Self-Financing Delta Hedging and Option P&L

# Black Scholes Stochastic Calc P&L


# Black Scholes Stochastic Calc P&L












I am reading The Volatility Smile by Emanuel Derman and Michael Miller, and am struggling with the technical details behind the following SDE for the P&L of a continuously delta-hedged long position in a European Call option under Black Scholes.

Roughly quoting the text, the P&L is modeled as follows:

> The value of the delta hedged portfolio at any time t, is given by: ${\Pi_t = C_t - \Delta_{BSM}*S_t}$. Where ${C_t}$ is the market price of the call option (evolves using Implied Volatility >${\sigma_I}$), while ${S_t}$ and ${\Delta_t}$ are computed using the Geometric Brownian Motion assuming realized volatility ${\sigma_R \neq\sigma_I}$. Hence, the incremental P&L at any time t is given >by: $${d(P\&L) = d(C_t)-d(\Delta_{BSM} S_t)}$$ $${=dC_t-\Delta_{BSM}dS_t-(C_t-\Delta_{BSM}S_t)rdt}$$ Where ${dC_t}$ is the incremental change in the value of the call option, ${\Delta_{BSM}dS_t}$ is the incremental change in the short position in the stock, and ${(C_t-\Delta_{BSM}S_t)rdt}$ is the incremental cost of borrowing to purchase the call position net of the cash earned by shorting position the stock.

From this, it seems that the implication is ${d(\Delta_{BSM}S_t) = \Delta_{BSM}dS_t + (C_t-\Delta_{BSM}S_t)rdt}$? How could this be derived from stochastic calculus? Is the incremental P&L possibly something other than ${d\Pi_t}$?

I've tried using Ito Calculus to say ${d(\Delta_{BSM}S_t) = d(\Delta_{BSM})S_t + \Delta_{BSM}dS_t + d(\Delta_{BSM})d(S_t)}$, and applying Ito's lemma to ${d\Delta_{BSM}}$, but I cannot get to the final P&L result provided by the text. And the text does not provide the steps, it only says it's obvious.

## Answer by Andrea (score 1, accepted)

https://quant.stackexchange.com/a/81079

The correct PnL is in integral form

$\int_0^T \Delta_tdS_t$

and not $\int_0^T d \left ( \Delta_t S_t \right ) = \Delta_T S_T - \Delta_0 S_0$

The strategy has to be self-financing.

Do it in discrete time and you will see the difference. You buy $N_t$ shares, and gain $N_t (S_{t+1}-S_t)$, then you rebalance to $N_{t+1}$ and do the same.

You will have to compute the increments day by day (and so the integral), the PnL of your portfolio is not only, $N_T S_T$, you will have to subtract the cost you paid to buy the $N_T$ shares.

This is the definition of self financing. Everything else would not make sense.

See as well: https://en.wikipedia.org/wiki/Self-financing_portfolio#Continuous_time