Deriving a Mean-Reverting Spot SDE from Futures Dynamics
Summary
The document derives a spot-price stochastic differential equation from a model for futures prices whose volatility decays with time to maturity. Setting the spot equal to the futures price at maturity, it first rewrites the log spot using the initial futures curve, an integrated variance adjustment, and a stochastic integral.
The derivation differentiates this expression, substitutes an identity for the stochastic integral, and applies Itô’s lemma to exponentiate the log-price dynamics. This yields a spot process with constant diffusion and a drift that depends on the initial futures curve and a linear mean-reversion term in log spot. The equations provide the supporting steps, but the result relies on the stated futures model and assumes the initial curve is differentiable; no empirical fit or market validation is presented.
Key ideas
- The spot price is identified with the futures price whose maturity is the current time.
- The log-price representation includes an integrated variance correction and a stochastic integral.
- Differentiating the log spot and applying Itô’s lemma produces the spot SDE.
- The resulting drift includes mean reversion in log spot and depends on the initial futures curve.
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# How were these SDE derived?
# How were these SDE derived?
Can anyone give me a detailed explanation of how below equations (3) and (4) are derived from (1) and (2)? \begin{align*} \frac{dF_{t,T}}{F_{t,T}} &=\sigma e^{-\lambda(T-t)}dB_t, \tag{1}\\ \ln(F_{t,T})&=\ln(F_{0,T})-1/2\int_{0}^{t}\sigma^2 e^{-2\lambda(T-s)}ds+\int_{0}^{t}\sigma e^{-\lambda(T-s)}dB_s.\tag{2} \end{align*} Given $\ln(S_t)=\ln(F_{t,t})$, we have: \begin{align*} \frac{dS_t}{S_t}=(\mu_t-\lambda \ln(S_t))dt+\sigma dB_t,\tag{3} \end{align*} where \begin{align*} \mu_t=\frac{\partial \ln(F_{0,t})}{\partial t} +\lambda \ln(F_{0,t})+\frac{1}{4}\sigma^2(1-e^{-2\lambda t}). \tag{4} \end{align*} Or anything related to them will be helpful.
## Answer by Gordon (score 11, accepted)
https://quant.stackexchange.com/a/27615
From $(2)$, \begin{align*} \ln S_t &=\ln F_{t, t} \\ &= \ln F_{0, t}-\frac{1}{2}\int_0^t\sigma^2 e^{-2\lambda (t-s)}ds+\int_0^t \sigma e^{-\lambda(t-s)} dB_s\\ &=\ln F_{0, t}-\frac{\sigma^2}{4\lambda} \left(1-e^{-2\lambda t}\right)+e^{-\lambda t}\int_0^t \sigma e^{\lambda s} dB_s. \end{align*} Then, \begin{align*} \lambda e^{-\lambda t}\int_0^t \sigma e^{\lambda s} dB_s = \lambda \ln S_t - \lambda \ln F_{0, t} + \frac{\sigma^2}{4} \left(1-e^{-2\lambda t}\right). \end{align*} Therefore, \begin{align*} d\ln S_t &= \left(\frac{\partial \ln F_{0, t}}{\partial t}-\frac{\sigma^2}{2}e^{-2\lambda t} - \lambda e^{-\lambda t}\int_0^t \sigma e^{\lambda s} dB_s\right)dt +\sigma dB_t\\ &=\left[\frac{\partial \ln F_{0, t}}{\partial t}-\frac{\sigma^2}{2}e^{-2\lambda t}+\lambda \ln F_{0, t} - \frac{\sigma^2}{4} \left(1-e^{-2\lambda t}\right) -\lambda \ln S_t\right]dt + \sigma dB_t\\ &=\left(\frac{\partial \ln F_{0, t}}{\partial t}+\lambda \ln F_{0, t} -\frac{\sigma^2}{4} - \frac{\sigma^2}{4} e^{-2\lambda t} -\lambda \ln S_t\right)dt + \sigma dB_t. \end{align*} Note that \begin{align*} d\langle \ln S, \ln S \rangle_t= \sigma^2 dt. \end{align*} By Ito's lemma, \begin{align*} dS_t &= de^{\ln S_t}\\ &= e^{\ln S_t} d \ln S_t + \frac{1}{2}e^{\ln S_t}d\langle \ln S, \ln S \rangle_t\\ &=S_t d \ln S_t + \frac{1}{2} \sigma^2 S_t dt\\ &= S_t\left[\left(\frac{\partial \ln F_{0, t}}{\partial t}+\lambda \ln F_{0, t} -\frac{\sigma^2}{4} - \frac{\sigma^2}{4} e^{-2\lambda t} -\lambda \ln S_t\right)dt + \sigma dB_t + \frac{\sigma^2}{2} dt \right]\\ &=S_t\big[\left(\mu_t - \lambda \ln S_t\right)dt + \sigma dB_t\big], \end{align*} where \begin{align*} \mu_t = \frac{\partial \ln F_{0, t}}{\partial t}+\lambda \ln F_{0, t} +\frac{\sigma^2}{4}\left(1- e^{-2\lambda t}\right). \end{align*}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.