Category Archives: Finance

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)…

Delta hedging, replicating portfolio, and risk neutral valuation pricing in a multiple time-step setting

The delta hedging and replicating portfolio perspectives for pricing options are important in that they enable us to think carefully as well as deeply concerning the economics of option pricing. For example, the delta hedging approach illustrates that an appropriately hedged portfolio consisting of either long-short call-share positions or long-long put-share positions is riskless and consequently must produce a riskless rate of return. Similarly, the replicating portfolio approach reminds us that a call option represents a “synthetic” margined stock investment, whereas a put option represents a “synthetic” short sale of the share combined with lending money. As we saw in our study earlier this semester of forward contracts, if there is any difference between the value of a derivative and its replicating portfolio, then one can earn profits with zero net investment and no exposure to risk. Thus, the “arbitrage-free” price for the derivative (option or forward) corresponds to the value of its replicating portfolio.

An important result shown in the Binomial Option Pricing Model (single-period) reading which we will cover tomorrow is that the delta hedging and replicating portfolio approaches to pricing options both imply that a “risk-neutral” valuation relationship exists between the derivative and its underlying asset.  This insight provides a deceptively simple pricing equation which enables us to price options by discounting the risk-neutral expected value of the option payoff (i.e., its certainty-equivalent) at the riskless rate of interest.  During tomorrow’s class meeting, I also hope to extend the pricing model from a single timestep to a multiple timestep setting. To accomplish this task, we rely upon backward induction. Backward induction involves beginning at the very end of the binomial tree and working our way back to the beginning. We’ll continue with the numerical example introduced during class today and illustrate the backward induction method for the delta hedging, replicating portfolio and risk neutral valuation approaches. We’ll discover that multiple timesteps imply that the delta hedging and replicating portfolio methods imply “dynamic” trading strategies which require portfolio rebalancing as the price of the underlying asset changes over time (cf. the Dynamic Delta Hedging Numerical Example (calls and puts) and Dynamic Replicating Portfolio Numerical Example (calls and puts) readings).

On the origins of the binomial option pricing model

In my previous posting entitled “Historical context for the Black-Scholes-Merton option pricing model,” I provide links to the papers in which Black-Scholes and Merton presented the so-called “continuous time” version of the option pricing formula. Both of these papers were published in 1973 and eventually won their authors (with the exception of Fischer Black) Nobel prizes in 1997 (Black was not cited because he passed away in 1995 and Nobel prizes cannot be awarded posthumously).

Six years after the Black-Scholes and Merton papers were published, Cox, Ross, and Rubinstein (CRR) published a paper entitled “Option Pricing: A Simplified Approach”. This paper is historically significant because it presents (as per its title) a much simpler method for pricing options which contains (as a special limiting case) the Black-Scholes-Merton formula. The reason why we began our analysis of options by first studying CRR’s binomial model is because pedagogically, this makes the economics of option pricing much easier to comprehend. Furthermore, such an approach removes much (if not most of) the mystery and complexity of Black-Scholes-Merton and also makes that model much easier to comprehend.

Historical context for the Black-Scholes-Merton option pricing model

Although we won’t get into the “gory” details on the famous Black-Scholes-Merton option pricing model until sometime later this semester when we cover Hull’s chapter entitled “The Black-Scholes-Merton Model” and my teaching note entitled “Derivation and Comparative Statics of the Black-Scholes Call and Put Option Pricing Equations“, I’d like to call your attention to the fact that the original papers by Black-Scholes and Merton are available on the web:

The Black-Scholes paper originally appeared in the Journal of Political Economy (Vol. 81, No. 3 (May – Jun., 1973), pp. 637-654). The Merton paper appeared at around the same time in The Bell Journal of Economics and Management Science (now called The Rand Journal). Coincidentally, the publication dates for these articles on pricing options roughly coincide with the founding of the Chicago Board Options Exchange, which was the first marketplace established for the purpose of trading listed options.

Apparently neither Black and Scholes nor Merton ever gave serious consideration to publishing their famous option pricing articles in a finance journal, instead choosing two top economics journals; specifically, the Journal of Political Economy and The Bell Journal of Economics and Management Science. Mehrling (2005) notes that Black and Scholes:

“… could have tried finance journals, but the kind of finance they were doing was outside the rubric of finance as it was then organized. There was a reason for the economist’s low opinion of finance, and that reason was the low analytical level of most of the work being done in the field. Finance was at that time substantially a descriptive field, involved mainly with recording the range of real-world practice and summarizing it in rules of thumb rather than analytical principles and models.”

Another interesting anecdote about Black-Scholes is the difficulty that they experienced in getting their paper published in the first place. Devlin (1997) notes: “So revolutionary was the very idea that you could use mathematics to price derivatives that initially Black and Scholes had difficulty publishing their work. When they first tried in 1970, Chicago University’s Journal of Political Economy and Harvard’s Review of Economics and Statistics both rejected the paper without even bothering to have it refereed. It was only in 1973, after some influential members of the Chicago faculty put pressure on the journal editors, that the Journal of Political Economy published the paper.”

References

Devlin, K., 1997, “A Nobel Formula”.

Mehrling, P., 2005, Fischer Black and the Revolutionary Idea of Finance (Hoboken, NJ: John Wiley & Sons, Inc.).

Availability of Problem set 4, along with a helpful hint

I have posted Problem set 4 on the course website.  This problem set is based upon the “Properties of Stock Options” reading, and it consists of four problems.  It is due at the beginning of class on Tuesday, February 12.

The fourth problem in this problem set references an Excel spreadsheet template called “Derivagem” which you can download from http://fin4366.garven.com/spring2019/DG300.xls. Open DG300.xls up in Excel, and you’ll encounter a dialog box which looks something like this:

Click on “Enable Macros”, and once the  spreadsheet is open, select the “Equity_FX_Index_Futures_Options” worksheet. The upper left corner of the spreadsheet is where you input the data for the problem.  For this problem, “Underlying Type” is “Equity”, “Option Type” is “Black-Scholes – European”, a $.50 dividend is paid in 6 months, which corresponds to .5 year, the current stock price is $41, volatility is 35%, and the risk-free rate of interest is 6%, the option’s life is 1 year, and the strike or exercise price is $40:

Once you have input these data, select the call button for the call option price and the put button for the put option price.  Depending on whether you select “Put” or “Call”, when you click on “Calculate”, the spreadsheet will report back put and call option prices based on the Black-Scholes-Merton (BSM) option pricing formula.  For now,  this formula may be somewhat of a black box for many (if not most) of you, which is a “problem” we will surely rectify in the not-too-distant future in Finance 4366.

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.