Basics – Sisi Tang

July 4, 2020March 8, 2021

Interest Rates Basic

Suppose $b>a>0$ . Let $P(a,b)$ be the discount factor discounting 1 dollar paid at $b$ to time $a$ . Let $T_1, T_2, \ldots, T_{N+1}$ be the floating rate reset date of the floating leg in a swap. For the case of LIBOR 3M, $T_i = \frac{i}{4}$ if we assume the day count convention is that 3 months is $\frac{1}{4}$ of a year. Define $\alpha_n = T_{n+1} - T_{n}$ .

Forward LIBOR Rate

Let L_n(t) be the forward LIBOR rate for period $[T_n, T_{n+1}]$ seen at time $t$ for $t < T$ . Then,

$L_n(t,T) := \frac{1}{\alpha_n}\left(\frac{P(t,T_n)}{P(t,T_{n+1})}-1\right).$

This definition makes sense if we seen it as:

$1+\alpha_nL_n(t) = \frac{1}{P(T_n, T_{n+1})} = \frac{P(t,T_n)}{P(t,T_{n+1})}.$

Instantaneous Forward Rate

Instantaneous forward rate is the forward rate with time interval goes to infinity.

$\begin{aligned}f(t,T)&=\lim_{\delta\rightarrow 0}\frac{1}{\delta}\left(\frac{P(t,T)}{P(t,T+\delta)}-1\right) \\ & = -\frac{\partial \log P(t,T)}{\partial T}\end{aligned}.$

Short Rate

Short rate $r(t)$ is an instantaneous interest rate such that if $r$ is deterministic

$P(t,T) = e^{-\int_t^Tr(s)\textnormal{d}s}.$

Or if $r$ is stochastic

$P(t,T) = \mathbf{E}\left[e^{-\int_t^Tr(s)\textnormal{d}s}|\mathcal{F}_t\right].$

Forward Par Swap Rate

In finance, ‘par’ usually means ‘equal’. Par swap rate is the fixed rate such that the floating leg and the fixed leg of the swap make the contract has value 0. We denote $S_n(t)$ by the forward par swap rate on tenor $[T_n, T_{N+1}]$ . Suppose the pay frequency of the fixed leg is the same as the floating leg (otherwise redefine the below annuity as one that matches the fixed leg pay frequency). Let $A_n(t)$ represents the present value of annuity, i.e.,

$A_n(t) = \sum_{k=n+1}^{N+1} \alpha_{k-1}P(t,T_k).$

Then the present value of the fixed leg is

$PV_{\mbox{fix}} = S_n(t)A_n(t)$

The present value of the floating leg is

$\begin{aligned}& PV_{\mbox{float}} \\ = & \sum_{k=n}^N L_k(t)\alpha_k P(t,T_{k+1}) \\ =& \sum_{k=n}^N \frac{1}{\alpha_k}\left(\frac{P(t,T_k)}{P(t,T_{k+1})}-1\right) \alpha_k P(t,T_{k+1}) \\ =& \sum_{k=n}^N \left(P(t,T_k) - P(t,T_{k+1})\right) \\ =& P(t,T_n) - P(t,T_{N+1}) \end{aligned}$

By letting $PV_{\mbox{fix}} = PV_{\mbox{float}}$ , we have that

$S_n(t) = \frac{P(t,T_n) - P(t,T_{N+1})}{A_n(t)}.$

December 17, 2019March 8, 2021

Feynman-Kac Formula

Feynman-Kac formula connects the solution to a SDE to the solution of a PDE. For example, the Black-Scholes formula is an application of Feynman-Kac.

Let $X(t)$ satisfies the following SDE driven by standard Brownian Motion:

$\textnormal{d}X_u = \beta(u,X_u)\textnormal{d}u + \gamma(u,X_u\textnormal{d}W_u).$

Let $h(y)$ be a Borel measurable function, and $t \in [0,T]$ . Define

$\begin{aligned} g(t,x) := \mathbf{E} [&\int_t^Te^{-\int_t^TV(\tau,X_{\tau})\textnormal{d}\tau}f(r,X_r)\textnormal{d}r \\ &+e^{-\int_t^TV(\tau,X_{\tau})}h(X_T)|X_t = x ] \end{aligned}.$

Then, $g(t,x)$ satisfies PDE

$g_t + \beta g_x + \frac{1}{2}\gamma^2 g_{xx} - Vg + f = 0.$

Proof.

To get an idea of how this formula is proved, we start with a base case. The general case is proved by the same manner with a little more complicated calculation.

Suppose $V=f=0$ .

Let $\mathcal{F}(t)$ be the natural filtration associated with the standard Brownian Motion in the SDE. By the Markov property of the solution to SDE, we have $\forall s \in [0,T]$ ,

$\mathbf{E}[h(X_T)|\mathcal{F}(s)] = \mathbf{E}[h(X_T)|\sigma(X_s)] = g(s,X_s).$

Then, for $0<s<t<T$

$\begin{aligned} \mathbf{E}[g(t,X_t)|\mathcal{F}(s)] & = \mathbf{E}[ \mathbf{E}[h(X_T)| \mathcal{F}(t) ] |\mathcal{F}(s)] \\ &= \mathbf{E}[h(X_T)|\mathcal{F}(s)] = g(s,X_s) \end{aligned}$

Hence, $g(t,X_t)$ is a martingale.

Then, we apply Ito’s formula to $g(t,X_t)$ . Its drift term should be zero because it is a martingale. This leads to the Feynman-Kac PDE.

$\begin{aligned} \textnormal{d}g(t,X_t) & = g_t\textnormal{d}t + g_x \textnormal{d} x + \frac{1}{2}g_{xx} \textnormal{d} X \textnormal{d} X \\ & = [g_t+\beta g_x + \frac{1}{2}\gamma^2 g_{xx}] \textnormal{d} t + \gamma g_x \textnormal{d} W\end{aligned}.$

By setting the $\textnormal{d} t$ term equal to $0$ , we get the PDE:

$g_t+\beta g_x + \frac{1}{2}\gamma^2 g_{xx} = 0.$

For the general case, we only need to factor out the $\int_0^t$ integral and make the integral inside the expectation only contains $\int_0^T$ term. We will get $G(t,X_t)$ is a martingale with $G(t,X_t)$ defined as:

$G(t,x):= e^{-\int_0^tV(\tau,X_{\tau})\textnormal{d}\tau}(g(t,x)+\int_0^t e^{-\int_t^rV (\tau,X_{\tau})\textnormal{d}\tau } f(r,X_r) \textnormal{d}r)$

Similarly, by applying Ito’s formula to above martingale and setting the drift term equal to 0, we will get the result.

February 4, 2019March 8, 2021

Cross Hedging

When our study group read John Hull’s Options, Futures, and Other Derivatives 10th Edition book section 3.4 Cross Hedging, the hedging ratio $h^*$ was given directly in (3.1). We filled in the derivation of it here.

Goal

We want to hedge an asset using a future contract whose underlying asset is different from the one being hedged. This usually happens for example in the case that the asset being hedged is not available in the future market. If we are going to sell an asset, then we take long position in a future contract; short otherwise.

Setting

$h$ : hedge ratio. The ratio of the size of the position taken in futures contracts to the size of the exposure.
$\Delta S$ : change of spot price of the asset to be hedged.
$\Delta F$ : change of price in the future contract.

Formulation

Mathematically, we want to minimize the variance of our portfolio value, i.e.,

$\begin{aligned}\mbox{Var}(\Delta S-h\Delta F) &= \mathbf{E}[(\Delta S-\overline{\Delta S}-h(\Delta F-\overline{\Delta F}))^2] \\&= \mathbf{E}[(\Delta S-\overline{\Delta S})^2] + h^2\mathbf{E}[(\Delta F-\overline{\Delta F})^2] \\ & \quad - 2h\mathbf{E}[(\Delta S-\overline{\Delta S})(\Delta F-\overline{\Delta F})] \\&= \sigma_{\Delta S}^2+ h^2\sigma_{\Delta F}^2 - 2h~\mbox{cov}(\Delta S,\Delta F) ,\end{aligned}$

where $\sigma_{\Delta S}, \sigma_{\Delta F}$ denote the standard deviation of $\Delta S$ , standard deviation of $\Delta F$ respectively. This is a quadratic fucntion in $h$ . The minimum of it is achieved at

$h=\frac{\mbox{cov}(\Delta S,\Delta F)}{\sigma_{\Delta F}^2}=\rho\frac{\sigma_{\Delta S}}{\sigma_{\Delta F}},$

where $\rho$ is the correlation of $\Delta S$ and $\Delta F$ . This formula for $h$ is exactly the slope of linear regression $\Delta S \sim \Delta F$ . Our derivation here also explains why linear regression is the minimum variance estimator.

Then, the variance

$\mbox{Var}(\Delta S-h\Delta F)=(1-\rho^2)\sigma_{\Delta S}^2.$