put-call parity is a financial concept that provides a relationship between the prices of European call and put options with the same strike price and expiration date. The formula is derived assuming there is no arbitrage opportunity in the market.

Identify the Call and Put Options

Consider a European call option with price \(C_t\) and a European put option with price \(P_t\). Both options have the same maturity \(T\) and strike price \(K\). We are to show the relationship between these options given the risk-free interest rate \(r\).

Consider Buying a Call and Selling a Put

One common method to derive the put-call parity is to consider a portfolio where you buy a call and sell a put option. The net investment required for this portfolio is \(C_t - P_t\).

Equivalence with a Synthetic Portfolio

Construct a synthetic portfolio that replicates this payoff: buying the underlying asset at current price \(S_t\) and borrowing \(K e^{-r(T-t)}\) (present value of strike price). The cost to set up this portfolio is also \(S_t - K e^{-r(T-t)}\).

Establish the Put-Call Parity Equation

Since both portfolios (own a call and sell a put, and hold the underlying while borrowing) should yield the same value at maturity under no-arbitrage conditions, their current values must also be equal. Therefore, \[ C_{t} - P_{t} = S_{t} - K e^{-r(T-t)} \]

Conclusion of Derivation

'Put-call parity' shows the interrelation between the option prices of a call and a put, given the underlying asset's price and the risk-free rate. This is essential for arbitrage-free pricing in efficient markets.

Key concepts

European call options

A European call option is a financial contract that grants the holder the right, but not the obligation, to purchase an underlying asset at a specified price, known as the strike price, on a predetermined expiration date. Unlike American options, European call options can only be exercised at maturity, not before.

This makes them simpler to analyze and evaluate. To understand them better, consider these aspects:

• **Right to Buy**: With a European call option, you can purchase the asset, but you don't have to if it's not favorable.
• **Strike Price**: This is the set price at which you can buy the asset when the option matures.
• **Expiration Date**: The date when the option can be exercised and expires if not acted upon.

Investors typically buy call options when they anticipate that the price of the underlying asset will rise above the strike price before or at the option's maturity. If correct, they can realize a profit by buying low at the strike price and potentially selling higher in the market.

European put options

European put options function as the converse of call options. They provide the holder the right, but not the obligation, to sell an underlying asset at a predetermined price, called the strike price, on a specific expiration date.

It's an attractive choice for traders who wish to hedge against potential downturns in the asset's price. Some key characteristics include:

• **Right to Sell**: European puts enable you to sell the asset at the agreed price, regardless of the market value at expiration.
• **Protect Against Decline**: Investors often purchase puts as a hedge if they expect the asset's price to decline.
• **Expiration Specificity**: Like European calls, European puts can only be exercised on the expiration date.

By understanding a European put option, an investor can tailor their strategy to protect against downside risk in their portfolio, while potentially profiting from anticipated price decreases.

arbitrage-free pricing

Arbitrage-free pricing is a foundational concept in financial markets that ensures securities prices are fair and aligned, preventing riskless profit opportunities from inconsistencies.

The principles of arbitrage-free pricing can be broken down as follows:

• **No Arbitrage Opportunity**: If price deviations arise, traders can exploit them through arbitrage—buying low and selling high across different markets, leading the prices to normalize.
• **Market Efficiency**: Markets are considered efficient when arbitrage opportunities are nonexistent or quickly exploited.
• **Price Equivalence**: Similar securities or portfolios must have comparable prices to avoid arbitrage. For example, with put-call parity, the combination of buying a call and selling a put matches the cost of holding the underlying asset, minus the present value of the strike price.

Arbitrage-free pricing frameworks, like put-call parity, are essential in pricing derivatives accurately. They ensure consistent valuations and enhance market stability.

risk-free interest rate

The risk-free interest rate is a theoretical rate of return of an investment with no risk of financial loss.

In practice, government bonds are often used as proxies for the risk-free rate, because governments are unlikely to default on their obligations. Here are some crucial points regarding the risk-free rate:

• **Benchmark Rate**: It serves as a baseline for assessing other investment risks. All other investments carry additional risk over this rate.
• **Time-bound**: Often reflected in different timeframes, such as short-term (Treasury bills) or long-term (Government bonds).
• **Influence on Options**: The risk-free rate is used in calculating the present value of the strike price in options pricing. Hence, it indirectly affects the valuation of options through models like the put-call parity.

Understanding this rate is crucial for calculating discounted cash flows and determining the present value of future financial commitments. It's a bedrock for establishing consistent, risk-averse investment strategies.