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pcaspers
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### Carr / Madan: A note on sufficient conditions for no arbitrage

I have a question on the paper "A note on sufficient conditions for no arbitrage" by Carr / Madan:

They say \sum_{i=1}^\infty q_{i,j} = 1. To me it seems this sums equals Q_{1,j}  = S_0 - C_{1,j} / K_1 != 1. Do we have add an additional strike, possibly K = K_0 = 0 and attach the probability 1 - Q_{1,j} to it to complete the probability distribution?

Less important, do we have to put an additional condition on the sequence of strikes to ensure (C_{i,j} - C_{i+1,j}) / (K_{i+1}-K_i) tends to zero when i goes to infinity, like "there is an epsilon > 0 s.t. K_{i+1}-K_i > epsilon for all i".

Cuchulainn
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### Re: Carr / Madan: A note on sufficient conditions for no arbitrage

Howdy Peter,
Here is article by Carr and Itkin, might be something to be gleaned?

https://engineering.nyu.edu/sites/defau ... ammaMo.pdf

And closer, MSc of Lykke Rasmussan, section 6 (BTW having difficulty reading your LATEX  ). She has 3 conditions for no-arbitrage.
I think Joerg knows thesis.
Attachments
Mar2012_MasterThesis_LykkeRasmussen.pdf
Last edited by Cuchulainn on December 14th, 2020, 9:52 pm, edited 1 time in total.
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Cuchulainn
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### Re: Carr / Madan: A note on sufficient conditions for no arbitrage

And I quote

$\frac{\partial C}{\partial K} \le 0$

$\frac{\partial^2 C}{\partial K^2} \ge 0$

$\frac{\partial C}{\partial \tau} \ge 0$

Does this make sense?
"Compatibility means deliberately repeating other people's mistakes."
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Alan
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### Re: Carr / Madan: A note on sufficient conditions for no arbitrage

I have a question on the paper "A note on sufficient conditions for no arbitrage" by Carr / Madan:

They say \sum_{i=1}^\infty q_{i,j} = 1. To me it seems this sums equals Q_{1,j}  = S_0 - C_{1,j} / K_1 != 1. Do we have add an additional strike, possibly K = K_0 = 0 and attach the probability 1 - Q_{1,j} to it to complete the probability distribution?

Less important, do we have to put an additional condition on the sequence of strikes to ensure (C_{i,j} - C_{i+1,j}) / (K_{i+1}-K_i) tends to zero when i goes to infinity, like "there is an epsilon > 0 s.t. K_{i+1}-K_i > epsilon for all i".

I agree with your sum and agree that an additional (unstated) condition seems to be needed. Since they are doing a lattice version of the continuum problem, let's first look at that. Fixing the maturity T, the call value is $C(K)$.  Breeden and Litzenberger say that $q(K) \equiv C_{KK}(K)$ can be interpreted, under some conditions, as the pdf of finding the stock price at strike K on maturity. What are the conditions? If the stock price cannot actually reach 0, then for mass preservation we need

$1 = \int_{0^+}^{\infty} C_{KK} \, dK = C_K |^{K=\infty}_{K=0^+}$

The upper end is not problematic: since C(K) is decreasing to 0, $C_K$ must be decreasing to 0 as well.

But, at the lower end, we need the condition that

(*)  $C_K(0^+) = -1$.

In the Carr-Madan lattice version this would be the condition that

$1 = \frac{C_0 - C_1}{K_1 - K_0} = \frac{S_0 - C_1}{K_1}$.

So you need to assume that $C_1 = S_0 - K_1$ as the lattice version of (*).

Cuchulainn
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### Re: Carr / Madan: A note on sufficient conditions for no arbitrage

Alan, If you differentiate Dupire PDE twice with respect to K you get a PDE with zero BC and Dirac payoff, so it's a pdf, yes?. We discussed for BS PDE in same vein. (but transform to (0,1)).a pdf is always >= 0 and that can be proved by PDE theory.

Is Dupire PDE just a common-or-garden PDE at the end of the day?
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Alan
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### Re: Carr / Madan: A note on sufficient conditions for no arbitrage

Dupire/Gyongy/Local vol model is somewhat specialized theory, in the sense that it requires a diffusion process (or at least an Ito process) for the asset price. But, yes, 2 $K$-derivatives should be a PDE for $q(K,T)$ if you know the local vol function. And (usually) 0's at the spatial boundaries, as you say.

Breeden-Litzenberger is more general with minimal assumptions. Basically, just need existence of a (norm-preserving) $q(K,T)$ at a given option expiration $T$. Doesn't require any type of dynamics: diffusions or whatever. The particle can just magically appear at expiration.

pcaspers
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### Re: Carr / Madan: A note on sufficient conditions for no arbitrage

In the Carr-Madan lattice version this would be the condition that

$1 = \frac{C_0 - C_1}{K_1 - K_0} = \frac{S_0 - C_1}{K_1}$.

So you need to assume that $C_1 = S_0 - K_1$ as the lattice version of (*).
Thanks. I am a bit reluctant to make this assumption, since $K_1 > 0$, so this would only hold if the stock has zero volatility?

pcaspers
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### Re: Carr / Madan: A note on sufficient conditions for no arbitrage

And I quote

$\frac{\partial C}{\partial K} \le 0$

$\frac{\partial^2 C}{\partial K^2} \ge 0$

$\frac{\partial C}{\partial \tau} \ge 0$

Does this make sense?
It does. I am setting up an arbitrage-checker on a discrete grid though and want some theoretical foundation for that.

Alan
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### Re: Carr / Madan: A note on sufficient conditions for no arbitrage

In the Carr-Madan lattice version this would be the condition that

$1 = \frac{C_0 - C_1}{K_1 - K_0} = \frac{S_0 - C_1}{K_1}$.

So you need to assume that $C_1 = S_0 - K_1$ as the lattice version of (*).
Thanks. I am a bit reluctant to make this assumption, since $K_1 > 0$, so this would only hold if the stock has zero volatility?
The continuum limit version would be something like
(**) $C(K) = S_0 - K + o(K), \quad \mbox{as} \quad K \rightarrow 0$.

Likely the precise nature of the sub-leading terms depends on whether or not  the stock price cannot reach the origin. It's probably been done carefully in somebody's paper; you might look for discussions of the Breeden-Litzenberger formula generalized to models that allow bankruptcy.  Another thing to check is the behavior implied by Roger Lee's no-arb condition for the implied vol at extreme strikes.

If you don't like my relation on the lattice, an alternative might be:
require only that $C_1 \ge S_0 - K_1$ and $C_1 > C_2$ and interpret any "missing mass" as the probability of finding the stock at the origin (or at least below $K_1$). This was more or less your original idea of how to handle it.

As a practical matter, if your arbitrage checker uses for $K_1$ some smallest quotable strike, then there's certainly a non-zero probability of finding the stock price below that value.

pcaspers
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### Re: Carr / Madan: A note on sufficient conditions for no arbitrage

That makes sense. Actually, in the paper, they start without making any model assumptions, they just impose some no-arbitrage conditions on a discrete set of observed option prices. From that, they construct a discrete probability distribution for $S_{T_j}$ at each maturity $T_j$ compatible with the observed prices. I think it's okay to have $P(S_{T_j} = 0) > 0$ in this model. In a second step they argue that under an additional calendar spread arbitrage condition there exists a discrete martingale generating the observed call prices, thereby proving that the call price matrix is free of (static) arbitrage.

For the arbitrage-checker all that does not really matter, the arbitrage conditions in the paper include the point $K_0=0, C_0=S_0$ in a reasonable way. I was just a little confused about the subsequent derivation of aribtrage-freeness.

Cuchulainn
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### Re: Carr / Madan: A note on sufficient conditions for no arbitrage

Dupire/Gyongy/Local vol model is somewhat specialized theory, in the sense that it requires a diffusion process (or at least an Ito process) for the asset price. But, yes, 2 $K$-derivatives should be a PDE for $q(K,T)$ if you know the local vol function. And (usually) 0's at the spatial boundaries, as you say.

Breeden-Litzenberger is more general with minimal assumptions. Basically, just need existence of a (norm-preserving) $q(K,T)$ at a given option expiration $T$. Doesn't require any type of dynamics: diffusions or whatever. The particle can just magically appear at expiration.
I went through the steps in deriving Breeden-Litzenberger, starting from the expectation for call C. Let's instead take the PDE for C and differentiate wrt K and KK to produce 2 pdes with a Heaviside and Dirac, respectively. I have done it for 'normal' delta and gamma by approximating it by fdm.

Some remarks and questions

1. I don't need to solve FPE to get the transition probability function. I could do it but haven't got around to it yet.
2. My approach looks like  the PDE version of the Derman/Kani probabilistic argument. I would say that they are essentially the same: SDE->Ito->PDE.
3. Computing theta using my PDE approach is not clea. $T$ is nowhere to be found in pde nor payoff. Unless we view $\sigma(T)$ as holding $T$.
4. I'm not sure when/how to pull $\sigma^{2}(K,T)$ out of the hat as it were.

The Continuous Sensitivity Equation (CSE) is discussed here.

https://www.datasim.nl/application/file ... hesis_.pdf

Dupire PDE on interval (0,1)
https://onlinelibrary.wiley.com/doi/epd ... wilm.10014
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

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http://www.datasim.nl

pcaspers
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### Re: Carr / Madan: A note on sufficient conditions for no arbitrage

On 5 I believe it's because they have a specific goal in mind, namely showing that the conditions they state are sufficient for no-arbitrage. The discrete model they construct is the most parsimonious and natural way to do that.