Mental Math and Pricing Tricks

Nobody prices a trade in their head. The system prices the trade, and it does it better than you will.

So why does every front-office interview contain some version of spot is 100, vol is 20, what is the three-month straddle worth?

Two reasons, and neither of them is arithmetic.

The first is operational. You need to know, within about two seconds, whether the number your screen just produced is plausible. Most of the real damage on a desk does not come from sophisticated errors. It comes from a wrong strike, a wrong maturity, a wrong ratio, a decimal in the wrong place — and from nobody in the chain having an expectation of what the answer should have looked like. The approximation is not a substitute for the model. It is the tripwire that catches the model when someone feeds it garbage.

The second is diagnostic, which is why interviewers like these questions so much. When you can produce a rough number quickly, you are demonstrating that you carry a working model of the market in your head: that you know volatility scales with the square root of time, that gamma P&L is quadratic, that a forward is a financing cost. That is what is being tested. The number is just the evidence.

What follows is the set of approximations I actually use, why each one works, and where each one breaks. Almost all of them come down to one constant and one square root.

Part 1 — The one number to memorise: 0.4

If you learn nothing else from this article, learn this:

ATM call ≈ 0.4 · S · σ · √T

ATM straddle ≈ 0.8 · S · σ · √T

That is the whole thing. A straddle is two options, so it is twice the call.

The worked example

Spot 100, implied vol 20%, one year to expiry, no rates or dividends.

0.4 × 100 × 0.20 × 1 = 8

The at-the-money call is worth about 8. Black-Scholes says 7.97.

Now the three-month straddle on the same name:

0.8 × 100 × 0.20 × √0.25 = 0.8 × 100 × 0.20 × 0.5 = 8

Also 8.

That coincidence is the point of a classic interview question: spot 100, flat vol 20%, which is cheaper — a three-month at-the-money straddle or a one-year at-the-money call? Neither. They are the same price, and a candidate who has the 0.4 in their head answers in four seconds while everybody else reaches for a pen.

Where 0.4 comes from

For an at-the-money-forward option, d₁ = σ√T / 2, which is small for any reasonable vol and maturity. Expand the Black-Scholes formula around that and almost everything cancels, leaving:

C ≈ S · σ · √T / √(2π)

And 1/√(2π) = 0.3989. Call it 0.4.

There is no magic and no memorised table. The constant is the height of the standard normal density at zero, which is why it turns up everywhere else in this article too.

What the straddle actually is

Here is the version of this that stays with you.

For a normal distribution, the expected absolute deviation is √(2/π) ≈ 0.798 standard deviations. Call it 0.8.

So the expected absolute move of a stock over a period T is roughly 0.8 · S · σ · √T — which is exactly the price of the straddle.

The at-the-money straddle costs the average move. That is not an approximation artefact, it is what the instrument is. You are paying for the expected distance travelled, and you make money if the stock travels further than that.

Once you see it that way you stop needing the formula. Someone asks what a one-year straddle is worth on a 20-vol name at 100, and the question becomes: how far do I expect this thing to move in a year? About 16 points. So the straddle is about 16, and the call is about 8.

Where it breaks

The approximation assumes at-the-money-forward and it ignores rates, dividends and skew. It is excellent for short and medium maturities on liquid names. It degrades when:

• The option is far from the money, where the whole expansion is invalid
• Maturity is long enough that the forward drifts meaningfully away from spot, which on a high-dividend name can be a year or less
• The skew is steep, which is exactly when a single-vol approximation is least appropriate

That is fine. You are not trying to be right to the cent. You are trying to know whether the screen showing 12 for that option is sensible or is telling you that somebody fat-fingered the maturity.

Part 2 — The rule of 16

The second thing to memorise is that there are about 252 trading days in a year, and:

√252 ≈ 15.87 ≈ 16

Which gives the most useful conversion on a trading floor:

Daily move ≈ annual vol / 16

A 16% vol name moves about 1% a day. That is the anchor. Everything else scales from it:

It works in both directions, and the reverse is the more useful one in practice. A stock has been moving 2% a day for a fortnight and you want to know what that implies. Multiply by 16: realised vol is around 32%. If the options are marked at 22%, you are looking at a book that is bleeding if it is short gamma, or a genuine opportunity if you can buy it.

This single conversion is how traders talk about volatility conversationally. When someone says “that name is a 2% a day stock”, they have said 32 vol without saying it.

Why √time and not time

Because volatility scales with the square root of time, not with time.

Variance is additive across independent periods. Volatility is the square root of variance. Two independent days of variance σ² give 2σ², so the volatility over two days is σ√2, not 2σ.

This is the single most load-bearing fact in the whole article, and it is worth being able to say why rather than just applying it. It is the reason the 0.4 formula has a √T in it. It is the reason a four-year option costs twice a one-year option rather than four times. And it is the reason a two-day event does not cost twice a one-day event.

Part 3 — Time: everything scales with √T

Because option value goes as √T, you can reprice for a different maturity without touching a model.

Halve the time remaining → multiply the price by 0.71

Quarter the time remaining → halve the price

The second one is the useful one, and it has a counter-intuitive consequence worth internalising: an at-the-money option still holds half its value when only a quarter of its life remains. Time decay is not linear and it is not spread evenly. It is slow for most of the life and violent at the end.

Which gives the theta shortcut. Differentiate C ∝ √T and you get:

Daily theta ≈ price / (2 × days remaining)

A one-year at-the-money call worth 8 decays about 8 / 730 ≈ 0.011 a day. The same option with ten days left and a price of 1.3 decays 1.3 / 20 ≈ 0.065 a day — six times faster in absolute terms, on a position worth a sixth as much.

Vega follows the same square root. An at-the-money option's sensitivity to one volatility point is:

Vega per vol point ≈ 0.004 · S · √T

For spot 100 and one year that is 0.4 per point, which you can also get for free from the 0.4 rule: the option is worth 8 at 20 vol, so each vol point is worth 8/20 = 0.4. At the money, price divided by vol in points is your vega per point. That reversibility is a good check that you have the right mental model rather than three unrelated memorised numbers.

Part 4 — Greeks in cash

Nobody on a desk says “my gamma is 0.03”. The number is meaningless until it is in euros, and converting is the first thing you should do to any greek that is handed to you.

Delta

Cash delta = δ × S × contract size

This is the euro amount of underlying you have to hold. Short 10,000 calls with a delta of 0.5 on a 100 stock with contract size 100 gives you a cash delta of 0.5 × 100 × 10,000 × 100 = 50 million euros of stock to buy. That number, not the 0.5, is what determines whether you can actually execute the hedge today.

Gamma

Cash gamma is conventionally defined as the change in your cash delta for a 1% move in spot. From which:

A 1% move pays half a percent of your cash gamma.

An x% move pays x² times that.

One million of cash gamma, spot moves 1%: ½ × 1% × 1,000,000 = 5,000 euros. The sign does not depend on direction — a delta-hedged long gamma book does not care which way the market went, only how far.

Now scale it, and this is where the quadratic bites:

Which answers the path question directly. Would you rather have three separate days of 3%, or one single day of 9%?

The single day, by a factor of three. Three days of 3% give you 3 × 45,000 = 135,000. One day of 9% gives you 405,000. Same total distance travelled, three times the money.

That is why clustered volatility makes a long gamma book and destroys a short one, and it is the same arithmetic that makes a gap through a barrier so much worse than a slow drift to the same level.

Vega

The conversion that catches people out, because vega comes out of most systems in points of premium rather than in cash:

Cash vega = notional × vega / spot

A 10 million notional down-and-in put on the Eurostoxx, spot 5530, vega quoted at 25.21:

10,000,000 × 25.21 / 5530 ≈ 45,600 euros per vol point

Which is the number that matters. A two-point move in implied volatility is ninety thousand euros. Nobody makes a risk decision on the 25.21.

Theta, and the break-even move

Theta is the mirror of gamma, and you can find where they cross without a model. Set the daily gamma P&L equal to the daily theta and the answer comes back to the rule of 16:

Break-even daily move ≈ σ / 16

On a 20-vol name you need about 1.25% a day just to pay for the time you are burning. If the stock is delivering 0.6% a day, being long gamma is costing you money regardless of how well you scalp it. That is the whole implied-versus-realised trade in one line, and it is available to you in your head, on the walk back from the coffee machine.

Part 5 — Delta and the digital, without a pricer

At-the-money delta ≈ 0.5

Slightly above for a call, because of the drift in the forward, but 0.5 is the number you say.

Two extensions worth having.

Volatility pulls every delta toward 0.5. Take an in-the-money call with a delta of 0.60 and raise the vol. The delta falls: 0.55, 0.52. High volatility widens the distribution until even an in-the-money option stops looking like a stock position, and you lose directional conviction. The mirror holds out of the money: a 0.40 delta rises toward 0.5. Across the whole observable range of volatility, every delta converges on the middle.

The corollary matters for hedging: if implied vol jumps ten points on your book, your deltas have all moved toward 0.5 and your hedge is stale even though spot has not moved at all.

A digital is a probability. A three-month at-the-money digital call is worth about 0.50, because a digital pays N(d₂) — the risk-neutral probability of finishing in the money — and at the money that is a coin flip.

But knowing the theoretical price of a digital is not the same as knowing where it trades, and the difference is the interesting part. A digital has infinite gamma at the strike at expiry, so it cannot be replicated. Desks hedge it with a real, tight call spread instead — long the 99, short the 101 — which necessarily over-prices it. Digitals always trade above their theoretical value, and the over-hedge is the margin.

Digital(K) ≈ ( Call(K−ε) − Call(K+ε) ) / 2ε

The price is the hedging cost. That is the general lesson, and it is worth more than the formula. Whenever a quoted price sits stubbornly away from a theoretical one, look for the part of the payoff that cannot be replicated cleanly.

Part 6 — Sanity checks by arbitrage

The strongest mental checks are not approximations at all. They are relationships that have to hold, in every model, or somebody gets free money.

Put-call parity

C − P = S·e^(−qT) − K·e^(−rT)

Which for short maturities and small rates is close enough to C − P = S − K.

Use it as a consistency test. If you think calls on a name look expensive and puts look fair, at the same strike and maturity, you are wrong about one of them or the forward you are using is not the forward the market is using — usually because you have the dividend or the repo wrong. Parity does not leave room for two independent views.

It also gives you a free price. If you know the call and you know the forward, you know the put. There is nothing to compute.

The forward

F = S · e^(r−q)T ≈ S · (1 + (r−q)T)

Spot 100, rates 5%, no dividends, one year: the forward is about 105. If the market shows 110, the forward is rich by five points and the trade is mechanical — sell the forward, borrow 100, buy the stock, carry it, deliver into the contract. Five points, no market risk.

The mental check is that the forward is a financing cost, never a forecast. If someone tells you the forward implies the market expects the stock to rise 5%, they have misunderstood the instrument.

Bounds

A call can never be worth more than the stock. A call can never be worth less than S − PV(K). A put can never be worth more than the present value of the strike. These sound trivial, and they catch real errors — a mispriced option is very often one that has violated a bound that nobody checked.

The scenario average

For questions that resist formulas, price three scenarios and average. It works, it is fast, and it is exactly what the risk-neutral expectation is doing anyway.

Which is more expensive: one call struck 100, or two calls struck 200?

Take three ending spot levels: 100, 200, 300.

The single lower-struck call, comfortably. You have just done a three-point numerical integration in your head, and the interviewer can see your reasoning rather than a memorised answer.

Part 7 — Rates and bonds in your head

Duration

The duration of a zero-coupon bond equals its maturity.

Exactly. There is one cash flow and all the risk sits at maturity. Every coupon bond has a duration shorter than its maturity, because some of the money comes back to you earlier.

The P&L of a rate move

P&L ≈ Notional × Duration × Δy

That is the only bond formula you need for a conversation. A one-year swap on 100 million with rates moving 10 basis points:

100,000,000 × 1 × 0.0010 = 100,000 euros

The same move on a five-year swap is five times that, half a million. Get the duration right before you get excited about the number, because it is the multiplier that changes.

And say the direction first. Paying fixed is being short a fixed bond, so you gain when rates rise. A candidate who computes before knowing the sign is guessing with extra steps.

Convexity, without the formula

Convexity is the bond version of being long gamma: you earn more when you win and lose less when you lose.

The mental picture that works: draw the present value of every cash flow as a bar on a timeline. The further the mass of value sits to the right, the more convexity. Which immediately gives you the standard answers without computing anything.

A zero-coupon has more convexity than a coupon bond of the same maturity, because all of its value is at the far end. A barbell of a five-year and a fifteen-year has more convexity than a ten-year bullet of the same duration, because the cash flows are dispersed rather than concentrated, so it falls less when rates rise. And a callable bond has negative convexity at low yields, because the issuer takes the upside away exactly when it would have been worth something — which is why it is cheaper than the equivalent straight bond.

Part 8 — Structured products in your head

Every structured product decomposes into stock, bonds and options. Once you write the decomposition, the mental pricing is arithmetic you already know.

Turbos

Price ≈ S − K

Leverage = S / (S − K)

Daily strike accrual: ΔK = K × r / 365

Residual at knock-out = max(0, (S at KO − K) × ratio)

Spot 100, strike 80: the certificate is worth about 20 for exposure to 100, so five times leverage. There is no volatility in any of that, which is the trap in the question — a turbo has delta 1 and no optionality at all, despite the documentation sometimes calling it an option.

Move the barrier closer to spot and the price falls while the leverage rises. Which answers which is more expensive, leverage five or leverage ten? Leverage five: the barrier is further away, the product is safer, and the client pays for that. Cheap certificates are cheap because they are close to dying.

Yield enhancement

Discount certificate = long stock − call struck at the cap

Reverse convertible coupon ≈ put premium / forward

The discount certificate decomposition answers a question people get wrong under pressure: a more expensive call makes the certificate cheaper, because you are subtracting more. Write the decomposition, read the sign, do not reason from intuition about what feels bullish.

And the coupon formula tells you where the money comes from. A 9% coupon means the client sold you roughly 9% of forward in put premium. He is not receiving interest, he is receiving an insurance premium, and he will discover this if the barrier breaks.

Capital protection

Guaranteed note = zero-coupon bond + call

A 100 investment at rates that make a 90 zero-coupon grow back to 100 leaves 10 to buy the option. Which lets you answer participation questions instantly: if the call you need costs 20 and you only have 10, the client gets half participation. More protection means less money for the option, and there is no version where he gets both.

Part 9 — The compounding traps

Two places where mental arithmetic goes wrong for everyone, including people who should know better.

Volatility drag

A stock rises 10%, then falls 10%. Where is it?

At 99, not 100. Because (1 + x)(1 − x) = 1 − x², and x² here is 1%.

Returns compound, they do not add. The gap between the arithmetic average return and the geometric one grows with volatility, which is why:

• A choppy market can leave an index flat while every leveraged product on it bleeds
• A daily-rebalanced 3× product loses money in a sideways market even when its underlying ends unchanged
• A fund with a spectacular average annual return can have a mediocre cumulative one

The single sentence to carry: the more it moves, the more the compounding costs you, whatever direction it moves in.

Doubling time

Rule of 72: years to double ≈ 72 / rate in percent

At 6%, twelve years. At 8%, nine. Useful whenever someone quotes an annualised return or a funding rate and you want to know what it means over the life of a five-year note, without opening anything.

Part 10 — Using this in an interview

Three habits, and they matter more than the formulas.

Say the method before the number. “At the money the call is about 0.4 times spot times vol times root T, so 0.4 times 100 times 0.2 — eight.” If the arithmetic is slightly off, the interviewer has still seen exactly what you know. If you say “eight” alone and it is wrong, you have shown them nothing to salvage.

Round aggressively, and say that you are. √252 is 15.87 and you should call it 16 out loud. Rounding openly signals that you know the precision you are working at, which is a trader's instinct. Producing 7.9683 signals the opposite — someone who has not worked out which digits matter.

Get the direction right before the magnitude. Does the position make or lose? Is the greek long or short? Almost every one of these questions is really a sign question with a number attached, and a candidate who reaches for the arithmetic before establishing the sign is telling you they do not have a picture of the position.

The card

None of this makes you a pricing engine, and it is not supposed to. What it gives you is an expectation — a number you were anticipating before the screen produced one.

That expectation is what makes a trader useful. It is the thing that fires when a quote comes back at 12 instead of 8, before anybody has worked out that the maturity was entered in months rather than years. And it is also, conveniently, the thing an interviewer is looking for when they ask you what a straddle is worth.

Learn the 0.4 and the 16. Almost everything else in this article is those two facts wearing a different hat.