Barrier Option Delta Under Heston: Zero-Correlation Shortcut and Monte Carlo
Summary
The document explains how to obtain the delta of a single-barrier up-and-in put under a stochastic-volatility model. In the special case of zero correlation between the asset and volatility shocks, and with zero interest rates and dividend yield, the barrier price can be related to a vanilla call price with transformed strike and a scaling factor. Since Heston vanilla call prices have semi-analytical Greeks, this relationship also gives a semi-analytical route to the barrier delta.
When correlation is nonzero, the response says Monte Carlo is needed, while pointing to a dimensionality reduction that can simplify simulation. It also cautions that nonzero rates or dividends invalidate the stated formula; one proposed workaround is to define the contract on a forward price, whose drift is zero under the stated setup. The result is limited to the specified barrier type and assumptions, although the author says other single barriers can be handled similarly. No numerical results or implementation details are given.
Key ideas
- With zero asset-volatility correlation and zero rates and dividends, an up-and-in put can be priced through a transformed vanilla call under stochastic volatility.
- A semi-analytical vanilla Heston delta can then be used to obtain the barrier option delta.
- Nonzero correlation requires Monte Carlo in the approach described, with a dimensionality reduction suggested to simplify simulation.
- The stated closed-form relationship assumes zero rates and dividends; forward-based contract design is offered as a possible workaround.
- The explanation focuses on a single up-and-in put and does not provide numerical validation.
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Full text
# Delta of a barrier option under Heston model
# Delta of a barrier option under Heston model
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?
## Answer by user34971 (score 2, accepted)
https://quant.stackexchange.com/a/70759
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:
- 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.
- 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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.