Since expected utility theory can not provide an adequate description of individ-ual choice, Kahneman and Tversky proposed cumulative prospect theory to explain the major violations of expected utility theory in choices between risky prospects.
In this chapter, we interest in seeking out optimal strategies, which make us to gain the maximal profit, in the sense of expected utility theory and cumulative prospect theory, respectively. Our main goal is to compare the optimal strategies in the sense of expected utility theory and cumulative prospect theory. Moreover, we also discuss the difference of optimal hedging strategies in these two senses.
3.1. Optimal Trading Strategy in One Period Model
A fundamental problem in the financial mathematics is to find out the opti-mal trading strategies which can reach the maxiopti-mal profit. In this section, we are interesting in comparing the optimal trading strategies under two different senses:
expected utility theory and cumulative prospect theory.
In the beginning, we set up the market model as follow: Consider a one-period market model in which time points are denoted by 0 and 1, and the market model has one risky asset (stock) and one riskless asset (bond). In time 1 there exist two market states: ω1 and ω2, with probability p1and p2, respectively. The current bond price B0 is 1, and the stock price S0 is s, and in time 1, the stock price denoted by
17
S1 is given by S1(ω1) = l and S1(ω2) = u where l < s < u.
Suppose that the interest rate at time 1 is 0, and initial wealth is x.
Let h = (h0, h1) be the trading strategy at time 0, where h0 and h1 means the number of shares invested in bond and stock, respectively. And denote the optimal strategy by h∗ = (h∗0, h∗1).
Theorem 1. In the sense of expected utility theory, if the mean of gains is greater than the mean of losses, the optimal inveatment amount of stock is greater than 0, i.e., h∗1 > 0 provided that p2(u − s) > p1(s − l). Moreover, the converse is also true. That is, h∗1 > 0 if and only if p2(u − s) > p1(s − l).
PROOF. Without loss of generality, suppose that the initial wealth is equal to 0.
In order to get the optimal trading strategy under expected utility theory, we must to find the strategy h = (h0, h1) such that f (h0, h1) = p1U (h0+ lh1) + p2U (h0+ uh1), reaches the maximal value, subjected to h0 + sh1 = 0, and where U is a concave function. Thus we can transform f (h0, h1) into
f (h1) = p1U ((l − s)h1) + p2(U (u − s)h1), then by first order derivative, we have
f0(h1) = p2(u − s)U0((u − s)h1) − p1(s − l)U0((l − s)h1).
Thus, if f0(h1) = 0,
p2(u − s)
p1(s − l) = U0((l − s)h1) U0((u − s)h1). Since p2(u − s) > p1(s − l), we get
U0((l − s)h1) U0((u − s)h1) > 1.
Hence h1 > 0 because of the property of concave function U . Moreover by second order derivative we acquire
f00(h1) = p1(l − s)2U00((l − s)h1) + p2(u − s)2U00((u − s)h1) < 0.
3.1. OPTIMAL TRADING STRATEGY IN ONE PERIOD MODEL 19
Therefore the optimal trading strategy is obtained as h∗1 > 0.
Next, we want to show that if h∗1 > 0 then p2(u − s) > p1(s − l). Suppose not, if p2(u − s) ≤ p1(s − l) and h1 > 0, then
f0(h1) = p2(u − s)U0((u − s)h1) − p1(s − l)U0((l − s)h1) ≤ 0.
This implies that f (h1) is a nonincreasing function, thus h1 > 0 can not be the optimal trading strategy which contradicts to h∗1 > 0. Therefore if optimal trading strategy occurs when h∗1 > 0 then p2(u − s) > p1(s − l). We complete the proof.
If a investor makes the decision under expected utility theory, he prefers risky asset to riskless asset only as the mean of gains is greater than the mean of losses.
After that we take cumulative prospect theory into account, and assume that the probability weighting functions of gains and losses are the same with form
w+(p) = w−(p) = pr
(pr+ (1 − p)r)1r. Theorem 2. In the version of cumulative prospect theory, if
(3.1) (w(p2)(u − s) − λw(p1)(s − l))(λw(p1) − w(p2)) > 0
holds, the optimal amount of stock is greater than 0. In fact, the converse is also true, i.e., h∗1 > 0 if and only if (3.1) must be hold.
Remark 1. We can separate (3.1) into two cases:
(3.2) w(p2)(u − s) > λw(p1)(s − l), where λw(p1) w(p2) > 1 and
(3.3) w(p2)(u − s) < λw(p1)(s − l), where λw(p1) w(p2) < 1.
PROOF. (Proof of theorem 2) Without loss of generality, suppose that the initial wealth is equal to 0. In the sense of cumulative prospect theory, if we want to
acquire the optimal trading strategy, we need to find out the strategy h = (h0, h1) such that
f (h1) = w(p1)v−((l − s)h1) + w(p2)v+((u − s)h1)
reaches the maximal value, where w is a probability weighting function and v−, v+ are the value functions of gains and losses, respectively. That is, v− is a convex function and v+ is a concave function. Use the property of value function, we can get
f (h1) = −λw(p1)v+((s − l)h1) + w(p2)v+((u − s)h1).
For simplicity, we denote v+= v. Then by first order derivative, we have
f0(h1) = −λw(p1)(s − l)v0((s − l)h1) + w(p2)(u − s)v0((u − s)h1),
and f0(h1) = 0, which implies
(3.4) λw(p1)(s − l)
w(p2)(u − s) = v0((u − s)h1) v0((s − l)h1). If (3.2) holds ,
v0((u − s)h1) v0((s − l)h1) < 1.
This indicates that (u − s)h1 > (s − l)h1. Moreover, due to u − s > s − l, we can get h1 > 0. Use the same argument we have if (3.3) holds, h1 > 0. Finally, we can check that
f00(h∗1) = −λw(p1)(s − l)2v00((s − l)h∗1) + w(p2)(u − s)2v00((u − s)h∗1) < 0,
where h∗1 satisfies the equation (3.4). Thus such h∗1 is the optimal trading strategy.
Conversely, since h∗1 > 0 is the optimal strategy, h∗1 must satisfies the equation λw(p1)(s − l)
w(p2)(u − s) = v0((u − s)h∗1) v0((s − l)h∗1). And we separate it into two cases.
3.1. OPTIMAL TRADING STRATEGY IN ONE PERIOD MODEL 21
In this theorem, we have a result that if a investor makes a decision in the sense of cumulative prospect theory, he is willing to buy the stock only when (3.1) holds.
However, it is independent of the form of vale function.
Next, we give a one-period model example that was given as above. Our main goal is that find out the optimal strategies in the senses of expected utility theory and cumulative prospect theory. The results of this example support Theorem 1 and Theorem 2, stated before.
Example 1. Consider the market model, sat up as before in the beginning of this section. For simplicity we denote it by b =
and D are called price vector and payoff matrix, respectively. Moreover, let p1 = p and then p2 = 1 − p.
Suppose that the value function which an investor takes is given by
v(x) =
Furthermore we assume that the probability weighting functions of gains and losses are the same, with form
w+(p) = w−(p) = pr (pr+ (1 − p)r)1r
In the sense of expected utility theory, we want to find out the strategy h = (h0, h1) such that
f (h0, h1) = p(1 − exp(−θ(h0+ lh1))) + (1 − p)(1 − exp(−θ(h0+ uh1))),
to reach the maximal value that subjects to h0+ sh1 = 0. It is not difficult to get that
h1 = 1
θ(u − l)ln(1 − p)(u − s) p(s − l) . By second order derivative test, we have
f00(h1) = −θ2(p(s − l)2exp(θh1(s − l)) + (1 − p)(u − s)2exp(−θh1(u − s))) < 0.
Therefore, the optimal trading strategy is
h∗ = (− s
θ(u − l)ln(1 − p)(u − s) p(s − l) , 1
θ(u − l)ln(1 − p)(u − s) p(s − l) ).
And we can find out that if (1 − p)(u − s) > p(s − l) holds, the optimal trading strategy h∗1 > 0, that is, an investor is willing to buy the stock if and only if the average of gains is greater than that of losses.
Next, we consider the same market model under cumulative prospect theory. Since in the version of cumulative prospect theory, it is defined over gains and losses relative to a specific reference point instead of final wealth. In this case, the optimal trading strategy is h = (h0, h1) such that
f (h1) = c(p)(1 − p)r(1 − exp(−θ(u − s)h1)) − λc(p)pr(1 − exp(θ(l − s)h1)),
where
c(p) = 1
(pr+ (1 − p)r)1r,
to reach the maximal value that constrains to h0+sh1 = 0. By first order derivative, we have
f0(h1) = c(p)θ((1 − p)r(u − s) exp(−θ(u − s)h1) − λpr(s − l) exp(−θ(s − l)h1)),
3.1. OPTIMAL TRADING STRATEGY IN ONE PERIOD MODEL 23
and if f0(h1) = 0 implies
ln (1 − p)r(u − s) − θ(u − s)h1 = ln λpr(s − l) − θ(s − l)h1. Thus the optimal trading strategy is
h∗ = (− s
θ(u + l − 2s)ln(1 − p)r(u − s)
λpr(s − l) , 1
θ(u + l − 2s)ln(1 − p)r(u − s) λpr(s − l) ).
And we can find out that if
(3.5) ((1 − p)r(u − s) − λpr(s − l))(λpr− (1 − p)r) > 0 holds, h∗1 > 0. In this case, it is easy to check f00(h∗1) < 0.
Remark 2. If (3.5) holds, one of
(1 − p)r(u − s) − λpr(s − l) > 0 where λpr
(1 − p)r > 1 or
(1 − p)r(u − s) − λpr(s − l) < 0 where λpr
(1 − p)r < 1 must be true.
According to this example, the results support the above two theorems. For investor who uses the version of expected utility theory to make a decision, he is willing to buy the risky asset if (1 − p)(u − s) > p(s − l) holds. In the sense of cumulative prospect theory, if (3.5) holds, the investor is willing to buy the risky asset more than riskless asset. Moreover, if we know the form of value function and probability weighting function that the investor takes, we can compute the optimal trading strategy. Then we want to find out the relation of optimal trading strategies in the sense of expected utility theory and cumulative prospect theory.
Remark 3. Suppose that we take the probability weighting function with the form given by
w+(p) = w−(p) = pr (pr+ (1 − p)r)1r
where r < 1, and assume that the sensitivity of losses is about 2.25, which proposed by Kahneman and Tversky. If 1
3 < p < 1
2 holds, we have λw(p)
w(1 − p) = λ( p
1 − p)r > 1.
In this situation, if w(1 − p)(u − s) > λw(p)(s − l) holds, (1 − p)(u − s) > p(s − l) must hold at the same time.
The main result is that if w(1 − p)(u − s) > λw(p)(s − l) where 1
3 < p < 1 2, an investor is willing to buy the risky asset in the sense of expected utility theory.
That is, an investor is willing to buy the stock in these two senses.
Up to now we only consider the market model with two states space, then we take four states market into account. And give an example as following.
Example 2. Suppose the market model is similar to the above example, that the value function and probability weighting function are the same as above, respectively.
But in this example we may assume that there are four states in time 1, called ω1, ω2, ω3 and ω4, with probability p1, p2, p3 and p4, respectively. The stock price at time 0 is S0 = s and price in time 1 is S1(ω1) = l1, S1(ω2) = l2, S1(ω3) = u1 and S1(ω4) = u2, where l1 < l2 < s < u1 < u2.
(1) In the sense of expected utility theory, let
f (h0, h1) = p1(1 − exp(−θ(h0+ l1h1))) + p2(1 − exp(−θ(h0+ l2h1))) +p3(1 − exp(−θ(h0 + u1h1))) + p4(1 − exp(−θ(h0+ u2h1)))
we want to find h = (h0, h1) such that f (h0, h1) reaches the maximum, which sub-jects to h0+ sh1 = 0. Then we transfer f (h0, h1) into
f (h1) = 1−exp(−θsh1)(p1exp(−θl1h1)+p2exp(−θl2h1)+p3exp(−θu1h1)+p4exp(−θu2h1)).
3.1. OPTIMAL TRADING STRATEGY IN ONE PERIOD MODEL 25
By first order derivative we have
f0(h1) = −θseθsh1(p1e−θl1h1 + p2e−θl2h1 + p3e−θu1h1 + p4e−θu2h1) +θeθsh1(l1p1e−θl1h1 + l2p2e−θl2h1 + u1p3e−θu1h1 + u2p4e−θu2h1)
= θeθsh1((l1− s)p1e−θl1h1+ (l2− s)p2e−θl2h1 +(u1− s)p3e−θu1h1 + (u2− s)p4e−θu2h1).
In order to obtain the extreme vale, f0(h1) = 0 must hold. In other words, if h∗1 is optimal strategy, it needs to satisfy the equation
(u1− s)p3e−θu1h∗1 + (u2− s)p4e−θu2h∗1 = (s − l1)p1e−θl1h∗1 + (s − l2)p2e−θl2h∗1.
Finally, we only need to check f00(h∗1) < 0. Since
f00(h1) = θ2seθsh1(−θ)((l1− s)l1p1e−θl1h1 + (l2− s)l2p2e−θl2h1 +(u1− s)u1p3e−θu1h1 + (u2− s)u2p4e−θu2h1)
and l1 < l2 < s < u1 < u2, we have f00(h∗1) < 0. Therefore h∗1 which satisfies the equation
(u1− s)p3e−θu1h∗1 + (u2− s)p4e−θu2h∗1 = (s − l1)p1e−θl1h∗1 + (s − l2)p2e−θl2h∗1
is the optimal trading strategy.
(2) Under cumulative prospect theory, we add the assumption, u1− s > s − l1, to this market model. Our main goal is to find the strategy h = (h0, h1) such that
f (h1) = (w(p1))(−λ(1 − eθ(l1−s)h1)) + (w(p1+ p2) − w(p1))(−λ(1 − eθ(l2−s)h1)) +(w(p3+ p4) − w(p4))(1 − e−θ(u1−s)h1) + (w(p4))(1 − e−θ(l1−s)h1)
reaches the maximal value, subjected to h0+ sh1 = 0. For convenience we give some notations as following: w(p1) = p1, w(p1+ p2) − w(p1) = p2, w(p3+ p4) − w(p4) =
p3, and w(p4) = p4. By first order derivative, we have
f0(h1) = λp1θ(l1− s)eθ(l1−s)h1 + λp2θ(l2− s)eθ(l2−s)h1 +p3θ(u1− s)e−θ(u1−s)h1 + p4θ(u2− s)e−θ(l1−s)h1.
Moreover, f0(h1) = 0 only if h1 satisfies the equation
p3(u1− s)e−θ(u1−s)h1+ p4(u2− s)e−θ(u2−s)h1
= λp1(s − l1)e−θ(s−l1)h1 + λp2(s − l2)e−θ(s−l2)h1.
At last, we need to check that if h∗1 such that
p3(u1− s)e−θ(u1−s)h∗1 + p4(u2− s)e−θ(u2−s)h∗1
= λp1(s − l1)e−θ(s−l1)h∗1 + λp2(s − l2)e−θ(s−l2)h∗1
holds, implies f00(h∗1) < 0. By second order derivative, we have
f00(h1) = θ2(λp1(s − l1)2e−θ(s−l1)h1 + λp2(s − l2)2e−θ(s−l2)h1
−p3(u1− s)2e−θ(u1−s)h1 − p4(u2− s)2e−θ(u2−s)h1).
Since u1− s > s − l1 implies s − l2 < s − l1 < u1− s < u2− s, we can get that
p3(u1 − s)2e−θ(u1−s)h1+ p4(u2− s)2e−θ(u2−s)h1
> λp1(s − l1)2e−θ(s−l1)h1 + λp2(s − l2)2e−θ(s−l2)h1,
i.e. f00(h∗1) < 0. Therefore h∗1 which satisfies the following equation
p3(u1− s)e−θ(u1−s)h∗1 + p4(u2− s)e−θ(u2−s)h∗1
= λp1(s − l1)e−θ(s−l1)h∗1 + λp2(s − l2)e−θ(s−l2)h∗1
is the optimal trading strategy.
3.2. REFERENCE POINT EFFECT 27
The main point of this section is that when we are making decision under ex-pected utility theory, the optimal strategy is different from the optimal strategy in the sense of cumulative prospect theory. Under expected utility theory we only concern about the final wealth and it says that investors are risk aversion in choice between risky investments. On the other hand, in the sense of cumulative prospect theory the investor’s evaluation of risk depends on gains or losses relative to a refer-ence point; furthermore, the value function is concave for gains and convex for losses, and steeper for losses than for gains. In other words, investors are risk aversion for gains and risk seeking for losses. Further, investors are more sensitive for losses then for gains. Therefore the optimization strategy in the sense of cumulative prospect theory depends on the degree of sensitivity of losses.
3.2. Reference Point Effect
Since the investor evaluates the prospect in the sense of cumulative prospect theory depending on gains and losses relative to a reference point rather than on final wealth; moreover, the investor’s attitudes toward risk are different from gains and losses. Therefore the reference point would affect the investor’s trading strategy.
Furthermore, the reference point is decided by investor’s subjective feeling, such as the past experience, and different from people to people. In this section we talk about the influence of reference point on trading strategy.
Consider a one-period market model in which time points are denoted by 0 and 1. Suppose there are two kinds of investors: A and B. Both of them take the same form of value function v(x) which is concave for gains and convex for losses, and x is the wealth change. Assume that before time 0, both A and B made wrong decisions and suffered a loss of w0. Furthermore suppose that both of them make decision under cumulative prospect theory. However, they take different attitude toward this prior loss. Investor A only takes the current wealth X0 into account and takes X0 as the reference point. But investor B cares about the prior loss and tries to make
up for the prior loss with gains in time 1. Hence investor B takes X0+ w0 as the reference point.
In such financial market model, there exists a simple lottery, denoted by (−x, p; y, 1−
p), where x > 0, y > 0, and 0 < p < 1. This means that the lottery has probability p to lose x, and gain y with probability 1 − p.
Then the value of this lottery for investor A, who takes the current wealth as the reference point is
F = w(p)v(−x) + w(1 − p)v(y),
where w(p) is a probability weighting function. If investor A does not take any action, his wealth change is 0 and thus v(0) = 0.
For investor B who takes the prior loss into account, in other words the reference point, he takes, is X0+ w0. The value of this lottery for investor B is
F−w0 = w(p)v(−x − w0) + w(1 − p)v(y − w0) − v(−w0) if y − w0 ≥ 0,
or
F−w0 = w(p)v(−x − w0) + (1 − w(p))v(y − w0) − v(−w0) if y − w0 < 0,
where v(−w0) is the value when investor B does not take any action.
The main point of this section is to compare the optimal amount of lotteries that investor A and investor B are willing to hold, respectively.
Theorem 3. If the gain of lottery is greater than two times of the prior loss, the value of this lottery for investor B is greater than for investor A. That is, if y > 2w0 holds, we can get F−w0 > F .
PROOF. Since y > 2w0, we have
F−w0 = w(p)v(−x − w0) + w(1 − p)v(y − w0) − v(−w0).
3.2. REFERENCE POINT EFFECT 29
And because of the property of probability weighting function
w(p) + w(1 − p) ≤ 1,
we can get
F−w0 ≥ w(p)v(−x − w0) + w(1 − p)v(y − w0) − (w(p) + w(1 − p))v(−w0)
= w(p)(v(−x − w0) − v(−w0)) + w(1 − p)(v(y − w0) − v(−w0)).
Without loss of generality, let v(0) = 0.
If w0 ≤ x, we can get
v(−x − w0) − v(−w0) − v(−x) = v(−x − w0) − v(−x) − v(−w0) + v(0).
By mean value theorem, there exist c1, c2, where −x − w0 < c1 < −x and −w0 <
c2 < 0 such that
v(−x − w0) − v(−x) = (−w0)v0(c1)
and
v(−w0) − v(0) = (−w0)v0(c2).
Since w0 ≤ x implies c2 > c1 and the value function is convex for losses, i.e. v0 > 0 and v00 > 0, we can acquire
v(−x − w0) − v(−x) − v(−w0) + v(0) = w0(v0(c2) − v0(c1)) > 0.
Therefore we have the main result
v(−x − w0) − v(−w0) > v(−x).
If w0 > x, we have
v(−x − w0) − v(−w0) − v(−x) = v(−x − w0) − v(−w0) − v(−x) + v(0).
Buy the same way, we can find c∗1 and c∗2, where −x − w0 < c∗1 < −w0 and −x <
c∗2 < 0, such that
v(−x − w0) − v(−w0) = (−x)v0(c∗1) and
v(−x) − v(0) = (−x)v0(c∗2)
Since w0 > x and the value function is convex for losses, we obtain v(−x − w0) − v(−w0) > v(−x).
Next use the similar argument and loss aversion property, −v(−w0) ≥ v(w0), we have
v(y − w0) − v(−w0) − v(y) = v(y − w0) − v(y) − v(−w0)
≥ v(y − w0) − v(y) + v(w0) = v(y − w0) − v(y) + v(w0) − v(0)
= v0(c3)(−w0) + v0(c4)(−w0) = w0(v0(c4) − v0(c3)),
where y − w0 < c3 < y and 0 < c4 < w0. Owing to y > 2w0 we have c4 < c3, and besides v is concave for gains. Then we can get
v(y − w0) − v(−w0) > v(y).
Therefore
F−w0 = w(p)v(−x − w0) + w(1 − p)v(y − w0) − v(−w0)
≥ w(p)v(−x) + w(1 − p)v(y) = F.
We complete the proof.
The main point of this theorem is that if the gain of the lottery is large enough, the value of this lottery for investor B who does not want to accept the reality of the prior loss is greater than for investor A who accepts the reality and takes the current wealth as the reference point.
3.2. REFERENCE POINT EFFECT 31
Then we interest in seeking out the difference of optimal trading strategies for investor A and investor B that takes different reference point into consideration.
Suppose that h∗ > 0 is the optimal amount of risky assets that investor A is willing to buy, and h∗−w0 is the optimal amount of risky assets that investor B is willing to hold. Our main goal is to compare the number of h∗ and h∗−w0. Before this we first give a lemma which describes one property of h∗−w
0. The statement is as following.
Lemma 1. If the optimal amount of risky asset that investor B is willing to hold is greater than 0, i.e. h∗−w
0 > 0, −w0+ yh∗−w
0 ≥ 0 must hold.
PROOF. Suppose that −w0 + yh∗−w0 < 0 holds, and we define ∆ > 0 to be a small unit of asset such that yh∗−w0 + ∆y − w0 < 0. Then we compare the profit of portfolio h∗−w
0 − ∆ with h∗−w
0. Since h∗−w
0 is the optimal strategy, we have
w(p)v(−xh∗−w0 + ∆x − w0) + (1 − w(p))v(yh∗−w0 − ∆y − w0)
≤ w(p)v(−xh∗−w0 − w0) + (1 − w(p))v(yh∗−w0 − w0)
By rearrangement, we get
(1 − w(p))(v(yh∗−w0− w0) − v(yh∗−w0 − ∆y − w0))
≥ w(p)(v(−xh∗−w0 + ∆x − w0) − v(−xh∗−w0 − w0)).
Because yh∗−w
0 + ∆y − w0 < 0 and v(x) is a convex function defined on x < 0, by the property of convex function and mean value theorem we have
v(yh∗−w
0 − w0) − v(yh∗−w
0 − ∆y − w0)
≤ v(yh∗−w0 + ∆y − w0) − v(yh∗−w0 − w0)
and
v(−xh∗−w0 + ∆x − w0) − v(−xh∗−w0 − w0)
≥ v(−xh∗−w0 − w0) − v(−xh∗−w0 − ∆x − w0).
So combining the above equations, we have
(1 − w(p))(v(yh∗−w0+ ∆y − w0) − v(yh∗−w0 − w0))
That is, investor B will prefer (h∗−w
0+ ∆) to h∗−w
0, which contradict to the fact that h∗−w0 is the optimal amount of risky asset that investor B is willing to hold.
Then we can get the main point of this section, and use the above lemma to complete the proof of the following theorem.
Theorem 4. If investor B is willing to buy a certain number of security, the optimal amount of risky asset that investor B is willing to hold is no less than that investor A is willing to hold, i.e., if h∗−w0 > 0, then h∗−w0 ≥ h∗.
PROOF. Suppose that h∗−w
0 < h∗ hold. Since h∗−w
0 > 0, by the above lemma we get −w0+ yh∗−w0 ≥ 0, and implies −w0+ yh∗ ≥ 0. Because h∗ is optimal amount of risky security that investor A is willing to hold, we have the following equation:
w(p)v(−xh∗) + w(1 − p)v(yh∗) ≥ w(p)v(−xh∗−w0) + w(1 − p)v(yh∗−w0).
3.2. REFERENCE POINT EFFECT 33
Rearranging the above equation we can get
w(1 − p)(v(yh∗) − v(yh∗−w0)) ≥ w(p)(v(−xh∗−w0) − v(−xh∗)).
If yh∗− w0 ≥ yh∗−w0 holds, we have v(yh∗−w
0) − v(yh∗−w
0 − w0) > v(yh∗) − v(yh∗− w0), because of the property of value function of gains. This implies
(3.6) v(yh∗) − v(yh∗−w0) < v(yh∗− w0) − v(yh∗−w0 − w0).
If yh∗− w0 < yh∗−w0 holds, by mean value theorem and concave function v, we have v(yh∗) − v(yh∗−w
0) < v(yh∗− w0) − v(yh∗−w
0 − w0).
Due to the similar argument and the property of value function of losses, we can obtain
(3.7) v(−xh∗−w
0) − v(−xh∗) > v(−xh∗−w
0 − w0) − v(−xh∗− w0).
By combining equation (3.6) and equation (3.7), we obtain
w(1 − p)(v(yh∗− w0) − v(yh∗−w0 − w0))
≥ w(1 − p)(v(yh∗) − v(yh∗−w0))
≥ w(p)(v(−xh∗−w0) − v(−xh∗))
≥ w(p)(v(−xh∗−w
0 − w0) − v(−xh∗− w0)) Therefore we can get
w(p)v(−xh∗− w0) + w(1 − p)v(yh∗− w0)
≥ w(p)v(−xh∗−w0 − w0) + w(1 − p)v(yh∗−w0 − w0),
which contradicts to the optimal amount of risky asset, h∗−w
0, which investor B is willing to hold.
The main point of this theorem is that if the investor who takes the prior loss into account, his attitude toward risk become risk seeking. In other words, if the reference point that investor B takes is greater than that investor A takes, investor B tends to increase the amount of risky asset which he holds.
In this section we know that the attitude of an investor toward risk may become risk seeking, if he takes the prior loss into account and wants to make up for the prior loss. That is, if an investor who does not accept the prior loss and takes X0 + w0 as the reference point, risk seeking preference would play a dominant role when he evaluates the lottery (−x, p; y, 1 − p).
3.3. Optimal Hedging Strategy in One Period Model
In this section, we compare the hedger’s optimal strategy in the future market in the sense of expected utility theory and cumulative prospect theory. Furthermore, we suppose that at time 0 the hedger made wrong decisions and suffered a loss of w0. Under expected utility theory the investor evaluates the prospect depending on the final wealth. But in the sense of cumulative prospect theory the investor may not accept the prior loss and then takes w0 as reference point.
First, we set up the market model as following: Consider a one period model in which time points are denoted by 0 and 1. At time 1, there are two market states, denoted by w1 and w2, with probabilities p and 1 − p, respectively. The current spot price is S0 = s, and at time 1, the spot price is given by S1(w1) = u and S1(w2) = l, where l < s < u. Suppose the futures market is unbiased and there exists no basis risk at time 1, thus the futures prices in time 0 and 1 are F0 = E(S1) and F1 = S1, respectively. Moreover, we suppose that the form of value function which the hedger takes is given by
v(x) =
1 − exp(−θx) x ≥ 0
−λ(1 − exp(θx)) x < 0
3.3. OPTIMAL HEDGING STRATEGY IN ONE PERIOD MODEL 35
where θ > 0 and λ > 1.
Suppose the position that the hedger holds in the futures market is h. The wealth change in time 1 is X1 = S1− s + h(F1− F0), i.e.,
In the beginning, we talk about the hedger’s optimal strategy h∗ in the sense of expected utility theory.
Theorem 5. Under expected utility theory, full hedging is the optimal trading strategy for a hedger. That is, h = −1 is the utility maximizing point.
PROOF. In the sense of utility theory, our main goal is to find out h∗ such that p(1 − exp(−θwu)) + (1 − p)(1 − exp(−θwl)), denoted by f (h), reaches maximum. By first order derivative, we get
f0(h) = (−p)(−θ)(1 − p)(u − l) exp(−θwu) − (1 − p)(−θ)(−p)(u − l) exp(−θwl)
= p(1 − p)θ(u − l)(exp(−θwu) − exp(−θwl))
In order to let f0(h) = 0 hold, we have wu = wl. This implies that h must be −1.
Moreover, by second order derivative we can get
f00(h) = p(1 − p)θ(u − l)(−θ(1 − p)(u − l) exp(−θwu) − θp(u − l) exp(−θwl))
= −p(1 − p)θ2(u − l)2((1 − p) exp(−θwu) + p exp(−θwl)) < 0.
Therefore, h = −1 is the optimal strategy for the hedger in the sense of expected
Therefore, h = −1 is the optimal strategy for the hedger in the sense of expected