## Questions And Problems

Which of the following can be estimated for an American option by constructing a single binomial tree delta, gamma, vega, theta, rho 14.2. Calculate the price of a 3-month American put option on a non-dividend-paying stock when the stock price is 60, the strike price is 60, the risk-free interest rate is 10 per annum, and the volatility is 45 per annum. Use a binomial tree with a time interval of 1 month. 14.3. Explain how the control variate technique is implemented when a tree is used...

## Mean Reversion

Rendleman and Bartter assume that the short-term interest rate behaves like a stock price. One important difference between the interest rates and stock prices is that interest rates appear over time to be pulled back to some long-run average level. This phenomenon known as mean reversion and is not captured by the Rendleman and Bartter model. When r is high, mean reversion tends to cause it to have a negative drift when r is low, mean reversion tends to cause it to have a positive drift. Mean...

## Stock Indices Currencies and Futures Contracts

In this chapter we tackle the problem of valuing options on stock indices, currencies, and futures contracts. As a first step, the analysis in Chapter 10 is extended to cover European options on a stock paying a continuous dividend yield. It is then argued that stock indices, currencies, and many futures prices are analogous to stocks paying continuous dividend yields. The basic results for options on a stock paying a continuous dividend yield can therefore be extended to value options on these...

## The Rendleman And Bartter Model

Rendleman and Bartter make particularly simple assumptions about m(r) and s(r) in Equation (15.10).4 They assume that m(r) - Mr and s(r) Sr where M and S are constants. This means that r follows geometric Brownian motion. It has a constant expected growth rate of M and a constant volatility of 5 in a risk-neutral world. It can be modeled using a binomial tree similar to the one used for stocks in Chapter 14. The parameters u, d, and p chosen are as follows 4See R. Rendleman and B. Bartter, The...

## Info

An interest cost of 2,500 is incurred in the first week. The stock price falls by the end of the first week to 48This reduces the delta to 0.458, and 6,400 of shares are sold to maintain the hedge. This realizes 308,000 in cash and the cumulative borrowings at the end of week 1 are reduced to 2,252,300. During the second week the stock price reduces to 471 and delta declines again and so on. Toward the end of the life of the option it becomes apparent that the option...

## The Lognormal Property Of Stock Prices

A variable has a lognormal distribution if the natural logarithm of the variable is normally distributed. It has just been shown that the model of stock price behavior developed in Chapter 9 implies that In ST - In S lt p ji - T - t , ojT -t 10.6 where St is the stock price at a future time T S is the stock price at the current time, t and 4 gt m, s denotes a normal distribution with mean m and standard deviation s. From the properties of the normal distribution, it follows from Equation 10.6...

## Rho

The rho of a portfolio of derivative securities is the rate of change of the value of the portfolio with respect to the interest rate.14 It measures the sensitivity of the value of a portfolio to interest rates. For a European call option on a non-dividend-paying stock, and for a European put option on the stock, where d2 is defined as in Equation 10.27 . These same formulas apply to European call and put options on stocks and stock indices paying a dividend yield at rate q, and to European...

## Summary

There are six factors affecting the value of a stock option the current stock price, the strike price, the expiration date, the stock price volatility, the risk-free interest rate, and the dividends expected during the life of the option. The value of a call generally increases as the current stock price, the time to expiration, the volatility, and the risk-free interest rate increase. The value of a call decreases as the strike price and expected dividends increase. The value of a put...

## Company S Cash Position Measured In Millions Of Dollars Follows A Generalized Wiener Process

Where the e,- i 1,2, , N are random drawings from a standardized normal distribution. From property 2 the e, 's are independent of each other. It follows from Equation 9.2 that z T z 0 is normally distributed with2 mean of z T - z 0 0 variance of z T - z 0 N At T standard deviation of z T z 0 Vt Thus in any time interval of length T, the increase in the value of a variable that follows a Wiener process is normally distributed with a mean of zero and a standard deviation of lt JT. It should now...