Question

Suppose that a stock paying no dividends is trading at $$\$ 15.60$$ a share. European calls on the stock with strike price $$\$ 15$$ and exercise date in three months are trading at $$\$ 2.83 .$$ The interest rate is \(r=6.72 \%\), compounded continuously. What is the price of a European put with the same strike price and exercise date?

Short answer

The price of the put option is $$1.98$$.

Step-by-step solution

Identify Known Variables

Let: \( S = 15.60 \) (current stock price), \( K = 15 \) (strike price), \( C = 2.83 \) (price of the call option), \( r = 0.0672 \) (interest rate), \( T = 0.25 \) (time to maturity in years).

Use the Put-Call Parity Formula

The put-call parity for European options states: \( C + Ke^{-rT} = P + S \) where \( P \) is the price of the put option.

Rearrange the Put-Call Parity Formula to Solve for P

Rearrange to get: \( P = C + Ke^{-rT} - S \)

Calculate the Present Value of the Strike Price

Compute the present value: \( Ke^{-rT} = 15e^{-0.0672 \times 0.25} \)

Simplify the Present Value Calculation

Calculate the exponent: \( -0.0672 \times 0.25 = -0.0168 \). Then compute: \( e^{-0.0168} \)

Compute the Exponential Term

Using a calculator, find: \( e^{-0.0168} \approx 0.9834 \)

Complete the Present Value Calculation

Multiply to get: \( 15 \times 0.9834 \approx 14.75 \)

Plug Values into the Rearranged Formula

Now substitute: \( P = 2.83 + 14.75 - 15.60 \)

Calculate the Price of the Put Option

Simplify the equation: \( P = 2.83 - 0.85 \). This gives: \( P = 1.98 \)

Key concepts

European options

European options are financial derivatives that grant the holder the right, but not the obligation, to buy or sell an underlying asset at a pre-agreed price (strike price) on a specified future date (expiration date). Unlike American options, European options can only be exercised at the expiration date, not before.
European options are commonly used in financial markets because they offer investors a way to hedge risks or speculate on future price movements.
They are simpler to model and analyze than American options, as their exercise is restricted to one specific date.

Present value

Present value (PV) is a fundamental concept in finance and economics. It refers to the current worth of a future sum of money or cash flows, discounted at a specific interest rate. This calculation is crucial because it allows us to compare the value of money received or paid at different times.
To calculate the present value, use the formula:
\[ PV = \frac{FV}{(1 + r)^n} \] where:

• PV = Present Value
• FV = Future Value
• r = interest rate
• n = number of periods until the payment or receipt

This equation helps in determining how much the future cash flows are worth today. In the context of European options, we use present value to determine the strike price adjusted for the interest rate over time.

Interest rate

The interest rate is a crucial factor in financial calculations, reflecting the cost of borrowing money or the return on invested capital. It is typically expressed as a percentage of the principal amount. In the context of European options and put-call parity, the interest rate is used to discount the strike price to its present value.
For continuous compounding, which is commonly used in financial mathematics, the present value of a future sum can be calculated using the formula: \[ PV = FV \cdot e^{-rT} \] where:

• FV = Future Value
• r = interest rate
• T = time to maturity in years

This continuous compounding approach provides a precise evaluation of how the interest rate affects the value of future cash flows over time.

Financial engineering

Financial engineering involves the use of mathematical techniques, financial theory, and engineering methods to solve complex financial problems, structure financial products, and create strategies to optimize financial outcomes. It encompasses areas like derivatives pricing, risk management, portfolio structuring, and hedging.
One key concept in financial engineering is put-call parity, which helps in understanding the pricing relationship between European call and put options. This principle is integral in creating synthetic positions, arbitrage strategies, and ensuring no mispricing occurs in the market.
Financial engineers leverage these principles to design new financial instruments, tailored to meet specific investment goals or hedge against potential risks in the financial markets.