Calculus on the Desk: Taylor Expansions and Convexity

You will almost never integrate anything in a trading interview. What you will do constantly is expand to second order and reason about curvature. Those two moves — Taylor and Jensen — are this lesson.

The second-order Taylor expansion

For a smooth function around a point:

$$f(x + \Delta) \approx f(x) + f'(x)\,\Delta + \tfrac{1}{2} f''(x)\,\Delta^2$$

Finance keeps renaming this formula:

• Options: price change $\approx \Delta \cdot dS + \tfrac{1}{2}\Gamma \cdot dS^2$ (+ time and vol terms). Delta is the first derivative, gamma the second. The P&L of a delta-hedged position is the pure second-order term $\tfrac12 \Gamma\, dS^2$ — always positive when long gamma, since $dS^2 \ge 0$ regardless of direction. That is what "long gamma profits from movement" means, mathematically.
• Bonds: price change $\approx -D \cdot \Delta y + \tfrac{1}{2} C \cdot \Delta y^2$ — duration and convexity, the same expansion in yield space.
• Mental math: $\sqrt{102} = 10\sqrt{1.02} \approx 10(1 + 0.01) = 10.1$; $(1.03)^{10} \approx 1 + 0.3 + \binom{10}{2}(0.03)^2 \approx 1.344$. First-order expansions power half the fast-arithmetic tricks in the trainer, and $\ln(1+x) \approx x - x^2/2$ underlies every "small return" approximation.

Jensen's inequality: convexity has a price

For a convex function, $E[f(X)] \ge f(E[X])$ — the average of the function exceeds the function of the average. Concave flips the sign. This one inequality generates a startling share of interview content:

• Why options have time value. An option payoff $\max(S-K, 0)$ is convex in $S$, so $E[\max(S_T - K, 0)] \ge \max(E[S_T] - K, 0)$: uncertainty adds value above intrinsic. More volatility, more value — vega is Jensen's inequality with a Greek letter.
• Volatility drag. Log wealth is concave, so a strategy that gains or loses 10% with equal chance loses in compound terms: $\sqrt{1.1 \times 0.9} \approx 0.995$ per period. In general, geometric growth $\approx \mu - \sigma^2/2$ — the arithmetic mean overstates what you compound at, by exactly half the variance. This term is why the Kelly lesson maximizes log wealth, and why levered ETFs decay sideways.
• $E[1/X] \ne 1/E[X]$ (and $E[X^2] \ge E[X]^2$ — that gap is variance). Any interview question that swaps an expectation through a nonlinear function is testing whether you flinch. You should flinch.

A worked classic

"A stock is 100 today. Tomorrow it's 110 or 90 with equal probability, then flat forever. A contract pays $10{,}000/S_T$. Price it."

The trap: $10{,}000/E[S_T] = 100$. Jensen says no — $1/S$ is convex:

$$E[10{,}000/S_T] = \tfrac{1}{2}\left(\frac{10000}{110}\right) + \tfrac{1}{2}\left(\frac{10000}{90}\right) = \tfrac{1}{2}(90.91 + 111.11) \approx 101.01$$

The convexity is worth about 1%. Structurally identical questions get asked with FX ("price a contract paying 1/exchange-rate" — the Siegel paradox) and with harmonic averages of prices. The reflex: nonlinear payoff + uncertainty ⇒ expand or enumerate, never pass the expectation through.

The interview versions

• "Delta-hedged long option, market moves ±2% either way — your P&L?" — Positive both ways: $\tfrac12\Gamma\,dS^2$; you paid theta for it.
• "A strategy makes +50%/−40% each year with equal odds. Long-run outcome?" — Compound factor $\sqrt{1.5 \times 0.6} = \sqrt{0.9} < 1$: it grinds to zero despite a +5% arithmetic mean. Volatility drag in one line.
• "Estimate $1.02^{30}$." — $e^{30 \times 0.0198} \approx e^{0.594} \approx 1.81$. Taylor on the way in ($\ln 1.02 \approx 0.0198$), and knowing $e^{0.6} \approx 1.82$ on the way out.