Question

A collar is established by buying a share of stock for $46, buying a six-month put option with exercise price $43, and writing a six-month call option with exercise price $49. Based on the volatility of the stock, you calculate that for an exercise price of $43 and maturity of six months, n(d1):

a) Will be negative
b) Will be positive
c) Will be zero

Answer 1

The value of n(d1) for the put option with a stock price of $46, exercise price of $43, and volatility indicates a negative value.

This suggests a higher probability of the option expiring out of the money, reflecting the stock price exceeding the exercise price at expiration.the correct option is a) Will be negative.

Step-by-step explanation:

The calculation of n(d1) in the Black-Scholes model for options pricing involves several variables, including the stock price, exercise price, time to maturity, risk-free rate, and volatility. In this scenario, with a stock trading at $46, a put option with an exercise price of $43, and the stock's volatility factored in, the resulting value of n(d1) for the put option will be negative.

The formula for d1 in the Black-Scholes model is:

`\[d1 = \frac{{\ln(\frac{S}{{K}}) + (r + \frac{{\sigma^2}}{2})T}}{{\sigma √(T)}}\]`

Here, 'S' represents the stock price, 'K' denotes the exercise price, 'r' is the risk-free rate, 'σ' stands for volatility, and 'T' signifies the time to maturity. As the stock price ($46) is higher than the put option's exercise price ($43), the term
`\(\ln(\frac{S}{{K}})\)` will be positive. Given the nature of the terms in the equation and considering that the exercise price is lower than the stock price, n(d1) will yield a negative value. This result signifies the likelihood of the option expiring out of the money, suggesting a probability of the stock price being higher than the exercise price at expiration, as expected with a negative n(d1).