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Commerce. Engage in arbitrage. E20.
创建日期：2024-04-24 15:54:56
标签： E20
Commerce. Engage in arbitrage

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Commerce. Engage in arbitrage. E20.

In finance, an arbitrage opportunity means the possibility of a risk-free trading profit. This could be achieved
by making a deal which leads to an immediate profit with no risk of future loss, or making a deal which has no
risk of loss but a non-zero probability of making a future profit.
An arbitrageur is a trader who exploits arbitrage possibilities. Such possibilities do not exist for long—because
they are effectively money-making machines!
The term short-selling or shorting means selling an asset that is not owned with the intention of buying it
back later. The mechanics are as follows: suppose client A instructs a broker to short sell S. The broker
then borrows S from the account of another client and sells S and deposits the proceeds in the account of A.
Eventually A pays the broker to buy S in the market who returns it to the account he borrowed it from. (It is
possible that the broker will run out of S to borrow and then A will be short-squeezed and be forced to close
out his position prematurely.)
Example 1.1a. Suppose an investor has a portfolio which includes security A and security B. The prices at time 0
are as follows: sA = 6 and sB = 11. He assesses the prices at time 1 will be SA1 = 7 and S
B
1 = 14 if the market goes
up and SA1 = 5 and S
B
1 = 10 if the market goes down.
time 0 time 1 time 1
(market rises) (market falls)
A 6 7 5
B 11 14 10
Check there is an arbitrage opportunity.
Solution. Suppose the investor sells two A and buys one B at time 0. Hence he makes a gain of 1 at time 0. At
time 1, whether the market rises of falls, there is no change in the value of his portfolio.
Clearly, investors will sell A and buy B. This implies the current price of A will fall relative to the price of B, until
sA = sB/2.
Example 1.1b. Suppose an investor has a portfolio which includes security A and security B. The prices at time 0
are as follows: sA = 6 and sB = 6. He assesses the prices at time 1 will be SA1 = 7 and S
B
1 = 7 if the market goes up
and SA1 = 5 and S
B
1 = 4 if the market goes down.
time 0 time 1 time 1
(market rises) (market falls)
A 6 7 5
B 6 7 4
Check there is an arbitrage opportunity.
Solution. Suppose the investor buys one A and sells one B at time 0. Hence he neither gains nor loses at time 0. At
time 1, there is no change in the value of his portfolio if the market rises; he gains 1 if the market falls.
Clearly, investors will buy A and sell B. The current price of A will rise relative to the price of B, until the arbitrage
opportunity is eliminated.
ST334 Actuarial Methods cR.J. Reed Sep 27, 2016(9:51) Section 1 Page 131

Page 132 Section 1 Sep 27, 2016(9:51) ST334 Actuarial Methods cR.J. Reed
1.2 Existence of an arbitrage. Suppose investment A costs the amount sA and investment B costs the
amount sB at time t0. Let NPVA(t0) denote the net present value at time t0 of the payoff from investment A.
Here are three situations where an arbitrage exists:
• If NPVA(t0) = NPVB(t0) and sA 6= sB then there is an arbitrage.
• If NPVA(t0) 6= NPVB(t0) and sA = sB then there is an arbitrage.
• If sA > sB and NPVA(t0)  NPVB(t0) or if sA < sB and NPVA(t0) NPVB(t0) then there is an
arbitrage.
Example (1.1a) is an example of the first formulation—apply it to two A and one B. Example (1.1b) is an
example of the second formulation. Now consider the third formulation: sA > sB andNPVA(t0)  NPVB(t0).
Selling one A and buying one B at time 0 produces a cash surplus at time 0 and increases the value of the
portfolio at time t = 1.
1.3 Option pricing.
Example 1.3a. Suppose a share has price 100 at time t = 0. Suppose further that at time t = 1 its price will rise
to 150 or fall to 50 with unknown probability. Suppose the effective rate of interest is r per annum.
Suppose C denotes the price at time t = 0 of a call option for one share for the exercise price of 125 at the exercise
time of t = 1. Determine C so there is no arbitrage.
Solution. The given information is summarised in the following table:
t = 0 t = 1
probability p1 probability 1 p1
share 100 150 50
call option at t = 1 for 125 C 25 0
Consider buying x shares and y call options at time t = 0. This will cost 100x + Cy at time t = 0. At time t = 1, this
portfolio will have value 150x + 25y if the share value rises to 150, or 50x if the share value falls to 50.
Hence if y = 4x, the value of the portfolio will be 50x whether the share price rises or falls.
Suppose an investor has capital z at time 0 where z > max{100, 4C}. Consider the following 2 possible decisions:
• Investment decision A: buy 1 share, invest the rest;
• Investment decision B: buy 4 options, invest the rest.
At time t = 0, the value of the portfolio is z in both cases. For decision A, the value of the portfolio at time t = 1 is
(z 100)(1 + r) +
n 150
50
= (z 100)(1 + r) + 50 +
n 100
0
For decision B, the value of his portfolio at time t = 1 is
(z 4C)(1 + r) +
n 4⇥ 25 = 100
0
For no arbitrage, we can equate these. Hence (100 4C)(1 + r) = 50 which implies
C =
1
4
✓
100 50
1 + r
◆
We can check this is the condition for no arbitrage as follows:
Suppose C > 14

100 50/(1 + r).
At time t = 0, sell 4 options for 4C and borrow 50/(1+r). The total amount received is z = 4C+50/(1+r) > 100.
Use 100 to buy one share and the rest is the arbitrage—the rest is the amount z 100.
At time t = 1 sell the share. If the share then has price 150, repay the 50 and use the remaining 100 for paying
for the 4 options; if the share has price 50, then the options are worthless and use the 50 to repay the loan.
Suppose C < 14

100 50/(1 + r).
At time t = 0, short sell one share for 100; purchase 4 options for 4C and put the rest of the money, which is
100 4C > 50/(1 + r), in the bank.
At time t = 1, if the share has price 150, then the 4 options are worth 25 each and there will be more than 50 in
the bank; if the share has price 50, then the options are worthless but there will be more than 50 in the bank. In
both cases, the obligation of repurchasing the share which was sold short is covered.
More succinctly, compare
t = 0 t = 1: probability p1 t = 1: probability 1 p1
share 100 150 50
four call options at t = 1 for 125 4C 100 0
If no arbitrage, we must have 4C + 50/(1 + r) = 100.
7 Arbitrage Sep 27, 2016(9:51) Section 2 Page 133
2 Forward contracts
2.1 Historical background. The spot price for a commodity such as copper is the current price for immediate
delivery. This will be determined by the current supply and demand.
Suppose a manufacturer requires a supply of copper. He cannot plan his production if he does not know how
much he will have to pay in 6 months’ time for the copper he then needs. Similarly, the copper supplier does
not know how to plan his production if he does not know how much he will receive for the copper he produces
in 6 months’ time. Both supplier and consumer can reduce the uncertainty by agreeing a price today for the
“6 months’ copper”.
2.2 Forward contracts. A forward contract is a legally binding contract to buy/sell an agreed quantity of
an asset at an agreed price at an agreed time in the future. 
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