Local Volatility Model
1 Local Volatility Models
In mathematical finance local volatility models are option pricing models that treat volatility as a function of both the asset price \(S_t\) and time \(t\). The asset price process \((S_t)\) has dynamics given by the SDE
\[ dS_t=(r_t-\delta_t)S_t\,dt+\sigma(t,S_t)S_t\,dW_t, \]
where \(r_t\) is the interest rate, \(\delta_t\) the dividend yield and \(W\) a Brownian motion. A derivation as for the Black-Scholes PDE shows that if \(V_t=u(t,S_t)\) is the value of the replicating portfolio then
\[ \partial_tu+rx\,\partial_xu+\frac{1}{2}\sigma^2(t,x)x^2\,\partial_{xx}u-ru=0. \]
Solving this PDE is inefficient for pricing a single option, since it requires the solution at all \((t,x)\) just to produce one number.
2 Dupire’s PDE
In the local volatility model there is a model-dependent result, Dupire’s formula, that gives call prices directly as a function of maturity and strike. Let \(C(T,K)\) be the price at time \(0\) of a call with maturity \(T\) and strike \(K\), and take constant \(r\) and no dividends. Then
\[ \frac{\partial C}{\partial T}(T,K)=-rK\frac{\partial C}{\partial K}(T,K) +\frac{1}{2}\sigma^2(T,K)K^2\frac{\partial^2C}{\partial K^2}(T,K), \]
with the natural initial condition \(C(0,K)=(S_0-K)_+\), since at \(T=0\) the option price is the immediate payoff. The equation can be read backwards to recover the local volatility from option prices,
\[ \sigma^2(T,K)=\frac{\partial_TC+rK\,\partial_KC}{\tfrac{1}{2}K^2\,\partial_{KK}C}, \]
which also yields the implied volatility surface \(I(T,K)\) of the model (see Volatility Surfaces).
3 The CEV Model
A special case is the constant elasticity of variance (CEV) model, with dynamics
\[ dS_t=\mu S_t\,dt+\sigma S_t^{\gamma}\,dW_t, \]
where \(\sigma>0\) and \(\gamma\ge 0\) is the elasticity parameter. This is a local volatility model with \(\sigma(t,x)=\sigma x^{\gamma-1}\), so the instantaneous variance of returns is \(\sigma^2S_t^{2(\gamma-1)}\), whose elasticity with respect to \(S_t\) is constant. Special cases are
- \(\gamma=1\): Geometric Brownian Motion, the Black-Scholes-Merton model,
- \(\gamma=0\): the arithmetic (Bachelier) model \(dS_t=\mu S_t\,dt+\sigma\,dW_t\).
For \(\gamma<1\) volatility rises when the price falls (the leverage effect), which produces a downward-sloping implied volatility skew, as is observed in equity markets. For \(\gamma>1\) the skew is upward-sloping.
4 Problems
A local volatility model is designed to produce an implied volatility surface \(I(T,K)\) that is not flat in \(T\) or \(K\), but it is still imperfect. It suffers from the so-called \((t,T,K)\)-problem.
Since \(\sigma=\sigma(t,x)\), volatility is always the same given the same time and stock price. If we calibrate at time \(0\) and obtain the surface \(I(0,T,K)\) from options maturing at \(T\), we therefore expect the same surface \(I(t,T+t,K)\) at a later time \(t\) from options maturing at \(T+t\). Empirically, these two implied volatility surfaces are not the same under time translation, which contradicts the model. This motivates stochastic volatility models.
5 References
- Dupire, B. (1994). Pricing with a smile. Risk, 7(1), 18-20.
- Cox, J. C. and Ross, S. A. (1976). The valuation of options for alternative stochastic processes. Journal of Financial Economics, 3(1-2), 145-166.