Efficient hedging with a power loss function

In the following we consider a more general hedging method, namely efficient hedging with a power loss function:

l(y) = y^p, y ≥ 0, p > 0.

In this case the optimal strategy φ^∗ for a given contingent claim H is defined E[l([H − V_T(φ^∗)]⁺)] = min

φ E[l([H − V_T(φ)]⁺)].

Again φ are all self–financing strategies with nonnegative values satisfying the budget restriction

V₀(φ) ≤_Tp_x· E^∗[H].

Efficient hedging proceeds similarly to quantile hedging, i.e., after a modified claim is found, super/perfect this modified claim. F¨ollmer and Leukert (2000) show that the solution for the efficient hedge exists and coincides with a perfect hedge for a modified contingent claim H_P with the following structure:

H_P = H − α_p dP^∗ different cases are distinguished according to the p–values. The implicit survival proba-bility has the form of

Tp_x =

The notation P in the nominator is taken to denote the modified contingent claim as in H_P. For the case of p = 1, at the first sight, the resulting expression for the modified contingent claim is different from the one obtained in the quantile hedging, but we prove these two expressions are the same later. I.e., we are back to the quantile hedging.

We start with the case of p > 1. For this purpose, the expression of dP^∗/dP is needed. Ac-cording to Equation (2.14) and the analogous derivation in Equation (2.18), the Random–

Nikodym density can be reformulated into

dP^∗

Substituting this density in the modified contingent claim, we obtain

H_P = H − α_p· Lemma 2.3.3. We claim that

(a) The value of α_p given in Equation (2.24) is larger than 0.

(b) If e^{c ∆t}is the solution of ^S(t_S(tⁱ⁺¹⁾

(a) Assume that α_p is smaller than 0, then it follows α_pR^p−11  S(t_i+1)

This contradicts the budget condition and implies that the equality that E^∗[HP] =

Tp_xE^∗[H] cannot hold. Therefore, α_p must be larger than 0.

(b) For α_p > 0, a unique solution is found for the equation

is a convex and decreasing function of ^S(t_S(tⁱ⁺¹⁾

i) . We denote e^c∆t this solution. Obviously e^c∆thave to be larger than e^g∆t. Consequently, α_p equals

With this cognition, the modified contingent claim HP is transformed into H_P =  S(t_i+1)

Consequently, the price for the modified contingent claim for the case p > 1 and α_p > 0 is

µ _Tp^∗_x σ _Tp^∗_x g _Tp^∗_x µ = 0.03 0.0729241 σ = 0.2 0.0592240 g = 0.02 0.0592240 µ = 0.05 0.0637018 σ = 0.3 0.0780432 g = 0.03 0.0625627 µ = 0.07 0.0548723 σ = 0.4 0.0968713 g = 0.04 0.0661808 µ = 0.09 0.0466340 σ = 0.5 0.1172620 g = 0.05 0.0701061

Table 2.3: Implied survival probability with parameters:  = 0.05, T = tM = 12, g = 0.02, µ = 0.06, σ = 0.2, p = 2.

p _Tp^∗_x  _Tp^∗_x

p = 2 0.0592240  = 0.01 0.0100341 p = 3 0.0515828  = 0.02 0.0214264 p = 4 0.0488637  = 0.03 0.0335235 p = 5 0.0474696  = 0.04 0.0461523

Table 2.4: Implied survival probability with parameters: T = t_M = 12, g = 0.02, µ = 0.06, σ = 0.2,  = 0.05, p = 2.

where the critical value e^c∆t is determined by the equation: Tpx = ^E_E^∗∗^[H[H]^P]. Of course this is only possible if the survival probability is already known. Still we can go one step further to determine the efficient hedging strategy by taking the derivative of E^∗[H_P] with respect to the stock price. However, here we follow the idea of Melnikov (2004b), i.e., as in the quantile hedging, we mainly analyze the implied survival probability after determining the critical value by fixing a constrained shortfall probability

P  S(t_i+1)

S(t_i) ≤ e^c∆t



= 1 − ,

i.e.,

c = 1

∆t



N⁻¹(1 − )σ√

∆t + (µ −1 2σ²)∆t



. (2.25)

In Tables 2.3 and 2.4, it is observed that all the implied survival probabilities are quite small and close if the insurance company bears its risk with a power loss function (p > 1).

This indicates that the company cannot accept many transfers between the financial risk to the insurance risk. Table 2.3 demonstrates how the implied survival probability depends on the market return of the asset, its volatility and the strike parameter g. Compared to the quantile case, it is observed that all the effects of these parameters are reversed. In

the quantile hedging, the survival probability is given by the ratio

which decreases with g. On the contrary, in the efficient hedging (p > 1), it is given by the sum of a similar ratio which is conditional on a counter–event and a term which increases with g. Therefore, a rise in g leads to a rise in the survival probability. Furthermore, a higher p value leads to a smaller implied survival probability. As p goes up, the degree of risk aversion increases, as a compensation, the insurance company would rather choose some safer older customers than young customers. The effects of  on the considered survival probability are listed in Table 2.4. As  goes up, more financial risk is borne, unexpectedly, the insurance company will sign contracts with young clients. Hence, no transfer from the financial risk to the mortality risk is possible.

We proceed with the second case 0 < p < 1, where the modified contingent claim for our case is of the form of

The event under the indicator function is equivalent to ( _dP probability of 1, consequently it leads to an equivalence between the modified contingent claim with the original claim. This contradicts the initial budget constraint and accord-ingly the idea of efficient hedging. Therefore, the only interesting case here is that αp

is larger than 0 and µ < σ²(1 − p). In this case the equation owns a unique solution.

µ _Tp^∗_x σ _Tp^∗_x g _Tp^∗_x  _Tp^∗_x µ = 0.03 0.702308 σ = 0.3 0.746807 g = 0.02 0.746807  = 0.01 0.93566 µ = 0.04 0.717634 σ = 0.4 0.704046 g = 0.03 0.739995  = 0.01 0.882329 µ = 0.05 0.732469 σ = 0.5 0.665453 g = 0.04 0.732897  = 0.01 0.833927 µ = 0.06 0.746807 σ = 0.6 0.628399 g = 0.05 0.725502  = 0.01 0.788996

Table 2.5: Implied survival probability with parameters:  = 0.05, T = tM = 12, g = 0.02, µ = 0.06, σ = 0.3, 0 < p < 1

Following this we obtain the price of the modified contingent claim:

E^∗[H_P] = E^∗



H1ⁿ_S(ti+1)

S(ti) <e^c∆t o



= N (c − ¹₂σ²)∆t σ√

∆t



− N (g − ¹₂σ²)∆t σ√

∆t



− e^g∆tN (c + ¹₂σ²)∆t σ√

∆t



+e^g∆tN (g +¹₂σ²)∆t σ√

∆t

 .

Again here we can determine the value of c and that of α_p through the relation _Tp_x =

E^∗[HP]

E^∗[H] , given that the survival probability is known. As before, we are more interested in the implied survival probability. Hence, as in the quantile hedging case c can be derived as a function of the given significance level:

P  S(t_i+1)

S(t_i) ≤ e^c∆t



= 1 − , i.e.,

c = 1

∆t



N⁻¹(1 − )σ√

∆t + (µ −1 2σ²)∆t

 .

Plugging this value in the expression for the modified contingent claim, we obtain the price we look for and the resulting survival probability for different hedge values.

Table 2.5 is displayed for the scenario µ < σ²(1−p), where a unique solution for c is found.

Above all, p < 1 indicates that the hedger is a risk–taking insurance company. If you look at the expression for the survival probability carefully, you will observe that it does not depend on p at all, i.e., same survival probabilities result for all p’s which satisfy the condition µ < σ²(1 − p). In this scenario, the transfer between the financial and insurance risk becomes possible again because you observe quite big survival probabilities overall.

When  is increased (more financial risks), as a consequence, the survival probabilities are decreased (more older safer customers are preferred). All of the other effects are the same as in the quantile hedging case.

Finally, the case of p = 1 can be constructed as follows: Here α_p is determined by

P (A^∗) = P

In fact, p = 1 reflects exactly the quantile hedging. Although the quantile price looks different from that in Equation (2.26), this gives precisely the same value obtained in Subsection 2.3.1. On the one hand, in the quantile hedging, the maximal success set is given by where X is a standard normal distributed random variable. On the other hand, the maximal success set in the efficient hedging for the case p = 1 is expressed by

 S(t_i+1)

S(t_i) > (R1αp)^−σ2µ



= {X > N⁻¹()}.

The above equation holds because it is valid that (R₁α_p)^−σ2µ = exp