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In the standard Black Scholes market there is only one stock. In the generealized market there can be a finite amount, but my impression is that there are few stocks in the market. The real world consists of thousands of stocks. Is it then a weakness that there are so few stocks in the stochastic models or do they give reasonable results despite this?

If there are more stocks are there more arbitrage possibilities? For example a stock may be arbitrage free in one model, but if we model it with more stocks we may create arbitrage? Or can this not happen?

Since these models are well known and used a lot I guess there is an explanation as to why it is ok to just have a few stocks, but what is the reason?

Update:

To make the quesiton clearer: In the Black-Scholes Model we have a risky asset and a bank. But why is it ok to only have one risky asset in the model, when in the real world there are thousands of risky assets. Is the model realistic with only one risky asset?

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  • $\begingroup$ @Kevin No, not really. Maybe my question is misunderstood but it is very simple: In the Black Scholes model we model a risky asset and a bank account. But why is it ok to only have one risky asset when the real world consists of many risky assets? Could there be that we miss arbitrage opportunities regarding the stock when we only model one risky asset? $\endgroup$ – user394334 May 9 at 19:49
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Not sure why so many people down-vote this question. It also has nothing to do with stochastic calculus in my opinion. If anything, the paper provided by @lays just validates the question (as it clearly states that in the BS economy there are only two assets).

Why do I like that question? Let's assume you have access to an option pricing tool that handles option payoffs in a flexible way (if you do not, you will simply have to believe me I am afraid). If you price two separate European call options on two different underlying's, each will be priced with BS (technically your system will offer more choices but usually there are more problems than benefits if you use complex models for vanilla options). Take the sum. That is your total cost.

Now, compute the price in one go (simply add them in your pricing tool, defined as two separate call options like before, but now priced simultaneously as the sum of these two - just like you summed up the two individual BS prices before). Interestingly, you will observe that your price does not match. How come? Since you have two underlying's, theory tells you that correlation between them is needed.

So which price is correct?

Let's start with a few assumptions in BS:

  • underlying follows geometric Brownian motion; exhibits lognormal distribution of returns (meaning log returns are normally distributed)
  • liquid underlying with continuous trading and no restrictions or market frictions like transaction costs, regulatory constraints (short selling and so forth), no taxes, continuous prices without jumps,...
  • no arbitrage (that is an assumption, not a result, so your question about arbitrage is simply ruled out - more on that later)
  • a continuously compounded risk free rate that is known and constant; due to no market frictions, borrowing and lending is allowed at this rate
  • volatility of the underlying is known and constant
  • yield of the underlying (if there is) is known and constant
  • “dynamic hedging” turns it into a risk-free instrument
  • complete markets

. . . Probably more but too lazy to get into the nitty gritty detail.

Now, start from the bottom:

  • If markets are complete, we do not need options (they are meaningless and add no economic value). If markets are inefficient, we may need options. However, if we need the assumption of complete markets to value an option, it is in fact impossible to price them. So either, we do not need them, or cannot price them. That's called the Hakansson's paradox.
  • Dynamic hedging is a mathematical impossibility
  • Nothing is constant
  • Nothing is frictionless
  • No arb is an issue, which is why many FX option pricing tools artificially ensure internal consistency by implying one of the 4 values in covered interest rate parity. The link shows how this is done in practice. Reason? Otherwise, it would matter what you start with and you could always choose the direction that makes you better off (but your counterparty may have a different opinion).
  • Well, prices jump, sometimes ridiculously (Gamestop...)
  • (Log) returns are not normal

BSM is a model. Developed by economists. Based on assumptions that make most people outside of formal (neoclassical) economics smile. However, what is the alternative?

Back to my example with two options. Now assume you trade hundreds of options a day. Ideally the model now accounts for not only the distribution of the individual pairs but also the crosses (e.g. EURUSD, USDCHF as main pairs, but you should also care about EURCHF). Now you generally face the problem that you have a correlation matrix that is not guaranteed to be positive-definite. A common approach to circumvent this is by shifting the eigenvalues of the correlation matrix. Solving a system of SDEs with hundreds or thousands options will reach your computational capacity in no time. Also, what if you add a new option, did you now misprice all previous products? Also, this is still all based on Bachelier's work (3rd and 2nd last paragraphs for some ideas). I deliberately left BSM out here as they mainly rebranded existing ideas into arguments that fit (financial) economics.

If you run some exercises, you realize that the difference is actually not that big (depending on correlation, tenor, vol, moneyness ...). Adding a few more, and you will reach timeout in your pricing engine. So that does not work either.

Ultimately, you either simplify or you do not trade options (which is rational after all, if the assumptions would hold). Equations (and diagrams) of formal economics are, ideally, just like scaffolding Krugman, P.6. They are used to help construct an intellectual edifice to a problem. Once that is achieved, the scaffolding can be stripped away, leaving only plain English behind.

With regards to options, BSM is mainly a mapping tool. You observe option prices (if listed like equity or commodity, you use it to compute Implied Vol surfaces) or you observe IVOL (many OTC markets like FX or IR options quote directly in IVOL) and use it to provide you with a price. Given you have a smile already shows that BSM is wrong. It does however make it easier to compare and to see where options are overpriced along moneyness/delta vs historical levels, etc.

These IVOL levels do not come from BSM but are influenced by many things, like supply and demand. One way to hedge away inventory risk in options is to use other options (and you mainly care about net positions). This in turn contradicts the BSM theory with geometric Brownian motion and continuous-time hedging where the demand and supply for options should not affect the price of options (they are redundant after all).

To answer your question: "Is the model realistic?" No, not in the slightest. However, is a map a realistic replication of a complex world? No, not in the slightest. Did it ever bother you when you looked at a Mercator map? Probably not. After all, you get an idea where continents and countries are. Moreover, it is representing north as up and south as down everywhere and preserves local directions and shapes. Let's just ignore that it distorts sizes, it still works quite well.

Edit:
@Antoine Conze, this is more an observation rather than a claim. If you have access to Bloomberg you can try DLIB BLAN, which loads with a single vanilla European equity option script as an example. It is easy to extend to two contracts, and using all(contract1,contract2) provides the sum. You will see that once you validate, the correlation tab shows up. Now, obviously, that is just an implementation issue and the code most certainly does not check what the combination of these two underlyings actually is. Yet, if you use this tool, you will get a different answer for the same product if you price individually or simultaneously.

Ignoring that this may (or may not) be a specification issue, there is an interesting thought experiment. I assume we believe weighted basket options need correlation to be priced. Now let's compute European options and only look at the last day, with assumed values to make it easy and model free.

Individual calls vs weighted basket (model free):

  • formula for (partial) performance (expressed as performance for ease of comparison among multiple underlyings) $$Performance = \frac{final\; value}{start\; value}$$
  • call value (strike expressed in performance) $$call = max(0, Performance - Strike)*Notional$$
  • payoff for individual calls are just summed up $$ sum(call1+call2)$$
  • performance for weighted basket $$Basket\; Perf. =Performance1*Weight1 + Performance2*Weight2$$
  • payoff basket $$ max(Basket\; Performance - Strike,0)* Total\; Notional$$

$Total\; notional = 1000$ hence per underlying 500 (equal weight)
$S_{t_o}=100$ for both underlyings (does not change anything) and assume
$S_{1t}=110$ and $S_{2t}=95$ (in whatever future period).
assume strike is ATMS (hence K=1)

  • individual calls $$ Call_1 = max\Big(\frac{S_{1t}}{S_{1t_o}}-1,0\Big) = max\Big(\frac{110}{100}-1,0\Big) = 0.1 $$ $$ Call_2 = max\Big(\frac{95}{100}-1,0\Big) = 0$$
  • sum of calls with each 50% of total Notional: $$0.1*500+0*500=50$$
  • weighted return of basket $$bskt =\frac{110}{100}*0.5 + \frac{95}{100}*0.5 = 1.025$$ $$Call = max\Big(bskt-K,0\Big)*N = max\Big(1.025-1,0\Big)*1000= 25$$

As expected, weighted basket is different from individual calls - which is why correlation matters and models account for this.
Now, what about forwards? Going long a call and short a put replicates a FWD synthetically. Generally, $$FWD = (Performance - Strike)*Notional$$

Back to our example:

  • put value (strike expressed in performance) $$put = max(0, Strike- Performance)*Notional$$ $$ put_1 = max\Big(1-\frac{S_{1t}}{S_{1t_o}},0\Big) = max\Big(\frac{110}{100}-1,0\Big) = 0 $$ $$ put_2 = max\Big(1- \frac{95}{100},0\Big) = 0.05$$
  • compute forward
    $$Fwd1 = call1*N/2-Put1*N/2 = 0.1∗500-0∗500 = 50$$
    $$Fwd2 = call2*N/2-Put2*N/2 = 0∗500-0.05∗500 = -25$$
  • sum of forwards $$Fwd1-Fwd2 = 50 +(-25) = 25$$

Now, for the basket, you still have the weighted return. Just a forward is defined as $$(Basket\; Performance - Strike)* Total\; Notional$$ $$(1.025-1)*1000=25$$

And now, there is a problem. If you need to model (take into account) correlation for a call or put basket option, there is no reason why you should not use one if you combine these individual calls and puts in whatever way. However, it was just shown that two individual forwards (which are unrelated to each other in BS) or a basket forward result in the same outcome. If the cashflows are identical, price needs to be identical (in theory). It is easy to run a simulation to check if that logic really applies for all sorts of performance values (as well as weights, start values ...).
I used Julia to generate arbitrary price vectors for two generic underlyings.
enter image description here enter image description here

Individual options on each underlying cover some cases, which are not protected by a basket option which is why they cost more than a basket. This is highlighted in red above. Now, let's consider the forwards.
enter image description here
Looks as if the difference is zero throughout. Let's use arbitrary weights and start values and run 1 million different price scenarios and compute the total difference of all outcomes.
enter image description here
enter image description here

Either way, ignoring whether one should or should not account for other securities in pricing options, VaR computations on (option) portfolios frequently do take correlation into account and it can be shown that not doing so is problematic.

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  • $\begingroup$ Thank you very much, it was a very informative answer! The example with the pricing of two options combined or separately is very interesting. $\endgroup$ – user394334 May 11 at 8:39
  • $\begingroup$ By the way, lets say you have an option that you price using a model with only one underlying stock, could it be that if you have a model with more underlying stocks, but have the same option only depending on that one stock, could the price then be different? Maybe your example already covered this, sorry if it did. $\endgroup$ – user394334 May 11 at 8:42
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    $\begingroup$ Why do you claim that when pricing two options on two different assets in one run the prices would depend on correlation between the assets ? $\endgroup$ – Antoine Conze May 11 at 11:17
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    $\begingroup$ @AKdemy the reason why BB Dlib produces different prices for the sum of two calls when priced separately or together is that it uses MC simulation, and when priced together it generates correlated brownian innovations applying a cholesky transform to uncorrelated gaussian deviates, so this is just an implementation issue. The PV of a call on $X_1$ only depends on the marginal of $X_1$, regardless of what the joint distribution of $X_1$ and $X_2$ is. $\endgroup$ – Antoine Conze May 13 at 10:32
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Your assumption is wrong.The Black & Scholes model isn’t only based on one stock.I think you are confusing Black and Scholes with risk neutrality and the assumption that every underlying must be priced with a risk free drift in the semi martingale process.

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  • $\begingroup$ Thank you, but could you please explain it a little further? In the Black-Scholes Market you have that the process is modelled by a Stochastic differential equation, but in many cases you have just one stochastic differential equation? Why is that ok and not for instance have a system of thousands differential equations since there are thousands of stocks in the world? $\endgroup$ – user394334 May 9 at 16:51
  • $\begingroup$ I think you might have a look a this so as to learn the basics behind the stochastic equations: frouah.com/finance%20notes/Black%20Scholes%20Formula.pdf $\endgroup$ – lays May 9 at 19:17
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    $\begingroup$ It says it right in your link that in the "Black-Scholes economy" there are two assets, a stock and a bond. The question is simply: why is it ok to have only one stock in the modelled economy when in the real world there are thousands of stocks? $\endgroup$ – user394334 May 9 at 19:33
  • $\begingroup$ You invest on whatever European option you want but under the BS assumption you will replicate your option price value using a bond and a stock(for this you can check the PDE of the BS equation).Basically that what it is meant. $\endgroup$ – lays May 9 at 20:01

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