At-the-Money Definitions in the Heston Model

Summary

The document sets out a stochastic volatility model with correlated Brownian drivers for the asset price and variance, then asks how to define an at-the-money call. It contrasts log moneyness measured against the initial asset price, for which zero log moneyness means the strike equals that price, with a definition based on the forward price. Its central issue is whether those conventions coincide in the model.

The post presents the model equations and the question but provides no answer, calculation, or option-pricing evidence. The distinction matters because the forward price can differ from the spot or initial price depending on the underlying’s carry and the time horizon. Consequently, “ATM” needs a stated reference price and valuation time; spot ATM and forward ATM are not interchangeable in general. The document is a conceptual question about option conventions, not a strategy or a full Heston derivation.

Key ideas

• The Heston setup models asset returns with stochastic variance and correlated Brownian shocks.
• Zero log moneyness relative to the initial asset price sets the strike equal to that price.
• Forward ATM instead sets the strike equal to the forward price for the relevant maturity.
• Spot-based and forward-based ATM conventions need not coincide, so the reference price should be specified.

Tags

# ATM strike Heston model


# ATM strike Heston model












I'm thinking about the heston model. price of the asset $S^1=(S_t^1)_{t \leq T}$ fullfills the differential equation $dS_t^1=S_t^1(\mu dt + \sqrt{V_t} dB_t^1)$ the stochastic volatility is given by $V=(V_t)_{t \leq T}$ $dV_t= \kappa (\theta-V_t)dt+ \sigma \sqrt{V_t} d B_t^V$

$dB_t^V dB_t^1=d[B^V,B^1]_t= \rho dt$

Now with an exchange of measure I get the following $d \tilde{S_t^1}=\tilde{S_t^1} \sqrt{V_t} d W_t^V$

$d V_t= \kappa (\theta- V_t)dt+ \sigma \sqrt{V_t} d W_t^V$

$dW^1 dW^V= \rho dt$

$\tilde{S_t^1}=S_0 e^{X_t}$

$dX_t=-\frac{1}{2}V_t dt+ \sqrt{V_t} dW_t^1$

Now I want to define that a call option is at-the-money $K=S_0 e^x$ with $x$ the log moneyness. So the option is ATM if $K=S_0$. I also saw the definition for ATM that $K=F_t$. My question now is are these two definitions for ATM the same or not?

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