A P&L at expiration is one line on a chart. Real positions are marked every day, and before expiration the P&L depends on three things at once: where the stock is, where implied volatility is, and how much time has passed. Those three move together, and usually in the direction that hurts. A stock that falls 5% in a day rarely does it with implied volatility unchanged. So the question is never "what is my P&L if the stock drops 5%". It is "what is my P&L if the stock drops 5%, vol rises 5 points, and a day passes". That is a scenario, and a grid of scenarios is how a desk looks at risk.
A scenario grid you can drag
The chart below is a short straddle on a $100 stock, sold with 30 days left at 30% implied volatility. The dark line is the P&L at expiry. The green line is the P&L at whatever date and volatility you set with the sliders, across every stock price. Each position of the two sliders is one column of a scenario grid; the stock price along the bottom is the rows.
Sold with 30 days left at 30% volatility. Drag volatility up with days held at 30 and the curve drops everywhere; drag days down and theta lifts the middle. The worst cells are a big move plus a vol rise.
Try three things. Leave days at 30 and push volatility from 30 to 40: the whole curve drops, even at $100, because the seller is short vega. Put volatility back and drag days toward zero: the curve rises in the middle as theta is earned, and bends toward the hard corners of the expiry line. Then combine a vol rise with a stock $8 lower. That corner is the worst place on the grid, and it is where markets tend to go.
Back of the envelope
You can estimate a scenario from the net Greeks without a model. Two pieces do most of the work:
- Vega times the vol change. A position with a vega of 0.46 per share gains 0.46 times 5, or $2.30 a share, if implied volatility rises 5 points. Short the position, that is a $2.30 loss.
- Half of gamma times the move squared. Gamma tells you how the delta changes as the stock moves, and the P&L from that curvature is about one half of gamma times the move squared. Gamma of 0.040 and a $10 move give 0.5 times 0.040 times 100, or $2.00 a share. Long gamma earns it; short gamma pays it.
Add the delta term (delta times the move) and a day of theta and you have a usable estimate of the whole scenario.
Hurt twice
Take the short straddle from the last lesson: stock at $200, one contract sold for about $1,600, net gamma minus 0.040, theta plus 0.26, vega minus 0.46, delta minus 0.06. Now run the scenario: the stock falls 5% to $190 and implied volatility rises 5 points, over one day.
| Piece | Working, per share | Per contract |
|---|---|---|
| Gamma | 0.5 x 0.040 x 10 x 10 = 2.00, paid | -$200 |
| Vega | 0.46 x 5 = 2.30, paid | -$230 |
| Delta | -0.06 x -10 = +0.60 | +$60 |
| Theta | one day at 0.26 | +$26 |
| Total | about -$344 |
The position gave back about a fifth of its premium in one day. Notice that neither hit alone was the story. The move cost $200 and the vol rise cost $230; the same position gets hurt twice, because a falling stock and rising implied volatility travel together. A grid makes this visible. Put price changes down the side, vol changes across the top, and the worst cell is almost always in the corner where the move and the vol change reinforce each other.
The same grid, fenced in
An iron condor carries the same short gamma and short vega, but the bought wings cap the loss. Drag the same sliders on the condor and the damage from a vol rise is smaller, and past the long strikes the curve flattens instead of falling away. The price of that protection is a smaller credit and less theta.
Short the 95 put and 105 call, long the 90 put and 110 call, sold with 45 days left. Push volatility up and compare the damage with the straddle; past $90 and $110 the loss stops growing.
Why the estimate drifts
The Greeks are measured at today's price, and they do not stay put. Move a short straddle's stock away from the strike and its vega and gamma both shrink. The envelope uses the starting Greeks for the whole trip, so on big moves it misses. Vega is a hump centered on the strike:
Net vega per share with 30 and 10 days left. It is most negative at the strike and fades on both sides, so after a big move a vol rise costs less than the starting vega suggests.
The maths: the Taylor expansion of P&L
For small changes, a position's P&L is approximately dV ≈ Δ·dS + ½·Γ·dS² + ν·dσ + θ·dt. dS is the change in the stock in dollars, dσ the change in implied volatility in vol points, dt the days passed, and Δ, Γ, ν (vega) and θ are the position's net Greeks per share. Each term is one line of the table above. The expansion also has cross terms, such as how delta changes with volatility and how vega changes with the stock, and higher powers of dS; dropping them is what makes it an estimate.
Check it against a full reprice. The $200 straddle, sold at 35% implied volatility with 30 days left, is worth $15.99 a share. Reprice both options at $190, 40% volatility and 29 days and they are worth $19.07. The short position loses $3.08 a share, about $308 a contract, against the estimate of $344. The gap is the Greeks moving along the way: at $190 the straddle's vega is 0.39, not 0.46, so the vol rise costs less than the starting vega said. For a quick check the envelope is fine; for a real risk number, reprice.
Reading a grid
A few habits. Look at the corners first, not the center. Check the row where the stock does not move but vol jumps, because that is the scenario people forget. And read the units: a vol shock can be quoted in points (40 becomes 45) or as a relative change (40 becomes 50 for plus 25%). The Terminal's Risk Manager uses the relative form, and it says so under the table.
Your turn
Start on the TSLA chain and get the two inputs for the estimate: the vega of the at-the-money call and the size of a one-standard-deviation move. Then take a short straddle on AMZN into the Risk Manager and read its net vega in dollars.