Question

Problem 1: Properties of Options

The price of a European put that expires in six months and has a strike price of $100 is $3.59. The underlying stock price is $102, and a dividend of $1.50 is expected in four months. The term structure is flat, with all risk-free interest rates being 8% (cont. comp.).

a. What is the price of a European call option on the same stock that expires in six months and has a strike price of $100? [1 marks]

b. Explain in detail the arbitrage opportunities if the European call price is $6.1. How much will be the arbitrage profit? [3.5 marks]

c. Explain in detail the arbitrage opportunities if the European call price is $8.8. How much will be the arbitrage profit? [3.5 marks]

Problem 2: Option Valuation

In this question, you need to price options with various approaches. You will consider puts and calls on a share. Please read following instructions carefully:

• The spot price of this share will be determined by your student number. You need to use the last digit of your student number. The spot price of the share will be $13

• The strike price of the options will be the share price you just calculated +2. For example, if the share price you calculated based on your student number is 56, the strike price of the options will be (56+2)=58

Based on this spot price and this strike price as well as the fact that the risk-free interest rate is 6% per annum with continuous compounding, please undertake option valuations and answer related questions according to following instructions:

Binomial trees:

Additionally, assume that over each of the next two four-month periods, the share price is expected to go up by 11% or down by 10%.

a. Use a two-step binomial tree to calculate the value of an eight-month European call option using the no-arbitrage approach. [2.5 marks]

b. Use a two-step binomial tree to calculate the value of an eight-month European put option using the no-arbitrage approach. [2.5 marks]

c. Show whether the put-call-parity holds for the European call and the European put prices you calculated in a. and b. [1 mark]

d. Use a two-step binomial tree to calculate the value of an eight-month European call option using risk-neutral valuation. [1 mark]

e. Use a two-step binomial tree to calculate the value of an eight-month European put option using risk-neutral valuation. [1 mark]

f. Verify whether the no-arbitrage approach and the risk-neutral valuation lead to the same results. [1 mark] g. Use a two-step binomial tree to calculate the value of an eight-month American put option. [1 mark]

h. Calculate the deltas of the European put and the European
call at the different nodes of the binomial three. **[1
mark]**

Black-Scholes-Merton model:

Using the information given above regarding the spot and strike price, risk-free rate of return and the fact that the volatility of the share price is 18%, answer following questions:

i. What is the price of an eight-month European call? [1
mark]

j. What is the price of an eight-month American call? [1 mark]

k. What is the price of an eight-month European put? [1 mark]

l. How would your result from k. change if a dividend of $1 is expected in three months? How would your result from k. change if a dividend of $1 is expected in ten months? [2 marks]

Note for calculations with the BSM model: Keep four decimal points for d1 and d2. Use the Table for N(x) with interpolation in calculating N(d1) and N(d2).

Finally, m. Compare the results you obtained for the prices of European puts and calls using binomial trees and Black-Scholes-Merton model. How large are the differences when expressed as a percentage of the spot price of the share? Provide a possible explanation for these differences. [2 marks]

Answer #1

In this question, you need to price options with various
approaches. You will consider puts and calls on a share.
Based on this spot price (36) and this strike price (38) as well
as the fact that the risk-free interest rate is 6% per annum with
continuous compounding, please undertake option valuations and
answer related questions according to following instructions:
Binomial trees:
Additionally, assume that over each of the next two four-month
periods, the share price is expected to go...

Based on the spot price of $26 and the strike price $28 as well
as the fact that the risk-free interest rate is 6% per annum with
continuous compounding, please undertake option valuations and
answer related questions according to following instructions:
Binomial trees:
Additionally, assume that over each of the next two four-month
periods, the share price is expected to go up by 11% or down by
10%.
Use a two-step binomial tree to calculate the value of an
eight-month...

The price of a European put that expires in six months and has a
strike price of $100 is $3.59. The underlying stock price is $102,
and a dividend of $1.50 is expected in four months. The term
structure is flat, with all risk-free interest rates being 8%
(cont. comp.).
What is the price of a European call option on the same stock
that expires in six months and has a strike price of $100?
Explain in detail the arbitrage...

A stock index currently stands at 300 and has a volatility of
20%. The risk-free interest rate is 8% and the dividend yield on
the index is 3%.
Use the Black-Scholes-Merton formula to calculate the price of
a European call option with strike price 325 and the price of a
European put option with strike price of 275. The options will
expire in six months.
What is the cost of the range forward created using options in
Part (a)?
Use...

Consider a non-divided-paying stock where the stock price is
$200, the strike price is $200, the risk-free rate is 5% per annum,
the volatility is 35% per annum, and the time to maturity is 5
months.
a) Value the European call and put options using the
Black-Schools-Merton formula
b) Do you think the values of the European call and put options
satisfy the put- call parity? Provide evidence for your answer.
c) Value the European call and put options using...

The price of a European call that expires in six months and has
a strike price of $28 is $2. The underlying stock price is $28, and
a dividend of $1 is expected in 4 months. The term structure is
flat, with all risk-free interest rates being 6%. If the price of a
European put option with the same maturity and strike price is $3,
what will be the arbitrage profit at the maturity?

Suppose that stock price moves up by 5% (u=1.05) and d=1/u. The
current stock price is $50. Dividend is zero. Compute the current
value of a European call option with the strike price of $51 in 3
months using both replicating portfolio valuation method and the
risk neutral valuation method. The risk free rate is APR 5% with
continuous compounding (or, 5% per annum)1. Draw the
dynamics of stock price and option price using the one step
binomial tree.
2. Draw...

A 3-month European call on a futures has a strike price of $100.
The futures price is $100 and the volatility is 20%. The risk-free
rate is 2% per annum with continuous compounding. What is the value
of the call option? (Use Black-Scholes-Merton valuation for futures
options)

~~~In Excel~~~
Question 2. 1-month call and put price for
European options at strike 108 are 0.29 and 1.70, respectively.
Prevailing short-term interest rate is 2% per year.
a. Find current price of the stock using the put-call
parity.
b. Suppose another set of call and put options on the same stock
at strike price of 106.5 is selling for 0.71 and 0.23,
respectively. Is there any arbitrage opportunity at 106.5 strike
price? Answer this by finding the amount of...

The price of a non-dividend paying stock is $45 and the
price of a six-month European call option on the stock with a
strike price of $46 is $1. The risk-free interest rate is 6% per
annum. The price of a six-month European put option is $2. Both put
and call have the same strike price. Is there an arbitrage
opportunity? If yes, what are your actions now and in six months?
What is the net profit in six months?

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