Options and Nonlinear Risk

Question 1

An options book is short gamma of 1,500 delta shares per dollar. Its initial +3,000 delta is fully hedged with stock. Spot rises $4. Assume constant gamma and use second-order approximation. What is the starting net delta?

Accepted Answer

Zero.

Question 2

An options book is short gamma of 1,500 delta shares per dollar. Its initial +3,000 delta is fully hedged with stock. Spot rises $4. Assume constant gamma and use second-order approximation. What change in option delta follows?

Accepted Answer

−6,000 shares.

Question 3

An options book is short gamma of 1,500 delta shares per dollar. Its initial +3,000 delta is fully hedged with stock. Spot rises $4. Assume constant gamma and use second-order approximation. What is net delta immediately after the move?

Accepted Answer

−6,000 shares.

Question 4

An options book is short gamma of 1,500 delta shares per dollar. Its initial +3,000 delta is fully hedged with stock. Spot rises $4. Assume constant gamma and use second-order approximation. What approximate gamma P&L follows?

Accepted Answer

−$12,000

Question 5

An options book is short gamma of 1,500 delta shares per dollar. Its initial +3,000 delta is fully hedged with stock. Spot rises $4. Assume constant gamma and use second-order approximation. What stock adjustment restores delta neutrality locally?

Accepted Answer

Buy 6,000 shares.

Question 6

An options book is short gamma of 1,500 delta shares per dollar. Its initial +3,000 delta is fully hedged with stock. Spot rises $4. Assume constant gamma and use second-order approximation. If spot instead fell $4, what gamma P&L would follow?

Accepted Answer

−$12,000

Question 7

An options book is short gamma of 1,500 delta shares per dollar. Its initial +3,000 delta is fully hedged with stock. Spot rises $4. Assume constant gamma and use second-order approximation. Why verify a large jump with full repricing?

Accepted Answer

Gamma can change across the move.

Question 8

An options book is short gamma of 1,500 delta shares per dollar. Its initial +3,000 delta is fully hedged with stock. Spot rises $4. Assume constant gamma and use second-order approximation. What does the stock hedge leave unaddressed?

Accepted Answer

Gamma and implied volatility risk.

Question 9

An options book is short gamma of 1,500 delta shares per dollar. Its initial +3,000 delta is fully hedged with stock. Spot rises $4. Assume constant gamma and use second-order approximation. What if stock liquidity disappears after the jump?

Accepted Answer

Stress delayed and costly rehedging.

Question 10

An options book is short gamma of 1,500 delta shares per dollar. Its initial +3,000 delta is fully hedged with stock. Spot rises $4. Assume constant gamma and use second-order approximation. What measure makes the breach report actionable?

Accepted Answer

Stress loss and required delta trade at stressed depth.

Question 11

The near-term sleeve has vega of −$9,000 per volatility point. The long-term sleeve has +$6,000 per point. An event raises near volatility 5 points and long volatility 1 point. Ignore other terms. What is net vega to a parallel one-point move?

Accepted Answer

−$3,000

Question 12

The near-term sleeve has vega of −$9,000 per volatility point. The long-term sleeve has +$6,000 per point. An event raises near volatility 5 points and long volatility 1 point. Ignore other terms. What is near sleeve event P&L?

Accepted Answer

−$45,000

Question 13

The near-term sleeve has vega of −$9,000 per volatility point. The long-term sleeve has +$6,000 per point. An event raises near volatility 5 points and long volatility 1 point. Ignore other terms. What is long sleeve P&L?

Accepted Answer

+$6,000

Question 14

The near-term sleeve has vega of −$9,000 per volatility point. The long-term sleeve has +$6,000 per point. An event raises near volatility 5 points and long volatility 1 point. Ignore other terms. What is combined event P&L?

Accepted Answer

−$39,000

Question 15

The near-term sleeve has vega of −$9,000 per volatility point. The long-term sleeve has +$6,000 per point. An event raises near volatility 5 points and long volatility 1 point. Ignore other terms. What if both rose one point?

Accepted Answer

−$3,000

Question 16

The near-term sleeve has vega of −$9,000 per volatility point. The long-term sleeve has +$6,000 per point. An event raises near volatility 5 points and long volatility 1 point. Ignore other terms. What does the parallel measure conceal?

Accepted Answer

The large short exposure at the near tenor.

Question 17

The near-term sleeve has vega of −$9,000 per volatility point. The long-term sleeve has +$6,000 per point. An event raises near volatility 5 points and long volatility 1 point. Ignore other terms. Which hedge directly addresses the near risk?

Accepted Answer

Buy near-tenor vega within liquidity limits.

Question 18

The near-term sleeve has vega of −$9,000 per volatility point. The long-term sleeve has +$6,000 per point. An event raises near volatility 5 points and long volatility 1 point. Ignore other terms. If near vega is cut to −$5,000, what event P&L remains?

Accepted Answer

−$19,000

Question 19

The near-term sleeve has vega of −$9,000 per volatility point. The long-term sleeve has +$6,000 per point. An event raises near volatility 5 points and long volatility 1 point. Ignore other terms. What should the analyst monitor after the event passes?

Accepted Answer

Whether the tenor twist and hedge need normalize.

Question 20

The near-term sleeve has vega of −$9,000 per volatility point. The long-term sleeve has +$6,000 per point. An event raises near volatility 5 points and long volatility 1 point. Ignore other terms. What scenario complements a near vol spike?

Accepted Answer

Long vol rises while near vol is flat or falls.

Question 21

A book has vega of −$7k per point in downside puts, +$5k at the money, and +$2k in upside calls. The adverse scenario raises put vol 6 points, ATM vol 3, and call vol 1. What is aggregate parallel vega?

Accepted Answer

Zero.

Question 22

A book has vega of −$7k per point in downside puts, +$5k at the money, and +$2k in upside calls. The adverse scenario raises put vol 6 points, ATM vol 3, and call vol 1. What is put P&L in the skew scenario?

Accepted Answer

−$42k

Question 23

A book has vega of −$7k per point in downside puts, +$5k at the money, and +$2k in upside calls. The adverse scenario raises put vol 6 points, ATM vol 3, and call vol 1. What is ATM P&L?

Accepted Answer

+$15k

Question 24

A book has vega of −$7k per point in downside puts, +$5k at the money, and +$2k in upside calls. The adverse scenario raises put vol 6 points, ATM vol 3, and call vol 1. What is upside call P&L?

Accepted Answer

+$2k

Question 25

A book has vega of −$7k per point in downside puts, +$5k at the money, and +$2k in upside calls. The adverse scenario raises put vol 6 points, ATM vol 3, and call vol 1. What is combined skew P&L?

Accepted Answer

−$25k

Question 26

A book has vega of −$7k per point in downside puts, +$5k at the money, and +$2k in upside calls. The adverse scenario raises put vol 6 points, ATM vol 3, and call vol 1. Which dashboard breakdown reveals this risk?

Accepted Answer

Vega by strike and maturity.

Question 27

A book has vega of −$7k per point in downside puts, +$5k at the money, and +$2k in upside calls. The adverse scenario raises put vol 6 points, ATM vol 3, and call vol 1. Why stress skew alongside spot?

Accepted Answer

A selloff can disproportionately lift downside vol.

Question 28

A book has vega of −$7k per point in downside puts, +$5k at the money, and +$2k in upside calls. The adverse scenario raises put vol 6 points, ATM vol 3, and call vol 1. What if downside puts are hard to trade?

Accepted Answer

Evaluate feasible sizing or alternate protection.

Question 29

A book has vega of −$7k per point in downside puts, +$5k at the money, and +$2k in upside calls. The adverse scenario raises put vol 6 points, ATM vol 3, and call vol 1. Which limit addresses the hidden exposure directly?

Accepted Answer

Loss under a defined skew shock.

Question 30

A book has vega of −$7k per point in downside puts, +$5k at the money, and +$2k in upside calls. The adverse scenario raises put vol 6 points, ATM vol 3, and call vol 1. What should the report avoid?

Accepted Answer

Calling zero parallel vega zero volatility risk.

Question 31

A book has zero net delta and zero net vega at the starting point. Use an illustrative cross term equal to $2,000 × spot move in dollars × vol move in points. Ignore other terms for these calculations. What cross P&L follows spot −$3 and vol +4?

Accepted Answer

−$24,000

Question 32

A book has zero net delta and zero net vega at the starting point. Use an illustrative cross term equal to $2,000 × spot move in dollars × vol move in points. Ignore other terms for these calculations. What cross P&L follows spot +$3 and vol +4?

Accepted Answer

+$24,000

Question 33

A book has zero net delta and zero net vega at the starting point. Use an illustrative cross term equal to $2,000 × spot move in dollars × vol move in points. Ignore other terms for these calculations. What if spot falls $3 but volatility is unchanged?

Accepted Answer

Zero cross P&L.

Question 34

A book has zero net delta and zero net vega at the starting point. Use an illustrative cross term equal to $2,000 × spot move in dollars × vol move in points. Ignore other terms for these calculations. What if vol rises four points but spot stays flat?

Accepted Answer

Zero from the cross term.

Question 35

A book has zero net delta and zero net vega at the starting point. Use an illustrative cross term equal to $2,000 × spot move in dollars × vol move in points. Ignore other terms for these calculations. Why might two independent stress tests miss this exposure?

Accepted Answer

Neither test activates both inputs.

Question 36

A book has zero net delta and zero net vega at the starting point. Use an illustrative cross term equal to $2,000 × spot move in dollars × vol move in points. Ignore other terms for these calculations. What should verify the approximation for a very large shock?

Accepted Answer

Full repricing on the joint surface.

Question 37

A book has zero net delta and zero net vega at the starting point. Use an illustrative cross term equal to $2,000 × spot move in dollars × vol move in points. Ignore other terms for these calculations. How should shocked vol levels be specified?

Accepted Answer

By tenor and moneyness with a consistent surface convention.

Question 38

A book has zero net delta and zero net vega at the starting point. Use an illustrative cross term equal to $2,000 × spot move in dollars × vol move in points. Ignore other terms for these calculations. What if the adjustment reduces vanna but increases negative gamma?

Accepted Answer

Compare joint and isolated downside before approving.

Question 39

A book has zero net delta and zero net vega at the starting point. Use an illustrative cross term equal to $2,000 × spot move in dollars × vol move in points. Ignore other terms for these calculations. Which reported figure helps explain model disagreement?

Accepted Answer

The gap between approximation and repriced loss.

Question 40

A book has zero net delta and zero net vega at the starting point. Use an illustrative cross term equal to $2,000 × spot move in dollars × vol move in points. Ignore other terms for these calculations. What belongs in the next review trigger?

Accepted Answer

A material change in cross exposure or the spot-vol relationship.

Question 41

A delta-hedged book has vega +$4k per point and volga −$1.2k per point squared. Use P&L = vega × move + 0.5 × volga × move². Spot is unchanged. What P&L follows +2 points?

Accepted Answer

+$5.6k

Question 42

A delta-hedged book has vega +$4k per point and volga −$1.2k per point squared. Use P&L = vega × move + 0.5 × volga × move². Spot is unchanged. What P&L follows +8 points?

Accepted Answer

−$6.4k

Question 43

A delta-hedged book has vega +$4k per point and volga −$1.2k per point squared. Use P&L = vega × move + 0.5 × volga × move². Spot is unchanged. What P&L follows −2 points?

Accepted Answer

−$10.4k

Question 44

A delta-hedged book has vega +$4k per point and volga −$1.2k per point squared. Use P&L = vega × move + 0.5 × volga × move². Spot is unchanged. At what positive move other than zero does the approximation break even?

Accepted Answer

About 6.67 points.

Question 45

A delta-hedged book has vega +$4k per point and volga −$1.2k per point squared. Use P&L = vega × move + 0.5 × volga × move². Spot is unchanged. What does positive starting vega fail to guarantee?

Accepted Answer

Profit from a large vol increase.

Question 46

A delta-hedged book has vega +$4k per point and volga −$1.2k per point squared. Use P&L = vega × move + 0.5 × volga × move². Spot is unchanged. What happens to local vega after a four-point increase in this approximation?

Accepted Answer

It becomes −$0.8k per point.

Question 47

A delta-hedged book has vega +$4k per point and volga −$1.2k per point squared. Use P&L = vega × move + 0.5 × volga × move². Spot is unchanged. What analysis should precede a hedge based on that local vega?

Accepted Answer

Reprice at the stressed surface and check executable depth.

Question 48

A delta-hedged book has vega +$4k per point and volga −$1.2k per point squared. Use P&L = vega × move + 0.5 × volga × move². Spot is unchanged. Why test both rises and falls in volatility?

Accepted Answer

Linear and quadratic terms have different symmetry.

Question 49

A delta-hedged book has vega +$4k per point and volga −$1.2k per point squared. Use P&L = vega × move + 0.5 × volga × move². Spot is unchanged. What if a modeled volatility shock would make a bucket negative?

Accepted Answer

Review scenario feasibility and underlying model conventions.

Question 50

A delta-hedged book has vega +$4k per point and volga −$1.2k per point squared. Use P&L = vega × move + 0.5 × volga × move². Spot is unchanged. What should risk communicate if the approximation gives unreliable large-shock results?

Accepted Answer

Use repriced results and explain the approximation error.

Question 51

A desk's hedge plan assumes it can trade immediately after a $5 gap. Delta is −8,000 shares after the gap. Actual depth allows only 3,000 shares before spot rises another $2. Use linear residual P&L over that second move. What stock side offsets the negative delta?

Accepted Answer

Buy.

Question 52

A desk's hedge plan assumes it can trade immediately after a $5 gap. Delta is −8,000 shares after the gap. Actual depth allows only 3,000 shares before spot rises another $2. Use linear residual P&L over that second move. What residual remains after the first 3,000-share hedge?

Accepted Answer

−5,000 shares.

Question 53

A desk's hedge plan assumes it can trade immediately after a $5 gap. Delta is −8,000 shares after the gap. Actual depth allows only 3,000 shares before spot rises another $2. Use linear residual P&L over that second move. What residual P&L follows the next $2 rise?

Accepted Answer

−$10,000

Question 54

A desk's hedge plan assumes it can trade immediately after a $5 gap. Delta is −8,000 shares after the gap. Actual depth allows only 3,000 shares before spot rises another $2. Use linear residual P&L over that second move. What would full immediate execution have left from this delta term?

Accepted Answer

Zero.

Question 55

A desk's hedge plan assumes it can trade immediately after a $5 gap. Delta is −8,000 shares after the gap. Actual depth allows only 3,000 shares before spot rises another $2. Use linear residual P&L over that second move. What hidden assumption does the original plan contain?

Accepted Answer

Enough immediate executable depth.

Question 56

A desk's hedge plan assumes it can trade immediately after a $5 gap. Delta is −8,000 shares after the gap. Actual depth allows only 3,000 shares before spot rises another $2. Use linear residual P&L over that second move. Which stress improves it?

Accepted Answer

Combine gaps with thinner depth and waiting time.

Question 57

A desk's hedge plan assumes it can trade immediately after a $5 gap. Delta is −8,000 shares after the gap. Actual depth allows only 3,000 shares before spot rises another $2. Use linear residual P&L over that second move. What if gamma changes during the second move?

Accepted Answer

Recompute delta dynamically.

Question 58

A desk's hedge plan assumes it can trade immediately after a $5 gap. Delta is −8,000 shares after the gap. Actual depth allows only 3,000 shares before spot rises another $2. Use linear residual P&L over that second move. What cost belongs alongside residual P&L?

Accepted Answer

Spread and impact of the hedge executions.

Question 59

A desk's hedge plan assumes it can trade immediately after a $5 gap. Delta is −8,000 shares after the gap. Actual depth allows only 3,000 shares before spot rises another $2. Use linear residual P&L over that second move. What does a result based only on a final mark omit?

Accepted Answer

The path and cash needed to reach it.

Question 60

A desk's hedge plan assumes it can trade immediately after a $5 gap. Delta is −8,000 shares after the gap. Actual depth allows only 3,000 shares before spot rises another $2. Use linear residual P&L over that second move. What action might reduce vulnerability before an event?

Accepted Answer

Reduce short convexity or reserve executable hedge capacity.

Question 61

Stress creates $3m of cash variation margin from an overlay and a $1m initial margin increase by noon. Cash securities gain $5m on paper. Free cash is $2.5m and sale proceeds of $2m arrive after noon. What cash is due by noon?

Accepted Answer

$4m

Question 62

Stress creates $3m of cash variation margin from an overlay and a $1m initial margin increase by noon. Cash securities gain $5m on paper. Free cash is $2.5m and sale proceeds of $2m arrive after noon. What is the noon funding gap?

Accepted Answer

$1.5m

Question 63

Stress creates $3m of cash variation margin from an overlay and a $1m initial margin increase by noon. Cash securities gain $5m on paper. Free cash is $2.5m and sale proceeds of $2m arrive after noon. Do the unrealized cash security gains meet the call directly?

Accepted Answer

No.

Question 64

Stress creates $3m of cash variation margin from an overlay and a $1m initial margin increase by noon. Cash securities gain $5m on paper. Free cash is $2.5m and sale proceeds of $2m arrive after noon. What after-noon balance would follow if timely bridge funding of $1.5m is still outstanding?

Accepted Answer

$2m free cash before bridge repayment.

Options and Nonlinear Risk

A delta-hedged short-gamma book

Vega by maturity

Skew exposure

Vanna under joint moves

Volga and large shocks

Gap and hedge delay

Margin from an options hedge

Barrier and discontinuity

Aggregating Greeks

Committee response