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?

  • $\begingroup$ What is $x$? Do you want to set the ATM strike to be the forward to the option maturity? $\endgroup$ – Gordon Jan 7 at 18:03
  • $\begingroup$ You may assume $\mu=r$ under the risk-neutral measure. How did you make the measure change? I would suggest you edit your question so that it is easier for understanding. $\endgroup$ – Gordon Jan 7 at 18:05
  • 3
    $\begingroup$ Moneyness is a concept independent of the underlying model. When we speak of ATM we usually specify ATM spot, or ATM forward (ATMF). The first is simply $K=S_t$ the other is $K = F_t(T)$, where $T$ is the maturity of the option considered since ATMF is maturity dependent. $t$ is today/current time. $\endgroup$ – ilovevolatility Jan 8 at 4:48
  • $\begingroup$ So I can not mix up these two definitions? $\endgroup$ – P.G. Jan 8 at 10:19
  • $\begingroup$ No you can't, unless the risk neutral drift for the spot process $S_t$ is zero in which case $S_t = F_t(T) \,\, \forall T$ $\endgroup$ – ilovevolatility Jan 10 at 0:53

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