Chapter 6
Appendix A
This Appendix is intended to prove that the invalid transition probability in the Derman-Kani tree indicates the existence of arbitrage opportunities. Also, to avoid the occurrence of invalid transition probability, each newly determined stock price is required to be within the range as indicated in Eq. (3.11).
The up-move transition probability in the Derman-Kani tree is defined as:
(A.1)
where is the up-move transition probability at node at time , is the forward price at time , and is the stock price at time .
There are two cases for the transition probability to be invalid, which are 1) the transition probability is greater than 1, and 2) the transition probability is less than 0.
We are going to discuss them separately.
Case 1
From Eq. (A.1),
(A.2) In addition, since is the up-move from , and is the down-move from , is bound to be greater than or equal to . That is, the following condition must be hold
. (A.3)
Combining with Inequities (A.2) and (A.3) gives rise to the following inequality:
(A.4) If the relationship between stock prices and forward prices holds as inequality (A.4), one can make a riskless arbitrage by taking a short position on the forward at time and to buy the stocks for settlement at time . The net payoff is 0 at time , but the net payoff is either or at time .
Both of these two payoffs are greater than 0, whereas the initial cost is 0. These indicate arbitrage opportunities. Therefore, if the inequality (A.4) holds, arbitrage opportunities exist.
Case 2
From (A.1),(A.5)
Also, since is the up-move from , and is the down-move from , is bound to be less than or equal to . That is, the following condition must be hold
. (A.6)
Combining with Inequities (A.5) and (A.6) gives rise to the following inequality:
(A.7) If the relationship between stock prices and forward prices holds as inequality (A.7), one can make a riskless arbitrage by taking a long position on the forward at time and to settle the forward contract to buy the stocks and sell the stock immediately at time . The net payoff is 0 at time , but the net payoff is either or at time . Both of these two payoffs are greater than 0, whereas the initial cost is 0. Therefore, if the inequality (A.7) holds, arbitrage opportunities exist.
From Case 1 and Case 2, if the probability is greater than 1 or less than 0, there are arbitrage opportunities. Therefore, the forward price has to be within the range as the following inequality to ensure no arbitrage opportunities:
(A.8)
Looking at and in inequality (A.8) simultaneously, there are two inequalities. They are:
(A.9) Combining these two inequities into one gives rise to the following inequality:
(A.10)
To rule out arbitrage opportunities, this inequality must also hold. Because the forward prices on time are known, if is less than or greater than , the opportunities exist.
In conclusion, invalid transition probabilities indicate arbitrage opportunities. To ensure arbitrage opportunities not occur, the stock prices in the nodes within the implied tree have to be within the following range:
(A.11) We have now proved that the invalid transition probability in the Derman-Kani tree indicates the existence of arbitrage opportunities. The following is to prove inequality (A.11) must hold for the transition probability to be valid.
Looking at and in inequalities (A.11) simultaneously, there are two inequalities, which are:
(A.12)
Combining these two inequities into one gives rise to the following inequality:
(A.13)
If the first inequality does not hold, the transition probability, will be less than 0; if the second inequality does not hold, the transition probability, will be greater than 1. The transition probabilities in both cases are invalid. The proof is as follows.
Case 1
If , then
(A.14)
Also, must hold in the tree, i.e., . Therefore, the up-move transition probability, , is less than 0. That is
(A.15) This is an invalid transition probability.
Case 2
If , then
(A.16)
Subtracting from both and , inequality (A.16) becomes
(A.17)
By arranging items in inequality (A.17), (A.17) therefore becomes
(A.18)
It is invalid for a transition probability to be greater than 1.
According to both cases discussed above, if the inequality (A.13) does not hold, i.e., arbitrage opportunity exists; the transition probability will be invalid.
Appendix B
This section is to prove that if the stock prices in the nodes within the Li tree are out of a specific range as indicated in inequality (4.14), arbitrage opportunities exist.
To rule out arbitrage opportunities, the stock prices have to be within the range:
(B.1) where is the forward price at time , and is the stock price at time
. In opposite to (B.1), if the relationship between the forward price and stock price is either or , then arbitrage opportunities exist. Let’s discuss these two cases separately.
Case 1
Suppose the relationship between the forward price and stock price is
(B.2) Also, since is the up-move form , and is the down-move from
, is bound to be greater than or equal to . That is, the following condition must hold
. (B.3)
Combining with inequities (B.2) and (B.3) gives rise to the following inequality:
(B.4) If the relationship between stock prices and forward prices holds as inequality (B.4), one can make a riskless arbitrage by taking a short position on the forward at time and to buy the stocks for settlement at time . The net payoff is 0 at time , but the net payoff is either or at time . Both of these two payoffs are greater than 0, whereas the initial cost is zero. These indicate arbitrage opportunities. Therefore, if inequality (B.4) holds, arbitrage opportunities exist.
Case 2
Suppose the relationship between the forward price and stock price is
(B.5) Also, since is the up-move from , and is the down-move from
, is bound to be less than or equal to . That is, the following condition must hold
. (B.6)
Combining with inequities (B.5) and (B.6), gives rise to the following:
(B.7) If the relationship between stock prices and forward prices holds as inequality (B.7), one can make a riskless arbitrage by taking a long position on the forward at time and to settle the forward contract to buy the stocks and sell the stocks immediately at time . The net payoff is 0 at time , but the net payoff is either
or at time . Both of these two payoffs are greater than 0. Therefore, if inequality (A.7) holds, arbitrage opportunities exist.
From Case 1 and Case 2, if the probability is greater than 1 or less than 0, there is an arbitrage opportunity. Therefore, the forward price has to be within the range as the following inequality to rule out arbitrage opportunities.
(B.8)
Looking at and in inequality (B.8) simultaneously, there are two inequities. They are:
(B.9)
Combining these two inequities into one gives rise to the following:
(B.10)
To rule out arbitrage opportunities, this must also hold. Because the forward prices on time are known, if is less than or greater than
, arbitrage opportunities exist.
In conclusion, invalid transition probabilities indicate arbitrage opportunities. To ensure arbitrage opportunities not happen, the stock prices in the nodes within the implied tree have to be within the following range:
(B.11)
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