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1. Consider the lower bound of the European call option. Use your intuition to try to explain why this must be the lower bound. (Hint: read the text- book and lecture notes) Searching for intuition essentally menas looking for an explanation that is easy to understand (aim for easy for a child to understand) but need not be complete or a proof.
2. Consider a $1978 loan to be paid in 8 equal installments every 6 months at 6.75% interest rate (annual). The maturity of the loan is 4 years. Fill in the chart. Per Payment PV Factor Formula Total.
3.
Your father is 50 years old and will retire in 10 years. He expects to live for 25 years after he retires, until he is 85. He wants a fixed retirement income that has the same purchasing power at the time he retires as $50,000 has today. (The real value of his retirement income will decline annually after he retires.) His retirement income will begin the day he retires, 10 years from today, at which time he will receive 24 additional annual payments. Annual inflation is expected to be 3%. He currently has $90,000 saved, and he expects to earn 10% annually on his savings. The data has been collected in the Microsoft Excel Online file below. Open the spreadsheet and perform the required analysis to answer the question below.
| Required annuity payments | |||
| Retirement income today | $50,000 | ||
| Years to retirement | 10 | ||
| Years of retirement | 25 | ||
| Inflation rate | 3.00% | ||
| Savings | $90,000 | ||
| Rate of return | 10.00% | ||
| Calculate value of savings in 10 years: | Formulas | ||
| Savings at t = 10 | #N/A | ||
| Calculate value of fixed retirement income in 10 years: | |||
| Retirement income at t = 10 | #N/A | ||
| Calculate value of 25 beginning-of-year retirement payments at t =10: | |||
| Retirement payments at t = 10 | #N/A | ||
| Calculate net amount needed at t = 10: | |||
| Value of retirement payments | #N/A | ||
| Value of savings | #N/A | ||
| Net amount needed | #N/A | ||
| Calculate annual savings needed for next 10 years: | |||
| Annual savings needed for retirement | #N/A | ||
How much must he save during each of the next 10 years (end-of-year deposits) to meet his retirement goal? Do not round your intermediate calculations. Round your answer to the nearest cent.
1.
The lower bound for prices of European call options
Claim
Let c be the price of an European call option with strike price K
and maturity T. Then the following inequality
S0 − Ke−rT ≤ c ≤ S0
holds under the assumption that no dividend is payed on the stock.
Proof
The inequality c ≤ S0 was already proved. Suppose that
c + Ke−rT < S0. We will show that somebody can make an
arbitrage profit in this case.
First Proof)
Suppose one trader has already the stock in his portfolio. Now he
sees at the market the offer for a call option for the stock with
strike price K and maturity T for a price c such that
c + Ke−rT < S0.
He immediately sells one stock, obtains S0 EUR, buys the call
option and puts the cash Ke−rT on a deposit. After the maturity
date T he then has K (EUR). At maturity date he exercises the
call option and buys the stock with the cash K, and the stock is
again in his portfolio.
Thus he earned from this business S0 − (c + Ke−rT ) EUR without
any risk.
Second Proof
We consider two portfolios.
Portfolio A : 1 European call and cash = Ke−rT .
Portfolio B : 1 share.
We shall compare the performance of these portfolios. After time
T we have
Portfolio A : The cash is worth K.
If ST > K, the option is exercised and the portfolio is worth ST .
If ST < K, option is not exercised and the portfolio is worth K.
Thus the portfolio is worth max(ST ,K).
Portfolio B : the portfolio is worth ST .
Here we use the assumption that no dividends are payed on the
stock.
We see that portfolio A is worth at least as much as portfolio B at
time T.
Thus the worth of portfolio A at time 0 must be also bigger or
equal than the worth of portfolio B, since otherwise we could make
an arbitrage profit. Therefore
c + Ke−rT ≥ S0
2.
We calculate the value of installments using the excel formula "PMT"
PMT(0.0338,8,-1978,,0) = 286.25
| Loan | 1978 | ||||
| Semi annual rate | 3.38% | ||||
| Per | Payment | PV Factor | Formula | ||
| 0.5 | 286.25 | 0.49 | (1/r) -[1/r*(1+r)^0.5] | ||
| 1 | 286.25 | 0.97 | (1/r) -[1/r*(1+r)^1] | ||
| 1.5 | 286.25 | 1.44 | (1/r) -[1/r*(1+r)^1.5] | ||
| 2 | 286.25 | 1.90 | (1/r) -[1/r*(1+r)^2] | ||
| 2.5 | 286.25 | 2.36 | (1/r) -[1/r*(1+r)^2.5] | ||
| 3 | 286.25 | 2.81 | (1/r) -[1/r*(1+r)^3] | ||
| 3.5 | 286.25 | 3.25 | (1/r) -[1/r*(1+r)^3.5] | ||
| 4 | 286.25 | 3.68 | (1/r) -[1/r*(1+r)^4] | ||
| Total | 2290.03 |
3.
Value of savings in 10 years = 90000(1+10%)^10 = $233436.8
Value of fixed retirement income in 10 yeras = 50000/(1.03)^10 = $ 37204.70
Value of 25 year begigning of retirement payments = (50000*25)*(1.03)^25 = $2617222.41
Value of Payments = 2617222.41
Value of Savings = (233436.70)
Net Amount needed = $2383785.71
Annual savings needed for next 10 years = $177376.85
