Stress Testing Options and Other Nonlinear Positions

Why Nonlinear Positions Need Stress Repricing

A sensitivity such as delta describes how a position responds to a small change in a market factor near the current market state. A stress scenario, by contrast, may involve a large discrete move that carries an option across a strike or changes more than one pricing input. In those circumstances, a local estimate can be materially different from the value obtained by repricing the position in the stressed market.

The central discipline is to apply the same scenario to the relevant market inputs and then value the position under the stressed assumptions. Sensitivities remain useful: they can explain the direction and approximate size of a result, and help identify which terms drive a difference. They are not a substitute for full repricing when the position’s value is materially nonlinear or the shock is large.

Definition: Let $V_0$ be a position’s current value and $V_s$ its value after applying a specified stress scenario and repricing under the stated valuation model. The stressed profit and loss is $\Delta V=V_s-V_0$. A positive $\Delta V$ is a gain to the position; the corresponding loss is $L=-\Delta V$. State the sign convention and valuation assumptions when reporting a result.

A stress is a specified change in one or more market inputs, such as an underlying price, an interest rate, or implied volatility. Full repricing means evaluating the position’s valuation function at the stressed inputs, rather than extrapolating only from its current value and local sensitivities. It does not remove model risk: the answer still depends on the valuation model, the stress definition, and which inputs are held fixed.

Full Repricing and Local Approximations

For a position whose value is a differentiable function $V(S)$ of an underlying price $S$, delta is $\Delta=\partial V/\partial S$. Gamma is $\Gamma=\partial^2 V/\partial S^2$. For a sufficiently small change $\Delta S$, a second-order approximation to the change in value is

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

Delta captures the local slope; gamma captures local curvature. If value also depends on other factors, such as volatility $\sigma$, a first-order approximation can include terms such as $\text{Vega}\,\Delta\sigma$, where $\text{Vega}=\partial V/\partial\sigma$. Such an approximation still omits higher-order terms and interactions unless they are explicitly included.

The justification is a Taylor expansion of the valuation function around the current state. For a twice differentiable one-factor function, the change can be written as

V(S+\Delta S)-V(S) = V'(S)\Delta S+\frac{1}{2}V''(S)(\Delta S)^2+R_3,

where $R_3$ is the remainder containing third- and higher-order effects. Identifying $V'(S)=\Delta$ and $V''(S)=\Gamma$ gives the delta-gamma approximation when $R_3$ is small enough to neglect. A large shock can make that remainder important. At a kink, such as an option payoff at its strike, the second derivative may not exist at the kink, so a Taylor approximation there is not justified.

The following examples use hypothetical positions and market inputs. Each states a valuation model and its assumptions; the numerical result is conditional on those assumptions, not a forecast of actual market behavior.

Worked Example: Repricing a Call After a Large Spot Shock

A hypothetical European call has strike $K=\$100$ and one period remaining. Value it with a one-period binomial model: the underlying is currently $S_0=\$100$; over the period it moves by a factor $u=1.2$ or $d=0.8$; and the risk-free gross return is $R=1.05$. Assume no dividends, frictionless trading, and a risk-neutral probability $q=(R-d)/(u-d)$. A stress raises the current underlying price immediately to $\$130$. For the stressed valuation, keep $u$, $d$, $R$, and the remaining period unchanged. Compare full repricing with a delta-only estimate.

Method. First calculate the model’s risk-neutral probability. Then value the call at the initial and stressed spots using the discounted risk-neutral expected payoff. For the local estimate, calculate the initial delta while the down-state payoff remains zero and the up-state payoff remains positive.

The probability is

q=\frac{1.05-0.8}{1.2-0.8}=\frac{0.25}{0.4}=0.625.

The model is internally admissible because $d<R<u$, or $0.8<1.05<1.2$. At $S_0=\$100$, the up-state underlying is $\$120$, giving a call payoff of $\$20$; the down-state underlying is $\$80$, giving a payoff of zero. Thus

V_0=\frac{0.625(20)+0.375(0)}{1.05} =\frac{12.50}{1.05} =\$11.9048.

At the stressed spot of $\$130$, the up-state underlying is $\$156$, so its payoff is $\$56$. The down-state underlying is $\$104$, so its payoff is $\$4$. Full repricing gives

V_s=\frac{0.625(56)+0.375(4)}{1.05} =\frac{35.00+1.50}{1.05} =\$34.7619.

Therefore, $\Delta V=\$34.7619-\$11.9048=\$22.8571$. The initial delta in this region is $qu/R$: the up-state payoff changes by $u$ dollars per dollar change in spot, while the down-state payoff remains zero. Hence

\Delta=\frac{0.625(1.2)}{1.05}=0.7142857.

The spot shock is $\Delta S=\$30$, so the delta-only estimate is $0.7142857(30)=\$21.4286$. It understates the full-repricing gain by $\$22.8571-\$21.4286=\$1.4285$, using displayed rounded values. The result checks: both stressed binomial outcomes are above the strike, whereas the initial down outcome was below it. The position’s response therefore changes over the shock. The common mistake is to treat the initial delta as the exact change per dollar throughout a large move.

When a Shock Crosses a Strike

A piecewise payoff makes the limits of local measures particularly visible. Consider a hypothetical call spread held to expiration, with payoff $V(S)=\max(S-95,0)-0.5\max(S-110,0)$. Here $S$ is the underlying price at settlement, the position is long one call with strike $\$95$ and short one-half of a call with strike $\$110$, and there are no other cash flows. This is a settlement-payoff calculation, not an estimate of the option’s value before expiration.

At $S=\$100$, the payoff is $\$5-0=\$5$. Between the two strikes, the payoff rises dollar for dollar with $S$, so the local slope at $\$100$ is 1 and the local curvature is zero. If the underlying settles at $\$125$, however, both calls pay: the long call pays $\$30$ and the half-sized short call costs $\$7.50$. The exact stressed payoff is $\$22.50$, an increase of $\$17.50$ from the initial payoff. A delta-only estimate based on the initial slope would give $1(\$125-\$100)=\$25$, overstating the change by $\$7.50$. The reason is that the initial slope does not account for the short-call exposure that begins above $\$110$.

Worked Example: A Call Spread Whose Stress Crosses Two Strikes

Use the hypothetical call spread just defined: long one call with strike $\$95$, short one-half of a call with strike $\$110$, and settlement payoff $V(S)=\max(S-95,0)-0.5\max(S-110,0)$. The underlying is initially $\$100$ and is stressed to $\$125$ at expiration. Calculate the initial and stressed payoffs, the exact change, and the estimate from the initial local slope.

Method. Evaluate each option payoff separately at both underlying prices, apply the half-unit short position, and compare the exact change with the initial slope estimate. This avoids applying a single local sensitivity beyond the strike where the position’s slope changes.

At the initial price of $\$100$, the long call pays $\max(100-95,0)=\$5$, while the short call’s underlying option pays $\max(100-110,0)=\$0$. The position payoff is $5-0.5(0)=\$5$.

At the stressed price of $\$125$, the long call pays $\max(125-95,0)=\$30$. The short call’s underlying option pays $\max(125-110,0)=\$15$, so the half-sized short position costs $0.5(15)=\$7.50$. The stressed position payoff is $30-7.50=\$22.50$, and the exact change is $\$22.50-\$5=\$17.50$.

At $S=\$100$, which lies strictly between the strikes, the local slope is 1: only the long call is in the money. The local estimate is $1(\$125-\$100)=\$25$. It exceeds the exact change by $\$25-\$17.50=\$7.50$. The result checks against the payoff formula: after $S$ passes $\$110$, the short half-call reduces the slope from 1 to 0.5. The common mistake is to retain the initial slope after the stress crosses the short call’s strike.

Shocks to More Than One Pricing Input

A market stress can change an underlying price and another input at the same time. For an option, a volatility shock can affect value even if the spot sensitivity is known accurately. A spot-only delta estimate cannot capture that effect. Full repricing should use the stressed values of all inputs specified by the scenario, while keeping other assumptions fixed only when the scenario explicitly calls for it.

The next example uses the same one-period binomial framework as the first, but changes the up and down factors to represent a different spread of possible outcomes. This is a discrete model comparison, not a continuous-volatility formula. The risk-neutral probability must be recalculated for the stressed factors; using the original probability would be inconsistent with the stated model.

Worked Example: Repricing a Call Under Spot and Volatility Shocks

A hypothetical European call has strike $\$100$ and one period remaining. Initially, $S_0=\$100$, $u=1.2$, $d=0.8$, and the risk-free gross return is $R=1.05$. Assume no dividends and use the one-period risk-neutral binomial model. The stress changes the spot to $\$120$, the up factor to $1.3$, and the down factor to $0.7$; $R$, strike, and remaining time are unchanged. Compare full repricing with the initial delta-only estimate for the spot move, and explain what that estimate omits.

Method. Recalculate the risk-neutral probability from the stressed factors, then calculate the stressed discounted expected payoff. Separately calculate the initial spot delta and apply it only to the spot change. The two calculations differ in scope: the local estimate excludes the factor change representing the volatility stress.

Initially, $q=(1.05-0.8)/(1.2-0.8)=0.625$, as in the first example. The initial call value is $\$11.9048$. Under stress, the risk-neutral probability is

q_s=\frac{1.05-0.7}{1.3-0.7} =\frac{0.35}{0.6} =\frac{7}{12} \approx 0.5833.

The stressed model is admissible because $0.7<1.05<1.3$. At $S=\$120$, the up-state underlying is $1.3(120)=\$156$, giving a payoff of $\$56$; the down-state underlying is $0.7(120)=\$84$, giving zero. Thus the full-repriced value is

V_s=\frac{(7/12)(56)+(5/12)(0)}{1.05} =\frac{32.6667}{1.05} =\$31.1111.

The full-repricing change is $\$31.1111-\$11.9048=\$19.2063$. The initial delta is $0.7142857$, and the spot change alone is $\$20$, so the spot-only delta estimate is $0.7142857(20)=\$14.2857$. It is $\$4.9206$ below the full-repricing change, using displayed rounded values. This difference reflects both nonlinear response to the spot move and the change in the model’s up and down factors. It would be incorrect to attribute the entire difference to volatility alone: the spot move is also large, and a local delta is only a first-order estimate. The common mistake is to apply the old risk-neutral probability after changing the binomial factors.

A Practical Stress-Test Sequence

A disciplined calculation makes the scenario, valuation method, and comparison explicit. The sequence below is useful for options and other positions whose value is nonlinear in market factors.

Specify the position and starting value.
Identify units, contract terms, valuation date, and the current value $V_0$. Keep the units consistent when scaling a per-contract result to a portfolio.

Define the complete scenario.
List the market inputs that change and those held fixed. A spot-only shock and a joint spot-and-volatility shock are different scenarios.

Revalue under stressed inputs.
Apply the same valuation framework to the stressed state to obtain $V_s$. Check that model assumptions remain valid and that every changed input is reflected consistently.

Calculate and interpret the change.
Report $\Delta V=V_s-V_0$, or the loss $L=-\Delta V$, with the sign convention stated. Compare with local estimates to understand, not erase, the effect of nonlinearity.

Common Mistakes and Interpretation

Extrapolating delta as though it were constant. Delta is local. A large move can change the slope, particularly when an underlying crosses one or more strikes.
Treating delta-gamma as exact. It is a second-order approximation, not a full repricing. Its accuracy depends on the shock size, smoothness of the valuation function, and omitted higher-order terms.
Leaving a scenario input out of the valuation. If volatility is stressed, a spot-only calculation does not capture the volatility effect. Clearly distinguish a partial sensitivity estimate from the scenario’s full result.
Changing model inputs inconsistently. For example, changing binomial up and down factors while retaining a risk-neutral probability calculated from the old factors violates the stated pricing setup.
Confusing position value with loss. A positive $\Delta V$ is a gain under the convention here; a loss is $-\Delta V$. State whether a reported figure is a value change or a loss.
Ignoring position scale. A per-option result must be multiplied by the number of contracts and any contract multiplier before it represents the portfolio result.

Full repricing is particularly important when a stress crosses payoff boundaries, changes multiple pricing inputs, or is large relative to the scale over which the current sensitivities are informative. Sensitivity comparisons remain valuable diagnostics: a large gap between a full-repricing result and a local estimate is a signal to examine curvature, strike crossings, omitted factors, and model assumptions. This approach is aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus.

Key takeaway: Apply the specified stress to the position’s valuation inputs and reprice to obtain the scenario value. Use delta and gamma as local comparison tools, not as exact substitutes for stressed valuation when nonlinearity or large discrete moves matter.

Check Your Understanding

For each hypothetical problem, state the requested value change or approximation and check that the model assumptions and sign convention are applied consistently.

A one-period European call has strike $\$105$, $S_0=\$100$, $u=1.25$, $d=0.75$, and risk-free gross return $R=1.10$. Assume no dividends and use a risk-neutral binomial model. The spot is stressed to $\$120$, with all other inputs fixed. Calculate the risk-neutral probability, initial value, stressed value, and full-repricing change. (Answer: $q=0.7000$, initial value $\$12.7273$, stressed value $\$28.6364$, change $+\$15.9091$.)
At expiration, a hypothetical position is long one call with strike $\$90$ and short one-quarter of a call with strike $\$120$. The underlying moves from $\$110$ to $\$150$. Calculate both payoffs, the exact change, and the initial-slope estimate. (Answer: initial payoff $\$20$, stressed payoff $\$52.50$, exact change $+\$32.50$, initial-slope estimate $+\$40$.)
A portfolio holds 50 option contracts, each with a multiplier of 100. Per option, the initial delta is 0.60 and gamma is 0.04 per dollar squared. For a spot shock of $-\$10$, estimate the portfolio value change using delta and gamma. Assume sensitivities remain the only terms in the approximation. (Answer: per option $-\$4$; portfolio estimate $-\$20{,}000$.)
A position has spot delta 0.80 and vega of $\$2{,}500$ per unit change in volatility. The stress is a spot move of $-\$12$ and a volatility increase of 0.04. Calculate the first-order delta-only estimate and the combined delta-vega estimate, and identify the added contribution. (Answer: delta-only estimate $-\$9.60$; combined estimate $+\$90.40$; volatility contribution $+\$100$.)

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