Reconciling Spot and Forward Volatility Across Maturities

# Connection between the $\sigma$ parameters of the spot price and the forward price


# Connection between the $\sigma$ parameters of the spot price and the forward price












It is well known, that under the Black-Scholes framework:

$$F\left(t,T\right)=\exp\left(r\left(T-t\right)\right)S\left(t\right),$$

where $S\left(t\right)$ is the spot price of an asset at time $t$, $F\left(t,T\right)$ is the forward price of the same asset at time $t$ with maturity $T$ and $r$ is the constant risk free rate. $S$ and $F$ both follow geometric Brownian motions with constant parameters.

As a consequence of the equation above, I would say that the $\sigma$ parameters are the same for $S$ and $F$ (, i.e. the SDE of $S$ and $F$ have the same $\sigma$ parameters).

However, most of time (in practice and in theory as well) the forward prices with different $T$ maturities are modelled separately with different SDEs. Other words, for each $T$ maturity there is a different SDE of the forward price:

$$dF\left(t,T\right)=\sigma_{F^{T}}F\left(t,T\right)dW_{F^{T}}\left(t\right)$$

under the risk neutral measure, where $\sigma_{F^{T}}$ are different constans for each $T$, and $W_{F^{T}}$ are different Wiener-processes for each $T$. As a result, the $\sigma$ parameter can't be the same for $S$ and $F$, since for all maturities the $\sigma_{F^{T}}$ should match with the $\sigma$ parameter of $S$, even though we assumed $\sigma_{F^{T}}$ parameters can differ.

In short, these two statments above are contradictional to me. How can it be resolved? What do I misunderstand?

## Answer by ir7 (score 1, accepted)

https://quant.stackexchange.com/a/77764

They are not contradictory, if we choose sigmas reasonably (time-dependent, as suggested in the comments).

If $$ F_t^T = e^{r(T-t)} S_t \; \; {\rm and} \; \; dS_t = rS_tdt + \sigma_t S_t dW_t, $$ with $r$ constant and $\sigma_t$ deterministic function of $t$, then, by Ito Lemma, we get: $$ d F_t^T =\sigma_t F_t^T dW_t.$$ If we focus on a forward maturity structure with only two points, $ t_0< T_1 <T_2$, we could choose $\sigma$ to be, for example, piecewise constant: $$ \sigma_t= \left\{ \begin{array}{ll} \sigma_1 & t_0\leq t \leq T_1\\ \sigma_2& T_1 <t\leq T_2 \end{array} \right. $$ As $F_t^{T_1}$ dies out (zero diffusion coefficient/volatility) once $t>T_1$, its dynamics uses only $\sigma_1$. Similarly, $F_t^{T_2}$ dies out once $t>T_2$ and uses both $\sigma_1$ and $\sigma_2$.

So, using your notations:

$$ d F_t^{T_1} =\sigma^{F^{T_1}}_t F_t^{T_1} dW_t,$$ $$ d F_t^{T_2} =\sigma^{F^{T_2}}_t F_t^{T_2} dW_t,$$ where $$ \sigma^{F^{T_1}}_t := \sigma_t = \sigma_1, $$ for all $t\in [t_0,T_1]$ (which is constant), and $$ \sigma^{F^{T_2}}_t:= \sigma_t = \left\{ \begin{array}{ll} \sigma_1 & t_0\leq t \leq T_1\\ \sigma_2& T_1 <t\leq T_2. \end{array} \right. $$ (which is not constant).

The model you propose also suggests taking control of the Brownian driver of the second forward, which can be done by introducing: $$ W^1_t := W_t \; \; {\rm and} \; \; W_t^2, \; \; {\rm with } \; \; dW^1_tdW^2_t=\rho dt$$ ($\rho$ constant).