In this chapter we introduce trading strategy with transaction costs in one pe-riod market model, which is derived from the model specified by Kabanov (2002).
Suppose that the financial market is one period model, and our portfolio is (h0, h1), which means that the number of shares of assets invested in bond and stock, respec-tively. Moreover, we assume that the initial wealth is x0 = h0+ sh1, composed of bond and stock.
Suppose that the investor need to pay the transaction costs when they sell the stock, and consider the model with constant proportional transaction costs, denoted by λ. Therefore, we find out that if we buy the stock at time 0, the value of the portfolio at time 0 after trading is v0 = h0+ sh1, and in this situation h1 > h1 must hold. If we sell the stock at time 0, the value of the portfolio at time 0 after trading is v0∗ = h0+ sh1− λs(h1− h1), and in this situation h1 < h1 must hold.
In the following of this section we assume that in this market model our portfolio values are only affected by the asset price fluctuation and transaction costs. Thus we get v0 = x0 and v∗0 = x0− λs(h1 − h1). Our main task is to find out optimal trading strategy with transaction costs in the sense of expected utility theory and cumulative prospect theory, respectively.
43
Example 3. (Constant absolute risk aversion CARA)
Under expected utility theory, we consider the risk-averse utility function given by
U (x) = 1 − exp(−θx),
where θ > 0 is absolute risk aversion and be a constant. Moreover, the market model is sat up as the first section in chapter 3.
(1)If we buy the stock at time 0, then the final wealth is
w =
wl = x0+ h1(l − s) with probability p wu = x0+ h1(u − s) with probability 1 − p.
Under expected utility theory, we have to maximize the function f (h1), defined by
f (h1) = pU (x0+ h1(l − s)) + (1 − p)U (x0+ h1(u − s)),
to get the optimal strategy h∗1. Using first order derivative, we have
f0(h1) = pθ(l − s) exp(−θ(x0+ h1(l − s))) + (1 − p)θ(u − s) exp(−θ(x0+ h1(u − s))),
and let f0(h1) = 0 we get
(4.1) h1 = 1
θ(u − l)ln(1 − p)(u − s) p(s − l) .
In order to guarantee h1 > 0, we should add the condition, (1 − p)(u − s) > p(s − l), to this market model. Moreover, by second order derivative, we get
f00(h1) = −θ2(p(s − l)2exp(−θwl) + (1 − p)(u − s)2exp(−θwu)) ≤ 0.
Therefore we can get that if we buy the stock at time 0, the optimal trading strategy is
h∗1 = 1
θ(u − l)ln(1 − p)(u − s) p(s − l) > h1.
4. COMPARISON OF OPTIMIZATION IN THE SENSE OF EUT AND CPT II 45
(2) If we sell the stock at time 0, then the final wealth is
w =
Suppose that l − s + λs ≤ 0. Under expected utility theory, we have to maximize the function f (h1), defined by Moreover, by second order derivative, we get
f00(h1) = −θ2(p(s − l − λs)2exp(−θwl) + (1 − p)(u − s + λs)2exp(−θwu)) ≤ 0.
Therefore we can get that if we sell the stock at time 0, the optimal trading strategy is
h∗∗1 = 1
θ(u − l)ln(1 − p)(u − s + λs) p(s − l − λs) < h1.
From (1), (2), we conclude that if an investor who wants to buy the stock at time 0, he may choose the strategy h∗1 = (4.1) at time 0 to reach the maximum profit, and if an investor who wants to sell the stock at time 0 he may choose the strategy h∗∗1 = (4.2) at time 0 to reach the maximum profit.
we have h∗∗1 > h∗1. Thus we can conclude that the investor will neither buy nor sell the stock at time 0 when the number of shares of the stock of their initial wealth is in the interval
[ 1
θ(u − l)ln(1 − p)(u − s) p(s − l) , 1
θ(u − l)ln(1 − p)(u − s + λs) p(s − l − λs) ], called ”no trading” interval.
The main point of this remark is that if the optimal trading strategy in the case of selling the stock at time 0 , h∗∗1 , is no less than the optimal strategy in the case of buying the stock at time 0, h∗1, then there must exist a ”no trading” interval.
The following theorem shows that no matter which utility function we choose in the sense of expected utility theory there must exist a ”no trading” interval, however there is not a specific form of utility function. In other words, even though the forms of value function which an investor takes are different, there is a interval in which an investor will neither buy nor sell the stock at time 0.
Theorem 7. Assume that l − (1 − λ)s < 0. Under expected utility theory, if h∗1 and h∗∗1 exist, the inequality h∗∗1 > h∗1 must be true. Thus there exists a ”no trading” interval.
PROOF. Suppose that h∗1 ≥ h∗∗1 . If we buy the stock at time 0 and the optimal strategy is h∗1, under expected utility theory, we have
pU (x0+ h∗1(l − s)) + (1 − p)U (x0+ h∗1(u − s))
≥ pU (x0+ h∗∗1 (l − s)) + (1 − p)U (x0+ h∗∗1 (u − s)),
where U is a concave function. Rearranging the inequality, we have
(1 − p)(U (x0+ h∗1(u − s)) − U (x0+ h∗∗1 (u − s)))
≥ p(U (x0+ h∗∗1 (l − s)) − U (x0+ h∗1(l − s))).
4. COMPARISON OF OPTIMIZATION IN THE SENSE OF EUT AND CPT II 47
Because of h∗1 ≥ h∗∗1 , h∗1 > h1 and h∗∗1 < h1, we have
U (x0+ h∗1(u − s) − λs(h1 − h∗1)) − U (x0+ h∗1(u − s)) = U0(c1)λs(h∗1− h1) > 0,
and
U (x0+ h∗∗1 (u − s) − λs(h1− h∗∗1 )) − U (x0+ h∗∗1 (u − s)) = U0(c2)(−λs(h1− h∗∗1 )) < 0,
where
c1 ∈ (x0+ h∗1(u − s), x0+ h∗1(u − s) − λs(h1− h∗1))
and
c2 ∈ (x0+ h∗∗1 (u − s) − λs(h1− h∗∗1 ), x0+ h∗∗1 (u − s)).
Hence we obtain the following inequality
U (x0+ h∗1(u − s) − λs(h1 − h∗1)) − U (x0+ h∗∗1 (u − s) − λs(h1− h∗∗1 ))
> U (x0+ h∗1(u − s)) − U (x0 + h∗∗1 (u − s)).
Next, use the similar argument, we have
U (x0+ h∗∗1 (l − s)) − U (x0+ h∗1(l − s))
> U (x0+ h∗∗1 (l − s) − λs(h1− h∗∗1 )) − U (x0+ h∗1(l − s) − λs(h1− h∗1)).
Therefore, we can get that
(1 − p)(U (x0 + h∗1(u − s) − λs(h1− h∗1)) − U (x0+ h∗∗1 (u − s) − λs(h1 − h∗∗1 )))
> (1 − p)(U (x0 + h∗1(u − s)) − U (x0+ h∗∗1 (u − s)))
≥ p(U (x0+ h∗∗1 (l − s)) − U (x0+ h∗1(l − s)))
> p(U (x0+ h∗∗1 (l − s) − λs(h1− h∗∗1 )) − U (x0+ h∗1(l − s) − λs(h1− h∗1))).
That is,
pU (x0+ h∗1(l − s) − λs(h1− h∗1)) + (1 − p)U (x0+ h∗1(u − s) − λs(h1− h∗1))
> pU (x0+ h∗∗1 (l − s) − λs(h1− h∗∗1 )) + (1 − p)U (x0+ h∗∗1 (u − s) − λs(h1− h∗∗1 ))
which is contradicted to that h∗∗1 is optimal strategy in the case of selling the stock at time 0. Thus we can obtain h∗∗1 > h∗1.
This theorem shows that if the initial wealth is composed of bond and stock, and investors also take transaction costs into account. For the investor who wants to find out the optimal strategy in the sense of expected utility theory, should consider two cases, buying or selling the stock at time 0. Furthermore, we know that in this situation there exists a ”no trading” interval.
Next, we discuss the optimal strategy with transaction costs in the version of cumulative prospect theory. Suppose that the market model is the same as before.
Example 4. We consider the value function given by
v(x) =
Under cumulative prospect theory, the value function is defined over gains and losses relative to reference point instead of final wealth. In this example we take initial wealth as reference point. Thus when we want to acquire the optimal strategy in the version of cumulative prospect theory, we need to maximize f (h1) to get the
4. COMPARISON OF OPTIMIZATION IN THE SENSE OF EUT AND CPT II 49
optimal strategy h∗1, where
f (h1) = w(p)(−λ(1 − exp(θh1(l − s)))) + w(1 − p)(1 − exp(−θh1(u − s))).
Due to first order derivative we have
f0(h1) = θ(λw(p)(l − s) exp(θh1(l − s)) + w(1 − p)(u − s) exp(−θ(h1(l − s)))), optimal strategy if we buy the stock at time 0 in the sense of cumulative prospect theory. can transfer the final wealth, w, into
w∗ =
where w∗l and w∗u represent losses and gains, respectively. Thus in the sense of cumulative prospect theory our main goal is to figure out h∗∗1 which maximize the function
f (h1) = pv(h1(l − s) − λs(h1− h1)) + (1 − p)v(h1(u − s) − λs(h1− h1)).
By first order derivative, we have strategy if we sell the stock at time 0 in the sense of cumulative prospect theory.
By this example we conclude that if the investor wants to buy the stock at time 0 he may choose the strategy
h∗1 = 1
θ(u + l − 2s)lnw(1 − p)(u − s) λw(p)(s − l)
to reach the maximum profit, and if the investor wants to sell the stock at time 0 he may choose the strategy
Remark 5. Different from expected utility theory, we discover the fact that under cumulative prospect theory, the optimal strategy with transaction costs in the case of selling the stock at time 0 is relative to h1.
Remark 6. Due to rearrange the following inequality 1
4. COMPARISON OF OPTIMIZATION IN THE SENSE OF EUT AND CPT II 51
and implies
(4.3) 1
θ(u + l − 2s)lnw(1 − p)(u − s + λs) λw(p)(s − l − λs) < h1.
Thus in above example if (4.3) holds, the investor will sell the stock at time 0 and the optimal strategy is
h∗∗1 = 1
θ(u + l − 2(1 − λ)s)lnw(1 − p)(u − s + λs)
λw(p)(s − l − λs) + 2λsh1
u + l − 2(1 − λ)s. Moreover we get that the investor will neither buy nor sell the stock when h1 is in the interval
[ 1
θ(u + l − 2s)lnw(1 − p)(u − s)
λw(p)(s − l) , 1
θ(u + l − 2s)lnw(1 − p)(u − s + λs) λw(p)(s − l − λs) ].
Following, we discuss the same question. If there exists a ”no trading” interval in the sense of cumulative prospect theory, without specific form of value function ?
Theorem 8. Assume that l − (1 − λ)s < 0. Under cumulative prospect theory, if h∗1 and h∗∗1 exist, the inequality h∗∗1 > h∗1 must be true. Therefore there exist a
”no trading” interval.
PROOF. Suppose that h∗1 ≥ h∗∗1 . If we buy the stock at time 0 and the optimal strategy is h∗1, we have
w(p)v−(h∗1(l − s)) + w(1 − p)v+(h∗1(u − s))
≥ w(p)v−(h∗∗1 (l − s)) + w(1 − p)v+(h∗∗1 (u − s)),
where v− is a convex function and v+ is a concave function. Rearranging the above inequality, we get
w(1 − p)(v+(h∗1(u − s)) − v+(h∗∗1 (u − s)))
≥ w(p)(v−(h∗∗1 (l − s)) − v−(h∗1(l − s))).
Due to h∗1 ≥ h∗∗1 , h∗1 > h1 and h∗∗1 < h1, we obtain
−λsh1+ h∗1(u − s + λs) > h∗1(u − s)
and
h∗∗1 (u − s) > −λsh1+ h∗∗1 (u − s + λs).
By mean value theorem, we have
v+(−λsh1+ h∗1(u − s + λs)) − v+(h∗1(u − s))
> v+(−λsh1+ h∗∗1 (u − s + λs)) − v+(h∗∗1 (u − s)).
That is,
v+(−λsh1+ h∗1(u − s + λs)) − v+(−λsh1+ h∗∗1 (u − s + λs))
> v+(h∗1(u − s)) − v+(h∗∗1 (u − s)).
Similarly, since l − (1 − λ)s < 0 we have
h∗∗1 (l − s) > −λsh1+ h∗∗1 (l − s + λs)
and
−λsh1+ h∗1(l − s + λs) > h∗1(l − s).
Using mean value theorem, we get
v−(h∗∗1 (l − s)) − v−(h∗1(l − s))
> v−(−λsh1+ h∗∗1 (l − s + λs)) − v−(−λsh1+ h∗1(l − s + λs)).
4. COMPARISON OF OPTIMIZATION IN THE SENSE OF EUT AND CPT II 53
Combining above inequality, we can obtain
w(1 − p)(v+(−λsh1+ h∗1(u − s + λs)) − v+(−λsh1+ h∗∗1 (u − s + λs)))
> w(1 − p)(v+(h∗1(u − s)) − v+(h∗∗1 (u − s)))
≥ w(p)(v−(h∗∗1 (l − s)) − v−(h∗1(l − s)))
> w(p)(v−(−λsh1+ h∗∗1 (l − s + λs)) − v−(−λsh1+ h∗1(l − s + λs))).
Thus we have
w(p)v−(−λsh1+ h∗1(l − s + λs)) + w(1 − p)v+(−λsh1+ h∗1(u − s + λs))
> w(p)v−(−λsh1+ h∗∗1 (l − s + λs)) + w(1 − p)v+(−λsh1+ h∗∗1 (u − s + λs))
which is contradicted to that h∗∗1 is optimal strategy if we sell the stock at time 0.
Therefore h∗∗1 > h∗1 must hold.
This theorem shows that under cumulative prospect theorem there exists a ”no trading” interval, too.
Corollary 2. When we want to find out the optimal strategy with transaction costs in the sense of expected utility theory or cumulative prospect theory, there exists a ”no trading” interval, in which we will neither buy nor sell the stock at time 0, no matter which value function we choose.
Next, we talk about the length of ”no trading” interval. If we consider the value function given by
v(x) =
1 − exp(−θx) if x ≥ 0
−λ(1 − exp(θx)) if x < 0 ,
we are able to contrast two lengths of ”no trading” interval in the sense of expected utility theory and cumulative prospect theory, respectively.
Remark 7. The length of the ”no trading” interval in the version of cumulative prospect theory is no less than which in the version of expected utility theory.
PROOF. Under expected utility theory, the ”no trading” interval is
[ 1
θ(u − l)ln(1 − p)(u − s) p(s − l) , 1
θ(u − l)ln(1 − p)(u − s + λs) p(s − l − λs) ].
Hence the length of ”no trading” interval is 1
θ(u − l)ln(u − s + λs)(s − l) (s − l − λs)(u − s).
And under cumulative prospect theory, the ”no trading” interval is
[ 1
θ(u + l − 2s)lnw(1 − p)(u − s)
λw(p)(s − l) , 1
θ(u + l − 2s)lnw(1 − p)(u − s + λs) λw(p)(s − l − λs) ].
Thus the length of ”no trading” interval is 1
The main point of this remark is that if we consider the specific form of value function, we are able to find out the ”no trading” interval so as to calculate the length of ”no trading” interval. Moreover if the form of value function given by
v(x) =
the attitude of an investor toward risk bases on cumulative prospect theory is more conservative than which bases on expected utility theory.
4. COMPARISON OF OPTIMIZATION IN THE SENSE OF EUT AND CPT II 55
Then let us consider another value function, which Kahneman and Tversky sug-gested, given by
In this case we take constant relative risk aversion (CRRA) function given by
U (x) = xα,
By first order derivative, we have
h∗1 = x0(a − 1)
u − s + a(s − l) > h1, where
a = ((1 − p)(u − s) p(s − l) )1−α1 .
(2) If we sell the stock at time 0, our task is to compute
maxh1
p(x0− λsh1+ h1(l − s + λs))α+ (1 − p)(x0− λsh1+ h1(u − s + λs))α.
By first order derivative, we have
h∗∗1 = (b − 1)(x0− λsh1)
u − s + λs + b(s − l − λs) < h1, where
b = ((1 − p)(u − s + λs) p(s − l − λs) )1−α1 .
Moreover h∗∗1 < h1 implies
(b − 1)x0
u − s + b(s − l) < h1. Therefore we have a conclusion that if x0(a − 1)
u − s + a(s − l) > h1, the investor is willing to buy the stock at time 0 and the optimal strategy is
h∗1 = x0(a − 1) u − s + a(s − l). If (b − 1)x0
u − s + b(s − l) < h1, the investor is willing to sell the stock at time 0 and the optimal strategy is
h∗∗1 = (b − 1)(x0− λsh1) u − s + λs + b(s − l − λs). Moreover, there is a ”no trading” interval
[ x0(a − 1)
u − s + a(s − l), x0(b − 1) u − s + b(s − l)].
Under cumulative prospect theory, suppose that the financial market is no lend-ing and l − s + λs ≤ 0.
(1) If we buy the stock at time 0, our task is to compute
maxh1
f (h1) := −λw(p)(h1(s − l))α+ w(1 − p)(h1(u − s))α. By first order derivative, we have
f0(h1) = αhα−11 (−λw(p)(s − l)α+ w(1 − p)(u − s)α).
Case 1: If
w(1 − p)(u − s)α > λw(p)(s − l)α,
f (h1) is an increasing function. So the optimal strategy is h∗1 = h1+ h0
s . Case 2: If
w(1 − p)(u − s)α < λw(p)(s − l)α,
4. COMPARISON OF OPTIMIZATION IN THE SENSE OF EUT AND CPT II 57
f (h1) is a decreasing function. So the optimal strategy is
h∗1 = h1.
The main point is that if w(1−p)(u−s)α > λw(p)(s−l)αholds, transforming all our money into stock makes the maximal profit. On the other hand, if w(1 − p)(u − s)α < λw(p)(s − l)α holds, we would not buy the stock at time 0.
(2) If we sell the stock at time 0, our task is to compute
maxh1
−λw(p)(λsh1+ h1(s − l − λs))α+ w(1 − p)(λsh1 + h1(u − s + λs))α.
By first order derivative, we have
h∗∗1 = λs(c + 1)h1
u − s + λs − c(s − l − λs) < h1, where
(4.4) c = (w(1 − p)(u − s + λs)
λw(p)(s − l − λs) )1−α1 . Moreover h∗∗1 < h1 implies u − s > c(s − l).
Therefore if w(1 − p)(u − s)α > λw(p)(s − l)α holds, the investor is willing to buy the stock at time 0 and optimal strategy is h∗1 = h1+h0
s . If u − s > c(s − l) holds, the investor is willing to sell the stock at time 0 and the optimal strategy is
h∗∗1 = λs(c + 1)h1
u − s + λs − c(s − l − λs). Moreover if
u − s
s − l ≤ min{( λw(p)
w(1 − p))α1, (w(1 − p)(u − s + λs) λw(p)(s − l − λs) )1−α1 }, the investor will neither buy nor sell the stock at time 0.
When investors evaluate investments based on different value functions, the port-folio they choose may be distinct. If the value function is given by
v(x) =
the optimal strategy in the case, buying the stock at time 0, is h∗1 = 1
θ(u + l − 2s)lnw(1 − p)(u − s) λw(p)(s − l) , and in the case, selling the stock at time 0, the optimal strategy is
h∗∗1 = 1
θ(u + l − 2(1 − λ)s)lnw(1 − p)(u − s + λs)
λw(p)(s − l − λs) + 2λsh1
u + l − 2(1 − λ)s. Moreover, there exists a ”no trading” interval. If the value function is given by
v(x) =
the result is much more complicated and there exists boundary condition of market model. Furthermore, if we buy the stock at time 0 and w(1 − p)(u − s)α > λw(p)(s − l)α, the optimal strategy is
h∗1 = h1+ h0
In financial market if we take transaction costs into consideration, we have to talk about two situations, buying the stock at time 0 and selling the stock at time 0. Furthermore, we find out under some restrictions there is a ”no trading” interval in which we will neither buy nor sell the stock at time 0.
CHAPTER 5
Conclusion
Cumulative prospect theory modifies the drawbacks of expected utility theory.
Under cumulative prospect theory we replace objective probability with probability weighting function which is nonlinear. Moreover, we replace utility function with value function which is concave for gains and convex for losses.
In the sense of expected utility theory if the mean of gains is greater than the mean of losses, optimal amount of risky asset is greater than 0. However, in the sense of cumulative prospect theory optimal strategy depends on the degree of sensitivity in facing loss, denoted by λ. The larger λ is, the more conservative the investors are.
If the investor who does not accept the prior loss he will become risk seeking as the gain of lottery is large enough. Furthermore if the prior loss is sufficiently large, the hedger who take the prior loss into account will take very large positions showing no consideration of risks in the sense of cumulative prospect theory. But full hedging is the optimal strategy for a hedger in the sense of expected utility theory.
Finally, we consider the market model with transaction cost. An investor must to pay constant proportional transaction cost when he sells the stock. In the sense of expected utility theory and cumulative prospect theory there exists a ”no trading”
interval. And the length of the ”no trading” interval in the version of cumulative prospect theory is no less than which in the version of expected utility theory.
Therefore the attitude of an investor toward risk bases on cumulative prospect theory is more conservative than which bases on expected utility theory.
59
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