Interest Rate Correlation and Stock Volatility in Option Pricing Dynamics

Summary

The document asks why a stock-price diffusion changes when a model includes correlation between interest rates and stock returns. In the uncorrelated case, the stock diffusion coefficient is constant; in the correlated case described, it is scaled by the square root of the short rate. The questioner wonders whether this introduces a rate-related component in the stochastic term, but remains uncertain about the rationale.

The setting is Fourier-transform option pricing with interest rates modeled either by a square-root process or an Ornstein–Uhlenbeck process. The question notes that the altered stock dynamics appear with the square-root rate model but not with the Ornstein–Uhlenbeck specification. However, no answer, derivation, or calibration evidence is included, so the document identifies a modeling issue rather than resolving it. In particular, it does not establish that correlation alone requires the diffusion change or explain how the differing rate processes affect the model specification.

Key ideas

• The question examines stock-price dynamics in option pricing when short rates and stock returns are correlated.
• The cited setup scales stock diffusion by the square root of the rate in a square-root rate model.
• The questioner is unsure whether this change represents a rate-linked stochastic component.
• The document contrasts square-root and Ornstein–Uhlenbeck rate specifications but provides no resolution or derivation.

Tags

# Change of the stock price dynamics while pricing using the Fourier transform techniques


# Change of the stock price dynamics while pricing using the Fourier transform techniques












Right now I am trying to understand how we can use the Fourier theorem in obtaining the formula for option pricing (from Zhu J., "Modular pricing of options").

While modeling the interest rate, he considers two cases: when interest rate and stock returns are correlated and when they are not. In the second case, the stock price has dynamics $dS_t=r(t)S_tdt+vS_tdW_t,$ but while considering the correlation case it is changed to $dS_t=r(t)S_tdt+v\sqrt{r(t)}S_tdW_t.$

I don't understand why is it changed.

I think it is because before we had $r(t)$ only as a drift and we need a interest rate term associated with the Wiener process. Am I right? It still doesn't convince me, so is there any better way to explain it? Also, the changed is made only while modeling the interest rate as a square root model and it doesn't happen while considering interest rate as an Ornstein- Uhlenbeck process. Here, I have no idea why.

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.