How would you replicate a claim of the form [$] f [ g(S(T_1),T_1) + g(S(T_2), T_2)] [$] where [$] f [$] is a non-linear function? Carr, Madan, and others have given the recipe for replicating claims of the form [$] g(S_T,T) [$], but I'm not sure how to extend that to the claim [$] f[g(S(T_1),T_1) + g(S(T_2), T_2)][$].

A thought on a quick and dirty approximation. For [$]0 \le t < T_1[$], make some kind of local linear approximation to [$]f[$], so the replication can be done by the method for [$]g(S_T,T)[$], presumably two portfolios of vanilla options. Once [$]t = T_1[$] is reached, the replication can be done exactly by owning one such portfolio, as then [$]S(T_1)[$] is known.

Then, the issue becomes how often to readjust the local linear approximation.

Then, the issue becomes how often to readjust the local linear approximation.

Assuming we are in a complete markets setting and S(t) is a Markov process, can’t we just solve for the value of the claim by integrating the payoff function scaled by the proper numeraire against the bivariate stock price density (under your favorite pricing measure) and differentiate the result wrt the initial stock price to get the delta? A martingale representation theorem should ensure self financing of the strategy over time. Perhaps somebody could work out an example for S(1)*S(2)?

I assumed the question was about replicating with vanillas. Valuation and delta hedging are straightforward I think.

Sorry, I should have been more specific, as otherwise the question is too general.

So, specifically, assuming for the moment that Black-Scholes assumptions hold, the question is can I replicate the following

[$] \sqrt{C^2(S_1,T_1,K_1,T_2) / S_1^2 + C^2(S_2,T_2, K_2,T_3) / S_2^2} [$]

with vanilla options. Under the square root are also vanilla options, with known strikes K_1 and K_2, and T_1, T_2 and T_3 are also known.

And as a next step, which is what I'm really looking at, can we replicate

[$] \sqrt{C^2(S_1,T_1,S_1,T_2) / S_1^2 + C^2(S_2,T_2,S_2,T_3) / S_2^2} [$]

with**forward start ** options. The options under the square root are now forward start options. Let's assume there's a reasonably liquid market in forward starts, which for main equity indices and FX pairs is an OK assumption I think.

EDIT: Clearly, even if this can be replicated using vanillas/forward starts, the weights are going to be very messy, but I'm not even sure if in theory it is possible to replicate these with vanilla/forward start options, and whether the replication is going to be static or dynamic.

[Maybe some of you can see what (hopeless thing) I'm trying to do/achieve with this.]

So, specifically, assuming for the moment that Black-Scholes assumptions hold, the question is can I replicate the following

[$] \sqrt{C^2(S_1,T_1,K_1,T_2) / S_1^2 + C^2(S_2,T_2, K_2,T_3) / S_2^2} [$]

with vanilla options. Under the square root are also vanilla options, with known strikes K_1 and K_2, and T_1, T_2 and T_3 are also known.

And as a next step, which is what I'm really looking at, can we replicate

[$] \sqrt{C^2(S_1,T_1,S_1,T_2) / S_1^2 + C^2(S_2,T_2,S_2,T_3) / S_2^2} [$]

with

EDIT: Clearly, even if this can be replicated using vanillas/forward starts, the weights are going to be very messy, but I'm not even sure if in theory it is possible to replicate these with vanilla/forward start options, and whether the replication is going to be static or dynamic.

[Maybe some of you can see what (hopeless thing) I'm trying to do/achieve with this.]

Interesting. There are various levels of possibility here. If you believed in a constant vol world then it's trivial. If you believe in asset-independent vol it's still trivial. So it comes down to what you think vol is going to do. Ideally you'd be looking for a vol model-independent replication. Seems unlikely --- but then so often in quant finance things I think are unlikely seem to work out, it's one of the fundamental laws that quant finance is v easy! Maybe you'd just want to get rid of as much model risk as possible and then delta hedge what's left over.

Yes, and yes it should be relatively easy, but unfortunately on the second point, 95% of the quant finance papers I read nowadays I have no clue what they're talking about and how it can be applied. But I guess that's my problem.Ideally you'd be looking for a vol model-independent replication. Seems unlikely --- but then so often in quant finance things I think are unlikely seem to work out, it's one of the fundamental laws that quant finance is v easy!

Thinking about the problem below (or above when not logged in) a bit more, I think dynamic replication in the sense of dynamically rebalancing a portfolio of options, is perhaps the way to approach it.

For instance, at T_0 I hold a portfolio of forward start options to replicate the value of

[$] \sqrt{C^2(S_1,T_1,S_1,T_2)/S_1^2} [$]

at [$] T_1 [$].

At [$]T_1[$] I liquidate the portfolio and receive the cash value, and set up the portfolio that replicates the value of

[$] \sqrt{C^2(S_1,T_1,S_1,T_2)/S_1^2 + C^2(S_2,T_2,S_2,T_3)/S_2^2} [$]

at [$] T_2 [$] using a forward start portfolio of option again, where now [$] C^2(S_1,T_1,S_1,T_2)/S_1^2 [$] is just the known constant cash amount, which I have replicated in [$] [0,T_1] [$], and so forth.

I don't know, do we think that this should or could work? I think the strategy is self-financing? If above dynamic replication strategy is correct, then the question is reduced to how to replicate [$] \sqrt{C^2(S_1,T_1,S_1,T_2)/S_1^2} [$] at t=0, since at T=1,2,3, etc it's basically the same type of replication (with some additional cash/constant value under the square root).

Last edited by frolloos on May 24th, 2018, 1:14 pm, edited 7 times in total.

I looked at something similar to your first form a while ago...

One thing I tried at was to introduce the bivariate density function (for S(T1) and S(T2)) and write it in terms of marginals and a copula function -- which I was quite happy to assume was normal with correlation given by the time overlap. Then you can replace the marginals with option derivatives as usual, but integrating by parts throws up a pile of derivatives of the copula function which are annoying...

One thing I tried at was to introduce the bivariate density function (for S(T1) and S(T2)) and write it in terms of marginals and a copula function -- which I was quite happy to assume was normal with correlation given by the time overlap. Then you can replace the marginals with option derivatives as usual, but integrating by parts throws up a pile of derivatives of the copula function which are annoying...

Might be relevant here: Robust Bounds for Forward Start Options

Interesting. There are probably some Frechet copulas in there somewhere in disguise...

amike, the bivariate density / copula approach works if the we have two time periods only (T1 and T2). But won't it become intractable if more time steps are involved? That said, if you have something you could share on what/how you did I'd be happy to look at it and learn from it.

Thanks for the relevant reference Alan.

I'm still struggling a lot with this problem, I think dynamic replication with portfolios of options could shed some light. Will post something if I make progress, in a year or so..

Thanks for the relevant reference Alan.

I'm still struggling a lot with this problem, I think dynamic replication with portfolios of options could shed some light. Will post something if I make progress, in a year or so..

Ok, I think I got it.

I'll take a simpler claim with the following payoff at [$] T_2 [$]:

[$] (S_2 + S_1 - K)_+ [$]

which is basically an Asian option with two observation dates.

At [$] T_0 [$] I need to replicate the following "payoff" at [$] T_1 [$]:

[$] C(S_1, K - S_1, T_1,T_2) = E_1 [(S_2 + S_1 - K)_+] [$]

Because if I replicate this payoff, then at [$] T_1 [$] that's exactly what I need to have to replicate the payoff at [$] T_2 [$].

But seen from [$] T_0 [$], the "payoff" [$] C(S_1, K - S_1, T_1,T_2) [$] is just a slightly different type of forward start option than the conventional forward start option. So [$] C(S_1, K - S_1, T_1,T_2) [$] should be able to be replicated with a strip of "vanilla" forward start options at [$] T_0 [$].

I think the dynamic replicating strategy I describe above is self-financing and works, doesn't it?

I'll take a simpler claim with the following payoff at [$] T_2 [$]:

[$] (S_2 + S_1 - K)_+ [$]

which is basically an Asian option with two observation dates.

At [$] T_0 [$] I need to replicate the following "payoff" at [$] T_1 [$]:

[$] C(S_1, K - S_1, T_1,T_2) = E_1 [(S_2 + S_1 - K)_+] [$]

Because if I replicate this payoff, then at [$] T_1 [$] that's exactly what I need to have to replicate the payoff at [$] T_2 [$].

But seen from [$] T_0 [$], the "payoff" [$] C(S_1, K - S_1, T_1,T_2) [$] is just a slightly different type of forward start option than the conventional forward start option. So [$] C(S_1, K - S_1, T_1,T_2) [$] should be able to be replicated with a strip of "vanilla" forward start options at [$] T_0 [$].

I think the dynamic replicating strategy I describe above is self-financing and works, doesn't it?

It becomes messy with two times already, and perhaps not really what you are after in hindsight... briefly:

Writing [$]H[$] as the bivariate copula function for the joint distribution of the asset at [$]T_1[$] and [$] T_2[$], the value of an option with payoff [$] g(S(T_1),S(T_2)) [$] is ([$] f_1(S) [$] is the marginal density function for the value at [$]T_1[$], etc.):

[$]

v=\int_0^\infty dK_1 \int_0^\infty dK_2 f_1(K_2) f_2(K_2) H(F_1(K_1),F_2(K_2)) g(K_1,K_2)

[$]

Then you replace the densities with derivatives of the vanilla option prices as usual: [$]f(K)=\partial_K^2C(K)=\partial_K^2P(K)[$], break up the domains of integration into regions below (use the put) and above (use the call), and integrate by parts a couple of times. Standard stuff.

What results does look like weighted sum over vanilla puts/calls, but products this time rather than as a simple weighted sum, which is perhaps not really what you are after... I was interested in an approximate valuation formula rather than a replicating portfolio, so this was ok for me.

If you continue you also end with derivatives of the quantity: [$]Hg[$], and when you take a derivative of the copula function as it appears, you get back the density again:

[$]

\partial_{K_1}H(F_1(K_1),F_2(K_2))=f_1(K_1)\partial_{u_1}H(u_1,u_2)|_{u_1=F(K_1),u_2=F(K_2)}

[$]

which you either have to approximate away or replace again (which would lead to more terms)...

Writing [$]H[$] as the bivariate copula function for the joint distribution of the asset at [$]T_1[$] and [$] T_2[$], the value of an option with payoff [$] g(S(T_1),S(T_2)) [$] is ([$] f_1(S) [$] is the marginal density function for the value at [$]T_1[$], etc.):

[$]

v=\int_0^\infty dK_1 \int_0^\infty dK_2 f_1(K_2) f_2(K_2) H(F_1(K_1),F_2(K_2)) g(K_1,K_2)

[$]

Then you replace the densities with derivatives of the vanilla option prices as usual: [$]f(K)=\partial_K^2C(K)=\partial_K^2P(K)[$], break up the domains of integration into regions below (use the put) and above (use the call), and integrate by parts a couple of times. Standard stuff.

What results does look like weighted sum over vanilla puts/calls, but products this time rather than as a simple weighted sum, which is perhaps not really what you are after... I was interested in an approximate valuation formula rather than a replicating portfolio, so this was ok for me.

If you continue you also end with derivatives of the quantity: [$]Hg[$], and when you take a derivative of the copula function as it appears, you get back the density again:

[$]

\partial_{K_1}H(F_1(K_1),F_2(K_2))=f_1(K_1)\partial_{u_1}H(u_1,u_2)|_{u_1=F(K_1),u_2=F(K_2)}

[$]

which you either have to approximate away or replace again (which would lead to more terms)...

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