Derivatives Markets

Robert L. McDonald

Chapter 11 Binomial Option Pricing: II - all with Video Answers

Educators

Chapter Questions

Problem 1

Consider a one-period binomial model with $h=1$, where $S=\$ 100, r=0$ $\sigma=30 \%,$ and $\delta=0.08 .$ Compute American call option prices for $K=\$ 70$ $\$ 80, \$ 90,$ and $\$ 100$.
a. At which strike(s) does early exercise occur?
b. Use put-call parity to explain why early exercise does not occur at the higher strikes.
c. Use put-call parity to explain why early exercise is sure to occur for all lower strikes than that in your answer to (a).

Breanna Ollech

Numerade Educator

00:41

Problem 2

Repeat Problem $11.1,$ only assume that $r=0.08 .$ What is the greatest strike price at which early exercise will occur? What condition related to put-call parity is satisfied at this strike price?

Amy Jiang

Numerade Educator

01:56

Problem 3

Repeat Problem $11.1,$ only assume that $r=0.08$ and $\delta=0 .$ Will early exercise ever occur? Why?

Joanna Quigley

Numerade Educator

01:08

Problem 4

Consider a one-period binomial model with $h=1$, where $S=\$ 100, r=0.08$ $\sigma=30 \%,$ and $\delta=0 .$ Compute American put option prices for $K=\$ 100$ $\$ 110, \$ 120,$ and $\$ 130$
a. At which strike(s) does early exercise occur?
b. Use put-call parity to explain why early exercise does not occur at the other strikes.
c. Use put-call parity to explain why early exercise is sure to occur for all strikes greater than that in your answer to (a).

Breanna Ollech

Numerade Educator

01:32

Problem 5

Repeat Problem 11.4 , only set $\delta=0.08$. What is the lowest strike price at which early exercise will occur? What condition related to put-call parity is satisfied at this strike price?

Niamat Khuda

Numerade Educator

01:30

Problem 6

Repeat Problem $11.4,$ only set $r=0$ and $\delta=0.08 .$ What is the lowest strike price (if there is one) at which early exercise will occur? If early exercise never occurs, explain why not.

Aidan Jan

Numerade Educator

04:23

Problem 7

Let $S=\$ 100, K=\$ 100, \sigma=30 \%, r=0.08, t=1,$ and $\delta=0 .$ Let $n=10$
Suppose the stock has an expected return of $15 \%$
a. What is the expected return on a European call option? A European put option?
b. What happens to the expected return if you increase the volatility to $50 \% ?$

Manasvee Singh

Numerade Educator

04:23

Problem 8

Let $S=\$ 100, \sigma=30 \%, r=0.08, t=1,$ and $\delta=0 .$ Suppose the true expected return on the stock is $15 \% .$ Set $n=10 .$ Compute European call prices, $\Delta,$ and $B$ for strikes of $\$ 70, \$ 80, \$ 90, \$ 100, \$ 110, \$ 120,$ and $\$ 130 .$ For each strike, compute the expected return on the option. What effect does the strike have on the option's expected return?

Manasvee Singh

Numerade Educator

07:43

Problem 9

Repeat the previous problem, except that for each strike price, compute the expected return on the option for times to expiration of 3 months, 6 months, 1 year, and 2 years. What effect does time to maturity have on the option's expected return?

John Lee

Numerade Educator

04:23

Problem 10

Let $S=\$ 100, \sigma=30 \%, r=0.08, t=1,$ and $\delta=0 .$ Suppose the true expected return on the stock is $15 \%$. Set $n=10$. Compute European put prices, $\Delta$, and $B$ for strikes of $\$ 70, \$ 80, \$ 90, \$ 100, \$ 110, \$ 120,$ and $\$ 130 .$ For each strike, compute the expected return on the option. What effect does the strike have on the option's expected return?

Manasvee Singh

Numerade Educator

07:43

Problem 11

Repeat the previous problem, except that for each strike price, compute the expected return on the option for times to expiration of 3 months, 6 months, 1 year,and 2 years. What effect does time to maturity have on the option's expected return?

John Lee

Numerade Educator

02:37

Problem 12

Let $S=\$ 100, \sigma=0.30, r=0.08, t=1,$ and $\delta=0 .$ Using equation (11.17) to compute the probability of reaching a terminal node and $S u^{\prime} d^{n-i}$ to compute the price at that node, plot the risk-neutral distribution of year- 1 stock prices as in Figures 11.8 and 11.9 for $n=3$ and $n=10$.

Kratika Bhadauria

Numerade Educator

01:16

Problem 13

Repeat the previous problem for $n=50 .$ What is the risk-neutral probability that $S_{1}<\$ 80 ? S_{1}>\$ 120 ?$

Gregory Higby

Numerade Educator

02:37

Problem 14

We saw in Section 10.1 that the undiscounted risk-neutral expected stock price equals the forward price. We will verify this using the binomial tree in Figure 11.4
a. Using $S=\$ 100, r=0.08,$ and $\delta=0,$ what are the 4 -month, 8 -month, and 1-year forward prices?
b. Verify your answers in (a) by computing the risk-neutral expected stock price in the first, second, and third binomial period. Use equation (11.17) to determine the probability of reaching each node.

Kratika Bhadauria

Numerade Educator

00:39

Problem 15

Compute the 1 -year forward price using the 50 -step binomial tree in Problem 11.13.

Alison Rodriguez

Numerade Educator

00:39

Problem 16

Suppose $S=\$ 100, K=\$ 95, r=8 \%$ (continuously compounded), $t=1$ $\sigma=30 \%,$ and $\delta=5 \% .$ Explicitly construct an 8 -period binomial tree using the Cox-Ross-Rubinstein expressions for $u$ and $d:$
\[u=e^{\sigma \sqrt{h}} \quad d=e^{-\sigma \sqrt{\hbar}}\]
Compute the prices of European and American calls and puts.

Hossam Mohamed

Numerade Educator

00:50

Problem 17

Suppose $S=\$ 100, K=\$ 95, r=8 \%$ (continuously compounded), $t=1$ $\sigma=30 \%,$ and $\delta=5 \% .$ Explicitly construct an 8 -period binomial tree using the lognormal expressions for $u$ and $d:$
\[u=e^{\left(r-\delta-S \sigma^{2}\right) h+\sigma \sqrt{h}} \quad d=e^{\left(r-\delta-.5 \sigma^{2}\right) h-\sigma \sqrt{h}}\] Compute the prices of European and American calls and puts.

Hossam Mohamed

Numerade Educator

02:50

Problem 18

Obtain at least 5 years' worth of daily or weekly stock price data for a stock of your choice.
a. Compute annual volatility using all the data.
b. Compute annual volatility for each calendar year in your data. How does volatility vary over time?
c. Compute annual volatility for the first and second half of each year in your data. How much variation is there in your estimate?

Linh Vu

Numerade Educator

01:07

Problem 19

Obtain at least 5 years of daily data for at least three stocks and, if you can, one currency. Estimate annual volatility for each year for each asset in your data. What do you observe about the pattern of historical volatility over time? Does historical volatility move in tandem for different assets?

Carson Merrill

Numerade Educator

02:05

Problem 20

Suppose that $S=\$ 50, K=\$ 45, \sigma=0.30, r=0.08,$ and $t=1 .$ The stock will pay a $\$ 4$ dividend in exactly 3 months. Compute the price of European and American call options using a four-step binomial tree.

Dilip Paruchuri

Numerade Educator