How do different models impact option Greeks?

Question

If I trade an option using delta, vega, Prob OTM, etc. these are derived from a model. How do leading models impact valuations in terms of the Greeks?

I suppose to form a baseline it would have to be relative to Black-Scholes.

CRS Process Model, Heston Model, [CGMY][1] Model. For example, does the Heston model generally depreciate the delta of a call option relative to what BSM would predict? Can any descriptions of the models be learned from these comparisons?

Answer 1

This is an interesting and not so easy question. Here's my 2 cents:

• First, you should distinguish between mathematical models for the dynamics of an underlying asset (Black-Scholes, Merton, Heston etc.) and numerical methods designed to calculate financial instruments' prices under given modelling assumptions (lattices, Fourier inversion techniques etc.). Typically, the lattice technique you are referring to is a numerical method originally designed to compute prices under Black-Scholes modelling assumptions (of course variants exist). Sticking to the Black-Scholes world, you could also price instruments using totally different methods such as Monte Carlo or Finite Difference. Each numerical method has its own pros and cons in terms of Greeks production and within a single method, there can actually be many variants (e.g. Monte Carlo pathwise / likelihood ratio / vibrato sub-types).

• For the Delta, loosely speaking, traders tend to think in terms of: $$ \Delta = \frac{\partial C(S_0;K,T)}{\partial S_0} + \left. \frac{\partial C(S_0;K,T)}{\partial \sigma(S_0;K,T)} \right\vert_{S_0=\text{cst}} \frac{\partial \sigma(S_0;K,T)}{\partial S_0} $$ where I have considered a European call option $C(S_0;K,T)$ of implied volatility $\sigma(S_0;K,T)$ without loss of generality. The above equation can be re-expressed as: $$ \Delta = \Delta_{BS} + \nu_{BS} \frac{\partial \sigma(S_0;K,T)}{\partial S_0} $$ where the second term in the RHS is sometimes called shadow delta: it is related to how the implied volatility (surface) is expected to move with the spot. The market often exhibits different such volatility regimes and practitioners have come up with various stickiness assumptions or rules of thumb to capture these, see the seminal paper of Emmanuel Derman.

• It turns out that making a particular modelling assumption (i.e. picking a certain mathematical model for the dynamics of the underlying), implicitly shapes the spot/implied vol dynamics. In Black-Scholes for instance, you are obviously sticky-strike because the model theoretically stipulates that the volatility (= IV) is constant: $$ \sigma(S_0+\Delta S_0;K,T) = \sigma(S_0;K,T) $$ such that $$\frac{\partial \sigma(S_0;K,T)}{\partial S_0} = 0$$ and $\Delta = \Delta_{BS}$. In more elaborate diffusion frameworks, this term is not zero. Typically, in equity markets (negatively skewed, i.e. $\partial \sigma/ \partial K < 0$) you will have: $$\Delta_{LV} \leq \Delta_{BS} \leq \Delta_{SV}$$ Because, in inhomogeneous local volatility models à la Dupire you can show $\frac{\partial \sigma(S_0;K,T)}{\partial S_0} < 0$ (sticky implied tree/ sticky local delta). On the other hand, in space homogenous stochastic volatility models (e.g. Heston) you have $\frac{\partial \sigma(S_0;K,T)}{\partial S_0} = -\frac{K}{S_0}\frac{\partial \sigma(S_0;K,T)}{\partial K} > 0$ (sticky moneyness).

• Where this can become confusing is that, under given modelling assumptions, you can set up your numerical method to calculate virtually any kind of Delta, i.e. not necessarily model-consistent Deltas (this is well explained in the latest book of Lorenzo Bergomi). This can be understood by looking at a simple bump & revalue approach to compute the Delta, e.g. centred finite difference approximation $$ \Delta = \frac{V(S_0+\Delta S;\theta) - V(S_0-\Delta S;\theta)}{\Delta S} $$ Using this approach, you will typically obtain different Delta values depending on what variable(s) amongst the generic model parameters $\theta$ move along with the spot in the pricing function $V(S_0;\theta)$. For instance, in Black-Scholes, along with bumping $S_0$ you can choose to make the implied volatility $\sigma$ evolve according to a certain stickiness assumption (which goes against the theoretical behaviour), or you can leave it unchanged which would consistent with what the model predicts.

• The calculation of so-called minimum variance Deltas (under given modelling assumptions) is yet a separate question. In that case, Delta should best be viewed as: $$ \Delta^{MV} = \frac{d \langle V,S \rangle_t}{d \langle S,S \rangle_t} $$ Doing the computations in the BS case, you end up with a regular Delta: $$ \Delta^{MV}_{BS} = \frac{\partial V}{\partial S_0} $$ In LV à la Dupire you have $$ \Delta^{MV}_{LV} = \frac{\partial V}{\partial S_0} + \frac{\partial V}{\partial \sigma_{LV}}\frac{\partial \sigma_{LV}}{\partial S_0} $$ and in SV à la Heston you have $$ \Delta^{MV}_{SV} = \frac{\partial V}{\partial S_0} + \frac{\partial V}{\partial v_0}\frac{\xi \rho}{S_0} $$ Some practitioners even argue that the relationship $\partial\Sigma/\partial S$ ought to be calibrated empirically as no theoretical model seems to decently capture all its intricacies. They then write $$ \Delta^{MV}_{emp} = \Delta^{BS} + \nu^{BS} f_{emp}(\text{regressors}) $$ see for instance Optimal Delta Hedging for Options by Hull & White.

• To conclude, notice that Vega should be understood as an out-of-model Greek. In other words, the Black-Scholes Vega should be seen as a sensitivity towards a violation of the BS modelling assumptions themselves. In that sense, it can be compared to a sensitivity with respect to Heston parameters in Heston for instance. That being said, if you want the sensitivity to a parallel shift of the full IV surface (which is what the BS Vega represents mathematically), it is still possible to obtain it in Heston and other models, but this can rarely be done analytically. It usually requires some kind of bump the IV surface (! arbitrage opportunities) + re-calibrate the model approach.