The volatility can be a discontinuous function of S while preserving the result, as long as it only depends on the contemporaneous stock price. That is, the local volatility can jump up when the stock price goes down without making the call delta negative, as long as it jumps back down again if the stock price goes up.

Is this a new creeping requirement? 

No.
First, of all, bearish carefully said "permanently double" in the counter-example.
Second, the behavior just asserted follows from simply having a generic (smooth or non-smooth) [$]\sigma(S,t)[$].

Part of the answer is seeing the distinction in these two cases. (counter-example v. local vol). 

BTW, there should be a nice closed-form call value formula in the counter-example (perhaps involving an integral), 
which could be checked for somewhere negative S-derivative. 

 

 

Yes, it can be represented as the classic down and out call with a constant barrier but a time dependent rebate given by the BS call value with the higher vol minus the call value with the original vol. It’s probably optimistic to get an explicit solution for the integral of that against the first passage time density, but since the stock price only enters through said density, it should be a relatively tidy problem.

Maybe this approach is useful

This paper discusses necessary and suffiicient conditons for a PDE solution to be convex when the initial condition is convex.
Theorem 2.1 seems to be the main result in terms of convexity of coefficient of second derivative.

https://www.researchgate.net/publicatio ... _equations

In a more general real analysis, 
. A twice-differentiable function of a single variable is convex if and only if its second derivative is nonnegative on its entire domain.
. A differentiable function of one variable is convex on an interval if and only if its derivative is monotonically non-decreasing on that interval. If a function is differentiable and convex then it is also continuously differentiable.

I suspect PDE methods can be applied here since we can use the maximum principle the PDEs for delta and gamma as discussed here.

chrome-extension://efaidnbmnnnibpcajpcglclefindmkaj/https://www.datasim.nl/application/file ... hesis_.pdf

Here was my take on BGW (which has been mentioned) from my first book:

I’m not even sure what to prompt (-: I can relatively easily imagine scenarios when volatility depends on the stock price and the option value changes up or down. What’s this problem asking about?

As I mentioned, the question is 

What are conditions on the underlying stock price process such that C is an increasing and convex function of S(t)? (Q1) 

And it seems bearisch has slipped in a corollary

what conditions on the underlying price process guarantee that the first derivative of the option price is bounded by the sup and inf of the slope of said payoff function. == Q2

Are Q1 and Q2 to be proved?

// BTW bearish, what's a slope?

So the price equation needs to be linear in the second derivative of C wrt S? And it doesn’t need to be *continuous* diffusion?
What’s the answer then?

What's the answer then? BGW + see my download.
bearish's counter-example 'works' (escapes BGW proof) because the volatility in that one is not Markov.

What works? 
Basically, (i) any 1D (Markov) diffusion with generic local vol; (ii) any 2D (Markov) stochastic vol diffusion with volatility SDE (+ correlation) independent of stock price.

Markov is kind of redundant, but the point is that a more general (non-Markov) Ito process (like the counter-example) is excluded.

Both cases seem to translate to a linear PDE condition. And the markov property guarantees no regime-switching.

Both cases seem to translate to a linear PDE condition. And the markov property guarantees no regime-switching.

Maybe I missed it but what is a linear PDE condition?

Theorem 2.1 of Lions and Musiela addresses the original question.. 

https://www.researchgate.net/publicatio ... _equations

Umm - I’m in a bit of a rush here, but by restricting yourself to a PDE with one spatial dimension, you have already excluded all the interesting cases.

Umm - I’m in a bit of a rush here, but by restricting yourself to a PDE with one spatial dimension, you have already excluded all the interesting cases.

I was trying first of all to understand what the requirements as you sketched them were. 

Both cases seem to translate to a linear PDE condition. And the markov property guarantees no regime-switching.

Maybe I missed it but what is a linear PDE condition?

Theorem 2.1 of Lions and Musiela addresses the original question.. 

https://www.researchgate.net/publicatio ... _equations

Sorry, I meant linear in the second derivative, partial^2 C / partial S^2. Then if I differentiate wrt S, I get an equation for Delta with a similar structure”linear” structure, ergo the same terminal condition holds.
I hope that makes any sense (-:

Umm - I’m in a bit of a rush here, but by restricting yourself to a PDE with one spatial dimension, you have already excluded all the interesting cases.

I was trying first of all to understand what the requirements as you sketched them were. 

I had, at some point, restricted the stock price process to be adapted to a filtration generated by a 1-d Brownian motion. That rules out what we usually refer to as stochastic volatility models, but puts only weak restrictions on the volatility process. In particular, it can depend on the past behavior of the stock price, making the stock price process non-Markovian, as in my little toy example. In this case, the option price is no longer determined by the simple parabolic PDE, and other methods of analysis are required.

Umm - I’m in a bit of a rush here, but by restricting yourself to a PDE with one spatial dimension, you have already excluded all the interesting cases.

I was trying first of all to understand what the requirements as you sketched them were. 

I had, at some point, restricted the stock price process to be adapted to a filtration generated by a 1-d Brownian motion. That rules out what we usually refer to as stochastic volatility models, but puts only weak restrictions on the volatility process. In particular, it can depend on the past behavior of the stock price, making the stock price process non-Markovian, as in my little toy example. In this case, the option price is no longer determined by the simple parabolic PDE, and other methods of analysis are required.

PDEs don't do memory in general. Are we talking about PIDEs?

The general solutions would be found via probabilistic methods, but I am sure there are special cases that yield to other approaches (like PDEs in the Markovian local volatility scenario).

At the risk of being annoying, I’m curious what I’m missing about the problem, which seems to almost state its own solution.

I differentiated the pricing PDE (1) with respect to S, obtaining the equation for Delta (2). I then noticed that if the diffusion term is linear, it will always “erase” any interior maxima or minima - just like in the heat equation, where temperature can’t spontaneously form a local spike beyond the initial or boundary values.

Therefore, a sufficient condition for Delta to remain bounded within [m, M] is that the price PDE is linear in its second derivative. This guarantees that the differentiated PDE for Delta is also linear, and hence the slope at earlier times cannot escape the bounds set by the boundary condition at expiry.

Hmm - your calculation reminds me of Peter Carr’s “Deriving derivatives of derivative securities” paper, which is explicitly set in a Black-Scholes framework. I’ll try to muster some mental energy to work out at least one little example of my point tomorrow.