Category Archives: Risk

Art meets chemistry meets physics meets finance

Equations (1) and (2) in my “Geometric Brownian Motion Simulations” teaching note represent examples of so-called “Ito diffusions”. Interestingly, when looking at graphs produced by random number generators (such as are utilized by the Brownian Motion spreadsheet model used for this teaching note), people tend to “see” patterns in data even when no such patterns actually exist.

Ito diffusions represent a specific type of reaction-diffusion process. The Wired Magazine article referenced below provides a layman’s explanation of reaction-diffusion processes in chemistry, which are characterized by reactive molecules that can diffuse between cells. A special case of a reaction-diffusion process is a “pure” diffusion process, where substances aren’t transformed into each other but nevertheless randomly spread out over a surface. While the reaction-diffusion process makes for much more aesthetically pleasing art, other so-called diffusion processes (e.g., diffusion of thermal energy as characterized by heat equations or movements of speculative asset prices as characterized by Ito diffusions) similarly generate (what appear to the naked eye to be) “patterns” from randomness.

Hypnotic Art

Hypnotic Art Shows How Patterns Emerge From Randomness in Nature

These digital canvases represent British mathematician Alan Turing‘s theory of morphogenesis.

Intuition about arithmetic and geometric mean returns in finance

I should have posted this last week when we covered this in class, but better late than never!

In sections 6-7 of the Hull’s “Wiener Processes and Ito’s Lemma” chapter and my teaching note entitled “Applying Ito’s Lemma to determine the parameters of the probability distribution for the continuously compounded rate of return“, it is shown (via the application of Ito’s Lemma) that T-period log returns are normally distributed with mean (μ-σ2/2)T and variance σ2T. In the geometric Brownian motion equation (equation (6) in Hull’s chapter),

dS/S = μdt + σdz,

μ corresponds to the expected return in a very “short” time, dt, expressed with a compounding frequency of dt; in other words, it corresponds to the arithmetic mean return). μ-σ2/2 on the other hand corresponds to the expected return in a “long” period of time, T-t, expressed with continuous compounding; i.e., it corresponds to the geometric mean return.

To see the difference between the arithmetic and geometric mean return, consider the following numerical example. Suppose that returns in successive years are r(1) = 15%, r(2) = 20%, r(3) = 30%, r(4) =−20% and r(5) = 25%. If you add these returns up and divide by 5, this gives you the arithmetic mean value of 14%. The arithmetic mean of 14% is analogous to μ . However, the annualized return that would actually be earned over the course of a five-year holding period is only 12.4%. This is the geometric mean return which is analogous to μ-σ2/2. It is calculated with the following equation:

[1(1.15)(1.20)(1.30)(.80)(1.25)](1/5) – 1 = .124.

The “problem” with volatility is that the higher the volatility, the more it lowers the 5-year holding period return. We can create a mean preserving spread of the (r(1) = 15%, r(2) = 20%, r(3) = 30%, r(4) =−20%, r(5) = 25%) return series by resetting r(1) to 0% and r(5) to 40% ; both return series have arithmetic means of 14% but the (r(1) = 0%, r(2) = 20%, r(3) = 30%, r(4) =−20%, r(5) = 40%) return series has a higher variance (.058 versus .039 for the original return series). This increase in variance results in a lower geometric mean:

[1(1)(1.20)(1.30)(.80)(1.40)](1/5) – 1 = .118.

On the other hand, if we lower volatility, then this increases the geometric mean return. To see this, instead of resetting r(5) in the original return series from 25% to 40%, let’s leave r(5) at 25% and instead reset r(4) in the original return series from -20% to -5%. This change generates the following return series: (r(1) = 0%, r(2) = 20%, r(3) = 30%, r(4) =−5%, r(5) = 25%), which has a 14% arithmetic mean and variance of .024. With lower variance, the new return series has a higher geometric mean:

[1(1)(1.20)(1.30)(.95)(1.25)](1/5) – 1 = .131.

Today’s class and what’s next

Notwithstanding the “mathiness” encountered during today’s class meeting, the result on the difference between arithmetic and geometric mean is of great practical significance.  As we analytically and numerically showed, the geometric mean return is particularly important in demonstrating the adverse effect that excess volatility has on the long-run value of an investment plan.

After covering the arithmetic/geometric mean topic, I attempted to analytically demonstrate how Ito’s Lemma can be used to infer the stochastic process for the “arbitrage-free” price of a forward contract.  This topic appears in the “Application to Forward Contracts” section of the Wiener Processes and Ito’s Lemma chapter. Specifically, given that the instantaneous change in the price (S) of the underlying asset evolves according to the geometric Brownian motion equation dS = \mu Sdt + \sigma Sdz, it follows that the instantaneous change in the price (F) of a forward contract that references S must evolve according to the following equation:

dF = (\mu - r)Fdt + \sigma Fdz.

The one page PDF document entitled “Determining the Stochastic Process for a Forward Contract from Ito’s Lemma” provides the analytic details as to why and how this result obtains.  Next Tuesday, I will begin class by covering this teaching note and then segueing into our initial foray of the “Black-Scholes-Merton Model” chapter.  We will discuss the Geometric Brownian Motion, Ito’s Lemma, and Risk-Neutral Valuation reading in some detail and finish our time together by working on Risk Neutral Valuation Class Problems.

Merton: Applications of Option-Pricing Theory (shameless self-promotion alert)…

Now that we are beginning to study in more detail the famous Black-Scholes-Merton option pricing formula, it’s time for me to shamelessly plug a journal article that I published early in my academic career which Robert C. Merton cites in his Nobel Prize lecture (Merton shared the Nobel Prize in economics in 1997 with Myron Scholes “for a new method to determine the value of derivatives”).

Here’s the citation (and link) to Merton’s lecture:

Merton, Robert C., 1998, Applications of Option-Pricing Theory: Twenty-Five Years Later, The American Economic Review, Vol. 88, No. 3 (Jun. 1998), pp. 323-349.

See page 337, footnote 11 of Merton’s paper for the reference to Neil A. Doherty and James R. Garven (1986)… (Doherty and I “pioneered” the application of a somewhat modified version of the Black-Scholes-Merton model to the pricing of insurance; thus Merton’s reference to our Journal of Finance paper in his Nobel Prize lecture)…

On the role of replicating portfolios in the pricing of financial derivatives in general

Replicating portfolios play a central role in terms of pricing financial derivatives. Here is what we have learned so far about replicating portfolios in Finance 4366:

  1. Buying forward is equivalent to buying the underlying on margin, and selling forward is equivalent to shorting the underlying and lending money. Like options, forwards and futures are priced by pricing the replicating portfolio and invoking the “no-arbitrage” condition. If the forward/futures price it too low, then one can earn positive returns with zero risk and zero net investment by buying forward, shorting the underlying and lending money. Similarly, if the forward futures price is too high, one can earn positive returns with zero risk and zero net investment by selling forward and buying the underlying with borrowed money. This is commonly referred to as “riskless arbitrage”; it’s riskless because you’re perfectly hedged, and it’s arbitrage because you are buying low and selling high.
  2. The replicating portfolio for a call option is a margined investment in the underlying. For example, in my teaching note entitled “A Simple Model of a Financial Market”, I provide a numerical example where the interest rate is zero, there are two states of the world, a bond which pays off $1 in both states is worth $1 today, and a stock that pays off $2 in one state and $.50 in the other state is also worth one dollar. In that example, the replicating portfolio for a European call option with an exercise price of $1 consists of 2/3 of 1 share of stock (costing $0.66) and a margin balance consisting of a short position in 1/3 of a bond (which is worth -$0.33). Thus, the value of the call option is $0.66 – $0.33 = $0.33.
  3. Since the replicating portfolio for a call option is a margined investment in the underlying, it should come as no surprise that the replicating portfolio for a put option consists of a short position in the underlying combined with lending. Thus, in order to price the put, you need to determine and price the components of the replicating portfolio; we will begin class tomorrow by determining the the relative weightings (delta and beta) for the put’s replicating portfolio.
  4. If you know the value of a call, the underlying, and the present value of the exercise price, then you can use the put-call parity equation to figure out the price for the put option; i.e., {C_0} + PV(K) = {P_0} + {S_0} \Rightarrow {P_0} = {C_0} + PV(K) - {S_0}. Since we know the price of the call ($0.33), the present value of the exercise price ($1), and the stock price ($1), then it follows from the put-call parity equation that the value of the put is also 33 cents. More generally, if you know the values of three of the four securities that are included in the put-call parity equation, then you can infer the “no-arbitrage” value of the fourth security.

VXX, the exchange-traded version of the CBOE Volatility Index (AKA “VIX”)

The first exchange-traded product that allowed investors to bet directly on future stock swings will expire this month. Here is a look at how the transition will work and how the end for VXX came to be.