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Adding Constant Interest to Andersen's Heston Scheme

Article Quant Q&A · Author: JosephConrad
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Summary

The document addresses whether a martingale-corrected Andersen discretization for the Heston model needs adjustment when the model has a nonzero constant interest rate. The answer endorses adding the risk-free accrual over each time step to the exponent while retaining the correction and variance-dependent terms from the zero-rate scheme.

The justification uses a forward price with a fixed maturity: under constant rates, this forward removes deterministic carry from the spot process. Applying the zero-rate Andersen scheme to the forward and converting back to spot produces the added interest-rate term. This is an algebraic argument rather than a numerical test, and it applies to the stated constant-rate setup; the document gives no extension to stochastic rates or other carry assumptions.

Key ideas

  • A constant interest rate contributes an accrual term to the log spot update over each simulation step.
  • The zero-rate Andersen correction terms can be applied to a forward price process.
  • Converting the simulated forward back to spot yields the proposed rate adjustment.
  • The argument relies on constant rates and does not establish a result for stochastic interest rates.

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# Martingale correction for Andersen scheme with Interest Rate


# Martingale correction for Andersen scheme with Interest Rate












I have implemented martingale correction to my Andersen scheme for Heston model, as it is in the paper (page 19-22):

http://www.ressources-actuarielles.net/EXT/ISFA/1226.nsf/0/1826b88b152e65a7c12574b000347c74/$FILE/LeifAndersenHeston.pdf

However, Andersen derived martingale correction for asset process without interest rate, but I have interest rate in my model and implementation.

I think that implementing martingale correction for Andersen scheme with non-zero interest rate like this:

$$\hat{X}(t + \Delta) = \hat{X}(t) * exp(r \Delta + K_0 + K_1 \hat{v}(t) + K_2 \hat{v}(t+\Delta) + \sqrt{K_3 \hat{v}(t) + K_4 \hat{v}(t+\Delta)} \cdot Z)$$

should be fine, since interest rate is constant, but I'm not 100% sure.

Could anyone advice on this?

## Answer by Gordon (score 1, accepted)

https://quant.stackexchange.com/a/22738

Your adjusted scheme is correct. Basically, taking a maturity $T$, you can consider the forward price process $F_t^T = S_t e^{r(T-t)}$. You apply the Andersen scheme to $F_t^T$ and then note that \begin{align*} S_{t+\Delta} &= F_{t+\Delta}^T e^{-r(T-(t+\Delta))}\\ &=F_t^T \exp(\ \Box \ ) e^{-r(T-(t+\Delta))}\\ &=S_t e^{r(T-t)}\exp(\ \Box \ ) e^{-r(T-(t+\Delta))}\\ &=S_t \exp(r\Delta + \ \Box \ ), \end{align*} where the terms included in $\ \Box \ $ are the terms in the Andersen scheme with zero interest rate.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.