Supply and demand is what ultimately drives everything! But where is the supply and demand parameter or variable in Black-Scholes?

In a nutshell, the problem lies with people being so fond of complete markets. Complete markets are essentially markets in which you can hedge away risk. You can see why people like to work with such models. It makes them believe that they are safe!^{[1]} Hence the popularity of the deterministic volatility model. And even those models which are clearly not in complete markets, such as stochastic volatility or jump diffusion, people try to make complete by calibration! Of which more later.

A trivial observation: The world is net long equities after you add up all positions and options. So, net, people worry about falling markets. Therefore people will happily pay a premium for out-of-the-money puts for downside protection. The result is that put prices rise and you get a negative skew. That skew contains information about demand and supply and not about the only 'free' parameter in Black-Scholes, the volatility.

The complete-market assumption is obviously unrealistic, and importantly it leads to models in which a small number of parameters are used to capture a large number of effects.

Whenever a quant calibrates a model to the prices of options in the market he is saying something about the information content of those prices, often interpreted as a volatility, implied volatility. But really just like the price of a pint of milk is about far more than the cost of production, the price of an option is about much more than simple replication.

The price of milk is a scalar quantity that has to capture in a single number all the behind-the-scenes effects of, yes, production, but also supply and demand, salesmanship, etc. Perhaps the pint of milk is even a 'loss leader.' A vector of inputs produces a scalar price. So, no, you cannot back out the cost of production from a single price. Similarly you cannot back out a precise volatility from the price of an option when that price is also governed by supply and demand, fear and greed, not to mention all the imperfections that mess up your nice model (hedging errors, transaction costs, feedback effects, etc.).

Supply and demand dictate everything. The role of assumptions (such as no arbitrage) and models (such as the continuous lognormal random walk) is to simply put bounds on the relative prices among all the instruments. For example, you cannot have an equity price being 10 and an at-the-money call option being 20 without violating a simple arbitrage. The more realistic the assumption/model and the harder it is to violate in practice, the more seriously you should treat it. The arbitrage in that example is trivial to exploit and so should be believed. However, in contrast, the theoretical profit you might think could be achieved via dynamic hedging is harder to realize in practice because delta hedging is not the exact science that one is usually taught. Therefore results based on delta hedging should be treated less seriously.

Supply and demand dictate prices, assumptions and models impose constraints on the relative prices among instruments. Those constraints can be strong or weak depending on the strength or weakness of the assumptions and models.

Lesson 3:Jensen's Inequality Arbitraje

Jensen's Inequality states that if f() is a convex function and x is a random variable then

E[f(x)] > f(E[xj).

This justifies why non-linear instruments, options, have inherent value.

Example You roll a die, square the number of spots you get, and you win that many dollars. How much is this game worth? (Assuming you expect to break even.) We know that the average number of spots on a fair die is 3| but the fair 'price' for this bet is not (3|)2.

For this exercise f(x)isx2, it is a convex function. So

The fair price is 15^.

Jensen s Inequality and convexity can be used to explain the relationship between randomness in stock prices and the value inherent in options, the latter typically having some convexity.

Suppose that a stock price S is random and we want to consider the value of an option with payoff P(S). We could calculate the expected stock price at expiration as E[ ST ], and then the payoff at that expected price P(E[ ST]). That might make some sense; ask yourself what you think the stock price will be at expiration and then look at the corresponding payoff.

Alternatively we could look at the various option payoffs and then calculate the expected payoff as E[P(St)]. The latter actually makes more sense, and is indeed the correct way to value options (provided the expectation is with respect to the risk-neutral stock price, of course).

If the payoff is convex then

We can get an idea of how much greater the left-hand side is than the right-hand side by using a Taylor series approximation around the mean of S.Write

Therefore the left-hand side is greater than the right by approximately

This shows the importance of two concepts

• f"(E[S]): This is the convexity of an option. As a rule this adds value to an option. It also means that any intuition we may get from linear contracts (forwards and futures) might not be helpful with non-linear instruments such as options.

• E[e2]: This is the variance of the return on the random underlying. Modelling randomness is the key to valuing options.

The lesson to learn from this is that whenever a contract has convexity in a variable or parameter, and that variable or parameter is random, then allowance must be made for this in the pricing.

Example Anything depending on forward rates. If you price a fixed-income instrument with the assumption that forward rates are fixed (the deterministic models of yield, duration, etc.) and there is some non-linearity in those rates, then you are missing value. How much value depends on the convexity with respect to the forward rates and forward rate volatility.^{[2]}

Example Some things are tricky to model and so one tends to assume they are deterministic. Mortgage-backed securities have payoffs, and therefore values, that depend on prepayment. Often one assumes prepayment to be a deterministic function of interest rates, but this can be dangerous. Try to quantify the convexity with respect to prepayment and the variance of prepayment.

[1] How many trillions must be lost before people realize that hedging is not perfect?

[2] By 'convexity with respect to forward rates' I do not mean the curvature in the forward rate curve, I mean the second derivative of the contract with respect to the rates.

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