Derivatives Pricing Theory

Module 2 in the curriculum. The instruments themselves --- calls, puts, payoff diagrams, strategies, Greeks conceptually, implied vol intuitively --- are already familiar. The gap is the pricing theory underneath: why does Black-Scholes work, what assumptions does it encode, and what happens when those assumptions break?

Why this module matters

Derivatives pricing is where probability theory, stochastic calculus, and financial economics converge. The risk-neutral pricing framework is arguably the single most powerful idea in quantitative finance: it says you can price any derivative by computing an expectation under an artificial probability measure, without needing to know anyone’s risk preferences.

Learning roadmap

Core ideas to internalize

From binomial trees to Black-Scholes

The binomial model builds intuition: at each node, construct a replicating portfolio of $\Delta$ shares and $B$ bonds that matches the derivative’s payoff. The price must equal the cost of that portfolio (no arbitrage). As $\Delta t\to 0$, the binomial model converges to Black-Scholes:

$C=S_0\Phi (d_1)-K{e}^{-rT}\Phi (d_2)$

where $d_1=\frac{\ln (S_0/K)+(r+{\sigma }^2/2)T}{\sigma \sqrt{T}}$ and $d_2=d_1-\sigma \sqrt{T}$.

The martingale approach

Under the risk-neutral measure $\mathbb{Q}$, the discounted stock price $e^{-rt}{S}_t$ is a martingale. Any derivative with payoff $h(S_T)$ can be priced as:

$V_0=e^{-rT}{\mathbb{E}}^{\mathbb{Q}}[h(S_T)]$

This connects directly to Girsanov’s theorem and the change-of-measure machinery.

When Black-Scholes breaks

The implied volatility surface --- the set of ${\sigma }_{\text{imp}}(K,T)$ values that make BS match market prices --- is not flat, contradicting BS’s constant-vol assumption. The smile/skew encodes:

• Fat tails (crash risk) via out-of-the-money put skew
• Stochastic volatility (vol-of-vol) via the smile’s curvature
• Jumps via short-dated smile steepness

Models like Heston ($dv=\kappa (\theta -v)dt+\xi \sqrt{v}d{W}^v$) and Dupire local vol address these failures.

Prerequisites

• Stochastic Processes --- Ito calculus, Girsanov theorem, martingale representation (Module 0.4 --- hard dependency)
• Probability Theory --- measure theory, conditional expectation
• Fixed Income --- discounting, term structure basics

Key references

• Hull --- Options, Futures, and Other Derivatives (the standard practitioner text)
• Shreve --- Stochastic Calculus for Finance I & II (rigorous treatment, binomial then continuous)
• Gatheral --- The Volatility Surface (for Units 2.5—2.6)