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Alan
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Re: Valuation of put with reflecting barrier

October 28th, 2020, 4:00 pm



1. What are sequence of binomial lattices,  suitable norm to the continuous model (that should be standard),??
2. I can't visualise what's going on
3. I can't see the maths/numerics. BTW what does the continuous model look like? SDE, PDE, analytic, whatever?


inter os atque offam
1. Take a sequence of standard binomial lattices for a GBM stock price process each with (boundary) nodes at S=100. The originating node could be S(0)>100,
say S(0)=110. Each sequence member has decreasing time step size. A typical convergence notion for lattice to continuum models is weak convergence of measures.

2. Visualize a reflecting GBM process with reflection at S=100, starting at, let's say S(0) = 110. On the lattice, visualize a binomial lattice process with reflection at a lower level.

3. For the transition density, reflecting GBM has standard BS pde generator (no killing term) with standard zero derivative bc for reflection at a level. For an option value function, standard BS pde with zero derivative bc. However, a caveat is that the option values may be ``illegitimate" in the presence of arbitrage opps. The transition density should be fine, as well as expectations under the "physical" process. The simplest setup might be a world with zero physical drift and zero interest rates.

The main point is the implications of a trading strategy (on the lattice) that buys the stock whenever it reaches the barrier. If you bought the stock with borrowed funds, this is obviously an arbitrage opp in each lattice world. But, can it be shown rigorously that it converges to an arbitrage opp in the continuum limit?  (The author of the paper at issue believes "no". bearish & I believe "yes").
 
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Re: Valuation of put with reflecting barrier

October 28th, 2020, 5:23 pm

The main point is the implications of a trading strategy (on the lattice) that buys the stock whenever it reaches the barrier.

Fair enough, but my point above is that there is a strategy (one which replicates the delta of a short put struck at the barrier) which need never involve buying at the barrier, and which ensures we always receive the time value of that put (clearly the payoff is always zero).
 
There is a deeper point here relating to my earlier question about sigma root t. Does it mean the standard deviation of the probability distribution of the logarithm of the asset price at expiration? But the probability distribution here is severely truncated. Or does it mean something else, given that we can derive the BS formula on the assumption of such a truncated distribution? Note, a formula which contains sigma root t.
 
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Re: Valuation of put with reflecting barrier

October 28th, 2020, 7:49 pm

On the last point, I don't see how you can "derive the BS formula on the assumption of such a truncated distribution". Maybe I am missing something. Can you really derive it?
 
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Re: Valuation of put with reflecting barrier

October 28th, 2020, 10:01 pm

On the last point, I don't see how you can "derive the BS formula on the assumption of such a truncated distribution". Maybe I am missing something. Can you really derive it?

To be specific, it is trivial to derive the BS formula consistent with the assumption of such a truncated distribution, or indeed any other distribution. The distribution at expiry is neither a necessary nor sufficient condition of the derivation.  Is that not obvious? (Perhaps not).
 
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Re: Valuation of put with reflecting barrier

October 29th, 2020, 12:54 am

On the last point, I don't see how you can "derive the BS formula on the assumption of such a truncated distribution". Maybe I am missing something. Can you really derive it?

To be specific, it is trivial to derive the BS formula consistent with the assumption of such a truncated distribution, or indeed any other distribution. The distribution at expiry is neither a necessary nor sufficient condition of the derivation.  Is that not obvious? (Perhaps not).
Suspect you mean the BS pde, not the BS put or call value formula?  
 
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Re: Valuation of put with reflecting barrier

October 29th, 2020, 9:10 am

On the last point, I don't see how you can "derive the BS formula on the assumption of such a truncated distribution". Maybe I am missing something. Can you really derive it?

To be specific, it is trivial to derive the BS formula consistent with the assumption of such a truncated distribution, or indeed any other distribution. The distribution at expiry is neither a necessary nor sufficient condition of the derivation.  Is that not obvious? (Perhaps not).
Suspect you mean the BS pde, not the BS put or call value formula?  
I mean both. You derive the PDE, then a fairly simple integration gives you the Formula. 

What really puzzles me is that there appears to be nothing wrong with the derivation given in the paper. As the author says, the process in (A.4) is a semimartingale, because it can be decomposed into a local martingale (the Wiener term) and two finite variation processes, the drift and the reflection. Then he integrates the intrinsic value of the put over the risk-neutral density, that is, we integrate to give the valuation formula in section 5 of the paper – see equations (3) to (8). 
 
He also argues that “The instantaneous nature of the reflection means that the price does not spend any finite time at the barrier, so no arbitrage opportunities are created (we can never buy at the barrier with certainty of a price rise).” The latter point is true, but as long as we trade the synthetic through to expiry, a profit is guaranteed. And in any case, there cannot be two formulas giving different values for the same option contract. Hence there seems to be some kind of paradox.
 
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Re: Valuation of put with reflecting barrier

October 29th, 2020, 2:46 pm




To be specific, it is trivial to derive the BS formula consistent with the assumption of such a truncated distribution, or indeed any other distribution. The distribution at expiry is neither a necessary nor sufficient condition of the derivation.  Is that not obvious? (Perhaps not).
Suspect you mean the BS pde, not the BS put or call value formula?  
I mean both. You derive the PDE, then a fairly simple integration gives you the Formula. 
1. The PDE has doubtful validity at a reflecting barrier.
2. Even if the PDE were valid, the Euro-style option value is the expected value of the payoff over the terminal distribution. Since the latter is *not* the standard BS lognormal distribution, the option value (put or call) is *not* the BS formula.

If there are paradoxes, it's because the model admits arbitrage, although we don't yet have a proof or a good literature cite. I still suspect it's long-resolved in some careful paper somewhere. Failing finding that, I may try to develop an argument from the lattice approach above when I get some time. 
 
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Re: Valuation of put with reflecting barrier

October 29th, 2020, 3:39 pm

Agram and Øksendal (https://arxiv.org/abs/1909.12578) quote Jarrow and Protter (JBF 2005) and Karatzas and Shreve (1998) to the effect that asset price dynamics on a Brownian filtration with singular drift at a barrier is complete and admits arbitrage. That’s good enough for me.
 
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Re: Valuation of put with reflecting barrier

October 29th, 2020, 4:22 pm

Good find. Following up, I found the Jarrow and Protter they cited. Looks pretty technical but, like you, I trust all these authors.
 
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Re: Valuation of put with reflecting barrier

October 29th, 2020, 4:29 pm

Many thanks for the references.
 
“the Euro-style option value is the expected value of the payoff over the terminal distribution.”
 
Which I question, if the value of a contingent claim is the mathematical value of the self-financing strategy with replicates it.
 
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Re: Valuation of put with reflecting barrier

October 29th, 2020, 4:45 pm

Many thanks for the references.
 
“the Euro-style option value is the expected value of the payoff over the terminal distribution.”
 
Which I question, if the value of a contingent claim is the mathematical value of the self-financing strategy with replicates it.
I don't know if it applies here, but I do know examples where there are two different plausible option values and it's not clear which one is 'right'. It happens when you have a call option when the stock price process is a strictly local martingale. 

One example is the lognormal SABR model with positive stock-vol correlation, which I discuss in my "Option valuation under Stochastic Volatility II" book, ch. 8.
You can find a table, showing the two values in an example. Give me a few minutes and I'll post it. Another is the CEV model [$]dS = S^p dW[$], with [$]p>1[$].which I discuss in my first book. (Also, here's some discussion of the issue by Peter Carr and coauthors)

In my table, the expected terminal value of the call = 'Low' value = minimal dynamic replicating value. The 'High' value is the one that preserves put-call parity.  The issue is only with the call  option; the put value is unique. The issue arises because [$]E[S_T] < S_0[$] when you have a strict local martingale, even though the SDE has no drift term. 
SABR.Lewis.jpg
 
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Re: Valuation of put with reflecting barrier

October 29th, 2020, 6:42 pm

Thanks!
 
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Re: Valuation of put with reflecting barrier

October 30th, 2020, 11:20 am

 
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Re: Valuation of put with reflecting barrier

October 30th, 2020, 12:14 pm

On the last point, I don't see how you can "derive the BS formula on the assumption of such a truncated distribution". Maybe I am missing something. Can you really derive it?

To be specific, it is trivial to derive the BS formula consistent with the assumption of such a truncated distribution, or indeed any other distribution. The distribution at expiry is neither a necessary nor sufficient condition of the derivation.  Is that not obvious? (Perhaps not).
I am getting lost is a sea of words.
What is BS formula with a 'truncated distribution'? Do you have an example?

BTW Are we talking about BS PDE on [$](0,\infty)[$] or  on [$](0,H[$] )? Is reflection ==> Neumann boundary condition at [$] S = H[$]?
My C++ Boost code gives
262537412640768743.999999999999250072597198185688879353856337336990862707537410378210647910118607313

http://www.datasimfinancial.com
http://www.datasim.nl
 
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Re: Valuation of put with reflecting barrier

October 30th, 2020, 4:35 pm

"What is BS formula with a 'truncated distribution'? "

My actual words were "derive the BS formula on the assumption of such a truncated distribution", which I followed up, to clarify, 

To be specific, it is trivial to derive the BS formula consistent with the assumption of such a truncated distribution, or indeed any other distribution. The distribution at expiry is neither a necessary nor sufficient condition of the derivation. 

I.e. we can derive both the PDE and the formula without making any assumptions about the terminal distribution.
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