Beyond Black–Scholes: local, stochastic and jump models

Everyone knows Black–Scholes is wrong. The interesting question is what you replace it with, and the answer depends on whether you need to *fit* today’s prices or to *predict* how they will move — because the models that do the first perfectly are the ones that do the second worst.

Model	What it adds	What it buys	What it costs
Local volatility (Dupire)	$\sigma(S, t)$ from the surface	Fits every quoted price exactly	Unrealistic dynamics — the smile flattens as spot moves
Stochastic volatility (Heston)	Volatility as its own process	Realistic smile dynamics; vol of vol	Cannot fit the whole surface exactly
Jump diffusion (Merton)	Poisson jumps in the price	Genuine fat tails and short-dated skew	More parameters; hedging is incomplete
SABR	A parametric smile with a vol-of-vol	A workable smile interpolation	A fitting tool more than a dynamic model
Local–stochastic (LSV)	Both at once	Exact fit plus plausible dynamics	Expensive to calibrate and to run

Table 5.1 · The successors. The industry answer for an exotics book is the last row, and the reason is exactly the tension in rows one and two.

Fitting is not the same as being right. Local volatility can reproduce every option price on the screen, which sounds like the end of the argument. It is not, because a model is also a statement about tomorrow. Dupire’s model implies the smile flattens as spot rises — and empirically the smile moves roughly *with* spot instead. So a book hedged on local volatility is hedged against a dynamic that does not happen. Fitting the surface is a constraint any usable model must satisfy; getting the dynamics right is what determines whether the hedge works.

Proposition 5.2

Why short-dated skew needs jumps

A diffusion cannot generate much skew at very short maturities: over a day, a continuous process simply cannot move far enough for the tails to matter, so the model-implied smile is nearly flat while the market’s is steep. Jumps fix this immediately, because a jump can happen in an instant regardless of horizon. This is the cleanest empirical argument that price paths are not continuous.

Holds when

Stochastic volatility generates skew that grows with maturity; jumps generate it at all maturities.
A model with both is the standard way to fit the whole term structure of skew.

Definition 5.3

Market incompleteness

Incomplete market — In Black–Scholes every option can be replicated exactly with stock and cash, so there is one arbitrage-free price. Add a second source of randomness — stochastic volatility, or a jump of random size — and stock alone can no longer replicate the payoff. The price is then no longer unique: it depends on a market price of risk for the new factor, which must be estimated rather than derived. Hedging becomes a matter of minimising residual risk rather than eliminating it.

Example 5.4

What a jump does to a wing

A stock at $\$100$ has $20\%$ diffusive volatility over a month. How likely is a $30\%$ fall under a pure diffusion, and what does that say about the $\$70$ put?

Show the worked solutionHide the worked solution

Worked solution

Formula $\sigma_{\text{month}} = 20\% \times \sqrt{1/12}, \qquad z = \frac{\ln(0.7)}{\sigma_{\text{month}}}$
Substitute $\sigma_{\text{month}} = 0.20 \times 0.2887 = 5.77\%$
Solve $\ln 0.7 = -0.3567$
$z = -0.3567/0.0577 \approx -6.2$
Answer $\text{a six-sigma event — essentially zero probability}$

Sanity check. A lognormal model prices that put at effectively nothing, while the market will quote a real bid. The gap is not a mispricing; it is the market saying the process is not a diffusion. Thirty per cent falls in a month happen far more often than once per never.

Beyond Black–Scholes: local, stochastic and jump models

After this lesson you should be able to

Why short-dated skew needs jumps

Market incompleteness

What a jump does to a wing

The rest of this lesson is in Premium