# 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 $\rho=0$. Jul 3 '14 at 15:40
• These are two completely separate questions - I suggest you form two separate questions out of it! Jul 3 '14 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. Jul 3 '14 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 $C(\sigma)-C^{Market}=0$. Jul 3 '14 at 16:50
• It wouldn't exist on the border of arbitrage opportunity, when intrinsic value is larger or equal to option price. Jul 3 '14 at 23:34