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I have the answers in word file. I need them to be transferred to Excel showing calculations with links and formulas. Each answer on a separate tab with tab number given as question number,
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Question 1
n = 2 periods
a = 30% per annum = 0.30
r = 1% per annum = 0.01
Calculate u and d:
U= ea √(T/n)= e 0.30 √(0.5)= e 0.30 √(0.5)= e 0.300.7071= e0.2121= 1.2365
d= 1/u= 1/1.2365= 0.8086
Calculating the potential stock prices at the conclusion of each phase
The stock prices can either rise/fall
S0= current price= $60
If stock prices rises: Su = S0 u = 60 1.2365 ~ $74.19
Stock prices falls: Sd = S0 d = 60 0.8086 ~ $48.52
Ending of the second period:
Stock prices rise: Suu = Su u = 74.19 1.2365 ~ $91.74
Stock prices falls afer the first: Sud = Su d = 74.19 0.8086 ~ $59.98
Stock price goes down after first movement: Sdd = Sd d = 48.52 0.8086 ~ $39.29
Determining the up and down variables
Estimate u and d using the risk-free rate, volatility, and number of periods
u ~ 1.2365, d ~ 0.8086.
Determining the maximum stock prices
Analyze the maximum stock price among every possbiel stock price
Maximum stock price = $91.74
Calculating the option layoff
Option payoff = Maximum stock price – Current stock price
$91.74- $60= $31.74
Discounting the option pay off to the current value
This is performed using the risk free rate
Present value of the option payoff = Option payoff / (1 + r)^T =
$31.74/ (1+0.01)1= $31.42
Early exercise privilege premium = Present value of the option payoff – Current price of the
option
= $31.42 – $0 (because the premium is unpaid) = $31.42
Thus the early exercise privilege premium of a one year American look back on the maximum
option is $31.42
Question 2
Regulating Stock Price for Ex-Dividend Date:
Because the ex-dividend date falls at the close of the second time frame, the stock price is
modified only following the second month. This implies that the payout is taken off the stock
price at the start of the third month.
Regulated price of the stock at the third month= $60- $1
= $59
Calculating Call Option Value at Maturity:
A European call choice enables the investor to purchase stock at the price of strike on the day it
is set to expire. Since the option matures at the conclusion of the third month, we use the updated
stock price.
Call Value at Maturity (Max (Adjusted Stock Price – Strike Price, 0)):
If the stock value is higher than the price at strike, the call option is in-the-money and worth the
distinction between the two.
If the stock value falls under the strike price, the call choice is considered out-of-the-money and
has no value.
The call value at maturity (CVM):
CVM= Max ($59-$60, 0)
CVM= $0= (out-of-the-money)
Discounting call value to the current price
Current value= call value at maturity e-rT
r = risk-free rate per period (0.02)
T = time to maturity (3 months)
Present value= $0e-0.023
= $0
With a 10% monthly appreciation or depreciation rate and a strike price of $60, the stock price is
likely to go under $60 by the date of maturity. This results in the call option being out of the
money and having an outcome of zero. While dividend payments lower stock prices, they have
no influence on the value of call options that are currently out of the money.
Thus, the current value of the 3-month 60 European call in this situation is zero.
Question 3
S0 = Initial stock price = $40
u = Element by which the stock price can rise = 1.10
d = Element by which the stock price can decline = 0.90
r = Risk-free rate per period = 2% per annum = 0.02 per month T = Time to maturity = 2 months
X = Exercise price (strike price) = Stock price at the time of shouting
Approximate stock values after two months:
If the stock improves in both times
Su= So U2= 401.102= 48.40
Stock increases in the first phase but decreases in the second term.
Sd= Soud= 401.100.90= 39.60
If the stock declines in both times
Sd= Sod2= 400.902= 36.36
expected payoff at time 1
Expected Payoff
1= e-=0.022 [0.5547(Su-X)+ (1-0.5547)0
To calculate the exercise price X, we look at two possible outcomes:
Stock rises in the two months:
X= Su= 48.40
Stock falls in both priods:
X= Sd= 36.36
Compute the estimated incentive at time one for each workout price situation.
If X= 48.40
The expected pay off time= 1=e−0.04×[0.5547×(48.40−48.40)]=0
If X= 36.36
Expected pay off time:
1= e-0.04 [0.554748.40-36.36)]= 4.270
Evaluating these payoffs with zero (since the shout option has no negative value), we discover
that the shout choice is only utilized.
When X= 36.36 leads to $4.270
The price of the shout choice is $4.270 when the exercise value has been adjusted to $36.36.
Question 4
Where 10% improves the stock value over the following two months.
Every month, there is a 1.1 rise
40*1.1*1.1
=$48.4
Here 1 month has 0.9 depreciates
=40*0.9*0.9
=$32.4
Here 1 month has 0.9 depreciates
=40*0.9*0.9
=$32.4
The possibility is 50%.
Then the stock =(0.5*48.4)+(0.5*32.4)
=24.2+16.2
= 40.4
Question 5
Here, the government guarantees that we will acquire a project at the outset.
u=180/104
d=60/104
=0.58
ert =e0.08×1
=1.0833
The government assures project acquisition at time one, similar to a put choice= e rt
=e0.08∗1=1.0833
P=e rt−d/u−d
where rt=risk free rate
=1.0833−0.58/1.73−0.58
1−p=0.5623
Value of option =e−rt (fu.p+fd(1−p))
=e−0.20 (0×(0.4377)+18×(0.5623))
=8.2867
The essential rate of return 20%
The value option is 8.2867 with a necessary rate of return of 20%
Question 6
In this example, the embedded option is a call option.
You have the option, but not the responsibility, to purchase the project for $100 (exercise price),
even if its value falls to $60.
Project Valuation based on Potential Outcomes:
Upside Case (Value goes to $180):
Project value = $180
Payoff = Project value – Initial investment – Loan repayment
=$180 – $44 – ($60 * 1.02) = $74.20
Downside Case (Value goes to $60):
Project value = $60
Exercise the call option: Payoff = Project value – Exercise price = $60 – $100 = -$40 (Loss)
Don’t exercise the call option: Payoff = Initial investment – Project value = -$44
Risk-Neutral Valuation:
Although the risk-free rate is 2%, we must discount the payoffs in both scenarios utilizing the
risk-neutral probability (RNP). Suppose that the RNP in both up and down situations is 50%.
Upside Case:
Discounted payoff = $74.20 * (1/1.02) = $72.75
Downside Case:
Exercise option: Discounted payoff = -$40 * (1/1.02) = -$39.22
Don’t exercise: Discounted payoff = -$44 * (1/1.02) = -$43.14
Embedded Option Value:
The expected payoff considering both cases is:
Expected payoff = (RNP * Upside payoff) + ((1 – RNP) * Downside payoff)
Expected payoff = (0.5 * $72.75) + (0.5 * -$39.22) = $16.76
Isolating the Embedded Option Value:
We know the project’s cost is $104 and the cost of borrowing is $60 * 2% = $1.2.
Therefore, the value of the fundamental asset (project without option) is $104 – $44 – $1.2 =
$58.8.
This predicted payment reflects the whole worth of the project, including the imbedded option.
Embedded Option Value Calculation:
Embedded option value = Expected payoff – Value of underlying asset
Embedded option value = $16.76 – $58.8 = -$42.04
The imbedded call option’s value is negative (-$42.04). This suggests that executing the option is
not viable at the current price ($100), given the anticipated future prices and borrowing costs. In
layman’s words, it’s not worth paying $100 to acquire the project if it may drop to $60, regardless
of whether you have the option to do so.
Question 7 (repeated question 4)
Question 8
Upside: If the charge increases by 20%, the destiny cost is 1000 * 1.20 = 1200.
The downside: If the price drops by 10%, the future cost will be 1000 * 0.90 = 900.
Compute the present value of debt repayment in each scenario:
As a 0-coupon bond, bondholders only get the face value upon adulthood (500). We will use the
chance-loose pricing of 4% to barter this down to its gift price.
Upside present value: PV_upside = 500/ (1+0.04)2= 462.96
Downside present value: PV_downside = 500/ (1+0.04)2= 462.96
Evaluate the equity price in each case.
Upside fairness: 1200 (future price) – 462.96(debt) = 737.04
Analyze the expected go back on equity
Expected Return of Equity= (0.514%) + (0.514%)= 14%
Thus, compute the yield on debt.
Create an equation where the present charge (one thousand) is equal to the estimated future
revenue price of the firm.
Rd = (0.14 – 0.04) * (833.33) / 500 = 0.2091
=20.91%
Question 9
1. Delta-Neutral Portfolio
Yes, the offered portfolio is already delta neutral.
Justification: The “Delta” value is zero, implying that the actual asset price variations in the near
term have no effect on the portfolio’s total price shift.
2. Delta-Neutral & Vega-Neutral Portfolio:
No, it is often impossible to construct a delta-neutral and vega-neutral portfolio with just one
traded choice.
Justification:
Options contain both delta and vega.
Modifying the position dimensions of just one choice will influence both its delta (the rate at
which
Disparity of price of option with fundamental price) and vega (sensitivity of option value) to
fluctuations in predicted volatility.
Because of their dependency, altering the position size of a single option cannot separately
modify its delta and vega, making it impossible to achieve both delta-neutrality and veganeutrality simultaneously.
3. Delta-Neutral, Gamma-Neutral, & Vega-Neutral Portfolio:
This is often not doable with a single traded option. Identical to the preceding situation, changing
the starting point size of one option affects all three Greeks (delta, gamma, and vega).
Alternatives to achieve neutrality
Option spreads: Combining many options with various strike prices and/or
time limits can assist obtain a near approximation to delta neutrality, gamma-neutrality, or veganeutrality individually but not all three simultaneously.
Advanced techniques using many legs (buying and selling different choices) can be devised to
meet particular neutrality goals. However, they need extensive skill and incur higher risks.
Justification:
Due to these Greeks’ dependencies, it is often impossible to build delta-neutral, gamma-neutral,
and vega-neutral portfolios with a single transacted choice; rather, more complex strategies or
choice mixtures are necessary for achieving neutrality.
Hence, Perfect neutrality among all three Greeks (gamma, vega, and delta) is not possible with a
single choice; however, intricate frameworks or choice extends can provide the proper neutrality
thresholds for risk control.
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