Break-Even Covariance and Average Delta-Hedging P&L

# Understanding assumption in Delta hedging P&L from Bergomi Chapter 1


# Understanding assumption in Delta hedging P&L from Bergomi Chapter 1












In Chapter 1 of Bergomi's Stochastic Volatility modelling book there is a derivation of the delta hedging P&L to get a black-scholes like formula.

The derivation in the multi asset case goes something like this.

Suppose there are $n$ assets $S_1,\ldots, S_n$ and there is a pricing function for an option $P(t,S_1,\ldots, S_n)$. Then the (daily) P&L of a delta hedge for a short position in this option has the form

$$P\&L = -(P(t+dt) - P(t)) + r\left(P - \sum_{i=1}^{n}\Delta_i S_i\right) + \sum_{i=1}^{n} \Delta_i \delta S_i + \sum_{i=1}^{n} q_iS_idt$$ Using the Taylor expansion for $P$ and setting $\Delta_i = \frac{\partial P}{\partial S_i}$ to remove the degree $1$ $dS$ term, we get $$P\&L = -\left(\frac{\partial P}{\partial t} + \sum_{i=1}^{n}(r-q_i) S_i \frac{\partial P}{\partial S_i} - rP\right)dt - \frac{1}{2}\sum_{i,j=1}^{n}\frac{\partial^2 P}{\partial S_i \partial S_j}S_iS_j\frac{dS_idS_j}{S_iS_j} = -Adt -\frac{1}{2}\sum_{i,j}\phi_{i,j}\frac{dS_idS_j}{S_iS_j}$$

The derivation continues a bit in this manner with diagonalizing $\phi$ and writing it in a nice form, but my question is related to a condtion Bergomi imposes: $A = -\frac{1}{2}\text{tr}(\phi C)$ where $C$ is a positive definite matrix. He later goes on to say "the condition that our model is usable – no situation in which our carry P&L is systematically positive or negative – is that there exists a positive break-even covariance matrix $C(t, S)$" such that the above equation holds.

I would like a justification or a motivation for why we should want a condition like this, that $$\frac{\partial P}{\partial t} + \sum_{i=1}^{n}(r-q_i) S_i \frac{\partial P}{\partial S_i} - rP = -\sum_{i,j} \frac{\partial^2 P}{\partial S_i \partial S_j} C_{i,j}$$

## Answer by Arshdeep (score 1)

https://quant.stackexchange.com/a/79621

Same as the scalar version, your PnL is a statistical quantity and you want its average to be 0, that is the only place where demand and supply will match

## Answer by i_delta (score 1)

https://quant.stackexchange.com/a/80034

Basically, you require that there exist a dynamics - whatever it is - such that average P&L vanishes.

Given any model for the $S_i$, the resulting realized covariance matrix $C^r_{ij} = \frac{1}{dt}E\left[\frac{dS_idS_j}{S_iS_j}\right]$ is positive.

If no positive $C$ exists such that your last equation holds, then there exists no instance of a realized dynamics for the $S_i$ such that the P&L averages out to zero.

If instead such a $C$ exists, then your average P&L during $dt$ reads: $$E[P\&L] = \frac{1}{2}\sum_{ij} \phi_{ij}(C_{ij} - C^r_{ij})dt $$ If realized covariances $C^r_{ij}$ happen to match implied ones $C_{ij}$, the P&L averages out to zero.