# Does Implied Volatility always exist?

I am considering a simple Heston Model Market with one risky and one riskless asset.

The dynamics of the riskless asset is simply $dB_t=r*B_t*dt$

The dynamics of the risky asset is as follows,

$dS_t=r*S_t*dt+\sqrt{V(t)}*S_t*dW_t, S_0>0$

$dV_t= \alpha*(\beta-V_t)*dt+\gamma*\sqrt{V_t}*dW^{\rho}_t, V_0=\sigma^2$

$W^{\rho}_t = \rho*W_t +\sqrt{1-\rho^2}*W^*_t$

where $W_t,W^*_t$ are independent standard one-dimentional Brownian Motion.

I want to ask firstly, whether there exist a explicit solution for $S_t$ and $V_t$. If yes, can you please tell me what is it and how to find it.

Secondly, when i simulate this market and compute the price of simple European call option on this risky asset using Monte Carlo $C_{Heston}$ and then compute the implied volatility in the Black and Scholes Market, i cannot find implied volatility for some values of the strike price.

this is the equation i am using to find implied volatility of Heston Model in Black and Scholes Market, $s*exp(r*T)*\Phi(\frac{(ln(s/K)+(r+(1/2)*\sigma^2)*T)}{(\sigma*\sqrt{T})})-K*\Phi(\frac{(ln(s/K)+(r-(1/2)*\sigma^2)*T)}{(\sigma*\sqrt{T})}) = C_{Heston}$

where $\Phi$ is CDF of Normal(0,1).

solving for $\sigma$ using computer algebra gives, $RootOf(-S_0*exp(r*T)*erf((1/4)*\frac{(T*Z^2+2*r*T+2*\ln(S_0/K))*\sqrt{2}}{(Z*\sqrt{T})})+erf((1/4)*\frac{(-T*Z^2+2*r*T+2*ln(S_0/K))*\sqrt{2}}{(Z*\sqrt{T})})*K-S_0*exp(r*T)+K+2*C_{Heston})$

but this is becoming complex for some values of $K$ in the model.

So my question is does the implied volatility of Heston Model in Black Scholes Model for European Call Option exist for all values of the strike price $K\gt 0$.

Please answer in easy to understand and elaborate manner as i am new to this subject.

• Your model is not a Black-Scholes Model, it is a Stochastic Volatility Model. You cannot use the Black-Scholes Formula for this. You may find a solved case by the Heston Model which assumes $\rho=0$. Commented Jul 3, 2014 at 15:40
• These are two completely separate questions - I suggest you form two separate questions out of it! Commented Jul 3, 2014 at 16:07
• Thank you for pointing that out. What i meant was price of the European Call Option is computed using Monte Carlo for Heston Model then its implied volatility is computed in the Black and Scholes Model. Commented Jul 3, 2014 at 16:20
• I am not sure if your method makes sense, but the implied volatility from Black Scholes Formula would always exist, just a closed-form expression does not. So you may use a numerical solver instead for $C(\sigma)-C^{Market}=0$. Commented Jul 3, 2014 at 16:50
• It wouldn't exist on the border of arbitrage opportunity, when intrinsic value is larger or equal to option price. Commented Jul 3, 2014 at 23:34