The swap market Bergomi model

The Bergomi stochastic volatility model uses a forward variance curve as its modelling object. A variance swap is a contract that pays the realised variance of the log return of a tradable asset $S$ less a strike rate ${\phi }_T(t)$ at maturity $T$. We assume that the payoff at $T$ is given as:

where ${\phi }_T(t)$ is set at $t$ to make the value of the variance swap zero. Namely, ${\phi }_T(t)$ must satisfy:

$E_t^Q$ is the $t$-conditional expectation under the risk-neutral measure. In this subsection, we assume the risk-free rate process $r_t$ to be deterministic.

A discrete forward variance swap rate ${\phi }_{T_1,T_2}(t)$ is defined (using ${\phi }_T(t)$) as:

We can confirm that ${\phi }_{T_1,T_2}(t)$ is a martingale under the risk-neutral measure; for , we obtain:

An infinitesimal forward variance swap rate is obtained by taking limit $\epsilon \to 0$ for ${\phi }_{T,T+\epsilon }(t)$; ${\xi }_t^T\triangleq {lim}_{\epsilon \to 0}{\phi }_{T,T+\epsilon }(t)$.

A continuous Bergomi model is specified by assuming lognormal dynamics for infinitesimal forward variance swap rates in a forward variance curve :

where ${\omega }_t^u$ is a static model parameter, $W_t^u$ is a Brownian motion under the risk-neutral measure and $T_e$ is the model terminal date. Note ${\xi }_t^T$ has zero risk-neutral drift because the forward variance swap rate is a martingale under the risk-neutral measure. The underlying process $S_t$ follows:

where $W_t^S$ is a Brownian motion under the risk-neutral measure. With (1) and (3), we obtain ${\xi }_t^t={lim}_{\epsilon \to 0}{\phi }_{t,t+\epsilon }(t)={({\sigma }^S)}^2$. Thus, we get ${\sigma }^S=\sqrt{{\xi }_t^t}$.

The Bergomi model assumes $S_t$ and ${\{{\xi }_t^u\}}_{t\le u\le T_e}$ to be model state variables. Consider that we are managing a derivatives contract using the underlying asset $S_t$ and the infinitesimal forward variance swaps on as our hedging instruments. We denote the value of the derivatives contract by $V(S_t,{\{{\xi }_t^u\}}_{t\le u\le T_e},t)$. The pricing equation is written as:

with the correlation functions:

Consider a hedged contract $V_t^{\mathrm{H}}$ that is a portfolio of a unit of the derivatives contract $V_t$, a bank account $B_{0,t}=\exp ({\int }_0^t{r}_{u}du)$, the underlying asset $S_t$ and the forward variance swaps on . We require $V_t^{\mathrm{H}}$ to satisfy ${\partial }_S{V}_t^{\mathrm{H}}=0$, ${\partial }_{{\xi }^u}{V}_t^{\mathrm{H}}=0$, and $V_t^{\mathrm{H}}=0$. Then, the P&L formula of the hedged contract $V_t^{\mathrm{H}}$ is computed using (4) as:

Equation (5) provides the term-wise breakeven condition for the P&L of $V^{\mathrm{H}}$. Therefore, when a non-negligible P&L trend appears, we can easily understand which term causes it and how we should modify the model assumptions. Note all of the hedging instrument price changes can be interpreted as variations of the state variables; thus, we can use the model parameters ${\rho }_t^u$, ${\rho }_t^{uv}$ and ${\omega }_t^u$ solely to control the breakeven condition.

Before proceeding to our discussion on the application of the Bergomi model to interest rate modelling, let us define the basic variables first. Consider the discrete time grids $T_i={\sum }_{u=0}^{i-1}{\delta }_u$, $T_0=0$, with accrual factors ${\{{\delta }_u\}}_{u=0,1,\mathrm{\dots },e-1}$, where $T_e$ is the terminal date of the model. Denote by $P(t,T_i)=P_t^i$ the discount factor at time $t$ with maturity date $T_i$. We denote a continuous bank account process by $B_{t,T}=\exp ({\int }_t^T{r}_{u}du)$, where $r_t$ is a risk-free rate process. Swap rates and associated annuity factors are given as:

respectively. We denote $T_u$ by $u$ when no confusion can arise, and we assume empty sums denote zero while empty products denote 1.

Here, we introduce an interest rate variance swap contract such that the variance swap rate will be a martingale under the associated annuity measure. Owing to the martingale property, we can build a two-factor model of a swap rate and a forward variance curve under the associated annuity measure. In addition, we are able to compute a European swaption price using the two-factor model. As we will see later, this simplifies the relationship between the forward variance curve and the market European swaption price, and the P&L interpretation in terms of the market observables becomes clear.

Consider a contract that, at $T_l$, pays the sum of quadratic variation of a swap rate $S_t^{i,j}$ for the period $[T_k,T_l]$, where $T_l\le T_i$. The quadratic variation is multiplied by $A^{i,j}/P^l$ at the end of each observation time grid and rescaled with the factor ${(T_l-T_k)}^{-1}$; the payoff at $T_l$ is given as:

Also consider a variance swap on a swap rate that pays $Q_{k,l}^{i,j}$ and receives $A_l^{i,j}{\psi }_{t,k,l}^{i,j}$ at $T_l$, where ${\psi }_{t,k,l}^{i,j}$ is fixed at $t\le T_k$ so the value of the contract is zero. This means:

where $E_t^{T_l}$ and $E_t^{A^{i,j}}$ denote the $t$-conditional expectation under a $T^l$ terminal measure and an $A^{i,j}$ annuity measure, respectively. From (6), we obtain:

This indicates ${\psi }_{t,k,l}^{i,j}$ is a martingale under the $A^{i,j}$ annuity measure. For , we get:

Next, we consider the dynamics of a swap rate process $S_t^{i,j}$. Because $S_t^{i,j}$ is a martingale and, hence, driftless under the $A^{i,j}$ annuity measure, we assume:

where $a_t^{i,j}$ is a stochastic process and $W_t^{(i,j),A^{i,j}}$ is a Brownian motion under the $A^{i,j}$ annuity measure. We define an infinitesimal variance swap rate ${\xi }_t^{i,j,T}$ as ${lim}_{\epsilon \to 0}{\psi }_{t,T,T+\epsilon }^{i,j}$. This leads to:

Thus, we get $a_t^{i,j}=\sqrt{{\xi }_t^{i,j,t}}$.

We assume lognormal dynamics for ${\xi }_t^{i,j,T}$, which is a martingale under the $A^{i,j}$ measure:

where ${\omega }^{i,j}$ and ${\kappa }^{i,j}$ are static model parameters.

In this article, we assume a variance curve is driven by a single Brownian motion $Z_t^{(i,j),A^{i,j}}$ for simplicity. The extension to the multi-factor setting is straightforward. In order to obtain a low-dimensional Markov representation, we introduce the Ornstein-Uhlenbeck (OU) state variable:

With $X_t^{i,j}$, the dynamics of ${\xi }_t^{i,j,T}$ is given as:

The swap market model is classified by the underlying swap rates to be modelled (Jamshidian 1997; Oya 2018). In this article, we consider the co-terminal swap market model for which the yield curve dynamics are determined by modelling swap rates that share a common terminal date $T_e$ (Jamshidian 1997). We will work with a $T_e$ terminal measure. To specify the co-terminal swap market model, we omit the end index for a swap for ease of notation, ie, $S_t^{i,e}=S_t^i$. We introduce the state variable vector as:

with $N_R=e-1$, $N_s=2N_R$.

The dynamics of $Y_t$ under the $T_e$ terminal measure are given by adding no-arbitrage drifts to (8) and (12):

where ${\{W_t^{(i),T_e}\}}_{i=1,\mathrm{\dots },N_s}$ are the Brownian motions under the $T_e$ terminal measure, defined as:

We define the correlation functions as ${⟨\mathrm{d}{W}^{(i),T_e},\mathrm{d}{W}^{(j),T_e}⟩}_t={\rho }_t^{Y,ij}\mathrm{d}t$.

In the previous subsection, we formulated the model dynamics using the state variable vector $Y\triangleq {(S^1,\mathrm{\dots },S^{N_R},X^1,\mathrm{\dots },X^{N_R})}^{\mathrm{T}}$. Remember the elements of the state vector are directly related to market observables. A forward swap rate $S_t^i$ is evaluated from spot starting swap rates, while $X_t^i$ is computed from a variance swap rate ${\xi }_t^{i,T}$ using (11). In reality, the interest rate variance swap contract is illiquid in the market; thus, we use European swaptions as our hedging instruments instead. For this purpose, we discuss how to compute ${\xi }_t^{i,T}$ using the market price of a European swaption.

We use the different computation procedures for $t=0$ and $t>0$. At $t=0$, the initial forward variance curve ${\xi }_0^{i,T}$ is computed in order for the model to reproduce the market prices of European swaptions assuming $X_t^i=0$. For $t>0$, European swaption market price changes are reflected in the state variable $X_t^i$. In other words, the model parameter calibration to market prices of European swaptions is performed only at $t=0$. After that, the market price changes are interpreted as model state variables variations.

The initial forward variance curve is parameterised as ${\xi }_0^{i,T}={({\sigma }_0^i)}^2{\mathrm{e}}^{{\theta }^{i}T}$. ${\theta }^i$ is the model parameter for the control of the curve shape of ${\xi }_0^{i,T}$. ${\sigma }_0^i$ is calibrated to a European swaption market price at $t=0$ and then fixed for $t>0$.

Consider the computation of ${\xi }_t^{i,T}$ at $t=T_s$. We will obtain ${\xi }_s^{i,T}$ so the model reproduces the market price of a European swaption on $S^i$ with strike $K$ using the efficient numerical computation scheme presented in Bergomi (2015). Here, we use a two-factor model for a swap rate $S^i$ and a state variable $X^i$ under the associated annuity measure for the computation of the model swaption price. Denote by $T_i$ the expiry time of a swaption on $S^i$. Note $S^i$ becomes driftless under the $A^i$ annuity measure:

where $Z_t^{(i),⟂,A^i}$ is a Brownian motion that is independent of $Z_t^{(i),A^i}$.

Next, let us look at a payer swaption price on a swap rate $S^i$:

where we use the tower rule. Because $Z_t^{(i),⟂,A^i}$ and $Z_t^{(i),A^i}$ are independent, $S_i^i$ conditioned on a path for $Z_t^{(i),A^i}$ follows a normal distribution with the following moments:

Thus, $E_s^{A^i}[{(S_i^i-K)}^{+}|{\{Z_u^{(i),A^i}\}}_{T_s\le u\le T_i}]$ can be calculated analytically using the normal Black-Scholes formula as $\mathrm{NBS}({\overline{S}}^i,{\widehat{\sigma }}^i,K,T_i-T_s)$ with:

where $N(x)$ is the standard cumulative normal distribution function and $N^{\prime }(x)=\mathrm{d}N/\mathrm{d}x$. Then, the swaption price is obtained as $A_s^i\times E_s^{A^i}[\mathrm{NBS}({\overline{S}}^i,{\widehat{\sigma }}^i,K,T_i-T_s)]$.

We assume the market swaption price for a European swaption on $S^i$ as of $T_s$ is given as a normal implied volatility ${\sigma }_s^{I,i}$. We require the model price of the swaption to be equal to the market price:

Equation (20) is solved using a one-dimensional root-finding algorithm in terms of ${\sigma }_0^i$ for $T_s=0$ and $X_s^i$ for $T_s>0$.

Here, we analyse the P&L formula with the co-terminal swap market Bergomi model (SMBM) and discuss how we could adjust our model parameters using the results of the P&L analysis. The derivation of the P&L formula proceeds in the standard manner, and we obtain a well-known form of the P&L formula. The novelty of the co-terminal SMBM approach is that this theoretical P&L formula can be used for risk management when both interest rate swaps and European swaptions are used as hedging instruments. On the other hand, the existing interest rate models need to rely on calibration procedures to reproduce the market prices of hedging European swaptions in such a context. Consequently, the theoretical P&L formula does not hold for the P&L number computed as the price difference of two different model parameter assumptions.

Assume we hold a derivatives contract and the value of that contract is denoted by $V(Y_t,t)$. The pricing equation of $V(Y_t,t)$ is given as:

where ${\mu }_t^{u,Q}$ is the no-arbitrage drift of $Y^u$ under the risk-neutral measure. We do not provide the explicit form of ${\mu }_t^{u,Q}$ here because it is irrelevant to the discussion of a hedged contract. ${\sigma }_t^{Y,i}$ is given as:

Next, we consider a hedged contract $V^{\mathrm{H}}$ for the derivatives contract:

where ${\{w_u\}}_{1\le u\le N_S+1}$ are the hedge weights. The prices of hedging instruments are given as:

Namely, we hedge the derivatives contract with interest rate swaps, payer European swaptions of strike $K^i$ and a bank account $B$. We build the hedged contract to satisfy the equations below:

The hedging conditions (26) and (27) are given as a linear system and solved using standard linear algebra.

Now, we consider the second-order P&L formula for the hedged contract $V^{\mathrm{H}}$. Using a Taylor expansion of order 2 in $\delta Y$, we obtain:

Equation (28) indicates the second-order P&L is given as the sum of the differences of the realised quadratic cross variations $\delta Y^u\delta Y^v$ and the breakeven level ${\rho }_t^{Y,uv}{\sigma }_t^{Y,u}{\sigma }_t^{Y,v}\delta t$ multiplied by the gamma term ${\partial }^2{V}^{\mathrm{H}}/\partial Y^u\partial Y^v$, as in (5). Therefore, we can carry out a term-wise analysis in the same manner as for the Bergomi model. The calibration procedure is not required for the co-terminal SMBM, and we can use the model parameters solely to control the breakeven levels. Let us summarise the model parameters as follows:

• •

  ${\theta }^i$ determines the shape of the initial forward variance curve ${\xi }_0^{i,T}$, which affects $\sqrt{{\xi }_t^{i,t}}$, which appears in the breakeven level expression as ${\sigma }_t^{Y,i}$ for $1\le i\le N_R$.

• •

  ${\omega }^i$ controls the lognormal volatility of ${\xi }_t^{i,T}$. In terms of the state variable, ${\omega }^i$ controls the scale of $X^i$. Because ${\sigma }_t^{Y,i}=1$ always holds for , the control of the scale of $X^i$ is equivalent to the control of the breakeven levels for $\delta X^i$.

• •

  ${\rho }_t^{Y,ij}$ controls the correlations between state variables.

Because an interest rate model often contains a large number of state variables, practitioners frequently use this model driven by a limited number of Brownian motions to reduce its complexity. This approach is also useful with the co-terminal SMBM to aid in intuitively understanding the economy of the product. In this article, we consider as an example the co-terminal SMBM for which swap rates and forward variance curves are driven by three Brownian motions:

where we parameterise ${\{c^{(i),u}\}}_{u=1,2}$ as $c^{(i),1}=\cos {\alpha }^{(i)}$, $c^{(i),2}=\sin {\alpha }^{(i)}$.¹¹ 1 In practical applications, the number of Brownian motions and coefficients is determined based on historical analysis of market movements (eg, principal component analysis) in order to make the P&L contribution from the second term of (33) small enough. We assume the correlations are given as ${⟨\mathrm{d}{W}^{C,i},\mathrm{d}{W}^{C,j}⟩}_t={\rho }_t^{C,ij}\mathrm{d}t$ for $i,j=1,2,3$, with ${\rho }_t^{C,12}=0$.

We consider the P&L formula associated with the above Brownian motion changes. Define the value function of a derivatives contract parameterised with reduced factors as $V^h(h,t)\triangleq V(Y_t+{\sum }_{u=1}^3{h}^u{m}^u,t)$, with $h={\{h^u\in ℝ\}}_{u=1,2,3}$ and $m={\{m^u\in {ℝ}^{N_S}\}}_{u=1,2,3}$. $m$ is given as:

With $V^h$, we can write the theta term for the hedged contract in (28) as:

where $V^{H,h}\triangleq V^{\mathrm{H}}(Y_t+{\sum }_{u=1}^3{h}^u{m}^u,t)$. Also, the gamma term in (28) becomes:

where $\delta h={\{\delta h^u\in ℝ\}}_{u=1,2,3}$ is defined so the $L_2$-norm of $\delta Y-{\sum }_{u=1}^3\delta h^u{m}^u$ is minimised:

With (31) and (32), we obtain the second-order P&L formula with reduced factors: