Interpreting Heston Model Greeks and Their Role in Option Pricing
Summary
The document explains option Greeks in the Heston stochastic volatility model, focusing on delta as the sensitivity of theoretical option value to the underlying price. It also defines gamma, rate sensitivity, theta, sensitivity to variance, vanna, and volga, then relates these derivatives through the Heston pricing equation.
The equation illustrates that the option value is shaped by several sources of risk and their associated sensitivities, not delta alone. Greeks can therefore help describe how a model’s price responds to changes in market inputs and support hedging analysis. The source offers mathematical definitions and a pricing relationship, but does not explain how to infer market conditions by comparing models, nor does it provide empirical examples or guidance on hedging implementation.
Key ideas
- Delta measures an option’s price sensitivity to the underlying asset.
- The Heston model includes sensitivities to variance and cross-effects between price and variance.
- The pricing equation connects option value with theta, delta, gamma, variance sensitivity, vanna, and volga.
- Greeks describe model-based sensitivities but do not, by themselves, establish conclusions about market conditions.
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# The Greeks of a stochastic volatility model: what's the purpose?
# The Greeks of a stochastic volatility model: what's the purpose?
let's take delta; What can the delta of a Heston model be used for? I know it can used for hedging strategies, but can we say something about the market and the model by looking at the delta. Can we conclude anything by comparing the deltas of two models? Or other Greeks
## Answer by user16651 (score 2)
https://quant.stackexchange.com/a/29550
Delta measures the sensitivity of an option's theoretical value to a change in the price of the underlying asset. It is normally represented as a number between minus one and one, and it indicates how much the value of an option should change when the price of the underlying stock rises by one dollar.Indeed $$\color{red}{\Delta \approx \frac{U(S+dS,v,t,T,K)-U(S,v,t,T,K)}{dS}}$$ where $U$ denotes the option price and $v_t$ is the stochastic volatility.
Mathematically speaking
In the Heston model, we have $$\frac{\partial U}{\partial t}+\,r{{S}_{t}}\frac{\partial U}{\partial S}+[\kappa (\theta -{{v}_{t}})-\lambda {{v}_{t}}]\,\frac{\partial U}{\partial v}-rU+\\\frac{1}{2}{{v}_{t}}{{S}_{t}}^{2}\frac{{{\partial }^{2}}U}{\partial {{S}^{2}}}+\rho \sigma \,{{v}_{t}}{{S}_{t}}\frac{{{\partial }^{2}}U}{\partial S \partial v}+\frac{1}{2}{{\sigma }^{2}}{{v}_{t}}\frac{{{\partial }^{2}}U}{\partial {{v}^{2}}}=0\tag 1$$ As you know $$\Delta =\frac{\partial U}{\partial S}\,,\,\Gamma =\frac{{{\partial }^{2}}U}{\partial {{S}^{2}}}\,,\,\rho =\frac{\partial U}{\partial r}\,,\,\Theta =\frac{\partial U}{\partial t},\\ \vartheta =\frac{\partial U}{\partial v},\operatorname{Vanna}=\frac{{{\partial }^{2}}U}{\partial S\partial v},\operatorname{Volga}=\frac{{{\partial }^{2}}U}{\partial {{v}^{2}}}\tag 2$$ $(1)$ and $(2)$ $$rU=\color{red}{\Theta} +\,(r-q)S\color{red}{\Delta} +[\kappa (\theta -v)-\lambda v]\color{red}{\vartheta} +\frac{1}{2}v{{S}^{2}}\color{red}{\Gamma} +\rho \sigma \,vS\,\,\color{red}{\operatorname{Vanna}}+\frac{1}{2}{{\sigma }^{2}}v\,\color{red}{\operatorname{Volga}}$$
Therefore the Greeks are so important.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.