# 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 $$\frac{\partial P}{\partial t} + \sum_{i=1}^{n}(r-q_i) S_i \frac{\partial P}{\partial S_i} - rP = -\sum_{i,j} \frac{\partial^2 P}{\partial S_i \partial S_j} C_{i,j}$$

## 2 Answers

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

• I don't follow this case. I am not able to see why this condition is related to the more intuitive condition "average P&L should be 0". Taking expectations on both sides of the second equation above wouldn't work directly Commented Jun 4 at 20:03
• No expectations needed, the delta hedged portfolio has deterministic PnL today until it is rebalanced. Commented Jun 4 at 21:00
• The only statistics that manifest ehre are $dS(i)dS(j)=cov(i,j)*dt$, which is applicable in the differential form. That makes PnL known if you know covariance. Commented Jun 4 at 21:07

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.