Hedging

Hedging reduces risk by taking an offsetting position sized by beta, correlation, or delta, depending on what type of exposure is being neutralized.

1. The idea

A beta hedge neutralizes market exposure by shorting an index position sized as $β \times (portfolio value)$ , since a stock or portfolio's expected sensitivity to the market is captured by its beta from the CAPM regression.

A minimum-variance hedge using a futures contract chooses the hedge ratio $h^{*} = ρ \frac{σ _{S}}{σ _{F}}$ (correlation between spot and futures returns times the ratio of their volatilities), which minimizes the variance of the hedged position — this generalizes the beta hedge to any two correlated instruments, not just an index.

A delta hedge neutralizes an option's exposure to small moves in the underlying by holding $- Δ$ shares against a long option position, but since delta changes with the stock price (gamma), it must be rebalanced periodically — a process called dynamic hedging.

2. Why it matters

Every risk desk hedges some exposure — equity beta, commodity price risk via futures, or options via delta — and knowing which hedge ratio applies to which type of risk is a core practical skill.

3. Core mathematics

Beta hedge size

Short index notional = β \times V_{p or t f o l i o}

Minimum variance hedge ratio

h^{*} = ρ \frac{σ _{S}}{σ _{F}}

Delta hedge

Shares to hold = - Δ \times (option contracts)

4. How to recognize it

Think of this when…

'How many futures contracts should be shorted to hedge this exposure'
Any question involving beta and hedging an equity portfolio
Delta-neutral or dynamic hedging setups for options books

5. Worked example

Worked example

A spot position has volatility 20%, correlated 0.8 with a futures contract whose volatility is 25%. Find the minimum-variance hedge ratio.

1
Apply the formula.
$h^{*} = ρ \frac{σ _{S}}{σ _{F}} = 0.8 \times \frac{0.20}{0.25}$
2
Compute.
$h^{*} = 0.8 \times 0.8 = 0.64$
Answer: Hedge ratio ≈ 0.64, meaning short 0.64 units of futures per unit of spot exposure

6. Common mistakes

Using correlation alone as the hedge ratio, ignoring the volatility ratio scaling
Applying a static delta hedge and forgetting to rebalance as the stock (and delta) moves
Confusing beta (regression slope against the market) with correlation directly

7. Quick check

The minimum-variance hedge ratio is:

A delta hedge needs rebalancing because of:

8. Practice

7 QuantSolve questions use this concept.

Practice this concept

9. Related methods

This lesson is conceptual; see the topic hub for methods.

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