In this section, we will study the optimal CVaR-based partial hedging with the assumption of sufficient hedging budget under the no arbitrage pricing functional. First we rewrite the
optimization problem (3.1) by assuming there is sufficient hedging budget min
f ∈LCV aRα(Tf(X)) (3.3)
We restate the assumptions imposed on the pricing function Π(·) in this section as follows
Assumption 3.2.1. (a) Π(·) admits no arbitrage opportunities. (b) Π(X) 6 α1 · e−rT
· EP[X] holds for any contract X, where r is the risk-free rate and
T is the time to maturity of contract X.
With the above assumptions on the market pricing function and the admissible set of hedged loss functions which is stated in (3.2), we are ready to construct the optimal hedging strategy. The first step of the construction is done by the following lemma. Lemma 3.2.1. For a given random variable X and any function f ∈ L1, we can construct
a function ˆgf as follows:
ˆ
gf(x) = (x + f (v)− v)+. (3.4)
where v = VaRα(X) and (x)+ equals to x if x > 0 and zero otherwise. Then ˆgf ∈ L1.
Proof: Obviously, the above function ˆgf is well defined. Therefore, it is sufficient to
show that the function ˆgf ∈ L1 holds for any function f ∈ L1.
Note that f ∈ L1, we have f (x) 6 x holds for any x. In particular, For a given random
variable X, f (v) 6 v where v = VaRα(X). Thus, ˆgf(x) = (x + f (v)− v)+ 6 x holds for
any x. It is clear that ˆgf(x) > 0 for any x. Then we can conclude that 0 6 ˆgf(x) 6 x for
According to the definitions of retained loss function, for any x, we have Rˆgf(x) = x, if 0 6 x 6 v− f(v), v− f(v), if x > v − f(v),
From the above expression, it is straightforward that Rˆgf(x) is nondecreasing and left
continuous with respect to x. Therefore, we can conclude that ˆgf ∈ L1 holds for any
function f ∈ L1, according to definition 3.2.
Remark 3.2.1. For a given random variable X and any function f ∈ L1, if we denote
d = v− f(v), then the constructed hedged loss function can be rewritten as ˆ
gf(X) = (X− d)+ (3.5)
Therefore, the partial hedging strategy corresponding to the hedged loss function ˆgf is to
construct a call on the contingent claim.
The following lemma provides a comparison between the constructed hedging function and the original hedging function.
Lemma 3.2.2. For a given contract X and any hedging function f ∈ L1, if the hedging
function ˆgf is constructed as in Lemma 3.2.1, then ˆgf(x) 6 f (x) when x 6 v, and ˆgf(x) >
f (x) when x > v.
Proof: When x 6 v− f(v), according to the definition of function ˆgf in Lemma 3.2.1,
we can claim that ˆgf(x) = 0. From the definition of L1 which is stated in (3.2), we know
When x > v− f(v), according to the definition of function ˆgf in Lemma 3.2.1, we have
ˆ
gf(x) = x + f (v)− v. In this case,
f (x)− ˆgf(x) = (v− f(v)) − (x − f(x))
= Rf(v)− Rf(x)
From the definition of L1 which is stated in (3.2), we know that Rf is a nondecreasing
function. Therefore, we have Rf(v)− Rf(x) > 0, which implies that ˆgf(x) 6 f (x), when
v− f(v) 6 x 6 v, and Rf(v)− Rf(x) 6 0, which implies that ˆgf(x) > f (x), when x > v
From the above analysis, we can conclude that ˆgf(x) 6 f (x) when x 6 v, and
ˆ
gf(x) > f (x) when x > v.
Now we can state the main result of this subsection, which says that for a given payoff X with maturity T and any partial hedging strategy f ∈ L1, the partial hedging strategy
ˆ
gf(X) will outperform the partial hedging strategy f (X) under Assumption 3.2.1.
Theorem 3.2.1. Assume that the market is complete and the market pricing function satisfies Assumption 3.2.1. For any contract X with maturity T and any hedging function f ∈ L1, we can construct a call hedging strategy ˆgf ∈ L1 as (3.4), or equivalently (3.5),
such that
CV aRα(Tgˆf(X)) 6 CV aRα(Tf(X))
Therefore, the call hedging strategy is the optimal form of hedging in the sense of minimizing CVaR of the total exposed risk at maturity of the investor.
Proof: For brevity of expressions, we assume that P(X = v) = 0 in the following proof. According to the definition of Tf(X) given in (2.1) and the translation invariance property
CV aRα(Tf(X)) =CV aRα(Rf(X) + erT · Π(f(X))) =CV aRα(Rf(X)) + erT · Π(f(X)) =CV aRα(Rf(X)) + erT · Π(f(X) · 1(X 6 v)) + erT · Π(f(X) · 1(X > v)) =1 αE P[R f(X)· 1(X > v)] + erT · Π(f(X) · 1(X 6 v)) + erT · Π(f(X) · 1(X > v))
where r is the risk-free rate. The third equality is because of the linearity of the pricing functional, which is implied by the no arbitrage property in Assumption 3.2.1. The fourth equality is due to the definition of CVaR risk measure.
According to Lemma 3.2.2, we know that ˆgf(x) > f (x) when x > v. Now we construct
a contract Y which matures at T and has payoff as follows
Y = 0, if X 6 v, ˆ gf(X)− f(X), if X > v,
It is clear that Y is nonnegative, thus Y is a well-defined contract. From Assumption 3.2.1, we know that
Π(Y ) 6 1 α · e
−rT
· EP[Y ]
According to the construction of Y , the above inequality is equivalent to Π [(ˆgf(X)− f(X)) · 1(X > v)] 6 1 α · e −rT · EP[(ˆg f(X)− f(X)) · 1(X > v)]
Due to the linearity of the expectation and the pricing functional, we can rewrite the above inequality as follows erT · Π [ˆgf(X)· 1(X > v)] + 1 α · E P[(X − ˆg f(X))· 1(X > v)] 6erT · Π [f(X) · 1(X > v)] + 1 α · E P[(X − f(X)) · 1(X > v)]
From Lemma 3.2.2, we know that ˆgf(x) 6 f (x) when x 6 v, which implies that
ˆ
gf(X · 1(X 6 v)) 6 f(X · 1(X 6 v)). Since the pricing functional Π(·) admits no
arbitrage, we can claim that
Π(ˆgf(X)· 1(X 6 v)) 6 Π(f(X) · 1(X 6 v))
From the above two inequalities, we have
erT · Π(ˆgf(X)· 1(X 6 v)) + erT · Π [ˆgf(X)· 1(X > v)] + 1 α · E P[(X − ˆg f(X))· 1(X > v)] 6erT · Π(f(X) · 1(X 6 v)) + erT · Π [f(X) · 1(X > v)] + 1 α · E P[(X − f(X)) · 1(X > v)] which is equivalent to CV aRα(Tgˆf(X)) 6 CV aRα(Tf(X))
Remark 3.2.2. (a) The comments we made in Remark 2.2.1 and Remark 2.2.4 for the solutions to the VaR-based optimal partial hedging problem in Chapter 2 are similarly ap- plicable here for the CVaR case. In particular, we draw the following conclusions.
(i) Theorem 3.2.1 indicates that the call hedging strategy is optimal among all the s- trategies in L1 under Assumption 3.2.1. We note again that the optimal call hedging
strategy is to construct a call on the risk X and not on the asset that underlies X. (ii) The optimality of call hedging is model independent. It does not depend on the dy-
namic of the underlying or the specific pricing functional.
(iii) If the call option written on the risk X is available in the financial market, then the optimal partial hedging can be achieved via a simple static hedging strategy.
(b) In this section, we study the CVaR-based optimal partial hedging problem by assum- ing there is sufficient budget constraint. The reason why we removed the budget constraint is because we have imposed a relatively mild assumption on the pricing functional. As we can see in the next section, when we assume the pricing functional preserves stop-loss order, we will be able to solve the CVaR-based optimal partial hedging problem with any given budget constraint.
(c) Although we assume the hedging budget is sufficient in this section, the cost of constructing the partial hedging strategy is included in the total risk exposure. Therefore, the tradeoff between the cost and effect of the hedging strategy is still being considered in the objective function.