22.2 Discrete Time Hedging Strategies and Dynamic
22.2.2 Local Expected Shortfall-Hedging and the Related
The expected shortfall of a self-financing hedging strategyH of a contingent claim with payoffVT is defined as
Ef.VT FT.H //Cg;
whereFT.H /is the terminal wealth of the self-financing hedging strategyH at the expiration dateT. Practitioners want to know whether there exists an optimal hedging strategy, denoted byHES, such that the expected shortfall risk is minimized with a pre-fixed initial hedging capitalV0, that is,
HESDarg min
H2SEf.VT FT.H //Cg;
where
SD fH jH is a self-financing hedging strategy with initial hedging capitalV0g:
Cvitani´c and Karatzas (1999) and F¨ollmer and Leukert (2000) pioneered the expected shortfall-hedging approach and showed the existence of this hedging strategy. Schulmerich and Trautmann (2003) proposed a searching algorithm to construct a hedging strategy which minimizes the expected shortfall risk in complete and incomplete discrete markets. But the searching algorithm often spends large of computation time. In order to overcome this time-consuming problem,Schulmerich and Trautmann(2003) further proposed a local expected shortfall-hedging strategy.
The idea of the local expected shortfall-hedging strategy is introduced in the following.
The first step is to find an optimal modified contingent claim X, which is a contingent claim that belongs to the setof all modified contingent claims, where
fX jX < VT andEQ.X=BT/V0
for all risk-netral probability measureQg;
and
XDarg min
X2E.VT X/: (22.2)
The above definition implies that the superhedging cost of any modified contingent claim is lower or equal than the initial hedging capitalV0. By Proposition 2 of Schulmerich and Trautmann(2003), the superhedging cost of the optimal modifined contingent claimXis identical to the shortfall risk of the hedging strategyHES, that is,
Ef.VT FT.HES//Cg DE.VT X/:
Therefore, one can determine the desired hedging strategyHES by the following two steps:
[Dynamic programming of expected shortfall-hedging]
1. Find an optimal modified contingent claimX2with criterion (22.2).
2. Construct a superhedging strategy forX.
Since Step-2 can be accomplished by the method introduced in the previous section, the main concern is the first step. In complete markets, the optimal modified contingent claimXis a direct consequence of a slight modification of the Neyman- Pearson lemma (seeF¨ollmer and Leukert 2000;Schulmerich and Trautmann 2003).
The solution of the optimal modified contingent claim is given in Proposition 4 of Schulmerich and Trautmann(2003), that is,
X.!/DVT.!/I.P .!/=Q.!/>c/CI.P .!/=Q.!/Dc/ (22.3)
withcESDmin!fP .!/=Q.!/gand
D fV0BT VTEQ.I.P .!/=Q.!/>c//g.
EQ.I.P .!/=Q.!/Dc//:
If the market is incomplete, the construction of the optimal expected shortfall hedging strategy is much more complicated than that in complete markets due to the fact that the risk-neutral probability measures are not unique. For continuous time models,F¨ollmer and Leukert(2000) showed that an optimal hedging strategy exists but didn’t provide an explicit algorithm to calculate it. As for the discrete models, an algorithm of the optimal expected shortfall-hedging is given in Proposition 5 of Schulmerich and Trautmann(2003). The basic idea of this algorithm is still based on (22.3). The main difficulty is to deal with the non-uniqueness of the equivalent martingale measures. Let Q denote the smallest polyhedron containing all the martingale measures. Since Qis convex, there exists a finite number of extreme points of the convex polyhedronQ, denoted byQ1; ; QL, and thus the criterion of choosing optimal modified contingent claimX, maxQ2QEQ.X=BT/ V0, could be simplified by
iD1;max;LEQi.X=BT/V0:
However, it consumes a lot of computational effort to check this condition.
Therefore,Schulmerich and Trautmann(2003) further proposed the following local expected shortfall-hedging strategy, denoted byHLES:
[Dynamic programming of local expected shortfall-hedging]
LetVT be a European type contingent claim andFtSHbe the corresponding superhedging values at timet D 1; ; T. Then find sequentially a self-financing strategy HLES D .H1LES; ; HTLES/withHtLESminimizing the local expected shortfall
Et1f.FtSHFt.H //Cg;
fort D1; ; T, whereEt1./denotes the conditional expectation under the dynamic probability measure given the information up to timet1.
In the following, we give two examples to illustrate the construction of HtLES. Example2gives a one-period trinomial case and Example3considers a two-period situation.
Example 2. Consider the same one-period trinomial model as in Example1. Let
!1, !2 and !3 denote the states of S1 D uS0, dS0 and jS0, respectively, and P denote the dynamic probability measure with P .!1/ D 0:55,P .!2/ D 0:40 andP .!3/ D 0:05. As shown in Example1, the setQof risk-neutral probability measures can be expressed as
QD f.q1; q2; q3/W 1
2 < q1< 2
3; q2 D23q1> 0andq3D2q11 > 0g:
LetQbe the smallest polyhedron containingQ, that is, QD f.q1; q2; q3/W 1
2 q1 2
3; q2D23q10andq3D2q110g:
ThenQ1.!1; !2; !3/ D .12;12; 0/andQ2.!1; !2; !3/ D .23; 0;13/be two extreme points of this convex polyhedronQ.
A practitioner is willing to set her initial hedging capital to be 6, which is less than the initial capital required by the superhedging strategy 203. Our aim is to determine a trading strategy minimizing the expected shortfall with the initial hedging capital. By Proposition 5 of Schulmerich and Trautmann (2003), since EQ2.V1/ > EQ1.V1/, we consider Q2 first. In order to determine the modified contingent claim, one can apply Proposition 4 of Schulmerich and Trautmann (2003). However, Proposition 4 ofSchulmerich and Trautmann(2003) can not be implemented directly to the trinomial model since trinomial model is not a complete market model. Nevertheless, due to the fact thatQ2.!2/D0, we can ignore the state
!2temporarily and only determine the modified contingent claim by the states!1
and!3:
X.!/DV1.!/I.P .!/=Q2.!/>c/CI.P .!/=Q2.!/Dc/;
where c D minfP .!/=Q2.!/ W ! D !i; i D 1; 3g and is chosen to ensureEQ2.X/ 6. By straightforward computation, we haveX.!1/ D 10and X.!3/D 2.
Next, construct the superhedging strategy forX.!i/,i D1; 3. By the same way introduced in Sect.22.2.1, one can obtain the hedging portfolioH0LES D .hQ00;hQ10/
to be (hQ00D 34
hQ10D0:4: ;
which satisfiesH1LES.!1/DX.!1/D10andH1LES.!3/DX.!3/D 1. Finally, we defined the value of the modified contingent claim of state !2 by X.!2/ D H1LES.!2/D 2. Note that for this modified contingent claim, we haveEQ2.X/D EQ1.X/ D 6 and since any risk-neutral probability measure can be expressed by Q D aQ1C.1a/Q2,0 < a < 1, thus for all risk-neutral probability measure Q 2 Q we conclude thatEQ.X/ D 6, and the corresponding minimal expected shortfall is
EfV1F1.H0LES/gC DE.V1X/C D0:1:
Example 3. In this example, we extend the one-period trinomial model discussed in previous examples to two period. In each period, given the stock priceSt,t D0; 1, let the stock prices at the next time point beStC1DuSt,dSt andjSt, with dynamic probability0:55,0:40and0:05, respectively. In Fig.22.3, the values ofS0,S1 and S2are set withS0 D 100, u D1:1,d D 0:9andj D 0:8. The payoff at time-2 is defined byV2 D .S2100/C, which is the payoff of a European call option with
a b
Fig. 22.2 (a) One period trinomial model in Example2; (b) Local expected shortfall-hedging strategy of the one period trinomial model
(S2, F2, X2) (S1, F1, X1)
S0
100
3 0,1 3,
Under Q2 2 (64,0,-30)
(72,0,-20) (80,0,-10)
(90,0, -2) (110,14,14)
(72,0,-6) (81,0,-4) (99,0,0) (88,0,0) (99,0,7) (121,21, 21)
(88,0,0)
Fig. 22.3 Local expected shortfall-hedging strategy of the two period trinomial model in Example3
expiration dateT D 2and strike priceK D100. As in Example2, the probability measuresQ1.!1; !2; !3/ D .12;12; 0/andQ2.!1; !2; !3/ D .23; 0;13/are the two extreme points of the convex polyhedronQ. Also assume that an investor’s initial hedging capital is set to be 6, which is still less than the initial capital required by the superhedging strategy
Q2QmaxEQ.e2rV2/DEQ2.e2rV2/D 28 3;
where the riskless interest rater is set to be 0. In the following, we illustrate how to obtain the modified contingent claimXt at each time pointt D 1; 2, and then construct the local expected shortfall-hedging strategy.
Since EQ2.e2rF2/ > EQ1.e2rF2/ whereF2 D V2, thus under the proba- bility measureQ2, compute the time-1 payoffF1 by the conditional expectation,
EQ2.erF2jS1/, given the stock priceS1. The first step is to find the one period optimal expected shortfall-hedging with initial hedging capital 6 and payoffF1
in the one period trinomial model. This step can be solve by similar way as in Example 2. Hence, we obtain the modified contingent claim X1 and the corresponding hedging strategy.
For the second period, given any stock priceS1, the problem can be treated as another one period trinomial hedging task, that is, find the one period optimal expected shortfall-hedging with initial hedging capital X1.S1/ and payoff F2. Therefore, we can still adopt similar way as in Example2 to obtain the modified contingent claim X2 and the corresponding hedging strategy. The values of the modified contingent claimXi,i D 1; 2, are given in Fig.22.3. Note that for the modified contingent claimX2, we haveEQ2.X2/D EQ1.X2/ D 6and since any risk-neutral probability measure can be expressed byQ D aQ1 C .1a/Q2, 0 < a < 1, thus for all risk-neutral probability measureQ 2 Qwe conclude that EQ.X/D6, and the corresponding expected shortfall isEŒ.F2X2/CD1:235: