Deriving the Convexity Adjustment Between Futures and Forward Prices
Summary
The document derives the relationship between futures and forward prices when the underlying asset and short rate are stochastic and correlated. It assumes a geometric process for the asset price and a mean-reverting short-rate process, then integrates the rate over the contract’s remaining life to obtain its conditional distribution. The futures price is the expected future spot price, while the forward price incorporates discounting by the cash account.
Using Gaussian expectations, the derivation gives the futures-to-forward price ratio as an exponential adjustment with two contributions: short-rate variance and covariance between asset returns and interest rates. The sign and size therefore depend on rate volatility, asset volatility, their correlation, and time to maturity through the mean-reversion function. The result is specific to the stated model and measure assumptions; it is not a universal adjustment formula. Care is also needed to distinguish the ratio’s direction when converting a futures quote into a forward quote.
Key ideas
- Futures and forwards differ because futures gains are settled over time while forwards are discounted to maturity.
- The derivation assumes a mean-reverting short rate and a correlated Brownian shock for the asset.
- The adjustment contains a short-rate variance term and an asset-rate covariance term.
- Mean reversion determines how the remaining life enters the adjustment.
- The formula depends on the model assumptions and on whether the ratio is futures over forward or its inverse.
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Full text
# Convexity Adjustment for Futures
# Convexity Adjustment for Futures
Let $B_t$ be the cash account numeraire. The future and forward prices at time t are expressed as:
$$ Fut = E_t^Q\left[S_T\right],$$ $$ Fwd = \frac{E_t^Q[S_T/B_T]}{E_t^Q[1/B_T]}.$$
Where $$ \frac{dS(t)}{S(t)} = \mu dt + \sigma dW_s^Q(t),$$ $$dr(t) = -Kr(t)dt+ \alpha dW_r^Q(t),$$ $$<dW_sdW_r> = \rho dt.$$
Where $K$ is the mean reversion of the short interest rate $r$.
How is the convexity adjustment calculated in order to express the forward price in terms of the future price?
## Answer by Gordon (score 13, accepted)
https://quant.stackexchange.com/a/33156
We assume that, under the probability measure $Q$, \begin{align*} dS_t &= S_t\big(r_t dt + \sigma dW_s(t)\big),\\ dr_t &= -k\, r_t dt + \alpha dW_r(t),\tag{1} \end{align*} where $d\langle W_s(t), W_r(t)\rangle_t = \rho dt$. From $(1)$, for $s\ge t$, \begin{align*} r_s = e^{-k(s-t)}r_t + \alpha\int_t^s e^{-k(s-u)} dW_r(u). \end{align*} Then, for $T\ge t$, \begin{align*} \int_t^T r_s ds &=\frac{r_t}{k}\left(1-e^{-k(T-t)} \right)+\alpha \int_t^T\!\!\!\int_t^s e^{-k(s-u)} dW_r(u) ds\\ &=\frac{r_t}{k}\left(1-e^{-k(T-t)} \right)+\alpha \int_t^T\!\!\!\int_u^T e^{-k(s-u)} ds dW_r(u) \\ &=\frac{r_t}{k}\left(1-e^{-k(T-t)} \right)+\alpha \int_t^T\frac{1}{k}\left(1-e^{-k(T-u)} \right) dW_r(u)\\ &=r_t\beta(t, T)+\alpha \int_t^T \beta(u, T) dW_r(u), \end{align*} where $$\beta(t, T)=\frac{1}{k}\left(1-e^{-k(T-t)} \right).$$ Therefore, \begin{align*} E^Q\left(\frac{1}{B_T} \mid \mathcal{F}_t\right) &=\frac{1}{B_t}E^Q\left(e^{-\int_t^T r_s ds} \mid \mathcal{F}_t \right)\\ &=\frac{1}{B_t} e^{-r_t\beta(t, T) + \frac{\alpha^2}{2} \int_t^T \beta^2(u, T) du}. \end{align*} Moreover, \begin{align*} E^Q\left(S_T \mid \mathcal{F}_t\right) &= S_t E^Q\left(e^{\int_t^T r_s ds - \frac{\sigma^2}{2} (T-t) + \sigma \int_t^T dW_s(u)} \right)\\ &=S_t E^Q\left(e^{r_t\beta(t, T)+\alpha \int_t^T \beta(u, T) dW_r(u) - \frac{\sigma^2}{2} (T-t) + \sigma \int_t^T dW_s(u)} \right)\\ &=S_te^{r_t\beta(t, T)+ \frac{\alpha^2}{2} \int_t^T \beta^2(u, T) du +\alpha \sigma \rho \int_t^T\beta(u, T) du}. \end{align*} Consequently, \begin{align*} C(t, T) &= \frac{Fut}{Fwd}\\ &=\frac{E^Q\left(S_T \mid \mathcal{F}_t\right)}{E\left(\frac{S_T}{B_T} \mid \mathcal{F}_t\right)/E^Q\left(\frac{1}{B_T} \mid \mathcal{F}_t\right)}\\ &=\frac{S_te^{r_t\beta(t, T)+ \frac{\alpha^2}{2} \int_t^T \beta^2(u, T) du +\alpha \sigma \rho \int_t^T\beta(u, T) du}}{\frac{S_t}{B_t} B_t e^{r_t\beta(t, T) - \frac{\alpha^2}{2} \int_t^T \beta^2(u, T) du}}\\ &=e^{\alpha^2\int_t^T \beta^2(u, T) du +\alpha \sigma \rho \int_t^T\beta(u, T) du}. \end{align*}
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