Weak assumptions

This post may be easier to read if you have some comfort with financial mathematics.

Thousands of people across the history of finance have dutifully memorized one of the most famous results in financial mathematics, the Black-Scholes formula for pricing a European option. For the sake of completeness (skip ahead if you like), here is the formula for pricing a European call (C) or put (P) on a non-dividend-paying asset, which you can also find in countless textbooks and on countless websites:

$C = SN(d_1) - Ke^{-rt}N(d_2)$

$P = N(-d_2)Ke^{-rt} - SN(-d_1)$

where

$d_1 = \frac{\ln(S/K) + (r + \sigma^2/2)t)}{\sigma \sqrt{t}}$

$d_2 = \frac{\ln(S/K) + (r - \sigma^2/2)t)}{\sigma \sqrt{t}}$

$(d_2 = d_1 - \sigma \sqrt{t})$

and S is the underlying asset price, K is the strike price of the option, t is the time to option expiry, r is the interest rate out to time t, σ is the volatility of the underlying asset, and N() represents the cdf of a standard normal distribution.

It is important to remember that while this is a ubiquitous formula used to price options, so much so that option prices are thought of by many traders in terms of their Black-Scholes volatility rather than their dollar price, it is only a mathematical model and is only correct insofar as its assumptions are met. And as with all models, real life matches the model assumptions imperfectly. You could come up with another option pricing model based off of different assumptions and in some sense it would be no more “right” or “wrong” than Black-Scholes; the area of debate would be how well those assumptions fit reality.

For example, let’s say that you had an option on a small pharmaceutical company that was awaiting FDA approval on its only product, a drug upon which the entire firm’s fortunes rested. If the FDA approved, the stock would go to $100, and if not, the stock would go to $0. In this case Black-Scholes’s assumptions about the dynamics of the stock price are very poorly met, and it would not be a great model to use.

Some financiers who are particularly dutiful have also memorized formulas for the basic Black-Scholes greeks. For example, the deltas (sensitivities to underlying asset price) of a call and a put are

$\frac{\partial C}{\partial S} = N(d_1)$

$\frac{\partial P}{\partial S} = N(d_1) - 1$

The relationship between the delta of a call and a put of the same strike and expiry is therefore: call delta – put delta = 1. The formulas for the deltas are strictly Black-Scholes; you can get them by taking the derivative of the Black-Scholes pricing formula, and they might not be accurate under a different option pricing model. But the relationship between the two is not, depending solely on put-call parity.

Put-call parity states that the price of a call minus the price of a put equals the discounted present value of the asset price minus the strike price. It is a much weaker assumption than those that underlie Black-Scholes. You don’t need to say anything about volatility, or Brownian motion, or continuous-time hedging. Not only that, it’s very intuitive and logical: if you have the right to buy a stock above $100 at some point in the future, and someone has the right to sell a stock to you below $100 at that same point in time, you essentially have a forward agreement to buy the stock at $100, which at that point in time will be worth the expected value of the stock less $100, and which today will be worth the stock price less the discounted value of $100 at expiry. It’s much harder to imagine scenarios in which put-call parity would be violated than in which Black-Scholes assumptions are violated (in fact Black-Scholes assumptions imply put-call parity).

What this means is that any options model that accepts the weak and almost always realistic assumption of put-call parity must have the same relationship between call delta and put delta. Let’s look at another slightly trickier example, regarding vega (sensitivity to volatility) and theta (sensitivity to the passage of time). The Black-Scholes formulas for vega and theta of a call are:

$\frac{\partial C}{\partial \sigma} = SN'(d_1) \sqrt{t}$

$-\frac{\partial C}{\partial t} = SN'(d_1) \frac{\sigma}{2 \sqrt{t}} - rKe^{-rt}N(d_2)$

(The negative sign in the theta is there because I have represented t as time to expiry, and theta is typically thought of as how value changes as time moves forward, in which case t would be decreasing.) Let’s further assume that the interest rate is zero, so that the theta simplifies to:

$-\frac{\partial C}{\partial t} = SN'(d_1) \frac{\sigma}{2 \sqrt{t}}$

In this case, the relationship between vega and theta is:

$\frac{\partial C}{\partial \sigma} \frac{\sigma}{2t} = -\frac{\partial C}{\partial t}$

This relationship, though under the further assumption of a zero interest rate, holds under a weaker assumption than Black-Scholes: it requires that your volatility parameter (however you define that) and your time to expiry are used in the price solely in the form of an intermediate parameter σ * sqrt(t). To see this mathematically, let’s write the call price as some unspecified function of this intermediate parameter:

$C = f(\sigma \sqrt{t})$

Then if we take derivatives with the chain rule:

$\frac{\partial C}{\partial \sigma} = f'(\sigma \sqrt{t}) \sqrt{t}$

$-\frac{\partial C}{\partial t} = f'(\sigma \sqrt{t}) \frac{\sigma}{2 \sqrt{t}}$

and you can see that the relationship holds. If interest rates are zero, Black-Scholes does satisfy this weaker assumption; if we define V = σ * sqrt(t), the d1 and d2 terms can be rewritten as:

$d_1 = \frac{\ln(S/K) + V^2}{2V}$

$d_2 = d_1 - V$

We might call V “total” volatility. The intuition behind tying σ and t together is that an option price depends on the probability distribution of the asset out to time t, which in turn depend on a) the value of t is and b) how “innately” volatile the asset is, represented by σ. A high-volatility asset will have a wider distribution than a low-volatility asset over the same time frame, but the low-volatility asset will have a wider distribution at some point if you examine it over a sufficiently longer time frame than the high-volatility asset. Combining the two parameters as V = σ * sqrt(t) is to say that you’ve defined your σ as a per-root-time measure of volatility, or, more simply, you’ve defined σ² as a per-time measure of volatility. For those who have taken some stochastic math, you’ll know that this is indeed true of standard Brownian motion: variance at time t is σ²t.

Why might you be interested in this (which otherwise seems like a small mathematical exercise to kick at financial interview candidates)? Of course, the fewer assumptions your models need, the better, and we can more broadly and confidently apply any aspects of our modeling framework that depend on only a subset of the full assumptions. It’s not simply that we need to worry that much less about matching assumptions and reality, but also that these aspects of the model will be robust to changes in a real-world environment. In times of financial crisis, certain assumptions that were a very strong fit to reality for a long time may suddenly fall apart. Rather than either relying on violable assumptions or throwing out a model that does actually work most of the time, we can assess what aspects of our models rely on exactly what assumptions and be aware of what will and will not hold up in a changing environment.

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Written by Andy

Fri 24 Aug 2012 at 1:27 am

Posted in Uncategorized

Tagged with black-scholes, finance, modeling, normal distribution, statistics

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