Category Archives: Economics

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.).

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.

Replicating portfolios for Call and Put options

As we discussed during today’s class meeting, the replicating portfolios for long calls and long puts resemble the replicating portfolios for long forwards and short forwards. Specifically, the replicating portfolio for a long call comprises a margined investment in the underlying, whereas the replicating portfolio for the long put involves shorting the underlying and lending money. The primary difference between the replicating portfolios for forward contracts vis-a-vis replicating portfolios for options is that replicating portfolios for options involve fractional long/short positions in the underlying combined with fractional short/long positions in a riskless bond, whereas replicating portfolios for forward contracts require unitary positions (i.e., 100% long/short position in the underlying combined with 100% short/long position in a riskless bond).

My teaching note entitled “A Simple Model of a Financial Market” lays out the logic behind these claims concerning replicating portfolios for calls and puts. Suppose two states of the economy occur 1 period from now: good (g) and bad (b). In state g, a share of stock is worth $2 whereas in state b, a share of stock is worth $0.50; today’s price for a share of stock is {S_0} = \$1. Also suppose that today’s and next period’s riskless bond price is {B_0} = {B_1} = \$1. Furthermore, a call option with an exercise price of K = $1 is worth {C_{1,g}} = \$1 and {C_{1,b}} = \$0. Let {\Delta} correspond to the number of shares of stock in the replicating portfolio, and {\beta} correspond to the number of bonds. It follows that the value of the replicating portfolio in state g, {V_{1,g}} = \Delta({S_{1,g}}) + {\beta}({B_1}) = \Delta(\$2) + {\beta}(\$1) = \$1, and the value of the replicating portfolio in state b, {V_{1,b}} = \Delta({S_{1,b}}) + {\beta}({B_1}) = \Delta(\$.5) + {\beta}(\$1) = 0. Thus we have two equations in two unknowns:

{V_{1,g}} = \Delta (\$2) + \beta(\$1) = \$1, and {V_{1,b}} = \Delta (\$0.50) + \beta(\$1) = \$0.

By subtracting the equation for {V_{1,b}} from the equation for {V_{1,g}}, we obtain \Delta = 2/3, and {\beta} = -1/3, which implies that {V_0} = \Delta ({S_0}) + \beta ({B_0}) = 2/3(1) - 1/3(1) = \$1/3. Since the payoff on the replicating portfolio is identical to the payoff on the long call position, it follows that {V_0} = {C_0} = \$1/3 represents the arbitrage-free price for the call option.

Next, consider the replicating portfolio for the put option with an exercise price of K = $1. Since {S_{1,g}} = \$2 and {S_{1,b}} = \$0.50, this implies that {P_{1,g}} = \$0 and {P_{1,b}} = \$0.50. In state g, the value of the replicating portfolio {V_{1,g}} = \Delta({S_{1,g}}) + {\beta}({B_1}) = \Delta(\$2) + {\beta}(\$1) = \$0, and the value of the replicating portfolio in state b, {V_{1,b}} = \Delta({S_{1,b}}) + {\beta}({B_1}) = \Delta(\$.5) + {\beta}(\$1) = \$0.50 Thus we have two equations in two unknowns:

{V_{1,g}} = \Delta (\$2) + \beta(\$1) = \$0, and {V_{1,b}} = \Delta (\$0.50) + \beta(\$1) = \$0.50.

By subtracting the equation for {V_{1,b}} from the equation for {V_{1,g}}, we obtain \Delta = -1/3, and {\beta} = 2/3, which implies that {V_0} = \Delta ({S_0}) + \beta ({B_0}) = -1/3(1) + 2/3(1) = \$1/3. Since the payoff on the replicating portfolio is identical to the payoff on the long put position, it follows that {V_0} = {P_0} = \$1/3 represents the arbitrage-free price for the put option.

The Index Fund featured as one of “50 Things That Made the Modern Economy”

Tim Harford also features the index fund in his “Fifty Things That Made the Modern Economy” radio and podcast series. This 9 minute long podcast lays out the history of the development of the index fund in particular and the evolution of so-called of passive portfolio strategies in general. Much of the content of this podcast is sourced from Vanguard founder Jack Bogle’s September 2011 WSJ article entitled “How the Index Fund Was Born” (available at https://www.wsj.com/articles/SB10001424053111904583204576544681577401622). Here’s the description of this podcast:

“Warren Buffett is the world’s most successful investor. In a letter he wrote to his wife, advising her how to invest after he dies, he offers some clear advice: put almost everything into “a very low-cost S&P 500 index fund”. Index funds passively track the market as a whole by buying a little of everything, rather than trying to beat the market with clever stock picks – the kind of clever stock picks that Warren Buffett himself has been making for more than half a century. Index funds now seem completely natural. But as recently as 1976 they didn’t exist. And, as Tim Harford explains, they have become very important indeed – and not only to Mrs Buffett.”

Warren Buffett is one of the world’s great investors. His advice? Invest in an index fund

Insurance featured as one of “50 Things That Made the Modern Economy”

From November 2016 through October 2017, Financial Times writer Tim Harford presented an economic history documentary radio and podcast series called 50 Things That Made the Modern Economy. This same information is available in book under the title “Fifty Inventions That Shaped the Modern Economy“. While I recommend listening to the entire series of podcasts (as well as reading the book), I would like to call your attention to Mr. Harford’s episode on the topic of insurance, which I link below. This 9-minute long podcast lays out the history of the development of the various institutions which exist today for the sharing and trading of risk, including markets for financial derivatives as well as for insurance.

“Legally and culturally, there’s a clear distinction between gambling and insurance. Economically, the difference is not so easy to see. Both the gambler and the insurer agree that money will change hands depending on what transpires in some unknowable future. Today the biggest insurance market of all – financial derivatives – blurs the line between insuring and gambling more than ever. Tim Harford tells the story of insurance; an idea as old as gambling but one which is fundamental to the way the modern economy works.”