Estimating an Option’s Conditional Value After a Spot Move

Summary

The answer gives a local approximation for the conditional expected value of an option after a specified spot move. It assumes a stochastic spot process and a volatility process whose shocks are correlated, with a separate independent volatility shock. Applying a first-order change in option value, it expresses the change through delta times the spot move and vega times the volatility move.

Conditioning on the spot change links the expected volatility change to the correlated component of the volatility shock. Under the stated dynamics, the resulting approximation adds a vega adjustment proportional to the correlation, volatility-of-volatility parameter, and spot move to the usual delta estimate. The independent volatility shock has zero conditional mean under these assumptions. This is a local approximation, not a general risk-neutral pricing result: it requires specified dynamics and parameters, and omits higher-order effects such as gamma and volga, time passage, and changes in the volatility surface.

Key ideas

• A first-order option value change combines delta exposure to spot and vega exposure to volatility.
• Correlated spot and volatility shocks imply an expected volatility response conditional on the spot move.
• The independent volatility shock contributes zero to the conditional mean under the stated assumptions.
• The approximation depends on a chosen stochastic model and omits higher-order option sensitivities.

Tags

# What is the risk neutral expectiation of an option price given a move in spot?


# What is the risk neutral expectiation of an option price given a move in spot?












Lets say we have a volatility surface for the SPX at time t with spot S. We consequently know the price of some call option at maturity T with strike K. What is the risk neutral expectation of the option price at t+1day, if spot moves 1% ( E[option price | S*1.01] ) ?

## Answer by Frido (score 1)

https://quant.stackexchange.com/a/75199

Your question is unclear / lacks relevant detail, but I suspect what you're really asking is what happens to the vol change when the spot change is given.

Assume, as an example, the following dynamics: \begin{align} dS_t &= \sigma_t S_t dW_t \\ d\sigma_t &= \alpha \sigma_t ( \rho dW_t + \sqrt{1-\rho^2} dZ_t) \end{align} with $dW dZ = 0$.

You'd like to calculate $E_t[ C(S_{t+dt},\sigma_{t+dt},K) | dS_t = c]$. Now $$ C(S_{t+dt},\sigma_{t+dt},K) = C(S_t,\sigma_t, K) + dC(S_t,\sigma_t,K) $$ with $$ dC(S_t,\sigma_t,K) = \Delta dS_t + \nu d\sigma_t $$ with $\Delta$ the delta of the option and $\nu$ the vega.

Given $dS_t = c$ then \begin{align} dS_t &= c\\ d\sigma_t &= \frac{\rho \alpha}{S_t} dS_t + (\cdot) dZ_t \\ & = \frac{\rho \alpha}{S_t} c + (\cdot) dZ_t \end{align} Thus \begin{align} E_t[ C(S_{t+dt},\sigma_{t+dt},K) |dS_t = c] &= C(S_t,\sigma_t,K) + c \Delta + \frac{\rho \alpha}{S_t} c \nu + E[ (\cdot) dZ_t] \\ &= C(S_t,\sigma_t,K) + c \Delta + \frac{\rho \alpha}{S_t} c \nu \end{align}

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