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In Dupire's local volatility model, the volatility is is a deterministic function of the underlying price and time, chosen to match observed European option prices. To be more specific, given a smooth surface $(K,T)\mapsto C(K,T)$ where K is the strike and T is time to maturity. Dupire equation implies that there exits an unique continuous function $\sigma_{loc}$ defined by $$\sigma_{loc}^{2}(K,T)=\frac{\partial_{T}C(K,T)+rK\partial_{K}C(K,T)}{\frac{1}{2}K^{2}\partial_{KK}C(K,T)}$$ for all $(K,T)\in(0,\infty)\times(0,\infty)$ such that the solution to the stochastic differential equation $dS_{t}/S_{t}=rdt+\sigma(t,S_{t})dW_{t}$ exactly generates the European call option prices.

What do the dynamics of the local volatility mean? Are dynamics equivalent to the volatility surface? Why the dynamics of local volatility model is highly unrealistic?

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A general model (with continuous paths) can be written $$ \frac{dS_t}{S_t} = r_t dt + \sigma_t dW_t^S $$ where the short rate $r_t$ and spot volatility $\sigma_t$ are stochastic processes.

In the Black-Scholes model both $r$ and $\sigma$ are deterministic functions of time (even constant in the original model). This produces a flat smile for any expiry $T$. And we have the closed form formula for option prices $$ C(t,S;T,K) = BS(S,T-t,K;\Sigma(T,K)) $$ where $BS$ is the BS formula and $\Sigma(T,K) = \sqrt{\frac{1}{T-t}\int_t^T \sigma(s)^2 ds}$. This is not consistent with the smile observed on the market. In order to match market prices, one needs to use a different volatility for each expiry and strike. This is the implied volatility surface $(T,K) \mapsto \Sigma(T,K)$.

In the local volatility model, rates are deterministic, instant volatility is stochastic but there is only one source of randomness $$ \frac{dS_t}{S_t} = r(t) dt + \sigma_{Dup}(t,S_t) dW_t^S $$ this is a special case of the general model with $$ d\sigma_t = (\partial_t \sigma_{Dup}(t,S_t) + r(t)S_t\partial_S\sigma_{Dup}(t,S_t) + \frac{1}{2}S_t^2\partial_S^2\sigma_{Dup}(t,S_t)) dt + \frac{1}{2}S_t\partial_S\sigma_{Dup}(t,S_t)^2 dW_t^S $$ What is appealing with this model is that the function $\sigma_{Dup}$ can be perfectly calibrated to match all market vanilla prices (and quite easily too).

The problem is that while correlated to the spot, statistical study show that the volatility also has its own source of randomness independent of that of the spot. Mathematically, this means the instant correlation between the spot and vol is not 1 contrary to what happens in the local volatility model.

This can be seen in several ways:

  1. The forward smile. Forward implied volatility is implied from prices of forward start options: ignoring interest rates, $$ C(t,S;T\to T+\theta,K) := E^Q[(\frac{S_{T+\theta}}{S_{T}}-K)_+] =: C_{BS}(S=1,\theta,K;\Sigma(t,S;T\to T+\theta,K)) $$ Alternatively, it is sometimes defined as the expectation of implied volatility at a forward date. In a LV model, as the maturity $T$ increases but $\theta$ is kept constant, the forward smile gets flatter and higher. This is not what we observe in the markets where the forward smile tends to be similar to the current smile.

This is because the initial smile you calibrate the model too has decreasing skew: $$ \partial_K \Sigma(0,S;T,K) \xrightarrow[T\to +\infty]{} 0 $$

  1. Smile rolling. In a LV model, smile tends to move in the opposite direction of the spot and get higher independently of the direction of the spot. This is not consistent with what is observed on markets. See Hagan and al. Managing Smile Risk for the derivation. This means that $\partial_S \Sigma_{LV}(t,S;T,K)$ often has the wrong sign so your Delta will be wrong which can lead to a higher hedging error than using BS.
  2. Barrier options. In FX markets, barrier options like Double No Touch are liquid but a LV model calibrated to vanilla prices does not reproduce these prices. This is a consequence of the previous point.

The LV model is a static model. Its whole dynamic comes from the volatility surface at time 0. But the vol surface has a dynamic that is richer than that.

There are alternatives using multiple factors like SV models, LSV models (parametric local vol like SABR or fully non parametric local vol), models of the joint dynamic of the spot and vol surface etc... but the LV model remains the default model in many cases due to its simplicity, its ability to calibrate the initial smile perfectly and its numerical efficiency.

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  • $\begingroup$ Thank you so much for explaining in detail. However, I still don't understand the following: (1) In the equation of $d\sigma_{t}$, line 13, is the $dt$ in the first term on the RHS unnecessary? (2) How this equation came from? Could you please give a reference? (3) "The problem is that volatility actually has independent sources of randomness independent of the one of the spot: the instant correlation between the spot and vol is not 1." Do you mean that the correlation between the spot and vol is 1 in LSV model which is not realistic? $\endgroup$ – cmd1991 May 19 '15 at 19:48
  • $\begingroup$ (4) " This is because the initial smile you calibrate the model too has decreasing skew" Can this be implied by the definition of $\sigma_{loc}$? $\endgroup$ – cmd1991 May 19 '15 at 19:49
  • $\begingroup$ 1) dt was a typo I corrected it, 2) That is just Itô applied to the local vol function, 3) Yes in the LV model, 4) No this is because the local vol surface we observe in the market all have this feature: flatter smile at long maturity. $\endgroup$ – AFK May 20 '15 at 19:58
  • $\begingroup$ I remember that forward smiles produced by local volatility model were too flat in comparison to what is observed on the market. For the example, even simpler than a double barrier : take an up & out atm call, barrier 120. This is quite sensitive to the forward smile, and has quite a chance to be mispriced by a calibrated LV. $\endgroup$ – ujsgeyrr1f0d0d0r0h1h0j0j_juj Jun 10 '15 at 13:30
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Here "dynamics" means the assumed future behaviour of the spot process, namely that it follows the SDE $$ dS/S = r dt + \sigma_{loc}(S,t) dW_t .$$

There are various ways to see that these dynamics are unrealistic.

One is to look for time homogeneity. In normal cases, you expect the market to follow the same rules in one week and in one year from today. But inevitably a local volatility model will be calibrated to have more steeply increasing $\sigma_{loc}(K,T)$ as a function of $K$ when $T$ is small compared to when $T$ is large. This non-homogeniety of the calibrated model reflects a mismatch between the model and reality.

Another is to look at the model calibrated to subsequent days' option prices. If the model is right, the recalibrated $\tilde{\sigma}_{loc}$ is obtained from $\sigma_{loc}$ by translation. In practice that does not hold.

A third is to study hedge performance of vanilla options. According to the local volatility model, continuous time delta hedging can exactly replicate any option payoff. In practice there are hedging errors, and backtesting will show they are significant for the local volatility model.

Finally, one can look at exotic option prices. For an example, the local volatility model tends to significantly underestimate the fair value of a double-no-touch option with symmetric barrier levels around the current spot and moderate (10-20%) probability of paying off.

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