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Risk Surfaces for Mapping Model Vega to Market Implied Volatility Risk

Article Quant Q&A · Author: James
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Summary

The document asks how a risk surface can turn sensitivities from a model’s implied volatility function into sensitivities to market quoted volatilities. The described method adds an interpolated surface of risk bumps to the model volatility. At the base valuation, every bump is zero, so the pricing inputs and value remain unchanged. The bumps are treated as independent risk coordinates, with values specified at a discrete set of strike and maturity knots; differentiating with respect to those coordinates gives sensitivities associated with the market volatility grid.

The key idea is a change of risk coordinates rather than a change to the priced model. The document quotes a passage from a computational finance book but does not provide a worked calculation, numerical evidence, or a full explanation of interpolation and calibration details. It is framed as a request for intuition, so it identifies the motivation and construction without resolving implementation choices such as grid design or how sensitivities should be reported between knots.

Key ideas

  • A zero valued risk surface leaves the original model implied volatilities and price unchanged.
  • The surface is interpolated from bumps defined at discrete strike and maturity knots.
  • Derivatives with respect to the risk bumps provide sensitivities aligned with market implied volatility points.
  • The technique separates the model used for valuation from the coordinates used to express volatility risk.

Tags

OptionsVolatilityDerivatives pricingRisk management
Full text
# From model vega matrix to market vega matrix


# From model vega matrix to market vega matrix












I'm reading Antonie Savine's fascinating book Modern Computational Finance AAD and Parallel Simulations. However, I got a bit confused while reading and couldn't make sense of how it works in his work. To be more specific, in the following paragraphs:

> We developed, in the previous chapter, functionality to obtain the microbucket $\frac{\partial V_0}{\partial \sigma(S,t)}$ in constant time. We check-point this result into calibration to obtain $\frac{\partial V_0}{\partial \hat{\sigma}(K,t)}$ , what Dupire calls a superbucket. We are missing one piece of functionality: our IVS $\hat{\sigma(K,T)}$ is defined in derived IVS classes, from a set of parameters, which nature depends on the concrete IVS. For instance, the Merton IVS is parameterized with a continuous volatility, jump intensity, and the mean and standard deviation of jumps. The desired derivatives are not to the parameters of the concrete IVS, but to a discrete set of implied Black and Scholes market-implied volatilities, irrespective of how these volatilities are produced or interpolated. To achieve this result, we are going to use a neat technique that professional financial system developers typically apply in this situation: we are going to define a risk surface: $$s(K,T)$$ such that if we denote $\hat{\sigma}(K,T)$ the implied volatilities given by the concrete IVS, our calculations will not use these original implied volatilities, but implied volatilities shifted by the risk surface: $$\sum(K,T) = \hat{\sigma}(K,T) + s(K,T)$$ Further, we interpolate the risk surface s(K,T) from a discrete set of knots: $$s_{ij} = s(K_i, T_j)$$ that we call the risk view. All the knots are set to 0, so: $$\sum(K,T) = \hat{\sigma}(K,T)$$ so the results of all calculations remain evidently unchanged by shifting implied volatilities by zero, but in terms of risk, we get: $$\frac{\partial}{\sigma(K,T)} = \frac{\partial }{\partial s(K,T)}$$ The risk view does not affect the value, and its derivatives exactly correspond to derivatives to implied volatilities, irrespective of how these implied volatilities are computed. We compute sensitivities to implied volatilities as sensitivities to the risk view: $$\frac{\partial V_0}{\partial s_{ij}}$$ ...

In my understanding, the risk surface added to the implied volatlity should be zero, but I don't see how this translates from model risk to market risk by adding the risk surface, can anyone provide any intuition behind this process?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.