At the beginning of the fall semester, I published a blog posting entitled “Milton Friedman on CEOs” which was highly critical of the Business Roundtable decision to throw Milton Friedman’s shareholder-centric model of corporate governance under the bus in favor of the more politically fashionable “stakeholder” model. A couple weeks later, I followed that blog posting up with a subsequent posting entitled “The Business Roundtable’s Recipe for Confusion” which links to a WSJ op-ed which argues, among other things, that when companies try to do the government’s job, inefficiency and uncertainty result. In yesterday’s WSJ, two professors from LSE and Columbia provide empirical support for these hypotheses in an article entitled “Is There Real Virtue Behind the Business Roundtable’s Signaling?“, claiming that ‘stakeholder capitalism’ fails on its own terms.”
Tomorrow’s coverage of the so-called underinvestment problem (where, in the absence of risk management, it may sometimes be “rational” to reject a positive net present value project) is based on the How Insurance Solves the Underinvestment Problem assigned reading. This reading is an unpublished appendix based on the following published (peer-reviewed) journal article:
Garven, James R. and Richard D. MacMinn, 1993, “The Underinvestment Problem, Bond Covenants and Insurance,” Journal of Risk and Insurance, Vol. 60, No. 4 (December), pp. 635-646.
The above-referenced article was recognized as the best (shorter) article published in the Journal of Risk and Insurance in the year that it was published, and according to Google Scholar, it has been cited 87 times.
See “Bridgewater Makes $1.5 Billion Options Bet on Falling Market” – this article details how Bridgewater has purchased $1.5 billion worth of put options written on the S&P 500 and Euro Stoxx 50 indices which would pay off if either the S&P 500 or the Euro Stoxx 50—or both—fall in value by this coming March. However, the notional value of this options position is small relative to the firm’s net long positions in equities, so apparently Bridgewater has for all intents and purposes merely purchased some insurance against downside risk.
Also see “Lenders Brace for Private-Equity Loan Defaults” on how financial institutions have been raising their default probability estimates for loans to private-equity-owned companies; article notes a 10% deterioration in credit risk in these so-called leveraged loans.
Here’s a copy of a spreadsheet that I authored which uses the Black-Scholes-Merton option pricing formula to price a call option (along with an otherwise identical (same underlying asset, same exercise price, same time to expiration) put option; you can bring up the spreadsheet by clicking on the screenshot below):In order to calculate the arbitrage-free price of a call option, we need to solve the following equation:
where and . The arbitrage-free price for the put may be obtained by applying the put-call parity equation. By doing so, we obtain the put pricing equation for an otherwise identical (same underlying, same exercise price, same time to expiration) put:
where and .
For calls and puts, we need five parameter values: S (current underlying asset price), K (exercise price, (volatility of underlying asset return), T (time to expiration, measured in number of years), and r (the annualized riskless rate of interest). These parameters all show up in cells B1:B5. The first step to solving call and put prices requires determining the values for and ; these values are listed in cells B7:B12. Once we obtain this information, it’s simply a matter of coding the equations for the BSM call and put prices; these prices show up in cells E1:E2.
Finally, I also include the components of the replicating portfolios for the call and put options. Of course, the values of these portfolios must be the same as the call and put option values; otherwise, there would be riskless arbitrage opportunities. As on pp. 17-22 of the Derivatives Theory, part 1 lecture note, one replicates a call option by buying delta units of the underlying asset on margin, whereas one replicates a put option by shorting delta units of the underlying asset and lending money. In the Black-Scholes-Merton pricing model, the call delta corresponds to , whereas the put delta corresponds to .
During class today, we showed how the multi-timestep binomial option pricing formula (also known as the Cox-Ross-Rubinstein, or CRR model) converges in the limit (as the number of timesteps n become arbitrarily large and the length of each timestep becomes arbitrarily small) to the famous (and Nobel-prize winning) Black-Scholes-Merton (BSM) option pricing formula. Speaking of BSM, allow me to shamelessly plug a journal article that I published early in my academic career which Professor 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 BSM model to the pricing of insurance; thus Merton’s reference to our Journal of Finance paper in his Nobel Prize lecture)…
Instead of Fahrenheit or Celsius, a metric called “degree days” is used to capture variability in temperature. The risk management lesson here is that this metric makes it possible to create risk indices which companies can rely upon for pricing and hedging weather-related risks with weather derivatives.
Replicating portfolios play a central role in terms of pricing financial derivatives. Here is a succinct summary from today’s class meeting:
- 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.
- The replicating portfolio for a call option is a margined investment in the underlying. During today’s class meeting, we priced a one timestep call option where the price of the underlying asset is $100, the exercise price is also $100, u = 1.05, d = .95, the interest rate r = 5%, and the timestep . Given these parameters, the payoff on the call is $5 at the up (u) node and $0 at the down (d) node. The replicating value consists of half a share that is financed by a margin balance of $47.30; thus the “arbitrage-free” price of the call option is (.5(100) – 47.30) = $2.70.
- 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, we need to determine and price the components of the replicating portfolio. We will begin class next Tuesday by reviewing our analysis of the replicating portfolio approach to pricing calls and puts, and move on to other pricing methods such as delta hedging and risk neutral valuation.
This is the closest thing to an obituary I have seen for some time for a financial services industry that apparently finds itself in somewhat of a free fall…
Our coverage of the Capital Market Theory topic has provided the following important insights:
- Borrowing and lending at the riskless rate of interest in combination with investing in (mean-variance efficient) risky portfolios enable investors to obtain superior risk-return trade-offs compared with investing only in mean-variance efficient risky portfolios. In the figure below (taken from page 2 of the Capital Market Theory lecture note), investors select portfolios along the Capital Market Line, which is given by the following equation: .In the above figure, corresponds to the optimal level of exposure to the market index which is labled as point M. When , the investor is fully invested in the riskless asset. When , the investor is partially invested in the riskless asset and in the market index; such portfolios are referred to as “lending” portfolios. When , the investor is fully invested in the market index. Finally, when , the investor funds her investment in the market index with her initial wealth plus borrowed money; such portfolios are referred to as “borrowing” portfolios.
- Given that investors select (based upon their level of tolerance for risk) portfolios that lie on the Capital Market Line, this behavior has implications for the pricing of risk for individual securities. Specifically, the Capital Market Line implies that for individual securities, the Security Market Line must hold. The equation for the Security Market Line (which is commonly referred to as the Capital Asset Pricing Model, or CAPM) is given by the following equation: , where
- According to the CAPM, the appropriate measure of risk for an individual stock is its beta, which indicates how much systematic risk the stock has compared with an average risk investment such as the market portfolio. Beta for security i () is measured by dividing the covariance between i and the market () by market variance (). If the investor purchases an average risk security, then its beta is 1 and the expected return on such a security is the same as the expected return on the market. On the other hand, if the security is riskier (safer) than an average risk security, then it’s expected return is higher (lower) than the same as the expected return on the market.
- If the expected return on a security is higher (lower) than the expected return indicated by the CAPM equation, this means that the security is under-priced (over-priced). Investors will recognize this mispricing and bid up (down) the under-priced (over-priced) security until its expected return conforms to the CAPM equation.
- According to the CAPM, only systematic (i.e., non-diversifiable) risk is priced. Systematic risks are risks which are common to all firms (e.g., return fluctuations caused by macroeconomic factors which affect all risky assets). On the other hand, unsystematic (i.e., diversifiable) risk is not priced since its impact on a diversified asset portfolio is negligible. Diversifiable risks comprise risks that are firm-specific (e.g., the risk that a particular company will lose market share or go bankrupt).
The portfolio and capital market theory topics rank among the most important finance topics; after all, the scientific foundations for these topics won Nobel prizes for Markowitz (portfolio theory) and Sharpe (capital market theory). Here’s a succinct outline of these topics (as covered in Finance 4335):
- Portfolio Theory
- Mean-variance efficiency
- Portfolio Mean-Variance calculations
- Minimum variance portfolio (n = 2 case)
- Efficient frontier (n = 2 case under various correlation assumptions)
- Capital Market Theory
- Efficient frontiers with multiple number (“large” n) of risky assets (aka the “general” case)
- Portfolio allocation under the general case
- degree of risk aversion/risk tolerance determines how steeply sloped indifference curves are
- indifference curves for investors with high (low) degrees of risk tolerance (aversion) are less steeply sloped than indifference curves than for investors with low (high) degrees of risk tolerance (aversion)).
- Optimal portfolios (i.e., portfolios that maximize expected utility) occur at points of tangency between indifference curves and efficient frontier.
- Introduction of a risk-free asset simplifies the portfolio selection problem since the efficient frontier becomes a straight line rather than an ellipse in space. The same selection principle holds as in the previous point (point 2); i.e., investors determine optimal portfolios by identifying the point of tangency between their indifference curves and the efficient frontier. This occurs on the capital market line (CML) where the Sharpe ratio is maximized; everyone chooses some combination of the risk-free asset and the market portfolio, and risk tolerance determines whether the point of tangency involves either a lending (low risk tolerance) or borrowing (high risk tolerance) allocation strategy.
- The security market line (SML), aka the CAPM, is deduced by arbitrage arguments. Specifically, it must be the case that all risk-return trade-offs (as measured by the ratio of “excess” return () from investing in a risky rather than risk-free asset, divided by the risk taken on by the investor () are the same. If not, then there will be excess demand for investments with more favorable risk-return trade-offs and excess supply for investments with less favorable risk-return trade-offs). “Equilibrium” occurs when markets clear; i.e., when there is neither excess demand or supply, which is characterized by risk-return ratios being the same for all possible investments. When this occurs, then the CAPM obtains: .