The Maths You Need for Derivative Pricing

Why It Matters—and Where to Learn It

If you’re targeting a Sales & Trading internship at an investment bank, you might assume that you’ll spend most of your time schmoozing clients and executing trades. While relationship-building remains crucial, the landscape is shifting: across all industries, even “sales” roles are becoming more technical, as routine tasks are automated by AI agents. To stay ahead, you’ll need not just a grasp of markets, but fluency in the mathematics that underpins how modern banks price—and hedge—derivatives.

Core Mathematical Foundations

Disclaimer: These tools live in a deterministic world—where outcomes follow exact rules—unlike the stochastic realm of random-market moves. You must master these deterministic methods first to appreciate how randomness alters every step when you move on to stochastic calculus and derivative pricing.

• Differential & Integral Calculus
  – Why:
  • Sensitivity analysis: Δ (Delta) = ∂Price/∂Underlying tells you how much an option’s value shifts for a tiny move in the underlying; Γ (Gamma) = ∂²Price/∂Underlying² measures how that sensitivity itself changes.
  • P&L aggregation: Integrals let you sum up instantaneous P&L contributions (like Δ·dS) over a period to get total gains or losses.
  – Learn: Khan Academy Differential Calculus & Integral Calculus

• Probability & Statistics
  – Why:
  • Expected payoff: Under risk-neutral pricing, the fair price = discounted expected payoff. You need expectation and variance to compute that, and distributions (normal/log-normal) to model returns.
  • Tail & co-movement: Knowing the shape of distributions reveals extreme-event risks; correlation lets you hedge across multiple assets or manage portfolio VaR.
  – Learn: Khan Academy Statistics & Probability

• Ordinary Differential Equations (ODEs)
  – Why:
  • Interest accrual & discounting: €1 grows at rate r by an amount r·dt over dt, yielding €1e^{rT} in T years; discounting uses e^{−rT} to bring future payoffs back to today.
  • Backward induction: Freezing the underlying price turns the Black–Scholes PDE into an ODE in time—solve from known terminal payoff (e.g. max(S–K,0)) backward to today’s price.
  – Learn: Khan Academy Differential Equations

From Foundations to Finance

Building on deterministic tools, we now introduce randomness—the key to real‐world derivative pricing.

• Stochastic Processes
  – Why: Real markets are assumed to move randomly. Processes like Brownian motion model continuous-time randomness, letting you simulate or derive the distribution of future asset paths.
  – Learn: Stochastic Processes I (MIT OCW) & Stochastic Processes II

• Itô Calculus
  – Why: Ordinary calculus breaks on stochastic paths. Itô’s lemma is the “chain rule” when your underlying follows dW noise, allowing you to derive how functions of random variables evolve.
  – Learn: Itô Calculus

• Stochastic Differential Equations (SDEs)
  – Why: Asset prices are assumed to satisfy SDEs like dSt=μSt dt+σSt dWt

  Solving or simulating these gives you the expected payoff under risk neutrality.
  – Learn: Stochastic Differential Equations

Conclusion

You now have the deterministic and stochastic building blocks—calculus, probability, ODEs, stochastic processes, Itô calculus, and SDEs—required to see how any derivative’s “fair” price emerges from a discounted expected payoff. Armed with these foundations, you’re ready to dive into analytical formulas like Black–Scholes or implement Monte Carlo and PDE engines to price real-world products.