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Mohammed Ameer K.
Graduate Student in Mathematical Finance at UNCC
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Basic Math
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Question:

Ned Stark is aged five times more than his son Brandon Stark. After 12 years, he would be 3 and half times Brandon's age. After further 10 years, how many times would he be Brandon's age?

Mohammed Ameer K.

One important point for solving the age problems is always assume the current age and add or subtract from current age to build the linear equations and solve them. So, lets assume the current age of Ned Stark is N And the current age of his son Brandon Stark is B The first point given in the question is that Ned is aged five times more than his son Brandon. That means the relation between the current age of Ned and Brandon can be written as follows: \$\$ N = 5*B ---------equation(1) \$\$ The second point given in the question is the relation between Ned and his son Brandon after 12 years , so it can be written as follows: \$\$ N+12 = 3.5*(B+12) ---------equation(2) \$\$ Now we have 2 linear equations in two variables and we can solve them to get the values of N and B. Subsituting equation(1) in equation(2), \$\$ N+12 = 3.5*(B+12) \$\$ \$\$ 5B +12 = 3.5*(B+12) \$\$ \$\$ 5B +12 = 3.5*B+ (3.5*12) \$\$ \$\$ 5B +12 = 3.5B+ 42 \$\$ \$\$ 5B -3.5B = 42-12 \$\$ \$\$ 1.5B = 30 \$\$ \$\$ B = 20 \$\$ Since \$\$ N = 5*B = 5*20 = 100 \$\$ So the current age of Ned Stark is 100 years and current age of Brandon Stark is 20 years. We need to find the realtion between the ages of Ned and Brandon further 10 years after intial 12 years. So ideally we are 12+10= 22 years in future. The ages of Ned and Brandon 22 years from now is Age of Ned will be 122 years Age of Brandon will be 42 years. So the Ned's age will be \$\$ 122/42 = 2.9047619 \$\$ times of Brandon's age. Afer rounding off, Ned's age will be \$\$ 2.91\$\$ times of Brandon's age.

Economics
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Question:

A company XYZ finds that the marginal product of capital is 70 and the marginal product of labor is 30. If the price of capital is \$8 and the price of labor is \$5, describe how the firm should adjust its mix of capital and labor? What will be the result?

Mohammed Ameer K.

The information given in the question is, a) Marginal product of capital is 70 b) Marginal product os labor is 30 c) price of capital is \$8 d) price of labor is \$5 We need to find how the firm adjust its mix of capital and labor. So this is a question of marginal decision rule. So what exactly is marginal decision rule? Let's throw some light on this. The marginal decision rule describes that if marginal benefit of an activity exceeds the marginal cost of that activity, we should increase that particular activity until the marginal cost is equal to the marginal benefit. The marginal benefit of this activity is the utility gained by spending an additional \$1 on the good. The marginal cost is the utility lost by spending \$1 less on another good. Now lets calculate the marginal productiviy for both capital and labor in our case. Marginal productivity for capital = \$\$ MP_{k} = MU_{k}/P_{k} = 70/8 = 8.75 \$\$ Marginal productivity for labor = \$\$ MP_{l} = MU_{l}/P_{l} = 30/5 = 6 \$\$ The above two marginal productivites for capital and labor specifies that \$1 additional spent on capital will increase the company's output by 8.75 units and whereas \$1 additional spent on labor will increase the company's output by 5 units. Since the marginal productivity of capital is more than marginal productivity of labor, company will choose to spend on capital. It will continue to do so until the marignal benefit is equal to the marginal cost. It will be captial intensive for a period of time.

Finance
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Question:

A stock price is currently \$100. Over each of the next two six-month periods, it is expected to go up by 10% or down by 10%. The risk-free interest rate is 8% per annum with simple compounding.What is the value of a one-year European call option with a strike price of \$100?

Mohammed Ameer K.

From the data given in the question, we know that, a) It is a European call option b) Current stock price which is denoted by \$\$S_{0}\$\$ = \$100 c) Risk-free interest rate, r = 8% = 0.08 d) Striker price, K = \$100 e) Up or down percentage = 10% f) Number of periods = 2 (six month periods) Since it is given that how the stock price changes at every period. Either it will go up or go down by 10%. We know the stock prices at every period and we have strike price to calculate the pay offs at every period. So its a question based on the no-arbitrage option pricing theory for a European call option. We have to find out the value of the one-year European call option with a strike price of \$100. Let's denote the value as \$\$V_{0}\$\$. The calculation of \$\$V_{0}\$\$ will involve three steps. Step 1: We will calculate the stock prices at different periods and the pay offs. It is given the stock price at time 0, \$\$S_{0}\$\$ = 100 At time 1: The stock price can go up or go down by 10%. Let's denote as \$\$S_{1}(H)\$\$ if it goes up or \$\$S_{1}(T)\$\$ if it goes down. If the stock price goes up by 10% at time 1, then \$\$S_{1}(H) = 100 * (1+0.1) = 111 \$\$ If the stock price goes up by 10% at time 1, then \$\$S_{1}(T) = 100 * (1-0.1) = 90 \$\$ At time 2: The stock price can again go up or go down by 10%. Let's denote as \$\$S_{2}(HH)\$\$ if it goes up or \$\$S_{2}(HT)\$\$ if it goes down for \$\$S_{1}(H)\$\$ and similarly for the other node as well. If the stock price goes up by 10% at time 2, then \$\$S_{2}(HH) = 111 * (1+0.1) = 121 \$\$ If the stock price goes up by 10% at time 2, then \$\$S_{2}(HT) = 111 * (1-0.1) = 99 \$\$ Similarly, for the other node, Let's denote as \$\$S_{2}(TH)\$\$ if it goes up or \$\$S_{2}(TT)\$\$ if it goes down for \$\$S_{1}(T)\$\$ and similarly for the other node as well. If the stock price goes up by 10% at time 2, then \$\$S_{2}(TH) = 90 * (1+0.1) = 99 \$\$ If the stock price goes up by 10% at time 2, then \$\$S_{2}(TT) = 90 * (1-0.1) = 81 \$\$ Now we know the stock prices at all possible nodes and we also know it is a European call option, hence it can only be exercised at the expiration i.e., at time 2. The strike price of the call option K, is given as \$100 and we know that the pay off of a call option is 0 if the stock price is less than strike price and the pay off is \$\$ S_{T} - K \$\$ So the pay for all possible 4 outcomes at time 2 are as follows: So the pay off when \$\$S_{2}(HH)\$\$ is given by \$\$V_{2}(HH) = S_{T} - K = 121-100 = 21\$\$ Similarly for the other three scenarios, the pay off is 0 since the stock price is less than the strike price. So \$\$V_{2}(HT) =V_{2}(TH) =V_{2}(TT) = 0 \$\$. We have calculated the stock prices at different periods and their respective pay offs as well. This is the end of step 1. Step 2: We have to calculate the values of p and q which we will use in the no-arbitrage option pricing model to calculate the value of the option at time 0. To calculate p and q, we need r, u, d. Risk-free interest rate, r = 8% per annum, so the six-month interest rate will be 4% or 0.04 The ratio of stock price when it has moved up is u, so \$\$ u=S_{1}(H)/ S_{0} \$\$ So, \$\$ u=110 / 100 = 1.1 \$\$ Similarly, the ratio of stock price when it has moved down is d, so \$\$ u=S_{1}(T)/ S_{0} \$\$ So, \$\$ d=90 / 100 = 0.9 \$\$ We know that, \$\$p = (1+r-d) / (u-d) \$\$ So, \$\$p = (1+0.04-0.9) / (1.1-0.9) = 0.7 \$\$ Since \$\$p + q = 1 \$\$, that implies, \$\$q = 1 - p = 1-0.7 = 0.3\$\$. Now we have calculated u,d,p,q. That is the end of step 2. Step 3: We have to calculate the value of the option at time zero using the no arbitrage option pricing model. We will start at the last period i.e., time 2 and then we track back to time zero. We know the payoffs of four different nodes at time 2, \$\$V_{2}(HH),V_{2}(HT) ,V_{2}(TH) ,V_{2}(TT) \$\$ . If we have to calculate the values of pay off \$\$V_{1}(H),V_{1}(T)\$\$ for time 1, we use the below formuale . \$\$ V_{1}(H) = 1 / (1+r) (pV_{2}(HH) + qV_{2}(HT) )\$\$ \$\$ V_{1}(H) =( 1 / (1+0.04)) (0.7*21 + 0.3*0 )\$\$ \$\$ V_{1}(H) = 14.135\$\$ Similarly, \$\$ V_{1}(T) = 1 / (1+r) (pV_{2}(TH) + qV_{2}(TT) )\$\$ \$\$ V_{1}(T) = (1 / (1+0.04)) (0.7*0 + 0.3*0 )\$\$ \$\$ V_{1}(T) = 0\$\$. Now we know that pay offs at time 1, we track back to time 0 to calculate the value of the option at time 0 using the below formuale. \$\$ V_{0} = 1 / (1+r) (pV_{1}(H) + qV_{1}(T) )\$\$ \$\$ V_{0} = (1 / (1+0.04)) (0.7*14.135 + 0.3*0 )\$\$ \$\$ V_{0} = 7.175\$\$. So, the value of a one-year European call option with a strike price of \$100 is \$7.175

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