problem solving in financial mathematics

Basic practice problem set 2 – one-period binomial pricing model

This practice problem reinforces the concept of binomial option pricing model discussed in the following two posts

found in this companion blog. The practice problems are basic and straightforward problem to price an option using a binomial tree based on a volatility factor $\sigma$.

_____________________________________________________________________________________

Practice Problems

Practice Problem 1
A 6-month call option on a stock is modeled as a binomial tree. Given the following information:

• The initial stock price is $65. • The strike price is$70.
• $r=$ 0.05.
• $\delta=$ 0.
• $\sigma=$ 0.3.

find the replicating portfolio and the price of this 6-month call option.

Practice Problem 2
This problem uses the same information as in Problem 1, except that the stock pays annual continuous dividends at the rate of $\delta=0.02$.

Practice Problem 3
A 6-month put option on a stock is modeled as a binomial tree. Given the following information:

• The initial stock price is $65. • The strike price is$60.
• $r=$ 0.05.
• $\delta=$ 0.02.
• $\sigma=$ 0.3.

use risk-neutral probabilities to price this 6-month put option.

$\text{ }$
_____________________________________________________________________________________

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

_____________________________________________________________________________________

Answers

1. Replicating portfolio: $\Delta=$ 0.43508237 shares (long), $B=$ -$22.8747876 (borrowing). $\text{ }$ Call option price = $\Delta S + B$ =$5.405566446

$\text{ }$

2. Replicating portfolio: $\Delta=$ 0.406303779 shares (long), $B=$ -$21.36173125 (borrowing). $\text{ }$ Call option price = $\Delta S + B$ =$5.048014419

$\text{ }$

• $\text{ }$

$p^*=$ 0.447164974

$1-p^*=$ 0.552835026

$uS=$ =$81.57471137 (up stock price) $dS=$ =$53.37034398 (down stock price)

Put option price = $p^*$ (0) + $(1-p^*)$ (6.629656019) = $3.574614265 $\text{ }$ _____________________________________________________________________________________ $\copyright \ 2015 \text{ by Dan Ma}$ Advertisements Pricing a put option, a practice problem This practice problem reinforces the concept of binomial option pricing model discussed in the following two posts in a companion blog. _____________________________________________________________________________________ Practice Problems Suppose the stock of a certain company is currently selling for$50 per share. The price per share at the end of one year is expected to increase to $60 (20% increase) or to decrease to$40 (20% decrease). The stock pays no dividends during the next year. The annual risk-free interest rate is 5%. Assume all the options discussed here are European options.

1. A put option on the stock specifies an exercise price (strike price) of $45 and is set to expire at the end of one year. What is the fair price of this put option? Use the one-period binomial option model. What is the portfolio replicating the payoff of this put option? 2. Consider a call option on the same underlying stock and with the same strike price and the same time to expiration as the put option in #1. Use the put-call parity to derive the price of this call option. 3. Price the call option described in #2 using the binomial option model. $\text{ }$ _____________________________________________________________________________________ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ _____________________________________________________________________________________ Answers 1. Replicating portfolio: $\Delta=$ -0.25 shares (short), $B=$$14.26844137 (lending).
$\text{ }$
Put option price = $\Delta S + B$ = $1.768441368 $\text{ }$ 2. Call option premium =$8.963117265
3. $\text{ }$

4. Replicating portfolio: $\Delta=0.75$ shares (long), $B=$ -$28.53688274 (borrowing). $\text{ }$ Call option price = $\Delta S + B$ =$8.96311726
$\text{ }$

_____________________________________________________________________________________
$\copyright \ 2015 \text{ by Dan Ma}$

Pricing a call option, a practice problem

This practice problem reinforces the concept of binomial option pricing model discussed in the following two posts

in a companion blog.

_____________________________________________________________________________________

Practice Problems

Suppose the stock of a certain company is currently selling for $50 per share. The price per share at the end of one year is expected to increase to$60 (20% increase) or to decrease to $40 (20% decrease). The stock pays no dividends during the next year. The annual risk-free interest rate is 5%. Assume all the options discussed here are European options. 1. A call option on the stock specifies an exercise price (strike price) of$55 and is set to expire at the end of one year. What is the fair price of this call option? Use the one-period binomial option model. What is the portfolio replicating the payoff of this call option?
2. Consider a put option on the same underlying stock and with the same strike price and the same time to expiration as the put option in #1. Use the put-call parity to derive the price of this put option.
3. Price the put option described in #2 using the binomial option model.

$\text{ }$
_____________________________________________________________________________________

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

_____________________________________________________________________________________

Answers

1. Replicating portfolio: $\Delta=$ 0.25 shares, $B=$ -$9.512294245 (borrowing). $\text{ }$ Call option price = $\Delta S + B$ =$2.9877
$\text{ }$
2. Put option premium = $5.305324103 3. $\text{ }$ 4. Replicating portfolio: $\Delta=-0.75$ shares (short), $B=$$42.8053241 (lending).
$\text{ }$
Put option price = $\Delta S + B$ = \$5.3053241
$\text{ }$

_____________________________________________________________________________________
$\copyright \ 2015 \text{ by Dan Ma}$