Using Options to Hedge Nonlinear Market Exposure

Why Options Matter in an Equity Hedge

An equity-linked asset or liability may respond nonlinearly to a change in the share price. A share position has a constant delta and zero gamma, but an option’s delta generally changes as the share price moves. Consequently, a share-only hedge may offset the current exposure locally and still leave a material residual after the market moves. Options can be used to address that curvature as well as the initial directional exposure.

The calculations in this tutorial are local: they use sensitivities evaluated at the current market conditions. They are not guarantees that a hedge will remain effective after a large move. The inputs in the examples are hypothetical, and each option’s stated delta and gamma are per contract at the valuation point.

Definition: For a position with value $V(S)$, where $S$ is the price of the relevant equity, its delta is $\Delta=\frac{\partial V}{\partial S}$, and its gamma is $\Gamma=\frac{\partial^2 V}{\partial S^2}=\frac{\partial \Delta}{\partial S}$. Delta measures the approximate change in value for a one-dollar change in $S$; gamma measures the approximate change in delta for a one-dollar change in $S$. In this tutorial, share holdings are signed: a long position is positive and a short position is negative.

A long call or long put typically has positive gamma under standard option valuation conditions. A short option position reverses the sign of both its delta and gamma. The actual Greeks depend on the option and market conditions; use the values supplied for the valuation point rather than assuming they stay constant.

Equations for a Local Option-and-Share Hedge

Suppose an existing position has delta $\Delta_P$ and gamma $\Gamma_P$. A traded option has delta $\Delta_O$ and gamma $\Gamma_O$ per contract. Let $n$ be the signed number of option contracts and $q$ the signed number of shares added. A positive $n$ means long options; a negative $n$ means short options. A share position has delta $q$ and zero gamma, so the combined sensitivities are:

\Delta_{\text{combined}}=\Delta_P+n\Delta_O+q,\qquad \Gamma_{\text{combined}}=\Gamma_P+n\Gamma_O.

To set both local sensitivities to zero, first choose $n$ to offset gamma, then choose $q$ to offset the resulting delta. If $\Gamma_O\ne 0$, the continuous-contract solution is:

n=-\frac{\Gamma_P}{\Gamma_O},\qquad q=-(\Delta_P+n\Delta_O).

When only whole contracts can be traded, the gamma hedge may not be exact. Select the permitted contract count that best meets the stated objective, calculate the resulting combined gamma, and then set the share position to neutralize delta if required. Hedging delta with shares does not change gamma.

Worked Example: Offsetting Negative Gamma With Calls

A hypothetical equity-linked liability has delta $-2{,}400$ shares and gamma $-48$ shares per dollar. A call has delta $+24$ shares and gamma $+0.6$ shares per dollar per contract. Find the call and share positions that set both local sensitivities to zero. Assume the stated Greeks apply at the current share price, and allow whole contracts.

Method. Hedge gamma with calls first. Since a share position has zero gamma, the calls must contribute $+48$ gamma:

$n=-(-48)/0.6=80$ calls. The 80 long calls contribute gamma $80(0.6)=+48$, so the combined gamma is $-48+48=0$.

Then hedge delta. Before adding shares, the combined delta is $-2{,}400+80(24)=-2{,}400+1{,}920=-480$ shares. Therefore $q=-(-480)=+480$: buy 480 shares.

Check. Combined delta is $-2{,}400+1{,}920+480=0$, and combined gamma is $-48+48=0$. The local hedge is long 80 calls and long 480 shares. A common sign error is to short the shares because the original liability has negative delta; the correct share direction is determined by the delta after adding the calls.

Why the Hedge Equations Work

For a small change $\Delta S$ in the underlying price, a twice-differentiable position value can be approximated by a second-order Taylor expansion around the current price $S$:

V(S+\Delta S)-V(S)\approx \Delta\,\Delta S+\frac{1}{2}\Gamma(\Delta S)^2.

The first term is the local linear effect. The second term captures the leading curvature effect. Differentiating the local approximation with respect to the price move, or applying a first-order expansion directly to delta, gives:

\Delta(S+\Delta S)\approx \Delta(S)+\Gamma(S)\Delta S.

These approximations explain the two-part hedge. If the combined delta and gamma are both zero at the valuation point, both displayed terms in the approximate value change vanish. If only delta is zero, the gamma term can remain. This is a local second-order argument, not an exact result for a finite move: higher-order effects and changes in other market inputs are not included.

Choose the option position.
Use the existing gamma and the option gamma to offset curvature, respecting any whole-contract constraint.

Recalculate delta.
Add the delta contributed by the selected signed option position to the existing delta.

Choose the share position.
Take the negative of the delta after the options are included. Shares adjust delta but do not adjust gamma.

Verify both totals.
Substitute the signed positions into the combined-delta and combined-gamma equations.

Worked Example: Hedging Positive Gamma With Short Puts

A hypothetical equity exposure has delta $+500$ shares and gamma $+12$ shares per dollar. A put has delta $-20$ shares and gamma $+0.4$ shares per dollar per contract. Use whole contracts to set gamma to zero, then hedge delta exactly with shares.

Method. The exposure has positive gamma, so the option position must have negative gamma. Each short put contributes $-0.4$ gamma. The required number is $12/0.4=30$, so take a position of $n=-30$ puts.

The short puts contribute gamma $(-30)(+0.4)=-12$. Combined gamma is $+12-12=0$.

Their delta contribution is $(-30)(-20)=+600$ shares. The delta before adding shares is therefore $+500+600=+1{,}100$. To neutralize it, take $q=-1{,}100$, or short 1,100 shares.

Check. Combined delta is $500+(-30)(-20)-1{,}100=500+600-1{,}100=0$. Combined gamma is $12+(-30)(0.4)=12-12=0$. The hedge is short 30 puts and short 1,100 shares. A common mistake is to calculate the share hedge from the original $+500$ delta alone; the put position adds another $+600$ delta first.

Why a Static Hedge Can Become Inadequate

A hedge is static when the option and share positions are left unchanged after they are established. But option delta and gamma can change with the share price and with other valuation conditions. A hedge that neutralizes delta and gamma at one point may therefore have nonzero sensitivities later. Even if the underlying moves only, the initial Greeks provide local estimates; refreshed market values and Greeks are needed to assess the new position.

The next example uses supplied refreshed Greeks after a price move. This avoids treating an initial Taylor approximation as if it were an exact repricing model. It also shows why checking the old positions at new market conditions is different from calculating a new hedge.

Worked Example: Reassessing a Hedge After a Price Move

At a share price of $80, a hypothetical equity-linked exposure has delta $+700$ shares and gamma $-16$ shares per dollar. A call has delta $+20$ shares and gamma $+0.4$ shares per dollar per contract. Establish a whole-contract local hedge. Later, the share price is $84. At that point, refreshed values are: exposure delta $+640$, exposure gamma $-15$, call delta $+21$, and call gamma $+0.3$ per contract. Find the new hedge and quantify how the old positions perform locally.

Initial hedge at $80. Offset the exposure’s $-16$ gamma with calls: $n=-(-16)/0.4=40$ long calls. These add $40(20)=+800$ delta. Before shares, delta is $700+800=+1{,}500$, so short 1,500 shares.

Check at $80. Combined delta is $700+800-1{,}500=0$. Combined gamma is $-16+40(0.4)=-16+16=0$.

Evaluate the old positions at $84. The 40 calls now contribute delta $40(21)=+840$ and gamma $40(0.3)=+12$. With the original short 1,500 shares, combined delta is $640+840-1{,}500=-20$, and combined gamma is $-15+12=-3$. Thus the old hedge is no longer locally neutral at the refreshed valuation point.

Rebalance at $84. To offset gamma $-15$, use $15/0.3=50$ long calls in total. This means buying 10 additional calls. Their total delta is $50(21)=+1{,}050$, so delta before shares is $640+1{,}050=+1{,}690$. Short 1,690 shares in total. Relative to the old short 1,500-share position, this requires shorting another 190 shares.

Check. The new combined delta is $640+1{,}050-1{,}690=0$; the new combined gamma is $-15+50(0.3)=0$. The old hedge’s residual delta of $-20$ and gamma of $-3$ demonstrate why a static hedge can become inadequate.

Practical Interpretation and Common Traps

A local delta-and-gamma hedge is a sensitivity match, not a promise of exact future offset. Its usefulness depends on the quality and consistency of the Greeks, the relevance of the chosen option to the exposure, and the size of the market move. In practice, the valuation point should be explicit: option Greeks and exposure Greeks must refer to compatible underlying prices and units.

Use signed positions. A short option has negative contract count. Multiply its signed count by both the option’s delta and gamma.
Recalculate delta after choosing options. Gamma hedging can add or subtract substantial delta; the share hedge must offset the resulting total.
Do not use shares to hedge gamma. Under the stated model, shares have zero gamma. They can correct delta only.
Do not interpret local neutrality as permanent neutrality. Greeks change as prices and other valuation conditions change, and a second-order approximation omits higher-order effects.
Keep units aligned. If option Greeks are per contract, multiply by the number of contracts. If an exposure is stated in dollars of value rather than share-equivalent delta, convert consistently before adding positions.

This framework is aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus because it connects nonlinear market exposure to practical hedge sizing and sensitivity interpretation. It complements the earlier material on market risk measurement by focusing on what option Greeks imply for an actual hedge decision.

Key takeaway: First use options to address gamma, then use shares to offset the delta left by the combined exposure. Verify both signed totals, and reassess the hedge when market conditions change because an initially neutral position need not remain neutral.

Check Your Understanding

For each hypothetical problem, use the stated Greeks at the valuation point and show signed option and share positions.

An exposure has delta $-900$ shares and gamma $-15$ shares per dollar. A call has delta $+18$ and gamma $+0.5$ per contract. Find the whole-contract option position and share position that neutralize both Greeks. (Answer: long 30 calls and long 360 shares.)
An exposure has delta $+300$ shares and gamma $+8$ shares per dollar. A put has delta $-16$ and gamma $+0.4$ per contract. Find the option position that neutralizes gamma and the share position that then neutralizes delta. (Answer: short 20 puts and short 620 shares.)
An exposure has delta $+1{,}100$ and gamma $-21$. A call has delta $+25$ and gamma $+0.6$ per contract. Use whole contracts to neutralize gamma, then neutralize delta with shares. Verify both totals. (Answer: long 35 calls and short 1,975 shares; combined delta and gamma are both 0.)
An exposure has delta $-400$ and gamma $+10$. A put has delta $-12$ and gamma $+0.5$ per contract. Find the option position and exact share hedge. If the share price rises by $2 and the combined gamma at the initial valuation point is zero, what does the second-order approximation predict for the combined value change from delta and gamma? (Answer: short 20 puts and long 160 shares; the approximate combined change is $0.)
At the initial valuation point, an exposure has delta $+600$ and gamma $-12$; a call has delta $+15$ and gamma $+0.3$ per contract. Later, refreshed values are exposure delta $+560$, exposure gamma $-10$, call delta $+16$, and call gamma $+0.25$. What were the initial hedge positions, what are the old positions’ refreshed delta and gamma, and what is the new hedge? (Answer: initially long 40 calls and short 1,200 shares; old-position refreshed delta is $0$ and gamma is $0$; the new hedge is long 40 calls and short 1,200 shares.)

What You'll Learn