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Quantile hedging

Quantile hedging maximises the chance of meeting a claim with limited capital. Compare physical probabilities, pricing costs and the shortfall left by a partial hedge.

Quantile hedging chooses a hedge to maximise the probability of meeting a liability when the available capital is below the cost of full protection. It makes the funding trade-off explicit: a cheaper hedge can cover the entire claim in selected scenarios while leaving a shortfall elsewhere.

For a nonnegative claim HH due at time TT, a budget xx and terminal hedge wealth VTV_T, the objective is

p(x)=sup⁡V∈A(x)P(VT≥H).p(x)=\sup_{V\in\mathcal{A}(x)}\mathbb{P}(V_T\ge H).
Here A(x)\mathcal{A}(x) contains self-financing strategies starting with at most xx whose wealth stays nonnegative throughout. This admissibility condition matters: the hedge cannot finance success in some states by promising unlimited losses in others. The probability uses the physical measure P\mathbb{P}; the cost uses traded prices. Föllmer and Leukert formulate this problem in §§2.1–2.2.

Buying protection state by state

Consider a fictional, frictionless, one-period complete market with three mutually exclusive terminal states. An Arrow security for each state pays USD 1 only when that state occurs. Their prices today are USD 0.30, USD 0.30 and USD 0.40. Fractional holdings are permitted, and the discount factor is one, so buying all three reproduces a dollar of cash. These three securities span every terminal payoff; this is not a one-stock binomial model.

Suppose a liability requires USD 100 in every state:

Terminal statePhysical probability PiP_iPricing probability QiQ_iCost of covering USD 100 in this state
A60%30%USD 30
B30%30%USD 30
C10%40%USD 40

The QiQ_i equal the Arrow state prices here because discounting is one. They form a risk-neutral measure consistent with the security prices. The PiP_i are stipulated beliefs about how often the states occur. They are not historical estimates, and they are not interchangeable with QiQ_i.

Full replication costs 100(0.30+0.30+0.40)=100100(0.30+0.30+0.40)=100 dollars. With only USD 60, buy 100 units of the A security and 100 units of the B security. The terminal wealth is
(VT(A),VT(B),VT(C))=(100,100,0).(V_T(A),V_T(B),V_T(C))=(100,100,0).

It costs USD 60 and succeeds with physical probability 0.60+0.30=90%0.60+0.30=90\%. Its pricing-measure probability of success is only 0.30+0.30=60%0.30+0.30=60\%. The first measures modelled coverage frequency; the second supplies the cost weights.

All eight success sets in a fictional three-state market. Covering A and B costs 60 dollars and succeeds with physical probability 90 percent, the highest probability at the 60-dollar budget. Covering all states costs 100 dollars and succeeds with probability 100 percent.
All eight success sets in a fictional three-state market. Covering A and B costs 60 dollars and succeeds with physical probability 90 percent, the highest probability at the 60-dollar budget. Covering all states costs 100 dollars and succeeds with probability 100 percent.
Original fictional example. Each point fully covers USD 100 in the labelled states and pays zero elsewhere. Physical probabilities are stipulated; the one-period discount factor is one. The shaded region meets the USD 60 budget.

This is optimal among all admissible portfolios in this toy market, not merely among the two portfolios just compared. If a nonnegative terminal portfolio succeeds on a set SS, its cost is at least

100∑i∈SQi.100\sum_{i\in S}Q_i.

Paying more than 100 in a successful state cannot create another success, and a partial payment below 100 in another state does not count as success. Enumerating all eight subsets therefore proves the result. The affordable sets are none, A, B, C, and A together with B; their success probabilities are 0%, 60%, 30%, 10% and 90%. Every remaining set costs at least USD 70.

Why a likelihood ratio appears

In a complete market, the construction is: choose a success event SS, then replicate the modified claim H1SH\mathbf{1}_S. In discounted units its cost is EQH1S]\mathbb{E}^{\mathbb{Q}}H\mathbf{1}_S]. The [Neyman–Pearson lemma selects states by their physical probability relative to their claim-weighted pricing cost. For H>0H>0, the relevant score is

dP/dQH.\frac{d\mathbb{P}/d\mathbb{Q}}{H}.

A threshold set solves the event problem when it attains the budget appropriately. In the table, the scores Pi/(QiHi)P_i/(Q_iH_i) are 0.020.02, 0.010.01 and 0.00250.0025 per dollar: A then B buys the most coverage probability per dollar. Their combined cost exactly exhausts USD 60. States with H=0H=0 are already successful under nonnegative wealth and need no protection.

Atoms need care. Atomic terminal states can prevent a threshold from using the exact budget; sorting ratios alone is not a general solution to the finite subset problem. The paper's §2.3 relaxes the indicator to a test 0≤φ≤10\le\varphi\le1 and maximises the expected success ratio. Replicating HφH\varphi with φ<1\varphi<1 in a state only partially funds that state. It does not give full-payment success there. An actual randomised full-payment event needs additional, implementable randomisation and a corresponding market model. These distinctions are explicit in §§2.2–2.4.

A high success rate can hide a large unpaid claim

In state C, the USD 60 hedge pays nothing against the USD 100 liability. It has a 10% shortfall probability, a USD 100 shortfall conditional on failure, and a physical expected shortfall amount of 0.10(100)=100.10(100)=10 dollars. The 90% coverage figure does not soften that remaining obligation.

Quantile hedging counts full coverage; it does not penalise the size of an uncovered amount. An alternative is to minimise

EP ⁣[ℓ((H−VT)+)]\mathbb{E}^{\mathbb{P}}\!\left[\ell\big((H-V_T)^+\big)\right]
for an increasing loss function ℓ\ell. A convex function can penalise larger shortfalls more heavily. This is the focus of Föllmer and Leukert's Efficient Hedging: Cost versus Shortfall Risk, §§1–2. “Expected shortfall amount” here means the ordinary expectation of an unpaid claim; it is not automatically the tail-average risk measure called Expected Shortfall.

The toy result also depends on complete spanning, frictionless implementation and correct physical probabilities. In an incomplete market, the selected modified claim may require super-replication, and a single chosen pricing measure need not give its protection cost. A modelled success probability is therefore conditional on both the probability model and the feasible trading strategies.

To work through the pricing foundation, SelfMadeQuant's Market Model and Risk-Neutral Valuation develops replication and the physical-versus-pricing distinction, then market completeness (§§2.2 and 2.6). The link has a public preview; the full lesson requires membership.

Sources

  • Hans Föllmer and Peter Leukert, Quantile Hedging, author-hosted prepublication manuscript of the article published in Finance and Stochastics 3 (1999), 251–273. §§2.1–2.4 give admissibility, success-set optimisation, the Neyman–Pearson construction and the success-ratio extension; §4.1 explains the incomplete-market superhedging cost.
  • Hans Föllmer and Peter Leukert, Efficient Hedging: Cost versus Shortfall Risk, author-hosted manuscript of the article published in Finance and Stochastics 4 (2000), 117–146. §§1–2 distinguish probability of failure from loss-weighted shortfall risk.
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