Question

Calculate the BSM price for 120-day options with strike of $25 on a stock that is currently priced at $30.00 and is expected to pay a dividend of $0.40 in 15 days, $0.40 in 85 days, and $0.50 in 175 days. The continuous annual risk-free rate is 5%, and the yield curve is flat. The volatility is 15%.

a. Use BSM formula to calculate the call premium

b. Use BSM formula to calculate the put premium

c. What is the value of a put option based on put-call parity?

Answer 1

The Black-Scholes model makes certain assumptions:

• The option is European and can only be exercised at expiration.
• No dividends are paid out during the life of the option.
• Markets are efficient (i.e., market movements cannot be predicted).
• There are no transaction costs in buying the option.
• The risk-free rate and volatility of the underlying are known and constant.
• The returns on the underlying are normally distributed.

Black-Scholes Formula

The formula, takes the following variables into consideration:

• current underlying price
• options strike price
• time until expiration, expressed as a percent of a year
• implied volatility
• risk-free interest rates

a) Call premium calculations:

Strike price - 25

Current value - 30

Time (days) - 120

Volatility - 15 %

Risk free int. rate - 5%

Option premium - 5.4159

b) Put premium calculations:

Strike price - 25

Current value - 30

Time (days) - 120

Volatility - 15 %

Risk free int. rate - 5%

Option premium - 0.0083

c)

It defines the relationship that must exist between European put and call options with the same underlying asset, expiration and strike prices. (It doesn't apply to American-style options because they can be exercised any time up to expiration.)

Put/call parity states that the price of a call option implies a certain fair price for the corresponding put option with the same strike price and expiration (and vice versa). Support for this pricing relationship is based on the argument that arbitrage opportunities would exist whenever put and call prices diverged.

Put/Call Parity Example

The most simple formula for put/call parity is Call – Put = Stock – Strike. So, for example, if stock XYZ is trading at $60 and you checked option prices at the $55 strike, you might see the call at $7 and the put at $2 ($7 – $2 = $60 – $55). That's an example of put/call parity. If the call were trading higher, you could sell the call, buy the put, buy the stock and lock in a risk-free profit. It should be noted, however, that these arbitrage opportunities are extremely rare and it's very difficult for individual investors to capitalize on them, even when they do exist. Part of the reason is that individual investors would simply be too slow to respond to such a short-lived opportunity. But the main reason is that the market participants generally prevent these opportunities from existing in the first place.