Exam 11: Option Pricing: an Introduction

In a one-period binomial model, assume that the current stock price is $100, and that it will rise to $110 or fall to $90 after one month. The risk-free gross return per time step is 1.001668. If a 98-strike call option is trading at $2, how much arbitrage profit can you make in present value terms?

Suppose that in a binomial model, the stock moves up by a factor $\mathcal { U } = e ^ { \Sigma \sqrt { h } }$ and down by a factor $d = e ^ { - \sigma - \sqrt { h } }$ , where $h _ { c }$ is time in years. Letting $h =$ one month and $u = 1.05$ , what is the annualized volatility $\sigma$ of the stock?

In a one-period binomial model, assume that the current stock price is $100, and that it will rise to $110 or fall to $90 after one month. If the risk-neutral probability of the stock going up is equal to 0.52, what is the one-month forward price of the stock?

You hold a portfolio consisting of 300 calls (short) and 200 puts (long) on a given stock. The delta of the calls is $+ 0.62$ and the delta of the puts is $- 0.57$ . To delta hedge this portfolio, you should

Pricing options in the risk-neutral world implies that

Which of the following statements best describes the range of possible values of the delta of a call option?

In a one-period binomial model, assume that the current stock price is $100, and that it will rise to $110 or fall to $90 after one month. If an investment of a dollar at the risk-free rate returns $1.001668 after one month, what is the price of an Arrow security in the state where the stock price moves up?

In a one-period binomial model, assume that the current stock price is $100, and that it will rise to $110 or fall to $90 after one month. If the risk-free rate is 0.1668% per month in simple terms, what is the price of a 99-strike one-month put option?

In a portfolio insurance strategy, when stock prices drop, the portfolio is rebalanced by

The current price of a stock is 40. After one period, it will be worth either 41.80 or 38.20 with probabilities 0.60 and 0.40, respectively. The risk-free rate of interest over this period, in gross terms, is 1.05 (i.e., a dollar invested at the beginning of the period returns 1.05 at the end of the period). The no-arbitrage value of a one-period call option on this stock with a strike of 40

Assuming all else is constant, which of the following statements best describes the delta of a put option?

You hold a portfolio of European options on a stock that is (i) long 200 at-the-money calls, each with a delta of $0.52$ , (ii) short 200 at-the-money puts, and (iii) long 100 shares of stock. The aggregate delta of your portfolio is

In a one-period binomial model, assume that the current stock price is $100, and that it will rise to $110 or fall to $90 after one month. What is the delta of a 99-strike one-month put option?

In a one-period binomial model, assume that the current stock price is $100, and that it will rise to $110 or fall to $90 after one month. If an investment of a dollar at the risk-free rate returns $1.001668 after one month, what is the standard deviation of monthly stock return under the risk-neutral probabilities?

You are long 300 at-the-money calls on Stock ABC, each with a delta of $+ 0.54$ , and short 200 in-the-money calls on Stock XYZ, each with a delta of $+ 0.66$ . Which of the following statements is most accurate in this context?

In a one-period binomial model, assume that the current stock price is $100, and that it will rise to $110 or fall to $90 after one month. If an investment of a dollar at the risk-free rate returns $1.001668 after one month, how many 100-strike one-month call options and how many units of the stock will be needed to be combined to obtain a riskless $1 portfolio?

In a one-period binomial model, assume that the current stock price is $100, and that it will rise to $110 or fall to $90 after one month. If the risk-neutral probability of the stock going up or down is equal, what is the one-month risk-free interest rate in continuously-compounded and annualized terms?

Which of the following statements best describes the delta of vanilla options?

In a one-period binomial model, assume that the current stock price is $100, and that it will rise to $115 or fall to $80 after three months. If risk-free investment has a gross return of 1.005 per three month time-step, what is the best replicating portfolio of one hundred 101-strike three-month put options from the following options?

In a one-period binomial model, assume that the current stock price is $100, and that it will rise to $110 or fall to $90 after one month. If the risk-free rate is 0.1668% per month in simple terms, which of the following choices best describes the replicating portfolio for one hundred 99-strike one-month call options?