Forward Variance Quotes as Martingales Under No-Arbitrage Pricing

# Generating a P&L that is linear in the variation on an underlying at no cost


# Generating a P&L that is linear in the variation on an underlying at no cost












I am actually reading Lorenzo Bergomi's "Stochastic Volatility Modelling" book, and came across this bit :

I understand everything up to (5.5) included. But I don't see the point in mentioning the vanishing pricing drift. What is "pricing drift" ? After that he defines continuous (instantaneous in fact) VS forward variance and writes that they are driftless as well (same argument as for discrete VS forward variance), and write that in a diffusive setting they are equal to $\left(\ldots\right) dW_t^T$ where I guess $W^T$ is a standard Brownian motion under the forward $T$ measure.

Is pricing drift defined outside a diffusive setting or does everything here takes place in a diffusive setting ? (The "in a diffusive setting" is unsettling.)

What is general in Bergomi's remark about pricing drift ? I mean, is there a way to define being a martingle through linearity of a certain P&L or ?

## Answer by Quantuple (score 1, accepted)

https://quant.stackexchange.com/a/33297

Short answer

He's basically making a parallel between a forward variance trade and a futures trade. In both cases you should have that the underlying quotes are martingales in the absence of arbitrage.

Long(er) answer

Under the physical measure $\Bbb{P}$, an arbitrage is a (self-financing) trading strategy $V$ - or rather the value of a portfolio implementing this strategy - for which there exists a time $T > 0$ such that $$ V_0=0,\,\, V_T \geq 0\,\, \Bbb{P}-\text{a.s. and } \Bbb{P}(V_T \ne 0) > 0$$

Suppose you define an equivalent probability measure $\Bbb{Q}\equiv\Bbb{P}$. Since by definition, both measures agree on null events, our arbitrage definition translates to $$ V_0=0,\,\, V_T \geq 0\,\, \Bbb{Q}-\text{a.s. and } \Bbb{Q}(V_T \ne 0) > 0 \tag{A}$$

Notice that if $\Bbb{Q}$ is further a martingale measure, that is if $(V_t)_{t\geq0}$ emerges as a $\Bbb{Q}$-martingale: $$ V_0 = \Bbb{E}_0^\Bbb{Q} [ V_T ] $$ then $(A)$ will never happen. This explains the central role of equivalent martingale measures in arbitrage pricing theory.

Putting that back into context, you've managed to identify a (self-financing) strategy (i.e. buying and selling forward variance swaps), which at no cost ($V_t=0$), allows you to earn a quantity $$V_{t'} = (T_2-T_1) \left( \hat{\sigma}_{VS,T_1T_2}^2(t') - \hat{\sigma}_{VS,T_1T_2}^2(t)\right)$$

Based on what we've said earlier, in the absence of arbitrage, there should exist a measure $\Bbb{Q} \equiv \Bbb{P}$ such that $$ \Bbb{E}^\Bbb{Q}_{t}[V_{t'}] = V_t$$ hence, using the definitions of $V_t$ and $V_{t'}$, $$ \Bbb{E}^\Bbb{Q}_{t}\left[ \hat{\sigma}_{VS,T_1T_2}^2(t') \right] = \hat{\sigma}_{VS,T_1T_2}^2(t) $$ hence forward variance swap quotes are martingales. Assuming a continuous paths process (= in a diffusive setting), by the martingale representation theorem we should then have $$ \hat{\sigma}_{VS,T_1T_2}^2(t) = ... dW_t^\Bbb{Q} $$ hence no pricing drift under $\Bbb{Q}$.