Quantile hedging

In a quantile hedging methodology, the hedging is implemented by two steps: first, find a modified contingent claim; second, super/perfect–hedge this modified claim. The problem of a quantile hedging is formulated as follows: to construct an admissible strategy φ^∗ such that V_T(φ^∗) is close enough to H = f (T, S)1{τ^x>T }, i.e.,

P (V_T(φ^∗) ≥ H) = max

φ P (V_T(φ) ≥ H) under the initial–capital–constraint

V0(φ) ≤Tpx· Π0 ≤ Π0.

How to transform this idea to our specific contract is the main concern in the rest of this subsection. Since S_ti+1/S_ti, i = 0, · · · M − 1 are independently and identically distributed,

it is sufficient to investigate how to quantile hedge one single European call option. I.e., every single European call option can be considered as a one–period option and it begins at time t_i and matures at t_i+1. From Equation (2.16), we obtain

Tpx

Therefore, the value in the left–hand side of Equation (2.17) is the bound of the initial available capital for the each relevant call option. In a quantile hedging, a maximal suc-cess set which satisfies the above requirements, i.e., it makes the value of the admissible strategy be close enough to the considered contingent claim and the initial capital re-quirement stay below the constraint. As mentioned, the goal of this section is to analyze the mortality risk implicitly. In other words, we are looking for the survival probability induced by the maximal success set:

Tp^∗_x E^∗

where A^∗ is the maximal success set. Before we discuss this set in detail, we come to the following reformulation at first: According to F¨ollmer and Leukert (1999), the maximal success set is given by

( _dP

Due to the above reformulation of

dP

, the maximal success set is transformed into ( S(t_i+1)

-6

S(ti+1) S(ti)

^µ

σ2

S(ti+1) S(ti)

constant ·h

S(ti+1)

S(ti) − e^g∆ti+

e^g∆t e^{c ∆t}

Figure 2.3:

S(ti+1) S(ti)

^µ

σ2 and constant ·

hS(ti+1)

S(ti) − e^g∆ti+

for µ ≤ σ²

Of course these two constants in (2.19) and (2.20) are not the same. The latter one is the former one times expn

−¹₂µ ∆t + ¹₂ ^µ_σ²2 ∆to

. For the case of µ ≤ σ², which implies that

_S(t

i+1) S(ti)

_σ2^µ

is a concave function of _S(t

i+1) S(ti)



, the equation

 S(t_i+1) S(t_i)

^µ

σ2

= constant · S(t_i+1)

S(t_i) − e^g∆t

+

(2.21)

has only one solution (here we consider ^S(t_S(tⁱ⁺¹⁾

i) as a variable). Assume e^{c ∆t} is the solution of the above equation,⁷ then the maximal success set in (2.20) is equivalent to

 S(t_i+1)

S(t_i) ≤ e^{c ∆t}

 ,

c.f. Figure 2.3.1. The green curve is above the red one for the area ofn_S(t

i+1)

S(ti) ≤ e^{c ∆t}o .

7Due to the lognormally distributed asset price, it makes sense to choose this expression.

Hence,

In the quantile hedging, the financial risk is described by the shortfall probability, i.e., the hedger constrains the shortfall probability (under the market measure) to a certain level and strives for a goal by regulating the insurance risk. Now we let  denote this constrained shortfall probability. Usually c is determined by the level of , i.e.,

P (A^∗) = P S(t_i+1)

is derived for a given , which accordingly results in the following proposition.

Proposition 2.3.1 (Implied survival probability). Given that the financial risk is charac-terized by the constrained shortfall probability  the insurer wishes, i.e., P

S(ti+1)

S(ti) ≤ e^{c ∆t}

= 1−, for the case of µ < σ², the implied survival probability resulting from quantile hedging for the contract given in Equation (2.11) has a form of

Tp^∗_x =

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 = 0.05 0.732469 σ = 0.4 0.704046 g = 0.03 0.739995  = 0.02 0.882329 µ = 0.07 0.760644 σ = 0.5 0.665453 g = 0.04 0.732897  = 0.03 0.833927 µ = 0.09 0.786803 σ = 0.6 0.628399 g = 0.05 0.725502  = 0.04 0.788996

Table 2.1: Implied survival probability with parameters:  = 0.05, T = tM = 12, g = 0.02, µ = 0.06, σ = 0.3

with E^∗

h

S(ti+1)

S(ti) − e^g∆ti+

1_A^∗   F_ti



determined in Equation (2.22) and c given in Equation (2.23).

It is noticed that by choosing the size of , the insurer chooses how much financial risk he is willing to bear. At the same time, the  determines the c–value, which consequently af-fects the size of the implied survival probability. Thus, by setting the –value, the insurer aims at controlling the value of the implied survival probability, namely, a transfer be-tween the financial and insurance risk becomes possible. On some level, we always obtain an optimal survival probability for a given financial risk. The lower the resulting optimal survival probability is, the more old and safe customers should the insurer acquire. On the contrary, the higher the resulting optimal survival probability is, the more younger customers are allowed to be taken. Furthermore, if an increase in  leads to a decrease in_Tp^∗_x, it implies there exists some transfer between the financial and insurance risk. In other words, when the insurer takes more financial risk, as a compensation, more safe old customers are preferred by the insurer.

In the following some numerical results for the implied survival probability are calculated.

Table 2.1 demonstrates how the implied survival probability changes with µ, σ and g.

There are some obvious effects, e.g., the positive influence of µ on the considered prob-ability and the negative effect of σ and g on the probprob-ability. As µ goes up, in the expectation, the insurer’s paying ability goes up, and it allows the insurer to accept more younger customers. While σ and g has reversed effects on the implied survival proba-bility. Furthermore, it is observed that the implied survival probability decreases with the significance level  for a given T . As  goes up, i.e, the hedger (the insurance com-pany) is ready to take more risk and hedge with a smaller success probability (1 − ), as a compensation, a smaller implied survival probability results. That means, some safer old customers should be chosen because with a higher probability they are not going to survive the maturity date, which is good for the insurer who just issues pure endowment contracts. Thereby you see that the insurance company transfers part of its financial risks to insurance risks.

Definition 2.3.2 (Reduction level). Let_Tp_x denote the empirical survival probability the

 = 0.01  = 0.03  = 0.05 T = 12 4.24894 14.6598 23.5753 T = 18 2.04975 12.6997 21.8200 T = 24 0.00000 9.2789 18.7565

Table 2.2: Reduction level (%) with parameters: g = 0.02, µ = 0.06, σ = 0.3, x = 30.

insurer uses for pricing and hedging. By using the quantile hedging method, there is a reduction in the premium and this reduction level is given by

reduction level = 1 − E^∗

Apparently, the resulting premium from using quantile hedging method owns a smaller value than that determined by arbitrage pricing because a certain shortfall probability is permitted in the quantile hedging. The reduction level indicates how many percents this premium is reduced by applying quantile hedging method. Only this empirical survival probability coincides with the above calculated implied one, no essential reductions oc-cur. In the following we use the benchmark death distribution as an empirical mortality introduced in Subsection 1.4.3 of Chapter 1 and some numbers are given in Table 2.2.

Obviously the premium is reduced to a much bigger extent as the insurance company is permitted to take more risk, namely the risk is increased that the company will fail to hedge successfully. While for given –values, the reduction becomes less apparent as the maturity of the contract is lengthened. Combined with small ’s, say  = 0.01, almost no reduction of the premium is possible.

If µ > σ², then_S(t might have one, two or no solutions. If there is no solution, i.e., the function 

S(ti+1) S(ti)

^µ

σ2

lies always above the constant

hS(ti+1)

Obviously, it contradicts the initial investment constraints. If there is only one solution, the analysis is in analogy to the case of µ ≤ σ². If there are two solutions, say e^c1∆t and

In the case, the price of the call option under the quantile hedging is given by As a consequence, the implied survival probability can be determined by the quotient of the above expression and the price of the call option. And again c₁ and c₂ can be determined by the shortfall probability the insurer chooses.