Pricing A Variance Swap

Let’s look at how risk-neutral pricing of a variance swap can be constructed with this elegant formula:

$$\sigma^2_{strike} = \dfrac{2e^{rT}}{T}\Bigg[\int_0^{S_0e^{rT}} \dfrac{1}{K^2} P(K) dK + \int^{\infty}_{S_0e^{rT}} \dfrac{1}{K^2}C(K)dK\Bigg]$$

Static Portfolio Replication

proposition Let $f:(0, \infty)\rightarrow\mathbb{R}$ be twice continuous differentiable so that for any $k>0$ and any $s>0$, then,

$$f(s) = f(k) + f'(k)(s-k) + \int_0^kf''(K)(K-s)^+dK + \int_k^{\infty}f''(K)(s-K)^+dK$$

proof We will prove the case in which $s>k$; similar proof can be sketched for the other case. When $s>k$, we have $\int_0^kf''(K)(K-s)^+dK = 0$. The the second integral can be simplified as:

$$\int_k^{\infty}f''(K)(s-K)^+dK = \int_k^sf''(K)(s-K)dK$$

Using integration by part:

$$du = f''(K)dK ,\;\; v = s-K \\ \rightarrow u = f'(K) ,\;\; dv = -dK$$

And,

$$\begin{align} \int_k^s vdu &= uv |^s_k - \int_k^s udv \\ &= f'(K)(s-K) |^s_k - \int_k^s -f'(K)dK \\ &= -f'(K)(s-k) + f(s) - f(k) \;\;\square \end{align}$$

The above formula shows that any twice-differentiable continuous time-T payoff $f(S_T)$ can be replicated using a static portfolio of:

$$f(k) \text{ zero-coupon bond} \\ f'(k) \text{ forward at } k \\\ f''(K) \text{ puts across a continuum of strike } K \text{ from } 0 \text{ to } k \\ f''(K) \text{ calls across a continuum of strike } K \text{ from } k \text{ to } \infty$$

In practice, discrete set of strike can be used to approximately replicate. For example, given $S_0=100$ and payoff function $f(S_T) = \log S_T$. Here we choose to replicate with $k=100$ (Though we can theoretically replicate the payoff $f$ with any $k\in(0, \infty)$, a $k$ close to $S_0$ is typically chosen) and a discrete set of option strikes with increment of $5$ capping at $200$. In this replicating portfolio we will hold:

$$\log 100 \text{ zero-coupon bond} \\ \dfrac{1}{100} \text{ forward at } 100 \\ -\dfrac{5}{97.5^2} \text{ put striked at } 97.5 \\ \vdots \\ -\dfrac{5}{2.5^2} \text{ put striked at } 2.5 \\ \text{and} \\ -\dfrac{5}{102.5^2} \text{ call striked at } 102.5 \\ \vdots \\ -\dfrac{5}{197.5^2} \text{ put striked at } 197.5 \\$$

Variance Swap

A variance swap is an over-the-counter financial derivative that allows a party to trade on the future variance of a given underlying security. For example, a trader would pay the realized variance of log-price changes in exchange of a fixed payment called variance strike, normalized by the vega notional into dollar terms. The payoff of the VS is:

$$\text{Payoff (VS)} = N(\sigma^2_{realized} - \sigma^2_{strike})$$

We can calculate the variance strike which results in a zero discounted expectation of the payoff (per unit of vega notional).

$$\sigma^2_{strike} = \mathbb{E}\sigma^2_{realized}$$

Assuming that the underlying process follows a geometric Brownian motion with local volatility $\sigma_t$:

$$\dfrac{dS_t}{S_t} = \mu_t dt + \sigma_t dW \\ \rightarrow d\log S_t = (\mu_t - \sigma_t^2/2)dt + \sigma_t dW \\ \rightarrow logS_T \sim \mathcal{N}(logS_0 + rT -\dfrac{1}{2} \int_0^T \sigma^2_tdt, \;[\dfrac{1}{T}\int_0^T \sigma^2_tdt]\;T)$$

Therefore the realized variance of $\log S_T$ is:

$$\sigma^2_{realized} = \dfrac{1}{T}\int_0^T \sigma_t^2dt$$

Furthermore, if we combine the SDE of $dS_t$ and $d\log S_t$,

$$\dfrac{dS_t}{S_t} - d\log S_t = \dfrac{\sigma^2_t}{2}dt$$

Taking integral on both sides,

$$\int_0^T\dfrac{dS_t}{S_t} - \log \dfrac{S_T}{S_0} = \int_0^T \dfrac{\sigma^2_t}{2}dt$$

Combining with the formula for the realizead variance of $\log S_t$,

$$\sigma^2_{realized} = \dfrac{2}{T}[\int_0^T\dfrac{dS_t}{S_t} - \log \dfrac{S_T}{S_0}]$$

Using the proposition proven above, we can create a static replication portfolio and replicate the $\log S_T$ payoff.

$$\begin{align} \sigma^2_{strike} &= \mathbb{E}\sigma^2_{realized} \\ &= \mathbb{E}\{\dfrac{2}{T}[\int_0^T\dfrac{dS_t}{S_t} - \log \dfrac{S_T}{S_0}]\} \\ &= \dfrac{2}{T}[rT - \log \dfrac{S^{\ast}}{S_0} - \dfrac{Se^{rT} - S^{\ast}}{S^{\ast}} - \int_0^{S^{\ast}} \dfrac{-1}{K^2} (K-S_T)^+dK - \int^{\infty}_{S^{\ast}} \dfrac{-1}{K^2}(S_T-K)^+dK] \end{align}$$

If we choose the cutoff as the forward price $S^{\ast} = S_0e^{rT}$, we can largely simplified the formula as follow:

$$\sigma^2_{strike} = \dfrac{2e^{rT}}{T}[\int_0^{S_0e^{rT}} \dfrac{1}{K^2} P(K) dK + \int^{\infty}_{S_0e^{rT}} \dfrac{1}{K^2}C(K)dK]$$

In conclusion, if we assume a GBM underlying process, the fair-value variance strike can be calculated as the sum of calls and puts across a continuum of strikes.

Practical Consideration

In practice, variance swap is costly to implement, requires constant hedging and an entire array of options. The advantage of a variance swap is that it is purely exposed to volatility risk, as oppose to an option which contains directional risk.

The P&L of a variance swap depends directly on the difference between realized and implied volatility. Since historically the implied volatility has been above realized volatility, a.k.a. variance risk premium, volatility arbitrage (rolling short variance trade) can be carried out with variance swaps.

Reference

• Variance and Volatility Swaps, FinancialCAD Corporation, http://docs.fincad.com/support/developerFunc/mathref/VarianceSwaps.htm