Control Variates for Heston Barrier and Basket Option Pricing
Summary
The document asks how control variates might improve Monte Carlo pricing for European and American down-and-in basket puts under a Heston model simulated with the QE scheme. It considers using a vanilla put as a control for a barrier payoff, while noting that a distant barrier may weaken correlation and that no closed-form Heston basket price is available. It also proposes constructing controls from individual asset prices and analytically priced Heston options, using the known control expectations to adjust the simulated basket estimate.
These are questions and candidate approaches rather than a resolved method. The document provides no numerical results or test of the proposed controls, and it leaves open how to choose effective controls for American exercise or how to handle the barrier. The cited basket-option paper and a deterministic-volatility alternative are pointers for further study, not evidence that either approach works for this setup.
Key ideas
- Control variates can reduce Monte Carlo variance when the control payoff is strongly correlated with the target payoff.
- A vanilla put may be a weak control for a down-and-in basket option if the barrier changes payoff behavior substantially.
- The document proposes using analytically priced single-asset Heston options as controls for a worst-of basket payoff.
- The proposed controls and their effectiveness are not evaluated in the document.
- The American barrier option case remains unresolved.
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Full text
# Control Variate Barrier Basket Option
# Control Variate Barrier Basket Option
I need to improve the speed of convergence of PRNG Monte Carlo. I'm opening a new thread for that purpose and I have question / need confirmation about the algorithm.
I'm pricing options with Heston, QE scheme.
- For an European Down and In Put on a basket, I was thinking about using a vanilla put as a control variate (CV). I read that the correlation between the initial and modified (CV one) random variables had to be highly correlated to produce significant reduction. In the case my barrier is -30% from the Spot, I'm not sure this will be very correlated... What do you think ? By the way, there's no closed form formula for the Heston basket. How should I proceed then ?
- For an American D&I put ? What's the CV ? Same thing, no closed form formula.
Edit: To be a little more precise
Let's say I'm dealing with a WorstOf basket: $$\text{Pay off}(T)=\max(0,K-\min(S^T_1,S^T_2))$$ What I was thinking as a CV is the following: $$CV_1 = \max(0,K-E(S^T_1)-S^T_2)$$ $$CV_2 = \max(0,K-E(S^T_2)-S^T_1)$$ Then this would be equivalent to pricing 2 vanilla put with new strikes $K_i=K-E(S^T_i)$ where $E(S^T_i)$ is given by the analytical Heston price.
Therefore the new payoff would be:
$$\text{New Payoff} = e^{-rt}\dfrac{1}{n}\sum\limits_{j=1}^{n}\text{Old Payoff} - CV_1^{(j)} - CV_2^{(j)} + E(S^T_1) + E(S^T_2)$$
The idea can be found in the paper below: https://www.researchgate.net/profile/Paolo_Pellizzari/publication/23744210_Efficient_Monte_Carlo_Pricing_of_Basket_Options/links/54a176970cf267bdb902bee9/Efficient-Monte-Carlo-Pricing-of-Basket-Options.pdf
An another idea could be as below but I don't know how to apply the methodology (pricing with a deterministic volatility as close as Heston vol) http://www.iaeng.org/IJAM/issues_v45/issue_1/IJAM_45_1_07.pdfShown 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.