Notes: Carr and Wu (2009)

1. Introduction

In this note, I outline the main results in Variance Risk Premiums (RFS, 2009) by Peter Carr and Liuren Wu for use in a 5min presentation in Prof. Sargent‘s reading group.

Carr and Wu (2009) develops a method for quantifying the risk premium that investors demand for holding an asset that has a higher return variance. This method is based on the concept of a variance swap—a zero cost, derivative contract between $2$ traders, call them Alice and Bob, in which Alice pays Bob a fixed swap rate over the life of the contract and Bob pays Alice the realized volatility of the underlying asset. In this example, Alice is buying return variance from Bob. While variance swaps are not commonly traded, their payouts can be synthesized via a linear combination of call option contracts at different strike prices. Using this basic derivative pricing framework, the Carr and Wu (2009) then use data on $35$ stocks and $5$ indicies to study the behavior of the variance risk premium during the period from $1996$ to $2003$ .

I chose to review this article because this synthetic derivative methodology is an interesting tool for thinking about asset pricing inequalities. For example, this same approach should allow you to synthesize Sharpe Ratio or Entropic¹ swap contracts via options prices to the study risk-return boundaries predicted by different theories.

2. Variance Swaps

I begin by defining a variance swap in more detail and showing how to replicate its state contingent menu of payouts using traded options with different strike prices. Just like a more standard interest rate swap, a variance swap is defined by $4$ objects:

Underlying
Swap rate
Maturity
Notional Amount

The underlying is the asset whose variance is being tracked, the swap rate is the fixed rate which is paid by the buyer of the variance to the seller each period and the maturity is the number of periods over which this transaction occurs. For instance, suppose that Alice and Bob enter into a $\$1\mathtt{M}$ , $31$ day, variance swap contract on the S&P 500 at a rate of $15\%$ . Then, on day $1$ , Alice would pay Bob $\$1,000,000 \times 0.15 = \$150,000$ with certainty and Bob would pay Alice $\$1,000,000 \times V_1$ where $V_1$ represents the realized variance of the S&P 500 on the first day of the contract. So, if the the first day happened to have a high annualized variance of $20\%$ , then Bob would have to pay Alice $\$200,000$ . Over the life of the contract, Bob will pay Alice a net amount $X_{t \to T}$ given by:

(1) $\begin{align*} X_{t \to T} &= \left( V_{t \to T} - S_{t \to T} \right) \cdot L \end{align*}$

where $L$ represents the notional amount of the contract which was $\$1\mathtt{M}$ in the example above.

Since the contract has $\$0$ cost at the onset, we can characterize the swap rate at the expected present value of the variance payments adjusted for risk by the prevailing stochastic discount factor:

(2) $\begin{align*} S_{t \to T} &= \mathbb{E} \left[ M_{t \to T} \cdot V_{t \to T} \right] \end{align*}$

Since variance swaps are not traded in a liquid market with public prices, I now show how to synthesize this contract using a riskless bond and call options on futures contracts. Suppose that the price of a futures contract follows a geometric Brownian motion process:

(3) $\begin{align*} dF_t &= \mu_t \cdot F_t \cdot dt + \sigma_t \cdot F_t \cdot dW_t \end{align*}$

Suppose that the $\sigma_t$ process is restricted to ensure that the futures price is always positive and is predictable² with respect to the filtration $\mathcal{F}_t$ . Suppose that the riskless rate process is deterministic so that the futures and forward prices are identical. Thus, we have that the annualized quadratic variation on the log futures return from time $t$ to time $T$ is given by:

(4) $\begin{align*} V_{t \to T} &= \frac{\int_t^T \sigma_s^2 \cdot ds}{T-t} \end{align*}$

Using this futures return formulation, I can characterize the swap rate using a linear combination of out of the money European options:

Proposition (Variance Swap Rate): Given no arbitrage, the time $t$ value of the quadratic variation of an asset from time $t$ to time $T$ can be approximated by a portfolio of out of the money, European call option prices for all strike prices $K > 0$ with the same $T - t$ maturity using the formula:

(5) $\begin{align*} \mathbb{E} \left[ M_{t \to T} \cdot V_{t \to T} \right] &\approx \frac{2}{T - t} \cdot \int_0^\infty \frac{Y_{t \to T}^{(k)}}{k^2 \cdot B_{t \to T}} \cdot dk \end{align*}$

where $B_{t \to T}$ denotes the time $t$ price of a bond that pays out $\$1$ at time $T$ and $Y_{t \to T}^{(k)}$ denotes the time $t$ price of an out of the money European option with strike price $k$ and end date $T$ where $Y$ represents a call option if $k > F$ and a put if $k < F$ .

Roughly speaking, if you can buy and sell a bunch of state contingent pay outs that deliver $\$1$ if and only if the return on the underlying crosses some threshold, then via the de Moivre-Laplace convergence of the binomial distribution to the Gaussian under the twisted risk neutral measure, we should be able to characterize the price of the realized variance using these same assets.

Proof: By Ito’s lemma under the risk neutral measure we have that:

(6) $\begin{align*} \ln \left[ F_T \right] &= \ln \left[ F_t \right] + \int_t^T \frac{1}{F_s} \cdot dF_s - \frac{1}{2} \cdot \int_t^T \sigma_s^2 \cdot ds \end{align*}$

Thus, we can compute $V_{t \to T}$ as:

(7) $\begin{align*} V_{t \to T} &= 2 \cdot \left( \frac{F_T}{F_t} - 1 - \ln \left[ \frac{F_T}{F_t} \right] \right) + 2 \cdot \int_t^T \left( \frac{1}{F_s} - \frac{1}{F_t} \right) \cdot dF_s \\ &= \frac{1}{T - t} \cdot \int_t^T \sigma_s^2 \cdot ds \end{align*}$

If we use a Taylor expansion of $\ln \left[ F_T \right]$ around $\ln \left[ F_t \right]$ , we get:

(8) $\begin{align*} \ln \left[ F_T \right] &= \ln \left[ F_t \right] + \left( \frac{1}{F_t} \right) \cdot \left( F_T - F_t \right) - \frac{1}{k^2} \cdot \int_0^{F_t} \left( k - F_T \right)^+ \cdot dk + \frac{1}{k^2} \cdot \int_{F_t}^\infty \left( F_T - k \right)^+ \cdot dk \end{align*}$

Substituting this expansion back into the formulation of the realized variance above yields:

(9) $\begin{align*} V_{t \to T} &= \frac{2}{k^2} \cdot \left\{ \int_0^{F_t} \left( k - F_T \right)^+ \cdot dk + \int_{F_t}^\infty \left( F_T - k \right)^+ \cdot dk \right\} + 2 \cdot \int_t^T \left( \frac{1}{F_s} - \frac{1}{F_t} \right) \cdot dF_s \\ &= \left\{ \text{Out of money options.} \right\} \cdot \left( \frac{2 \cdot dk}{k^2} \right) \\ &\qquad \qquad \qquad + \int_t^T \left\{ \text{Reciprocal futures price spread.} \right\} \cdot \left( 2 \cdot B_s \right) \end{align*}$

3. Empirical Results

In the section above, I showed how to synthesize the payout to a variance swap rate using options on futures contracts. In the section below, I show how to empirically compute the swap rate and realized variance rates. First, consider the swap rate $S_{t \to T}$ . Since options contracts on futures are generally traded at $30$ day horizons with most contracts starting on the 15th of every month, there will not generally be a set 0f futures contracts with the appropriate maturity at each date. Thus, the authors interpolate the intermediary dates using the formula below:

(10) $\begin{align*} S_{t \to T} &= \frac{1}{T - t} \cdot \left( \frac{S_{t \to T_1} \cdot \left( T_1 - t \right) \cdot \left( T_2 - T \right) + S_{t \to T_2} \cdot \left( T_2 - t \right) \cdot \left( T - T_1 \right)}{T_2 - T_1} \right) \end{align*}$

The realized variance is a direct application of the theoretical value listed above using futures prices:

(11) $\begin{align*} V_{t \to T} &= \frac{365}{T-t} \cdot \sum_{i=1}^{T-t} \left( \frac{F_{(t + i) \to T} - F_{(t + i - 1) \to T}}{F_{(t + i - 1) \to T}} \right)^2 \end{align*}$

The figure below gives the relationship between the mean and standard deviation of the swap rate and realized variance rates of $35$ stocks and $5$ indicies over the period from $1996$ to $2003$ :

Pairwise scatterplot depicting the mean and standard deviation of the realized variance v, variance swap rate s, and log variance risk premium ln(v/s) for the 40 assets presented in Tables 2 and 3 of Carr and Wu (2009).

The authors also investigate the relationship between the log variance risk premia $\ln \left[ V_{t \to T,j} / S_{t \to T,j} \right]$ computed as the spread between the log realized variance and the log swap rate and various market factors denoted by $Z_{t \to T}$ using the regression specification below:

(12) $\begin{align*} \ln \left[ \frac{V_{t \to T,j}}{S_{t \to T,j}} \right] &= \alpha_j + \beta_j \cdot Z_{t \to T} + \varepsilon_{t \to T,j} \end{align*}$

Carr and Wu (2009) find that investors price variance risk premia as measured by $Z_{t \to T}$ equal to the realized variance on the S&P 500; however, classic risk factors such as the excess return on the market or the Fama and French (1993) $3$ -factor model do not account for the excess returns on this portfolio as the variance risk premia displays both a negative market $\beta$ as well as statistically significant negative $\alpha$ ‘s.

See Backus, Chernov and Zin (2010). ↩
In the stochastic calculus sense. ↩