Derivative Securities Assignment
Question 1 -
The S&P 500 spot level is 2,383. The 1-year at-the-money call on is selling at $150. The risk-free rate is 4% and the index pays a dividend yield of 3%. The S&P 500 options are European.
(a) What is the theoretical price of the 1-year at-the-money put price?
(b) The 1-year put is selling at $120 on the market, show how you can benefit from this arbitrage opportunity. Show all details.
(c) If the S&P 500 options were American, will there be an arbitrage opportunity?
Question 2 -
Consider a 6-month bull call spread on MDLZ with strikes of $35 and $50. TSLA spot price is $43 and its volatility is 20%. The risk-free rate is 4% per annum continuously compounded. We assume that MDLZ is not expected to pay any dividend.
(a) Use a 6-step binomial tree to price the spread (note: up and down movements need to match the volatility. Show all the tree parameters).
(b) What are the break-even point(s), the maximum profit and maximum loss for this strategy?
(c) Without using the binomial tree, what is the premium of the bear put spread with the same strike prices? Explain.
Question 3 -
A European derivative instrument on MDLZ has the following payoff structure at the maturity date in 3 years:
a) ST if ST < 30
b) 30 + 2 * (ST - 30) if 30 < = ST <= 50
c) 70 if 50 <= ST <= 70
d) ST if 70 <= ST
where ST is the price at the maturity date. The spot price is 44 and the volatility is 20%. The risk-free interest rate is 4% and we consider a 6-step binomial tree.
(a) Use Excel to draw this payoff pattern for the following price interval [0 , 100] with a step of 5.
(b) Based on the graph in (a), explain briefly how the premium of this derivative security should compare to MDLZ spot price.
(c) Price this contract using a 6-step binomial tree and confirm your findings in (b). Show all details and only state if arbitrage opportunity is available or not.
Question 4 -
Consider a 1-year European put on AMZN with a strike price of $900. AMZN spot price is $850 and its volatility is 20%. AMZN is expected to pay no dividend. The risk-free rate is 4%.
(a) Use Black-Scholes-Merton (BSM) model to price this put option.
(b) What is the put delta? If the short position decides to delta-hedge, what would the net cash flow be?
(c) Without using BSM model, what are the 1-year 900-strike straddle price and its delta?
Question 5 -
We consider the following options strategy on GOOG and its total payoff at the expiry date of the short-term option.
|
Option
|
Type
|
Position
|
Remaining Life
|
Strike
|
|
1
|
Call
|
Short x 1
|
0
|
750
|
|
2
|
Call
|
Long x 2
|
0.5
|
800
|
|
3
|
Call
|
Short x 1
|
0
|
850
|
For the options with a remaining life, we use BSM model to determine their values. We assume a volatility of 30% and a risk-free rate of 4%.
(a) Compute the total payoff for S = 750, 800 and 850.
(b) Use Excel to draw the payoff pattern for prices between $500 and $1000 with a step of $10.