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According to my understanding the derivation of the Black-Scholes PDE is based on the assumption that the price of the option should change in time in such a way that it should be possible to construct such a self-financing portfolio whose price replicates the price of the option (within a very small time interval). And my question is: Why do we assume that the price of the option has this property?

I will explain myself in more details. First, we assume that the price of a call option $C$ depends on the price of the underlying stock $S$ and time $t$. Then we use the Ito's lemma to get the following expression:

$d C = (\frac{\partial C}{\partial t} + S\mu\frac{\partial C}{\partial S} + \frac{1}{2}S^2 \sigma^2 \frac{\partial^2C}{\partial S^2}) dt + \sigma S \frac{\partial C}{\partial S} dW$ (1) ,

where $\mu$ and $\sigma$ are parameters which determine the time evolution of the stock price:

$dS = S(\mu dt + \sigma dW)$ (2)

Now we construct a self-financing portfolio which consist of $\omega_s$ shares of the underlying stock and $\omega_b$ shares of a bond. Since the portfolio is self financing, its price $P$ should change in this way:

$dP(t) = \omega_s dS(t) + \omega_b dB(t)$. (3)

Now we require that $P=C$ and $dP = dC$. It means that we want to find such $\omega_s$ and $\omega_b$ that the portfolio has the same price that the option and its change in price has the same value as the change in price of the option. OK. Why not? If we want to have such a portfolio, we can do it. The special requirements to its price and change of its price should fix its content (i.e. the requirement should fix the portion of the stock and bond in the portfolio ($\omega_s$ and $\omega_b$)).

If we substitute (2) in (3), and make use of the fact that $dB = rBdt$ we will get:

$\frac{\partial C}{\partial t} + S\mu\frac{\partial C}{\partial S} + \frac{1}{2}S^2 \sigma^2 \frac{\partial^2C}{\partial S^2} = \omega_s S \mu + \omega_b r B$ (4)

and

$\sigma S \frac{\partial C}{\partial S} = \omega_s S \sigma$ (5)

From last equation we can determine $\omega_s$:

$\omega_s = \frac{\partial C}{\partial S}$ (6)

So, we know the portion of the stock in the portfolio. Since we also know the price of the portfolio (it is equal to the price of the option), we can also determine the portion of the bond in the portfolio ($\omega_b$).

Now, if we substitute the found $\omega_s$ and $\omega_b$ into the (4) we will get an expression which binds $\frac{\partial C}{\partial t}$, $\frac{\partial C}{\partial S}$, and $\frac{\partial^2 C}{\partial S^2}$:

$\frac{\partial C}{\partial t} + rS \frac{\partial C}{\partial S} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 C}{\partial S^2} = rC$

This is nothing but the Black-Scholes PDE.

What I do not understand is what requirement binds the derivatives of $C$ over $S$ and $t$.

In other words, we apply certain requirements (restrictions) to our portfolio (it should follow the price of the option). As a consequence, we restrict the content of our portfolio (we fix $\omega_s$ and $\omega_b$). But we do not apply any requirements to the price of the option. Well, we say that it should be a function of the $S$ and $t$. As a consequence, we got the equation (1). But from that we will not get any relation between the derivatives of $C$. We, also constructed a replicating portfolio, but why its existence should restrict the evolution of the price of the option?

It looks to me that the requirement that I am missing is the following:

The price of the option should depend on $S$ and $t$ in such a way that it should be possible to create a self-financing portfolio which replicates the price of the option.

Am I right? Do we have this requirement? And do we get the Black-Scholes PDE from this requirement? If it is the case, can anybody, please, explain where this requirement comes from.

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in theory, yes, but then you need to readjust your position dynamically intraday and very often the cost and slippage of this replication will be so high that in practice it doesn't work. –  RockScience Jan 3 '12 at 10:45
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2 Answers

I feel that the best way to answer your question is to first quote your problematic idea and then carefully explain the subtle alternative. :)

The derivation of the Black-Scholes PDE is based on the assumption that the price of the option should change in time in such a way that ... And my question is: Why do we assume that the price of the option has this property?

The derivation of B-S PDE doesn't require the assumption above, though it will make derivation easier indeed. Besides, this is not an assumption. This is a natural consequence that we can derive. We don't need to assume option price's properties (i.e. should change in time in such a way ... ). Instead, we derive it, the way option price must follow.

To construct such a self-financing portfolio whose price replicates the price of the option (within a very small time interval).

You probably somewhat had this chicken-and-egg mystery in your mind: how can I replicate the price of the option before I ever know it? The truth is that (at least to me), for most of the time, my self-financing portfolio is trying to do/replicate something else irrelevant to the mysterious option price, and I found that the option price had better follow what I am doing, not the other way around.

Now we require that P=C and dP=dC.

Let's think a little deeper about the equation P=C: who forces it? why? how they do it? The answer is somewhat subtle: it depends on when you ask these questions.

  • At expiry (T): P$_T$ = C$_T$ is enforced by contract/exchange/law. If your counter-party default, buyer can sue the writer, or exchange will handle it and guarantee buyer's right.

  • Anytime between now and expiry: P$_t$ = C$_t$ is not forced by contract/exchange/law. If there are markets for options, stocks, and bonds, B-S arbitragers will jump in to "help" (instead of force it) by trading against any price deviation. Arbitrager will monetize the deviation by constantly rebalancing a self-financing portfolio P$_t$ that will eventually replicate the option's payoff at expiry.

See the difference? P$_T$ = C$_T$ is a requirement, i.e. boundary condition, enforced by law in reality, but P$_t$ = C$_t$ is not. P$_t$ = C$_t$ is more like a consequence than a requirement.

It means that we want to find such ωs and ωb that the portfolio has the same price that the option and its change in price has the same value as the change in price of the option. OK. Why not?

No, again we find the portfolio for something else. It's the option price that had better attach to our specially designed portfolio, not the other way around.

Alright, then what is the self-financing portfolio supposed to do? It's easier to answer the question step-by-step backward. Let's live in a discrete world for a moment without losing generality.

  • At $T-1$: The self-financing portfolio P$_{T-1}$ is designed to replicate the option's value at $T$, i.e. pay-off.

  • At $T-2$: The self-financing portfolio P$_{T-2}$ is designed to replicate the most near future self-financing portfolio's value (which has been determined in the previous step) $P$ $_{T-1}$ that will replicate the option's pay-off in the further future $T$. There is nothing to do with $C$ $_{T-2}$.

  • At $T-3$: The self-financing portfolio $P$ $_{T-3}$ is designed to replicate the most near future self-financing portfolio $P$ $_{T-2}$ that will replicate $P$ $_{T-1}$ that will replicate the option's pay-off at $T$. Again, C$_{T-3}$ is irrelevant.

  • ...

  • At $T-n$: The self-financing portfolio $P$ $_{T-n}$ is designed to replicate the most near future self-financing portfolio $P$ $_{T-n+1}$ that will replicate $P$ $_{T-n+2}$ that will replicate ... (eventually option's pay-off at $T$).

If I do every step correctly, I can hang on long enough to the expiry to let the contract/exchange/law enforce the arbitrage/convergence. Note that this is the only convergence (requirement) guaranteed in reality. C$_t$ = P$_t$ is not guaranteed. It is helped by the market/arbitrager, who are willing to (but they don't have to, nobody has to) diligently trade and hedge in order to secure the profit from the price difference.

In other words, we apply certain requirements (restrictions) to our portfolio (it should follow the price of the option). As a consequence, we restrict the content of our portfolio (we fix ωs and ωb). But we do not apply any requirements to the price of the option.

Now you should be able to answer yourself. In any cases, your portfolio is not restricted by the option price. It's the other way around: the market/arbitrager help restrict the option price using the self-financing portfolio.

Well, we say that it should be a function of the S and t. As a consequence, we got the equation (1). But from that we will not get any relation between the derivatives of C. We, also constructed a replicating portfolio, but why its existence should restrict the evolution of the price of the option?

Let me give you another analogy. If I tell you that at $T$, apple$_T$ is guaranteed to equal to orange$_T$. Now at $t$, an orange$_t$ = \$5, what do you think about apple$_t$'s value? Should the existence of orange and its evolution restrict apple's price? What will you do if their price are different?

To make my analogy more similar, let me also tell you this: "Hey, you got to do something to your orange$t$! Otherwise, it will become banana{t+1}!" There is no guarantee between apple and banana at expiry $T$.

What I do not understand is what requirement binds the derivatives of C over S and t. It looks to me that the requirement that I am missing is the following:

The price of the option should depend on S and t in such a way that it should be possible to create a self-financing portfolio which replicates the price of the option.

Am I right? Do we have this requirement? And do we get the Black-Scholes PDE from this requirement? If it is the case, can anybody, please, explain where this requirement comes from.

Is there still a missing requirement to you in my apple and orange analogy? Do you need any? :)

Again, the market/arbitrager help option price depend on S and t. Their trading activities make things work like that. However, this is not a pre-request to create the self-financing portfolio.

Now let me try to revise your argument:

The price of the option had better depend on S and t in such a way that it follows the value of a specially designed self-financing portfolio by volatility arbitrager. The arbitrager's self-financing portfolio is designed to replicate the most near future self-financing portfolio that will eventually replicate the option's payoff at expiry.

In conclusion, the missing requirement you originally thought is actually, a natural consequence. :)

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Will highly appreciate if someone can kindly teach me how to make P\$_{T−2}\$ work. P\$_{T−2}\$ display properly in edit mode (and comments), but not after post? Why? Orz –  楊祝昇 Jan 6 '12 at 1:39
    
@chrisaycock: Thanks a lot for the attempt! :) Yeah, it feels like debugging, not fun. I found an acceptable alternative using \$P\$ \$_{T-3}\$. It seems that \$_{x}\$ ... z\$_{y}\$ doesn't work but \$_{x}\$ ... \$z\$ \$_{y}\$ will work, Orz. –  楊祝昇 Jan 6 '12 at 1:40
    
Orz, debugging makes my answer a community wiki. Thus, I deleted and re-post it. –  楊祝昇 Jan 6 '12 at 1:43
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You are quite correct that there are further assumptions in the replicating argument. Once you are assuming your equation (2), that is, that $$ dS = S(\mu dt + \sigma dW) $$ along with the determinism of interest rates, the rest of the replication argument necessarily follows because you have constructed a mathematical world with nothing else in it. Hence $S, t, r, \sigma$ and $q$ are sufficient to price the option.

Nothing in finance forces any of the assumptions to be true, and as a matter of fact they are false to certain degrees. Consider for example a world in which $B(t)$ it itself stochastic, which of course we necessarily have to treat for itnerest rate options. In that case, an American call option will have a pricing formula depending on $S$, $t$, and various interest rate variables.

Alternatively, consider the case where $$ dS = S(\mu dt + \sigma dW + dJ) $$ for some jump process $J$. Depending on how $J$ behaves it may be theoretically impossible to replicate the option. One can still get at valuations using diversification arguments and the like, but those valuations will depend on other parameters.

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thank you for the answer. My problem is not in the assumptions but in the logic which combines these assumptions to obtain the Black-Scholes PDE. It is OK for me to assume that the stock prices is given by the diffusion process (equation 2) and that the bond is deterministic (i.e. not stochastic). I also can accept the fact that it is always possible to construct a self-financing replicating portfolio. what is not clear is why all that restricts the price of the options (as function of S and t). –  Roman Jan 5 '12 at 18:10
    
Well, what it means is that there is a "replication price" for options based solely on $S, t$ and the other variables, such that trading the option is unnecessary to achieve teh same payoff as the option itself. Actual option prices can differ, but any difference enables arbitrage and would hence disappear quickly (at least theoretically) –  Brian B Jan 5 '12 at 19:01
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