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I try to understand the derivation of the Black-Scholes equation based on the "constructing a replicating portfolio".

From mathematical point of view it looks simple. We assume that:

  1. Stock prices is given by Ito's process.
  2. The bond price grows exponentially.
  3. The derivative price is a function of time and stock price.

Then we apply Ito's lemma to get the stochastic differential equation for the derivative price. Then the logic is as following:

  1. As far as I understand, we consider a derivative as a mixture (in certain proportions) of stock and bond.
  2. Then the differential of the derivative price can be expressed through the differential of the stock and bond prices.
  3. Then this equation is compared with the differential of the derivative price obtained from the Ito's lemma. We require that the two equations for the differential of the derivative price should be the same and get the Black-Scholes equation.

What I do not understand, is why this procedure fixes the proportion of the stock and bond in the derivative. Why we cannot have a derivative consisting of an arbitrary amount of stock and bonds? We cannot have it because in this case the price of the derivative will not be given as a function of the stock price and time? So, why it is a problem? Why the derivative price should be a function of the stock price and time? I think that the root of my problem is that I do not know the concept of the self-financing and replicating portfolio. Could anybody, please, clarify this issue for me?

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Please see the FAQ –  SRKX Dec 23 '11 at 18:11
    
I think this question is pretty reasonable since the OP has tried to understand it himself, cited his knowledge, and pointed to a very specific issue that he has trouble with. This is exactly the kind of beginner-level question that would be good for us to answer. –  chrisaycock Dec 23 '11 at 18:29
    
@chrisaycock: I thought we decided that "if you aren't earning a living at this, it's probably off topic". I was trying to be consistent. –  SRKX Dec 23 '11 at 22:15
    
@chrisaycock: but I agree with you in the sense that the question is well formulated. –  SRKX Dec 23 '11 at 23:05

1 Answer 1

up vote 5 down vote accepted

Specifically, we have a generic conditional claim, $C$, that is a function of the diffusion process for the underlying, $S(t)$, and time $t$ so $C = C(S(t), t)$. As you pointed out, $C$ is an Ito process becuase it is a function of a stochastic process so we use Ito's Lemma to determine how the contingent claim varies as a function of the diffusion process $S(t)$.

Actually deriving the Black-Scholes PDE helps a lot in understanding how this all comes together, by the way. I will spare this group the derivation as one can easily find it elsewhere on the Web.

The reason we want a replicating portfolio is to hold fast the no-arbitrage assumption. This replicating portfolio is made up of a money market (or bond) that accrues interest as a function of time, $m(S(t), t)$ and a position in the stock as a function of time: $h(S(t), t)$ number of shares multiplied by $S(t)$ the underlying price. Note that the reason $m$ and $h$ are functions of the stock process is the dynamic nature of the replication. As the stock process diffuses, the number of shares and position in the money market changes. Also note that the function $h$ is called the hedge ratio, or delta ($\frac{\partial C}{\partial S}$) because it represents the number of shares of stock required to replicate, or hedge changes in the price of the claim.

So to answer some of your questions more directly:

why this procedure fixes the proportion of the stock and bond in the derivative?

It doesn't fix the proportion. This is a dynamic replicating portfolio where the hedge ratio, $h$ changes with the diffusion process. The money market (or bond) is there to dynamically finance the purchase (or sales) of shares based on the hedge ratio as the stock diffuses.

Why we cannot have a derivative consisting of an arbitrary amount of stock and bonds?

In essence, it kind of is arbitrary. The quantity of shares, determined by the stock process and denoted by the hedge ratio, $h$, depends on the random diffusion of the stock process. Geometric brownian motion is assumed for Black-Scholes:

$$dS = \mu s dt + \sigma s dW$$

Where $dW$ is a random Wiener process.

We cannot have it because in this case the price of the derivative will not be given as a function of the stock price and time? So, why it is a problem?

I'm not entirely sure what you mean here... but think of the replicating portfolio as a closed system. You have an initial deposit of shares and cash to dynamically replicate the claim through time. We want the no-arbitrage assumption to hold, which is why we do this.

Why the derivative price should be a function of the stock price and time?

This is the definition of a derivative, a contract that's value is derived from some other asset.

I think that the root of my problem is that I do not know the concept of the self-financing and replicating portfolio. Could anybody, please, clarify this issue for me?

Self-financing means we have an initial deposit of shares and cash which is meant to dynamically replicate the option through time. The self-replicating portfolio is holds the no-arbitrage argument.

If a call option was trading for 0.50 and the replicating portfolio (a portfolio that replicated the value through time) was trading at 0.55, one could buy the option and sell the portfolio to reap a risk-less profit (arbitrage).

These are the fundamental ideas of derivatives pricing: no-arbitrage and replicating portfolios.

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