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In the Vanna-Volga approach to pricing first generation exotics, such as single barriers, as I understand it the pricing is as follows:

Let $K,S_t < B$. I'll choose the ATM IV $I_{ATM}$ as the reference volatility for the VV price. Then, $$ UIP(S_t,K,B) = UIP^{BS} (S_t,K,B, I_{ATM}) + p \times \text{Hedge cost} $$ where $UIP^{BS} (S_t,K,B, I_{ATM})$ is the UIP price in a Black-Scholes world with a flat volatility equal to $I_{ATM}$, and $p$ is the touch probability. If I were considering an Up and Out (UOP) then $p$ would be the no-touch probability, right?

So here is my question: the UIP price in a Black-Scholes world with flat vol $I_{ATM}$ is $$ UIP^{BS} (S_t,K,B, I_{ATM}) = \frac{K}{B} C^{BS}(S_t, B^2/K, I_{ATM}) $$

The hedge cost would be $$ \sum_{i=1}^3 x_i \left( C(S_t,K_i) - C^{BS}(S_t,K_i,I_{ATM}) \right) $$ where the $C(S_t,K_i)$ are the market prices of options (i.e. using the actual IVs of the strikes $K_i$), and $x_i$ are determined such that \begin{align} \frac{\partial}{\partial\sigma} UIP^{BS} (S_t,K,B, I_{ATM}) &= \sum_{i=1}^3 x_i \frac{\partial}{\partial\sigma}C^{BS}(S_t,K_i,I_{ATM}) \\ \frac{\partial^2}{\partial\sigma\partial\sigma} UIP^{BS} (S_t,K,B, I_{ATM}) &= \sum_{i=1}^3 x_i \frac{\partial^2}{\partial\sigma\partial\sigma}C^{BS}(S_t,K_i,I_{ATM}) \\ \frac{\partial^2}{\partial S_t\partial\sigma} UIP^{BS} (S_t,K,B, I_{ATM}) &= \sum_{i=1}^3 x_i \frac{\partial^2}{\partial S_t\partial\sigma}C^{BS}(S_t,K_i,I_{ATM}) \end{align}

Is my understanding, in particular regarding the touch probability, correct?

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  • $\begingroup$ I don't have time at the moment to look at this properly. I think this answer might help. $\endgroup$
    – AKdemy
    Sep 13, 2023 at 5:55
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    $\begingroup$ @AKdemy I've seen that answer, and I've read the papers by Fisher, Wystup, etc. They mostly treat KO options, and then they use the survival probability (no touch probability), which makes sense I suppose. But I'm looking at a KI option, and logic would dictate I should then use the touch probability, but for some reason it doesn't feel right to me. $\endgroup$
    – Frido
    Sep 13, 2023 at 7:45

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I do not think you should (can) use the opposite probability (going from p touch to p no touch) because there exists a so called In-out parity: $$European \ vanilla\ option = European\ KI + European\ KO$$

The justification is simple:

  • assume you hold both a KI and KO option
  • if the barrier is untouched, the KO pays a vanilla payoff at expiry
  • if the barrier is touched, the KI pays a vanilla payoff at expiry
  • since the payoff is identical to a vanilla option, its price must also be equal due to no arbitrage

This is also what is mentioned in the Bloomberg article that is linked in the answer I linked in the comment. I'll just quote the relevant section below:

The compromise adjustment described in Equation 11 is justified for a knock-out option, with the vanna portion of the adjustment forced to zero in the limit of the option certainly being knocked out. For knock-in options, one could use a similar formula, replacing psym in Equation 11 with 1− psym, which is the probability of hitting the barrier. Unfortunately, this will not satisfy the no-arbitrage condition that a knock-out option plus a knock-in option equals a vanilla option. With this in mind, we will instead price knock-in options as the difference between the vanna-volga price of a vanilla option and the vannavolga price of the knock-out option.

Equation 11 prices a KO option and looks as follows: enter image description here

and provides the adjustment needed to the Black-Scholes price. I have replicated this in computer code (more or less since I used Bloomberg's OVML to do the bulk work of getting the various values needed to compute the options values). If needed, I can for sure find that somewhere or redo again (not in the near future I am afraid). The weights are chosen to match market prices (at the time this model was built) and are a compromise from opposing views where vega and volga should either be unweighted or weighted by a function that goes to zero as the barrier is approached.

Now, what the paper suggest is to decompose a KI option into: $$ KI\ option = BS\ - KO\ option$$

The Bloomberg article mentions several such adjustments and the same can be found in Wystup: Vanna-volga pricing.

For options with strike K, barrier B and type φ = 1 for a call and φ = −1 for a put, we use the following pricing rules which are based on no-arbitrage conditions. KI is priced via KI = vanilla − KO.

To summarize, I think you are right and you should not use the touch probability. Instead, you should decompose it and compute it via the KO option logic.

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    $\begingroup$ Thanks, this helps. I needed another pair of eyes to look at this. I'll implement it using KIKO parity. Reason I am looking at this is to compare the VV prices to prices obtained here: papers.ssrn.com/sol3/papers.cfm?abstract_id=4556330 $\endgroup$
    – Frido
    Sep 14, 2023 at 5:31
  • $\begingroup$ Glad it helped a bit. Will have to look at that paper some day. Have a good day. $\endgroup$
    – AKdemy
    Sep 14, 2023 at 6:18

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