BUS614: International Finance

9a. Value put options and call options using the Black-Scholes formula

• How is the Black-Scholes formula used?

The Black-Scholes formula is generally used to calculate option prices. It generally utilizes different variables to find the option's price. These variables include asset price, strike price, interest rates, time to expiration, and volatility.

The formula is exclusively used for European Options since the assumption is that the option can only be exercised at expiration and assumes that markets are efficient, among others. However, the formula cannot be used on US options since these could be exercised before the expiration date.

To calculate call options, the equation that is used most often is:

$C=S_{0} e^{-q T} N\left(d_{1}\right)-K e^{-r T} N\left(d_{2}\right)$

$S_0$ is stock price
$e$ is exponential number
$q$ is dividend yield percentage
$T$ is term
$K$ (sometimes rendered as $X$) is the strike price (exercise Price)
$r$ is risk-free rate
$σ$ is the standard deviation of long returns (volatility)

The first part of the formula, $S_0N(d_1)$, is what you will get. The second part of the formula, $Ke^{-rT}N(d_2)$, is what you will pay. $e^{-rT}$ is the discounted price to the present value.

$N(d_2)$ and $N(d_1)$ is the probability we will exercise the option. They are the cumulative distribution functions for a standard normal distribution with the following formula:

$\begin{aligned}d_{1} &=\frac{\ln \frac{S_{0}}{K}+\left(r+\frac{\sigma^{2}}{2}\right)(T-t)}{\sigma \sqrt{T-t}} \\d_{2} &=d_{1}-\sigma \sqrt{(T-t)} \\&=\frac{\ln \frac{S_{0}}{K}+\left(r-\frac{\sigma^{2}}{2}\right)(T-t)}{\sigma \sqrt{T-t}},\end{aligned}$

To calculate the put option, the equation used is:

$P=K e^{-r t} N\left(-d_{2}\right)-S_{0} e^{-q T} N\left(-d_{1}\right)$

To review, see Introduction to the Black-Scholes Formula.

9b. Describe the payoff of a financial future instrument 

• What are the payoffs of a financial future?

The loss or gain could happen when a change in the price of the underlying asset occurs. Generally, future contracts are standardized contracts traded in exchanges, making them liquid instruments. The gains and losses of these contracts are settled daily. Thus, when initiated, futures contracts are zero net present value contracts and are marked to market, making them good for hedging or speculating.

To review, see Forward and Futures and Contracts.

9c. Identify the benefits of entering into an "interest rate swap" and the nature of a currency swap 

• What are the benefits of entering into an Interest Rate Swap?
• How would you describe the nature of Currency Swap?

An interest rate swap is a derivative contract traded over-the-counter (OTC). This derivative helps reduce interest rate risk as it is used for hedging unfavorable interest rate fluctuations and helps reduce uncertainty. One of its advantages is that it can have longer horizons than financial futures and reduce the cost of loans.

A currency swap is a contract where two parties agree to exchange two currencies at a set rate and then re-exchange these currencies at an agreed-on rate at a fixed date in the future. It consists of two transactions: a spot transaction and a forward transaction.

One of its main advantages is that it is a riskless collateralized borrowing/lending investment where the investor can utilize their funds in a particular currency to fund obligations denominated in a different currency without incurring foreign exchange risk.

To review, see Interest Rate Swaps and Currency Swaps.

Unit 9 Vocabulary

This vocabulary list includes the terms that you will need to know to successfully complete the final exam.

• Currency Swap
  • It is a contract where two parties agree to exchange two currencies at a set rate and then re-exchange them at an agreed-on rate at a fixed date.
• Interest Rate Swap
  • It is a contract between two parties where one party receives a fixed amount periodically in exchange for the London Interbank Offered Rate (LIBOR) linked floating payments to the counterparty.
• Call Option
  • This refers to the right to buy an asset at a fixed date and price
• Put Option
  • This refers to the right to sell an asset at a fixed date and price
• Strike Price
  • This refers to the asset price at which an option can be exercised.