Chapter 27: More on Models and Numerical Procedures

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Why Alternatives to Black–Scholes–Merton?

The volatility smile/skew is direct evidence that the lognormal assumption is wrong. Alternative models aim to reproduce observed market prices (the volatility surface).

Constant Elasticity of Variance (CEV) Model

The CEV model introduces a parameter $α\alpha$ relating volatility to the stock price:

$\mu S \, dt + \sigma S^{\alpha} \, dz$

$α=1\alpha = 1$ : Standard Black–Scholes–Merton (lognormal)
$α<1\alpha < 1$ : Volatility increases as price falls (leverage effect) → produces volatility skew
$α<1\alpha < 1$ with $\alpha \leq 1$ : Equity-like skew

The CEV model produces a volatility skew naturally without assuming stochastic volatility.

Merton's Mixed Jump–Diffusion Model

Combines continuous diffusion with discrete jumps:

$dz+dq\frac{dS}{S} = (\mu - \lambda k) \, dt + \sigma \, dz + dq$

where $d q$ represents jumps arriving at rate $λ\lambda$ with jump size drawn from a lognormal distribution (mean $k$ ).

The possibility of a sudden large drop creates a volatility smile. Before the 1987 crash, jump-diffusion produced a symmetric smile; after the crash, the model better fits the observed skew.

Variance Gamma Model

A pure jump process where the time change follows a gamma process. The stock price follows:

$St=S0exp⁡(μt+X(t;σ,ν,θ))S_t = S_0 \exp(\mu t + X(t; \sigma, \nu, \theta))$

where $X$ is a variance gamma process. Can create both skew and smile through the $θ\theta$ parameter (negative $θ\theta$ = negative skew).

Stochastic Volatility Models

Volatility is not constant — it follows its own stochastic process.

Heston Model (1993)

The most popular stochastic volatility model:

$\mu S \, dt + \sqrt{V} S \, dz_1$

$\kappa(\theta - V) \, dt + \xi \sqrt{V} \, dz_2$

where:

$V$ = instantaneous variance (stochastic)
$κ\kappa$ = mean reversion speed
$θ\theta$ = long-run variance
$ξ\xi$ = volatility of variance ("vol of vol")
$ρ\rho$ = correlation between $dz_1$ and $dz_2$ ( $ρ<0\rho < 0$ produces equity skew)

The Heston model has a semi-analytic solution for European options (characteristic function approach).

SABR Model (2002)

Popular for interest rate options:

$\alpha F^{\beta} \, dz_1$

$dz2d\alpha = \nu \alpha \, dz_2$

where $α\alpha$ is stochastic volatility. Parameters $β\beta$ , $ν\nu$ , $ρ\rho$ allow flexible smile modeling. A closed-form approximation exists (Hagan formula).

Rough Volatility Models

Recent research (Gatheral, Jaisson, Rosenbaum) shows that log-volatility behaves like fractional Brownian motion with $\approx 0.1$ (very rough). Rough volatility models fit the volatility surface remarkably well with very few parameters.

The IVF (Implied Volatility Function) Model

Rather than specifying a stochastic process for the underlying, the IVF model makes volatility a deterministic function of stock price and time $σ(S,t)\sigma(S, t)$ , calibrated to exactly match all observed European option prices (the volatility surface).

This approach ensures perfect calibration to the market but has limitations for hedging and dynamics.

Convertible Bonds

A convertible bond gives the holder the right to convert the bond into a predetermined number of shares. It combines:

A straight corporate bond
An American call option on the stock (with special features)

Valuation requires modeling:

Stock price (stochastic, with volatility)
Interest rates (optionally stochastic)
Credit risk (the company may default)
Conversion, call, and put features (complex early exercise conditions)

Numerical methods: typically trinomial trees or finite difference with credit-adjusted discount rates. The bondholder will convert when the conversion value exceeds the straight bond value.

Path-Dependent Derivatives

Lookback Options

Floating lookback: terminal payoff depends on min/max of path. Can be priced using binomial trees where each node tracks both the current price and the min/max so far.

Asian Options

Average price options. The key challenge is that the number of possible averages explodes in a tree (non-recombining in the average dimension). Solutions:

Monte Carlo simulation (simple but slow)
Analytic approximations
PDE methods in two dimensions ( $S$ and $A$ , the running average)

Barrier Options in Trees

In a standard binomial tree, the barrier may fall between nodes, leading to mispricing. Solutions:

Adjust the tree so nodes land on the barrier
Use interpolation at the barrier
Use trinomial trees with flexible node placement

Monte Carlo and American Options

Monte Carlo faces a fundamental difficulty for American options: you need to know the early exercise decision at each time, which requires knowing the continuation value — but this is what you're trying to compute.

Solutions:

Least Squares Monte Carlo (Longstaff–Schwartz, 2001): Regress continuation value on basis functions of the state variables at each exercise date
Stochastic mesh methods
Dual approaches (give upper bounds on the option price)