The following is for a mathematical finance course, so showing things algebraically will be greatly appreciated.

Suppose the stock price is $50, and you bought 10 contracts (each
contract is 100 shares) of 3-month call option with strike $50 in the hope that
the stock will go up in three months. However, the call option comes with a
volatility risk, namely the option price will drop if the volatility drops. On the
other hand, you do not believe that the stock is going to go above $60 so you
decide to take a short position in a call with strike $60.

(a) How many contracts of this call should you sell in order that the vega of the portfolio is zero?

(b)Once the vega of the portfolio is zero, you will not worry about the volatility moves just
for the moment. But you still want the portfolio to be delta-neutral. What can
you do to maintain the portfolio to be delta-neutral? You can assume r = 0 and
= 30% at the time of pricing.

The Wheat producers of North America are expecting to match their supply to meet the change in the market demand during coming year that is expected to change. The current year market supply and supply function is assumed to be:

Q_D= 62.5 - 0.125P

Q_S= 0.5P - 100

Please predict either a rise or fall in the demand of wheat as a percentage change from the current year demand based on your observations from the website and derive a new demand function with the predicted percentage change (EITHER INCREASE OR DECREASE) in demand, (i.e. the quantity demanded will increase or decrease by THE PERCENTAGE for each level of price). (Please restrict your change in demand within ± 25%)

Please compute the following:

1. Market Equilibrium Price for current year and next year with percentage change.
2. Market Equilibrium Quantity for current year and next year with percentage change
3. Producer Surplus and Consumer Surplus for current year and following year with percentage change

Swifty Corporation manufactures a product with a unit variable cost of $100 and a unit sales price of $176. Fixed manufacturing costs were $480000 when 10000 units were produced and sold. The company has a one-time opportunity to sell an additional 1000 units at $145 each in a foreign market which would not affect its present sales. If the company has sufficient capacity to produce the additional units, acceptance of the special order would affect net income as follows:

Income would increase by $45000.

Income would increase by $3000.

Income would increase by $145000.

Income would decrease by $3000.

Coronado Industries is using the target cost approach on a new product. Information gathered so far reveals:

What is the target selling price per unit?

$0.48

$0.35

$0.70

$0.83

produces several types of pickled vegetables. The company budgets for each quarter in the last month of the previous quarter. In early March, Karen is preparing the budget for pickled beets. Budgeted sales are 13200 jars for April, 16500 jars for May, and 21000 jars for June. Each jar requires 1.3 pounds of beets. The pickling process takes 60 minutes for 20 jars. Because pressurized cooking is used, the processing is monitored by an employee at all times. Each jar of pickled beets sells for $15.00.

Karen requires ending Finished Goods inventory equal to 30% of the following month’s sales. Other information is as follows:

What is Karen’s direct labor budget for May?

Suppose ?(?, ?) = ?3?

a) (3 pts) Write down the formula and calculate the marginal rate of substitution (MRS) of X for Y.

b) (1 pts) Denote the price for good X is ? , the price for good Y is ? , and income is I.?? Write down the budget constraint.

c) (5 pts) Derive the individual demand for good X and good Y as a function of prices and income. (hint: solve the utility maximization problem)

d) (2 pts) Explain the following statement: Individual demand function is homogeneous of degree 0 with respect to prices and income.

e) (2 pts) Verify the homogeneity property with your demand functions derived in part d).

f) (4 pts) Now suppose ? = 3, ? = 4, ? = 100. Find the optimal consumption bundle. ??

g) (2 pts) Show whether or not the consumer can afford the bundle (15, 20).

h) (2 pts) What is the maximum utility level?

Agnieszka ‘s Opera House can sell tickets to two types of customers: music lovers and tourists. Assume for simplicity that each customer will purchase one ticket only. Assume (for simplicity) that the cost of providing the ticket is zero. The valuation or willingness to pay ($ per ticket) each type of buyer places on both types of tickets is presented below:

Which of the following pricing schemes yields the highest profits if the firm cannot identify the type of buyer who is making the purchase? Assume that if the customer is indifferent between buying a ticket or not, he buys it.
Note, P_N is the price of the normal ticket and P_V the price of the VIP ticket.

P_N=30; P_V=30.

P_N=300; P_V=400.

P_N=300; P_V=300.

P_N=50; P_V=351.

P_N=50; P_V=349.

JTC purchased call options on Flynn common shares on July 7, 2020, for $200 as a speculative investment. The call options give JTC the right to buy 100 shares at a strike price of $20 each. The options expire on January 31, 2021.

The following data is observed through 2020:

a. At September 30, 2020, the options are on JTC's balance sheet at a value of ? Muliple Choice: ["$350", "$150", "$200", "$1,950"].

b. In the fourth quarter (October - December) of 2020, JTC records a loss in time value of? Muliple Choice: ["$150", "$90", "$110", "$60"].

c. At December 31, 2020, the options are on JTC's balance sheet at a value of? Muliple Choice: ["$260", "$350", "$460", "$490", "$200", "$290"].

1. Assume the following information concerning two stocks that make up an index. What is the value-weighted return for the index? (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places.)

Index Level:

1. Assume that Roberts’s utility from consuming good X and good Y is given by the following function:

U = 5X^0.2Y^0.8

Where X is the quantity of good X while Y is the quantity of good Y.

Assume the price of X (P_X) is £5, the price of Y (P_Y) is £8 and he has a budget of £100 to spend on the two goods.

1. Using the Lagrangian multiplier, calculate the quantities of good X and good Y Robert should purchase to maximise his utility.
2. How much utility does Robert receive at his optimal consumption bundle?
3. Calculate l at the optimal consumption bundle and provide an explanation of its value.
4. Using your previous answers, sketch the budget constraint and indifference curve at the optimum consumption point. Clearly explain and label (i) the quantities of X and Y (ii) the intercepts of the budget constraint (iii) the slope of the budget constraint and (iv) the level of utility.

Consider the following bonds: · Bond “LUKE” is a 3-year bearing coupon bond of 5% every six-months with a face value of $1,000. Coupon payments of $50 are made every 6 months. · Bond “ROUGE” is a 3-year bearing coupon bond of 12% per year with a face value of $1,000. Coupon payments of $120 are made every 12 months. · Suppose that the yield on the bond is 6% per annum with continuous compounding.

a) Calculate the bond’s price, duration & convexity.

b) Regarding TSIR: explain the meaning of duration. According to the “Liquidity Preference Theory”, which is the best option for a trader for a 3-years investment? “LUKE” or “ROUGE”? Justify your answer.

c) How would you price the bond “LUKE” if the yield curve fell down 100 basis points (continuous compounding)?

d) How much is the modified duration regarding a rate in semi-annual compounding?