Under the Heston model with zero interest/dividend rate, is there a concrete example where the theta [$]\frac{\partial C}{\partial t}>0[$] where [$]C[$] is the price of a European call?

I easily found one with the OptionCity calculator. Just take a Black-Scholes case where you get that sign: say S = 2 K, r = 0.05, q = 0.10 (i.e., deep itm with a large dividend). Then, do the Heston model with those parms + same vols (instantaneous & long-run), rho=0, small vol of vol. You'll get essentially the same theta. 

I easily found one with the OptionCity calculator. Just take a Black-Scholes case where you get that sign: say S = 2 K, r = 0.05, q = 0.10 (i.e., deep itm with a large dividend). Then, do the Heston model with those parms + same vols (instantaneous & long-run), rho=0, small vol of vol. You'll get essentially the same theta. 

Thank you, Alan. But by "those params", do you mean still keeping "r = 0.05, q = 0.10"? I would like to have an example where r=q=0. Is there an easy way to do this?

By the way, thanks for introducing me to the nice app of yours, OptionCity. I have bookmarked it.

You're welcome. 

Yes, keep q and r as I gave. Or, in any event, pretty sure you need [$]q > r[$] to find a positive theta. The idea is, to get a positive theta, you need the option value (say for large S), to be below the intrinsic value S - K. Then, as time passes, the value will tend to increase.

If you insist on r=q, my guess is that theta will always be negative, but this is a guess, not a proof. 

If you insist on r=q, my guess is that theta will always be negative, but this is a guess, not a proof. 

Yes, I am insisting on zero rates (interest and dividend). I do not think the negativity is inadvertent. As a matter of fact, I have just encountered positive theta's in my simulation of option price under the time dependent inverse Gamma process with zero rates with the logarithmic moneyness reasonably close to [$]0[$]. I want to see if it is more common than the result of some stringent conditions. Theta can not be positive with one dimensional continuous path diffusion and [$]r>q[$]. With stochastic volatility under Heston, if the [$]\eta^2v\frac{\partial^2 C}{\partial v^2}[$] where [$]\eta[$] is the volatility of the volatility and [$]v[$] the variance, is negative enough, it may counteract the positive [$]vS^2\frac{\partial^2 C}{\partial S^2}[$] to give a positive [$]\Theta[$]. So for zero rates, we may need high volatility of volatility to achieve this.

Interesting conjecture. But I just ran some numerical tests that make me believe even more in my original intuition. Fixing all the other parameters with r=q=0, I took larger and larger [$]\eta[$]. For all test cases, I found negative theta. Indeed, it looks like, as [$]\eta \rightarrow \infty[$], 

[$]C - (S - K)^+ \sim \frac{f_1}{\eta}[$]     and  [$]C_t  \sim -\frac{f_2}{\eta}[$],

where [$]f_1[$] and [$]f_2[$] are positive functions independent of [$]\eta[$].  

I tried various moneynesses and rho, and Feller ratios as small as 0.00005 ([$]\eta = 200[$]).

Maybe I am missing something, but would a positive dC/dt not lead to a non well-posed PDE, i.e. on that does not satisfy and energy inequality/norm?
And Heston is (just) a parabolic PDE, albeit is a very popular one? 

No, positive examples have already been given. Also, consider any one-factor bond model for [$]B(t,r)[$], the zero coupon bond price. They typically have [$]B_t > 0[$]. The positive call option examples I gave are just parameter setups where the call is behaving like a bond because it is deep in-the-money and below parity. Another well-known example of options with positive theta are deep itm puts with r>0, so again, a price below parity (K-S) and behaving like a discount bond.

Interesting conjecture. But I just ran some numerical tests that make me believe even more in my original intuition. Fixing all the other parameters with r=q=0, I took larger and larger [$]\eta[$]. For all test cases, I found negative theta. Indeed, it looks like, as [$]\eta \rightarrow \infty[$], 

[$]C - (S - K)^+ \sim \frac{f_1}{\eta}[$]     and  [$]C_t  \sim -\frac{f_2}{\eta}[$],

where [$]f_1[$] and [$]f_2[$] are positive functions independent of [$]\eta[$].  

I tried various moneynesses and rho, and Feller ratios as small as 0.00005 ([$]\eta = 200[$]).

Let me do some more investigation and check my computation. One scenarios that may worth trying is [$]\eta=0[$], large [$]\lambda[$] mean reversion, low [$]\bar v[$] variance at infinity, varying time to maturity [$]T-t[$]. The idea is to make [$]\lambda(v-\bar v)\frac{\partial C}{\partial v}>-\frac12vS^2\frac{\partial^2 C}{\partial S^2}>0[$].

Maybe I am missing something, but would a positive dC/dt not lead to a non well-posed PDE, i.e. on that does not satisfy and energy inequality/norm?
And Heston is (just) a parabolic PDE, albeit is a very popular one? 

No. The sign of [$]\frac{\partial C}{\partial t}[$] is a local property and has no direct dependency on the (dissipative) energy norm which is a spatial global (integration) property. As a simple example, a concave payoff for a geometric Brownian motion with zero rates makes theta positive and the total energy is still dissipative. In fact, the sign depends on time and the spot when the payoff has both convexity and concavity at different spot values.

Interesting conjecture. But I just ran some numerical tests that make me believe even more in my original intuition. Fixing all the other parameters with r=q=0, I took larger and larger [$]\eta[$]. For all test cases, I found negative theta. Indeed, it looks like, as [$]\eta \rightarrow \infty[$], 

[$]C - (S - K)^+ \sim \frac{f_1}{\eta}[$]     and  [$]C_t  \sim -\frac{f_2}{\eta}[$],

where [$]f_1[$] and [$]f_2[$] are positive functions independent of [$]\eta[$].  

I tried various moneynesses and rho, and Feller ratios as small as 0.00005 ([$]\eta = 200[$]).

Let me do some more investigation and check my computation. One scenarios that may worth trying is [$]\eta=0[$], large [$]\lambda[$] mean reversion, low [$]\bar v[$] variance at infinity, varying time to maturity [$]T-t[$]. The idea is to make [$]\lambda(v-\bar v)\frac{\partial C}{\partial v}>-\frac12vS^2\frac{\partial^2 C}{\partial S^2}>0[$].

A few numerical checks suggest that last idea isn't going to work either. Also, if [$]\eta = 0[$], all s.v. models reduce to a deterministic volatility model with some non-negative [$]\sigma^2(t)[$]. But then the theta will reduce to the Black-Scholes theta (with r=q=0) times an extra [$]\partial/\partial t[$] of [$]U(t,T) \equiv \int_t^T \sigma^2(s) \, ds[$]. But since [$]U_t = -\sigma^2(t) \le 0[$], all these theta's will have the same sign (negative), regardless of the specifics of the s.v. V-drift. 

To check this case numerically, I suggest the OptionCity calculator using the [$]\xi-[$] series with [$]\xi = 0[$] (my symbol for the vol-of-vol). The results there should be exact (the other methods will fail), and it is likely easier than trying to take a degenerate case of the Heston model formula. (Or you can just use the B-S formula with the equivalent deterministic vol).

Finally, thinking some more about what can be proved. Doesn't the absence of calendar spread arbitrage (Gatheral and Jacquier, or here) imply that (for r=q=0), [$]C_T \ge 0[$] in any arbitrage-free model, where T is the maturity date? If so, then this means [$]C_t \le 0[$] for any time-homogeneous model, where t is the time. So there's nothing special about these results for Heston. Now, you said earlier you had found a positive [$]C_t[$] in a non-time-homogeneous model, so we don't have a contradiction there. But, the no-arbitrage result suggests it will be futile to seek a positive theta (with r=q=0) in any time-homogeneous setup. Pretty sure that resolves the thread title issue.

@Alan:

I read your numerical result. Your theoretical rationale is not completely clear to me, for the time being, and I have some doubt, for the time-homogeneous case. I will explain my doubt later. Or it will be clear to me later when I take some time to think it through. 

Two questions for now:
1. Does your assertion regarding the time-homogeneous case include the stochastic volatility case, when the volatility of volatility is not zero?
2. More importantly, how does the time-inhomogeneity of the mean reversion and volatility of volatility impact the result?

1. Yes, assertion covers an arbitrarily complicated risk-neutral diffusion process (n-factor diffusion), requiring only i) r=q=0, and ii) no time dependence in any parameter. (so the process is time-homogeneous). The proof of it is easy: The martingale inequality proved in the stack exchange link I gave proves [$]C_T \ge 0[$]. Time-homogeneity implies [$]C(t,T,\mbox{other parms}) = C(T-t, \mbox{other parms})[$]. Net result: [$]C_t \le 0[$].

(Actually a stronger assertion could be made: same proof seems to work if [$]\{S_t, \mbox{other state variables}_t\}[$] is any time-homogeneous n-dimensional Markov process where [$]S_t[$] is also a martingale. So requiring a 'diffusion' is not necessary; could be an n-factor jump-diffusion or whatever).

2. If parameters have time-dependence, I am not making any general assertion. 

Oh, yes, Alan. You are absolutely right about the time-homogeneous case. I was distracted and blindsided by other factors like the convexity of the second partial derivative of the volatility and thus embarrassingly obtuse.

Anyway, I have the answer to the time-inhomogeneous case now. The answer is affirmative. Let us examine the following simple formulation

[$]\frac{dS}{S} = s(t)\sqrt{v(t)} dB_S[$]

[$]v(t)[$] is some time-homogeneous positive Ito process with its coefficients independent of [$]S[$]. [$]s(t)[$] is a time-dependent deterministic positive function. Now take the example of Heston with volatility of volatility equal to zero and constant positive [$]\lambda[$]

[$]dv = -\lambda(v-1)dt[$]

[$]v(t)[$] is a time-homogeneous process. The total variance
 
[$]\text{var}(t)=\int_t^T s(\tau)^2 v(\tau-t) d\tau.[$] 

So long as [$]v(t)[$] decreases fast (for the specific example of above, [$]v(t=0)>1[$] and [$]\lambda[$] is somewhat large), and [$]s(t)[$] increases fast enough, var[$](t)[$] increases with [$]t[$] and so does the call [$]C(t,S,\sigma)[$] at least for some [$]t[$]. In essence, the mechanism is simply the following. Given positive tuples [$](v_1,v_2)[$] and [$](s_1,s_2)[$] we want 

[$]v_1s_1+v_2s_2<v_1s_2\Longleftrightarrow \big( (\frac{v_2}{v_1}+\frac{s_1}{s_2}<1) \wedge (v_2<v_1) \wedge (s_1<s_2)\big).[$] 


In conclusion, theta of a zero rate European call is always nonpositive for a time-homogeneous process, but can be positive for a time-inhomogeneous process.

I fail to see how that convolution-type integral arises for your var(t), as the model is apparently just [$]dS_t/S_t = \beta(t) dB_t[$] for some deterministic function [$]\beta(t)[$]. What is the call option value formula for [$]C(t,T,S_t)[$] in terms of my [$]\beta(t)[$]? If we can agree on the formula for that, we will agree on [$]C_t[$].