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as the question stated. I want to find a way to calculate the delta for a barrier option under the Heston model. Is there any closed-form solution? All I can find is:

  • Delta of a barrier, but under BSM
  • Delta under Heston, but vanilla

If there's no closed-form solution, I guess I have to use Monte Carlo to calculate that Greek?

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  • $\begingroup$ Added some additional comments. Just making sure you see them as they are relevant. $\endgroup$
    – user34971
    May 5, 2022 at 20:08

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I will discuss only the single barrier up-and-in put (UIP) with barrier $B \geq K, S(t)$. Other single barriers treated similarly.

The answer is it depends: If correlation between the instantaneous stochastic volatility and the asset price is zero then the price of an UIP under Heston model (or any other stoch vol model without jumps in the asset price) is $$ UIP(t) = \frac{K}{B} C^{SV} \left(S(t), \frac{B^2}{K} \right). $$ Here $C^{SV}$ stands for the vanilla call option price under stochastic volatility model (Heston or SABR or your SV model of choice).

Hence, since under Heston there is a semi-analytical expression for the delta of a vanilla, you also have a semi-analytical expression of the delta of a UIP (and similarly for other single barriers).

If correlation is non-zero, then you indeed have to do a Monte Carlo. However, take a look at equation (16) in a recent note of mine that simplifies the MC simulation (reduces dimensionality by one).

Important note:

  1. The closed form formula I wrote above holds when interest rate and dividend yield are zero. When they are nonzero you would still need to do MC unfortunately.
  2. The solution to this could be by product design: instead of quoting barrier options on spot prices $S(t)$, quote them on futures / forward prices $F(t) = S(t) e^{(r-q)(T-t)}$ which are drift less, and hence you can apply the closed form formula.
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    $\begingroup$ I believe that for pricing barrier options one could solve the PDE in which case model Delta and Gamma are calculated (for free) from the finite difference grid. In the Monte Carlo method, model Delta and Gamma will require regenerating the paths with the bumped spot as initial condition. $\endgroup$
    – AKdemy
    May 5, 2022 at 20:29
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    $\begingroup$ @AKdemy Yes, good point. In fact, for the OP the following paper might be very useful as it treats specifically Heston: sciencedirect.com/science/article/pii/S0898122112003215 $\endgroup$
    – user34971
    May 5, 2022 at 20:36

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