Unformatted text preview: ess the future value of the premium and it occurs at a terminal stock
price of zero.
f)
Short Put
We have no control over the exercise decision when we write a put. The buyer of the put option
decides whether to exercise or not, and he will only exercise if he makes a profit. As we have the
opposite side, we will never make any money at the expiration of the put option. Our profit is
restricted to the future value of the premium, and we make this maximum profit whenever the
stock price at expiration is greater than the strike price. However, we lose money whenever the
stock price is smaller than the strike, hence the largest loss occurs when the stock price attains its
smallest possible value, zero. We lose the strike price because somebody sells us an asset for the
strike that is worth nothing. We are only compensated by the future value of the premium we
received. 9 Part 1 Insurance, Hedging, and Simple Strategies Question 12.14. a)
In order to be able to draw profit diagrams, we need to find the future values of the put
premia. They are:
i)
ii)
iii) 35strike put: $1.53 × (1 + 0.08) = $1.6524
40strike put: $3.26 × (1 + 0.08) = $3.5208
45strike put: $5.75 × (1 + 0.08) = $6.21 We get the following payoff diagrams: 10 Chapter 2 An Introduction to Forwards and Options We get the profit diagram by deducting the option premia from the payoff graphs. The profit
diagram looks as follows: b)
Intuitively, whenever the 35strike put option pays off (i.e., has a payoff bigger than
zero), the 40strike and the 35strike options also pay off. However, there are some instances in
which the 40strike option pays off and the 35strike options does not. Similarly, there are some
instances in which the 45strike option pays off, and neither the 40strike nor the 35strike pay
off. Therefore, the 45strike offers more potential than the 40 and 35strike, and the 40strike
offers more potential than the 35strike. We pay for these additional payoff possibilities by
initially paying a higher premium. It makes sense that the premium is increasing in the strike
price. 11 Part 1 Insurance, Hedging, and Simple Strategies Question 2.16. The following is a copy of an Excel spreadsheet that solves the problem: 12 Chapter 3
Insurance, Collars, and Other Strategies
Question 3.2.
This question constructs a position that is the opposite to the position of Table 3.1. Therefore, we
should get the exact opposite results from Table 3.1. and the associated figures. Mimicking Table
3.1., we indeed have:
S&R Index S&R Put Payoff
(Cost +Interest) Profit
900.00 100.00 1000.00
1095.68
95.68
950.00
50.00 1000.00
1095.68
95.68
1000.00
0.00 1000.00
1095.68
95.68
1050.00
0.00 1050.00
1095.68
45.68
1100.00
0.00 1100.00
1095.68
4.32
1150.00
0.00 1150.00
1095.68 54.32
1200.00
0.00 1200.00
1095.68 104.32 On the next page, we see the corresponding payoff and profit diagrams. Please note that they
match the combined payoff and profit diagrams of Figure 3.5. Only the axes have different
scales.
Payoffdiagram: 13 Part 1 Insurance, Hedging, and Simple Strategies Profit diagram: Question 3.4.
This question is another application of PutCallParity. Initially, we have the following cost to
enter into the combined position: We receive $1,000 from the short sale of the index, and we
have to pay the call premium. Therefore, the future value of our cost is:
($120.405 − $1,000) × (1 + 0.02) = −$897.19 . Note that a negative cost means that we initially
have an inflow of money. 14 Chapter 3 Insurance, Collars, and Other Strategies Now, we can directly proceed to draw the payoff diagram: We can clearly see from the figure that the payoff graph of the short index and the long call
looks like a fixed obligation of $950, which is alleviated by a long put position with a strike price
of 950. The following profit diagram, including the cost for the combined position, confirms this: 15 Part 1 Insurance, Hedging, and Simple Strategies Question 3.6. We now move from a graphical representation and verification of the PutCallParity to a
mathematical representation. Let us first consider the payoff of (a). If we buy the index (let us
name it S), we receive at the time of expiration T of the options simply ST .
The payoffs of part (b) are a little bit more complicated. If we deal with options and the
maximum function, it is convenient to split the future values of the index into two regions: one
where ST < K and another one where ST ≥ K . We then look at each region separately, and hope
to be able to draw a conclusion when we look at the aggregate position.
We have for the payoffs in (b):
Instrument
Get repayment of loan
Long Call Option
Short Put Option ST < K = 950
$931.37 × 1.02 = $950
max (ST − 950, 0) = 0
− max ($950 − S T , 0) ST ≥ K = 950
$931.37 × 1.02 = $950
ST − 950
0 Total = S T − $950
ST ST We now see that the total aggregate position only gives us ST , no matter what the final index
value is – but this is the same...
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 Spring '14
 NguyenThiMaiLan
 Derivatives, simple strategies

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