Exam 15: Mathematics of Black-Scholes

Consider a stock that is trading at $50, has a volatility of 0.5, and pays no dividends. The risk-free rate is 4%. If the beta of the stock is 1.1, what is the beta of a 52-strike, one-year call option on this stock?

Consider a stock that is trading at $50. A six-month at-the-money put option on the stock has a price of 2.21 and a delta of $- 0.40$ . The stock volatility is 20%, the risk-free rate is 4%, and the beta of the stock is 1.1. What is the beta of the put?

Option pricing models are based on Ito processes. Which of the following statements best describes Ito processes? Ito processes $Y _ { t }$ are

Which of the following properties of a put option's beta is most valid?

Consider a stock that is trading at $50. A six-month at-the-money call option on the stock has a price of 3.45 and a delta of 0.60. The stock volatility is 20%, the risk-free rate is 4%, and the beta of the stock is 1.1. What is the beta of the call?

Given that $d Y _ { t } = b d W _ { t }$ and $X _ { t } = Y _ { t } ^ { 3 }$ , what is $d X _ { t }$ ?

The fundamental asset pricing partial differential equation (PDE) is used to derive the Black-Scholes formula. Which of the following statements is not true about the fundamental PDE?

Option pricing in continuous time makes use of Wiener processes. Which of the following is not a property of a Wiener process $W _ { t }$ , given $W _ { 0 } = 0$ ?

Given the following Ito process for a stock: $d S _ { t } = 0.2 S _ { t } d t + 0.4 S _ { t } d W$ , what is the expected value of the stock after 3 years if the current price of the stock is $50?

Given that $d Y _ { t } = b d W _ { t }$ and $X _ { t } = e ^ { Y t }$ , what is $d X _ { t }$ ?

Which of the following is not a characteristic of a price process $Y _ { t }$ that follows a geometric Brownian motion (GBM)?

Which of the following is necessary in order to solve the fundamental PDE to obtain the price of a derivative security?

A call option in the Black-Scholes model is a function of the stock price and time, i.e., $C ( S , t )$ . Which of the following statements is valid with regards to the change in the option price over time, i.e., $d C ( S , t )$ ?

Given that $d Y _ { t } = b d W _ { t }$ and $X _ { t } = \ln \left( Y _ { t } \right)$ , what is $d X _ { t }$ ?

The Black-Scholes model is based on a posited stochastic process for stock prices, where the movements in the stock are represented mathematically by a stochastic differential equation (SDE). Which of the following statements is most valid?