2. Option definitions, payoffs, and boundaries
Liuren Wu
We consider one “underlying” risky security (it can be a stock or exchange rate), and we use S to denote
its price, with S0 = 100 being its current price and ST being its future price at time T . Ignore bid-ask spread
and transaction cost and assume that you can buy or sell any amount of the security at the price S0 = 100.
1. Plot the payoff of a long call option position with strike K = 80 and expiry T .
Answer: The payoff is max(0, ST − K), or you can write it as (ST − K)+ , where the positive sign means
that you only take the positive value. If (ST − K) is negative, you don’t exercise and hence get zero.
60
50
Payoff = max(ST−K,0)
40
30
20
10
0
−10
−20
60
70
80
90
100
ST
110
120
130
140
(a) Also plot the payoff of a short call, a long put, a short put, respectively, all at the same strike and
expiry. Answer: I’ll plot all 4 cases together for comparison:
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Short call
60
40
40
20
20
Payoff = −max(ST−K,0)
Payoff = max(ST−K,0)
Long call
60
0
−20
−40
−60
60
0
−20
−40
70
80
90
100
ST
110
120
130
−60
60
140
70
80
90
60
40
40
20
20
0
−20
−40
−60
60
110
120
130
140
110
120
130
140
110
120
130
140
Long put
60
Payoff = max(K−ST,0)
Payoff = −max(K−ST,0)
Short put
100
ST
0
−20
−40
70
80
90
100
ST
110
120
130
−60
60
140
70
80
90
100
ST
(b) Plot a forward with delivery price K = 80.
I plot the payoff of both long forward and short forward position:
Short forward
60
40
40
20
20
Payoff = K−ST
Payoff = ST−K
Long forward
60
0
0
−20
−20
−40
−40
−60
60
70
80
90
100
ST
110
120
130
140
−60
60
70
80
90
100
ST
Show how to replicate the forward contract with (i) the underlying and bond and (ii) with calls
and puts at the same strike and expiry.
(i) You can do the buy-and-carry argument as we did in the first homework, using the underlying
and bond. By “bond,” I mean you can borrow money and save money in the bank. Borrowing
money is equivalent of selling a bond. Whereas saving the money in the bank is similar to buying
a bond.
(ii) Long call and short put at the same strike will create a long forward. This is shown by
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comparing the long forward payoff with the payoff from Long call + short put (the first column of
the previous four charts). The short forward position can be similarly created by short call+ long
put (the second column of the four charts).
(c) Assume zero rates and no dividends. If the call and the put (at K = 80 and same T ) are quoted at
$26 and $5, respectively. Is there an arbitrage?
Long call and short put creates a long forward. This portfolio costs 26 − 5 = 21. You can think
of this as the value of a “synthetic” forward, a forward created/synthesized from long a call and
short a put.
On the other hand, we know the value of this forward should be erT (F − K). The forward price is
100 (as there is zero rates or dividends). The value of the forward should be 100 − 80 = 20 (zero
rates). This value is generated based on buy and carry.
Since these two are the same contracts (payoffs), but difference prices, we should buy the forward
at $20, and sell the synthetic (sell the call, buy the put) for a selling price of $21. Thus, we net $1.
2. Assume zero rates and no dividends, is there an arbitrage trade if the call (at K = 80) is quoted at $101?
The call price looks high. The highest price of the call cannot be higher than the present value of the
forward price, which is 100. Hence, 101 is too high. To trade against this, sell the call for 101, and buy
the forward with zero delivery price for 100. You make one dollar right away and at expiry, the payoff
is ST − max(0, ST − 80). When ST > 80, we have ST − ST + 80 = 80. When ST < 80, we have ST . The
payoff is always positive, even though you received $1 to get into this portfolio.
Is there an arbitrage if the call is quoted at $19?
This looks low. I would check the lower bound, which should be the value of the forward with the
same strike (delivery price K − 80). The value of this forward is 100-80=20. And the option value
cannot be lower than the forward value of 20. So we buy the option for 19 and sell the forward for
20, and net $1. At expiry T , the payoff is max(ST − K, 0) − (ST − K). When ST > K = 80, the payoff
is ST − 80 − (ST − 80) = 0. When ST < 80, the payoff is 0 − (ST − K) = 80 − ST > 0. Hence, always
positive.
3. Assume zero rates and no dividends, is there an arbitrage if a put at K = 110 is quoted at $111? Is there
an arbitrage if it is quoted at $9?
The put payoff is (K − ST )+ . The price of $111 looks high because even if the stock price drops zero,
the payoff is only K − 0 = 110. Given zero rates, the present value of this payoff is $110. So the put
price cannot be higher than $110 as the stock price cannot drop below zero. The arbitrage would be
to just sell the put and say the sales receipt of $111 in the bank. At expiration, the worst is if the stock
price drops to zero an you pay $110, which will still leave you with one dollar. You’ll have more left in
the bank if the stock price ends above zero.
The price of $9 looks low, because the current stock price is S0 = 100 (and hence the forward is 100
since there is no rates and dividends)and the intrinsic value of the put option is e−rT (K − F) = 110 −
100 = 10. Remember the option value is equal to time value (optionality value)+intrinsic value. When
the option value is lower than the intrinsic value, there is an arbitrage. Another way of saying it is:
The short position in a forward with the same strike is worth e−rT (K − F) = 10 and the option is worth
more than the forward because of the optionality. The arbitrage is to long the forward (short the short
position) at strike $110, which costs you $10, and buy the put at $9. You net one dollar today. At
expiration, you cannot possibly lose money. Specifically, if the stock price is below strike, you exercise
the option to get the payoff of K − S, which cancel with the long forward position payoff S − K. If the
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stock price is above strike, you make money from the forward S − K > 0, but you don’t need to exercise
the put option.
4. How will the answers be changed if the underlying security is a stock that pays $1 per share dividend
every quarter if the option has a one-year maturity? (still zero rates)
This changes your forward valuation. The forward price becomes S − 4 = 100 − 4 = 96. The forward calculation becomes very easy this time because the rates are zero and hence the future value of
dividends are just the raw dividend amount (no need for compounding or discounting).
Let’s focus on question 3. It does not affect the answer if the put is worth $111 because we are comparing with stock price dropping to zero. Now, if the put price is $9, the arbitrage is larger because the
value of the short forward contract is worth more now at e−rT (K − F) = 110 − 96 = 14. The put has to
be higher than $14 to exclude arbitrage.
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