Pricing Models Beyond Black-Scholes: Binomial Trees, Local Volatility, Heston, and Jumps

Professionals price options beyond Black-Scholes with four families of models: binomial trees, which handle early exercise; local volatility, which fits every quoted option exactly; stochastic volatility, which lets volatility itself move randomly; and jump models, which allow sudden gaps. Each one relaxes a different Black-Scholes assumption, and each one is wrong in a different way, which is why desks pick the model to fit the product rather than looking for one true model.

In short: Black-Scholes assumes continuous trading, constant volatility, no jumps, and European exercise. Cox, Ross and Rubinstein (1979) replaced continuous time with a tree of up and down moves, which prices American options and converges to Black-Scholes. Dupire (1994) showed that one volatility function of price and time can reproduce every European option price on the market. Heston (1993) made variance a random, mean-reverting process correlated with the stock, which produces skew. Merton (1976) added random jumps, which fattens the tails and raises the price of short-dated out-of-the-money options. On a hypothetical $50 stock, a three-step tree prices a three-month at-the-money American put at $3.03 against $2.98 for the European version, and with 1,000 steps the European values match Black-Scholes to within a tenth of a cent.

This deep dive assumes you know the Black-Scholes formula (see Black-Scholes explained) and what implied volatility means (see implied volatility explained). It describes how models work and recommends no trade.

Why one model is not enough

If Black-Scholes were right, one implied volatility would price every option on an underlying. Since the 1987 crash, out-of-the-money index puts have traded at higher implied volatilities than at-the-money options, and implied volatility also differs by expiry (the volatility surface, see the volatility surface and skew). So there are two problems: a practical one (Black-Scholes prices only European options, and most US single-stock options are American) and an empirical one (market prices are inconsistent with one constant volatility).

Model	Assumption it relaxes	What it is good at	Main weakness
Binomial tree (Cox-Ross-Rubinstein 1979)	Continuous time; European exercise	American options, dividends, intuition	Still one constant volatility
Local volatility (Dupire 1994)	Constant volatility, replaced by a function of price and time	Fits all vanilla prices exactly	Predicts unrealistic smile dynamics
Stochastic volatility (Heston 1993)	Volatility is known	Skew from spot-vol correlation, realistic dynamics	Struggles with very short-dated skew
Jump diffusion (Merton 1976)	Continuous price paths	Short-dated wings, crash risk	Jump risk cannot be hedged with stock

Binomial trees: Cox, Ross and Rubinstein (1979)

John Cox, Stephen Ross and Mark Rubinstein published “Option Pricing: A Simplified Approach” in the Journal of Financial Economics in September 1979 (volume 7, pages 229-263). The authors credit William Sharpe with suggesting the discrete-time approach. Their abstract makes the case plainly: the model “requires only elementary mathematics, yet it contains as a special limiting case the celebrated Black-Scholes model,” and it gives “a simple and efficient numerical procedure for valuing options for which premature exercise may be optimal.”

How the tree works

Split the option’s life into n steps of length dt. In each step the stock either rises by a factor u or falls by a factor d. Cox, Ross and Rubinstein matched the tree to a volatility sigma with:

u = e^(sigma x sqrt(dt))      d = 1 / u
R = e^(r x dt)                 p = (R - d) / (u - d)

The number p is not the real probability of an up move. It is the risk-neutral probability: the weight that makes a portfolio of stock and borrowing replicate the option, so that the option’s value is the discounted weighted average of its two possible next values. Starting from the payoffs at expiry, you work backward one step at a time. For an American option, at each node you also compare the value of holding with the value of exercising now, and keep the larger.

Worked example: a three-step tree

Hypothetical Maple Widgets Inc. trades at $50. Price a three-month (T = 0.25) option struck at $50, with volatility 30% and a 4% continuously compounded rate, using three steps of one month each:

dt = 0.25 / 3 = 0.08333
u = e^(0.30 x sqrt(0.08333)) = 1.090463, d = 1 / u = 0.917042
R = e^(0.04 x 0.08333) = 1.003339
p = (1.003339 - 0.917042) / (1.090463 - 0.917042) = 0.497616

The stock tree, with the American put’s value at each node in brackets:

Step	Highest node			Lowest node
0 (today)	50.00 [3.03]
1 (1 month)	54.52 [1.04]	45.85 [5.01]
2 (2 months)	59.46 [0.00]	50.00 [2.08]	42.05 [7.95]
3 (expiry)	64.83 [0.00]	54.52 [0.00]	45.85 [4.15]	38.56 [11.44]

Look at the node in bold. At $42.05 with one month left, holding the put is worth (0.497616 x 4.15 + 0.502384 x 11.44) / 1.003339 = $7.79. Exercising immediately is worth 50 - 42.05 = $7.95. Exercising wins, so the American put is worth more there than the European one, and that extra value flows back to today: the three-step American put is $3.03 against $2.98 for the European put. The call, by contrast, is never worth exercising early on a stock with no dividends, so the American and European calls are both $3.48 on this tree.

The intuition is interest: exercising a deep in-the-money put collects the strike now, and once little optionality remains, interest on $50 is worth more than the chance of a rebound.

Convergence to Black-Scholes

Three steps is too coarse. Here is the same option as the number of steps grows, against the Black-Scholes values of $3.2297 for the call and $2.7322 for the European put:

Steps	European call	European put	American put
3	3.4809	2.9834	3.0251
10	3.1563	2.6588	2.7216
25	3.2593	2.7618	2.8021
50	3.2149	2.7174	2.7610
100	3.2223	2.7248	2.7659
500	3.2283	2.7307	2.7699
1,000	3.2290	2.7315	2.7704

The values oscillate as they converge, because the strike sits on a node for some step counts and between nodes for others; practitioners average adjacent step counts or place the strike on a node. And the early-exercise premium of the American put settles at about 3.8 cents ($2.7704 minus $2.7315), or about 1.4% of the option’s value. In their own numerical example, Cox, Ross and Rubinstein found binomial values within about a penny of Black-Scholes at 150 steps.

The early-exercise premium grows with moneyness and rates. For a one-year put on the same $50 stock at a 5% rate (1,000 steps):

Strike	European put	American put	Early-exercise premium
50	4.6756	4.9344	0.2587
55	7.3263	7.8084	0.4821
60	10.5267	11.3414	0.8147

This is why American puts cannot be priced with the Black-Scholes formula, and why put-call parity becomes an inequality for American options (see put-call parity and arbitrage). The exercise rules themselves are covered in American vs European options.

What trees do not fix

A Cox-Ross-Rubinstein tree still uses one volatility for every node, so it inherits Black-Scholes’ failure to price the skew. The next three models change the volatility assumption itself.

Local volatility: Dupire (1994)

In the January 1994 issue of Risk (volume 7, pages 18-20), Bruno Dupire asked whether a single process for the underlying could be compatible “with the observed smiles at all maturities” while keeping the model complete, meaning every option can still be hedged with the underlying alone. His answer was yes, if the instantaneous volatility is allowed to be a deterministic function of both the price level and time, sigma(S, t). He called this restricting attention to risk-neutral diffusions, and showed that under that restriction the process is unique.

Dupire opened with a term-structure example on the Nikkei 225: if six-month at-the-money options imply 20% and one-year options imply 18%, constant volatility cannot hold for both. Making volatility a function of time resolves it, because implied variance is the average of instantaneous variance. The arithmetic: one year at 18% is total variance 0.18^2 x 1 = 0.0324; six months at 20% uses 0.20^2 x 0.5 = 0.0200; so the second six months carries 0.0124 of variance, which is an instantaneous volatility of sqrt(0.0124 / 0.5) = 15.7%. The strike dimension is harder, and that is what Dupire’s equation solves. With zero interest rates it reads:

sigma_local(K, T)^2 = 2 x (dC/dT) / (K^2 x d^2C/dK^2)

The denominator is (up to discounting) the risk-neutral probability density, which comes from butterfly spreads, and the numerator from calendar spreads. Dupire noted that both are positive if prices are free of arbitrage. In other words, you can read local volatility directly off a smooth surface of call prices.

The same idea appeared at the same time as implied trees, in Emanuel Derman and Iraj Kani’s Goldman Sachs note “The Volatility Smile and Its Implied Tree” (January 1994) and Mark Rubinstein’s “Implied Binomial Trees” (Journal of Finance, July 1994, volume 49, pages 771-818).

Worked example: local volatility is roughly twice as steep

Take a hypothetical index at 100 with zero rates and a three-month implied volatility that falls 0.5 volatility points per strike point, flat across expiries: 25% at the 90 strike, 20% at 100, 15% at 110. Computing Dupire’s formula numerically from the resulting call prices:

Strike / level	Implied volatility	Dupire local volatility
90	25.0%	31.5%
95	22.5%	25.7%
100	20.0%	20.3%
105	17.5%	15.4%
110	15.0%	11.2%

Near the money, local volatility falls about 1 point per index point, roughly twice the implied slope. Derman explains the rule of thumb in “Regimes of Volatility” (Goldman Sachs, January 1999): implied volatility is roughly the average of local volatilities between today’s level and the strike, so local volatility must change twice as fast, just as forward rates rise twice as fast as yields along a linear yield curve.

Where local volatility fails

Local volatility fits today’s prices by construction, so it is the standard tool for pricing exotics consistently with vanillas. Its weakness is its forecast of tomorrow. Patrick Hagan, Deep Kumar, Andrew Lesniewski and Diana Woodward, in “Managing Smile Risk” (Wilmott magazine, September 2002), found that local volatility models predict smile dynamics “opposite of observed market behavior”: when the underlying falls, the model’s smile shifts to higher prices. They warned that delta and vega hedges derived from such a model “can be unstable and may perform worse than naive Black-Scholes’ hedges.” Their response was the SABR stochastic volatility model, which they applied to US dollar interest-rate options.

Stochastic volatility: Heston (1993)

Steven Heston’s “A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options” appeared in the Review of Financial Studies in 1993 (volume 6, number 2, pages 327-343). Earlier stochastic volatility models, he noted, either lacked closed-form solutions or assumed volatility was uncorrelated with the stock, which “cannot capture important skewness effects.” His model lets the variance v follow a mean-reverting square-root process with its own random shock, correlated with the stock’s:

dS = mu x S dt + sqrt(v) x S dz1
dv = kappa x (theta - v) dt + xi x sqrt(v) dz2
correlation(dz1, dz2) = rho

Here kappa is the speed of mean reversion, theta the long-run variance, xi the volatility of volatility, and rho the correlation. Heston solved for European prices with characteristic functions, which made the model fast enough to calibrate and is a large part of its influence.

Heston’s own conclusions are specific. Correlation between volatility and the spot “is necessary to generate skewness.” Without it, “stochastic volatility only changes the kurtosis,” raising far-from-the-money prices relative to near-the-money ones. And Black-Scholes produces prices “virtually identical” to stochastic volatility models for at-the-money options, which, he observed, explains some of the empirical support for Black-Scholes, since most trading is near the money.

Worked example: correlation makes the skew

We priced three-month options on a hypothetical index at 100 (4% rate) with Heston’s formula, using current and long-run variance of 0.04 (20% volatility), kappa = 2 and xi = 0.6, then converted each price back to a Black-Scholes implied volatility. Changing only the correlation:

Strike	rho = -0.7	rho = 0	rho = +0.5
80	27.4%	23.1%	18.3%
90	23.4%	20.4%	17.1%
95	21.3%	19.3%	17.3%
100	19.0%	18.8%	18.6%
105	16.7%	19.0%	20.3%
110	15.1%	19.8%	21.9%
120	14.6%	21.7%	24.8%

Exactly as Heston described: zero correlation gives a roughly symmetric smile (kurtosis only), negative correlation gives the downward equity-style skew, and positive correlation tilts it the other way. The at-the-money volatility barely moves across the three columns. We checked the formula against a 400,000-path Monte Carlo simulation; prices agreed to within about 2 cents.

Holding rho at -0.7 and raising the volatility of volatility steepens the skew: the 80-to-120 spread is 5.8 points at xi = 0.2, 12.9 points at xi = 0.6, and 15.2 points at xi = 1.0.

Mean reversion also shapes the term structure. With rho = -0.7, the gap between the 90% and 110% (forward moneyness) implied volatilities shrinks with maturity:

Expiry	90% strike	100% strike	110% strike	90-110 skew
1 month	24.2%	19.4%	15.4%	8.8 points
3 months	23.1%	18.5%	14.9%	8.2 points
1 year	20.3%	17.5%	15.1%	5.1 points
2 years	19.4%	17.6%	16.0%	3.3 points

The weakness is at the very short end. Measured on fixed strikes, the Heston skew levels off as expiry shrinks: in the same model, the 95-to-105 strike spread is 4.6 points at three months, 5.0 at one month, and 5.1 at one week. Over a few days a continuous process cannot travel far enough to make a 5% move likely, so the model cannot make short-dated puts much dearer. Equity index markets show the opposite pattern; as Derman’s Columbia lecture notes on the smile put it, implied volatilities are “steepest for small expirations as a function of strike.” That is the gap jumps fill.

Jumps: Merton (1976)

Robert Merton’s “Option Pricing When Underlying Stock Returns Are Discontinuous” appeared in the Journal of Financial Economics in 1976 (volume 3, pages 125-144). His starting point is that Black-Scholes replication requires a stock price with a continuous sample path. He added a Poisson process: on most days the stock diffuses normally, but occasionally “important new information” arrives and the price jumps.

The key modelling choice is about who bears jump risk. Merton assumed jumps are firm-specific (“the discovery of an important new oil well or the loss of a court suit”), so they are diversifiable and earn no risk premium. Under that assumption, and with lognormal jump sizes, the option price is a probability-weighted sum of Black-Scholes prices:

Price = sum over n of [ e^(-lambda' T) x (lambda' T)^n / n! ] x BS(n)

Each BS(n) is a Black-Scholes price conditional on exactly n jumps, with variance raised by the jumps and an adjusted rate, and the weights are Poisson probabilities. Merton also warned that “one can hardly claim strong empirical evidence to support these assumptions.” For index options the diversifiability assumption is clearly wrong: a market crash is the definition of a jump that cannot be diversified away, which is one reason index skew carries a risk premium.

Merton predicted the pattern of mispricing. With jumps, “deep-out-of-the-money options” become more valuable because “the possibility of a large, finite jump in price significantly improves” the chance of finishing in the money, and the differences “will be magnified as one goes to short-maturity options.”

Worked example: jumps steepen the short end

We priced options on a hypothetical index at 100 with Merton’s formula: diffusion volatility 15%, on average one jump every two years (lambda = 0.5), mean log jump -10% with a 10% standard deviation, and a 4% rate. Converted to Black-Scholes implied volatility:

Expiry	80% strike	90% strike	100% strike	110% strike	120% strike
1 month	32.5%	23.3%	16.5%	16.0%	19.2%
3 months	23.5%	19.5%	17.1%	16.3%	16.2%
1 year	19.2%	18.3%	17.7%	17.3%	17.0%

The one-month 80% put trades at 32.5% implied volatility even though the diffusion is 15% and the total annual volatility including jumps is about 18%. As expiry lengthens, many small diffusive moves dominate the occasional jump and the smile flattens toward the total volatility. The short end behaves the opposite way to Heston: the 95-to-105 spread is 1.4 points at three months, 2.6 at one month, and 6.7 at one week. Jumps steepen the near expiries, which is why short-dated index options are usually modelled with a jump component. The three-month 90% put is worth $0.54 under these Merton parameters; a 1,000,000-path Monte Carlo check gave $0.54 as well.

The cost of this realism is hedging. A jump moves the stock instantly, so no delta hedge with the stock can replicate the option. Jump models are therefore incomplete: the price depends on the compensation the market demands for jump risk.

How desks actually use these models

In practice the choice is pragmatic:

Listed American equity options: binomial or trinomial trees (or finite-difference grids), fed with a strike- and expiry-specific implied volatility and discrete dividends. The tree handles exercise; the volatility surface handles the skew.
Index exotics and structured products: local volatility to match the vanilla surface, sometimes blended with stochastic volatility (“local-stochastic volatility”) so that the forward smile dynamics are not absurd.
Volatility products and forward-start risk: stochastic volatility models, because those products depend on how the smile evolves, which local volatility gets wrong.
Very short-dated index options: models with jumps, or stochastic volatility models with a jump term added, because diffusion alone cannot produce the observed short-dated wings.

Models that reproduce the same quoted surface can disagree about everything not quoted. Derman’s 1997-1998 S&P 500 study found the market switching between regimes, sometimes consistent with local volatility and sometimes not. Model choice is therefore a hedging decision as much as a pricing one: the models agree on today’s vanilla prices and disagree on tomorrow’s deltas.

What can go wrong

Calibration is not validation. A model fitted exactly to today’s prices can still hedge badly, as Hagan and co-authors showed for local volatility. A perfect fit says nothing about whether the model’s dynamics are right.
Parameter instability. Heston parameters recalibrated each day often jump around, which means the model’s own hedge ratios change for reasons that have nothing to do with the market.
Discretization error. Trees oscillate, Monte Carlo has sampling and time-stepping error, and finite-difference grids have boundary effects. Small errors on one option become large on a book.
Unhedgeable risks. Jumps and volatility shocks cannot be hedged with the underlying alone. A model can tell you the price of that risk under its assumptions; it cannot remove it.
Wrong exercise assumptions. Pricing an American option with a European formula understates puts (by 3.8 cents in our three-month example, and by more than 80 cents for the deep in-the-money one-year put) and misses early exercise before dividends on calls.

The short version

Black-Scholes is the common language; the models beyond it are dialects for specific problems. Cox, Ross and Rubinstein’s tree handles early exercise and converges to Black-Scholes. Dupire’s local volatility reproduces every vanilla price but implies smile movements markets do not follow. Heston’s stochastic volatility generates skew from spot-volatility correlation and is weakest at very short expiries. Merton’s jumps fix the short end at the cost of risk that cannot be hedged with stock. Professionals use each where its weaknesses matter least and treat the gap between models as a measure of risk.