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From the Wikipedia page, we know call option as an example is price through delta hedging.

$$\Pi=-V+V_SS$$

and over $[t,t+\triangle t]$

$$\triangle\Pi=-\triangle V+V_S\triangle S$$

My questions are:

  1. Is it mathematically rigorous to use $\triangle$ notation in deriving BS formula?

  2. Is Delta-Hedging always the best trading strategy in the BS model? How about the other option/derivative models? If not, how to actually find the best trading strategies?

  3. If yes, how to implement it statistically?

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  • $\begingroup$ As you are asking about rigour. Note that a portfolio that has a constant short position of one unit of the derivative is not self-financing. See e.g. Chapter 3.1.10 in Musiela and Rutkowski (2005) "Martingale Methods in Financial Modelling". $\endgroup$ – LocalVolatility Dec 26 '16 at 11:22
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1.) in textbooks usually stochastic differential $d$ is used which is rigourous

2.) delta heding in Black Scholes worldis perfect as it's the only way to eliminate risk completely in non friction market with continuous hedging

3.) in practice usually discrete hedging is used by delta-hedging and gamma-hedging , example of statistical implementation in python is here: delta hedging simulation with python

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The $\Delta$ just define the time difference you are considering. In order to formulate the differential equation correctly you need to send $\Delta$ to zero. i.e $lim_\Delta\rightarrow 0 $

In order to understand the meaning of Delta Hedging, you need to understand the composition of the portfolio. As you stated, it is a short position in Derivative and $\Delta$ long position in the underlying asset. This portfolio cancels exactly the random term $dS$ with Wiener-process each other within the time intervall $\Delta t$. However $\Delta$ depends on the time intervall you are considering and this may vary during the time evolution. Therefore you need also hedge the time dependent change of $\Delta$, which is called as Gamma-Hedging.

The Hedging-Method with other Models are similiar. Since the change of the derivative and the underlying are generated by the same Wiener-Process, you can always design a hedging method in order to cancel the random-term, i.e. For the asset: $$dS=\mu dt+ \sigma dW$$ For the derivative: $$dV=\mu_V dt+ \sigma_V dW$$ Consider the Derivative V(i.e. Options) does not need to be a derivative(in mathematical sense). But this should demonstrate the idea. The idea is just to find a portfolio to cancel the $dW$.

So you can see, we completely canceled our uncertainty(in ideal case), under risk neutral measure, the change of your portfolio value is completely based on the time value of your money. If you are risk neutral, this is the best trading strategy.

The most common implementation is Finite Difference Methods and Monte-Carlo-Simulation. As the $\Delta$ is a derivative with respect to a Ito-Process, you need first solve this Differentialequation by finite difference methods and then simulate its path by Monte-Carlo-Simulation.

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