Category Archives: Finance

Cox-Ross-Rubinstein (CRR) option pricing spreadsheet

Here’s a spreadsheet which calculates call and put prices using the CRR framework for n = 1, 2, 3, and 4 timesteps.  For discussion of this method, see pp. 9-11 of http://fin4335.garven.com/fall2018/lecture15.pdf lecture note which we covered in class yesterday.

The key idea here involves determining option payoffs without having to recreate the entire stock binomial tree which is shown on page 9 of the above referenced lecture note.  The way this is done involves determining for each call option the minimum number of up moves required in order to determine the nodes at which the option will be in-the-money.  By rounding up to the nearest integer “a” the value of \ln (K/S{d^n})/\ln (u/d) for n = 1, 2, 3, and 4 (see rows 11 and 12 below), we determine that the 1 timestep call will only be in the money at the u node, the 2 timestep call will only be in the money at the uu node, the 3 timestep call will only be in the money at the uuu and uud nodes, and the 4 timestep call will only be in the money at the uuuu and uuud nodes.  This greatly simplifies the calculations for call prices because we know that the call payoffs will all be zero at all other nodes (see row 13 below).  Having determined the arbitrage-free call prices, then we find the arbitrage-free put prices by applying the put call parity equation P = C + K{e^{ - rn\delta t}} - S (you can bring up the spreadsheet by clicking on the screenshot below):

Black-Scholes-Merton option pricing spreadsheet

Here’s a copy of the spreadsheet that I created during the section 1 (11-12:15) meeting of Finance 4335 today (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:

C = SN({d_1}) - K{e^{ - rT}}N({d_2}),

where {d_1} = \displaystyle\frac{{\ln (S/K) + (r + .5{\sigma ^2})T}}{{\sigma \sqrt T }} and {d_2} = {d_1} - \sigma \sqrt T .. 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:

P = K{e^{ - rT}}N( - {d_2}) - SN( - {d_1}),

where N( - {d_1}) = 1 - N({d_1}) and N( - {d_2}) = 1 - N({d_2}).

For calls and puts, we need five parameter values: S (current underlying asset price), K (exercise price, \sigma (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 {d_1}, {d_2}, N({d_1}), N({d_2}),N( - {d_1}), and N( - {d_2}); 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 we showed way back during class on Tuesday, October 23, 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 N({d_1}), whereas the put delta corresponds to N({-d_1}).

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

During class on Thursday, we will show how the multi-timestep binomial option pricing formula which we studied during today’s class meeting converges into the famous (and Nobel-prize winning) Black-Scholes-Merton option pricing formula. Speaking of the Black-Scholes-Merton option pricing formula, 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 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

Replicating portfolios play a central role in terms of pricing financial derivatives. Here is a succinct summary from yesterday’s class meeting:

  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. During yesterday’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 \delta t = 1/12. 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.
  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, we need to determine and price the components of the replicating portfolio. We will begin class tomorrow 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.

Synopsis of last week’s Capital Market theory topic

Last week’s coverage of the Capital Market Theory topic provided the following important insights:

  1. 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 13 of the Capital Market Theory lecture note), investors select portfolios along the Capital Market Line, which is given by the following equation: E({r_p}) = {r_f} + \left[ {\displaystyle\frac{{E({r_m}) - {r_f}}}{{{\sigma _m}}}} \right]{\sigma _p}.Capital Market LineIn the above figure, \alpha corresponds to the optimal level of exposure to the market index which is labled as point M. When \alpha = 0, the investor is fully invested in the riskless asset. When 0 < \alpha < 1, the investor is partially invested in the riskless asset and in the market index; such portfolios are referred to as “lending” portfolios. When \alpha = 1, the investor is fully invested in the market index. Finally, when \alpha> 1, the investor funds her investment in the market index with her initial wealth plus borrowed money; such portfolios are referred to as “borrowing” portfolios.
  2. 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: E({r_i}) = {r_f} + \left[ {E({r_m}) - {r_f}} \right]{\beta _i}, where {\beta _i} = {\sigma _{i,m}}/\sigma _m^2.
  3. 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 ({\beta _i}) is measured by dividing the covariance between i and the market ({\sigma _{i,m}}) by market variance (\sigma _m^2). 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.
  4. 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.
  5. 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).

A (non-technical) Summary of Portfolio Theory and Capital Market Theory

During our last three Finance 4335 class meetings, we devoted our attention to portfolio theory and capital market theory.  Work in these particularly important areas of finance subsequently won Nobel Prizes for their proponents; the portfolio theory topic won Harry Markowitz the Nobel Prize in Economics in 1990, and William F. Sharpe shared the 1990 Nobel Prize with Markowitz for his work on capital market theory.

One of the better non-technical summaries of portfolio theory and capital market theory that I am aware of appears as part of an October 16, 1990 press release put out by The Royal Swedish Academy of Sciences in commemoration of the prizes won by Markowitz and Sharpe (see http://www.nobelprize.org/nobel_prizes/economic-sciences/laureates/1990/press.html). I have included an appropriately edited version of that press release below (it is important to also note that Merton Miller was cited that same year along with Markowitz and Sharpe for his work on the theory of corporate finance; I include below only the sections of the Royal Swedish Academy press release pertaining to the work by Messrs. Markowitz and Sharpe on the topics of portfolio and capital market theory):

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Financial markets serve a key purpose in a modern market economy by allocating productive resources among various areas of production. It is to a large extent through financial markets that saving in different sectors of the economy is transferred to firms for investments in buildings and machines. Financial markets also reflect firms’ expected prospects and risks, which implies that risks can be spread and that savers and investors can acquire valuable information for their investment decisions.

The first pioneering contribution in the field of financial economics was made in the 1950s by Harry Markowitz who developed a theory for households’ and firms’ allocation of financial assets under uncertainty, the so-called theory of portfolio choice. This theory analyzes how wealth can be optimally invested in assets which differ in regard to their expected return and risk, and thereby also how risks can be reduced.

A second significant contribution to the theory of financial economics occurred during the 1960s when a number of researchers, among whom William Sharpe was the leading figure, used Markowitz’s portfolio theory as a basis for developing a theory of price formation for financial assets, the so-called Capital Asset Pricing Model, or CAPM.

Harrv M. Markowitz
The contribution for which Harry Markowitz now receives his award was first published in an essay entitled “Portfolio Selection” (1952), and later, more extensively, in his book, Portfolio Selection: Efficient Diversification (1959). The so-called theory of portfolio selection that was developed in this early work was originally a normative theory for investment managers, i.e., a theory for optimal investment of wealth in assets which differ in regard to their expected return and risk. On a general level, of course, investment managers and academic economists have long been aware of the necessity of taking returns as well as risk into account: “all the eggs should not be placed in the same basket”. Markowitz’s primary contribution consisted of developing a rigorously formulated, operational theory for portfolio selection under uncertainty – a theory which evolved into a foundation for further research in financial economics.

Markowitz showed that under certain given conditions, an investor’s portfolio choice can be reduced to balancing two dimensions, i.e., the expected return on the portfolio and its variance. Due to the possibility of reducing risk through diversification, the risk of the portfolio, measured as its variance, will depend not only on the individual variances of the return on different assets, but also on the pairwise covariances of all assets.

Hence, the essential aspect pertaining to the risk of an asset is not the risk of each asset in isolation, but the contribution of each asset to the risk of the aggregate portfolio. However, the “law of large numbers” is not wholly applicable to the diversification of risks in portfolio choice because the returns on different assets are correlated in practice. Thus, in general, risk cannot be totally eliminated, regardless of how many types of securities are represented in a portfolio.

In this way, the complicated and multidimensional problem of portfolio choice with respect to a large number of different assets, each with varying properties, is reduced to a conceptually simple two-dimensional problem – known as mean-variance analysis. In an essay in 1956, Markowitz also showed how the problem of actually calculating the optimal portfolio could be solved. (In technical terms, this means that the analysis is formulated as a quadratic programming problem; the building blocks are a quadratic utility function, expected returns on the different assets, the variance and covariance of the assets and the investor’s budget restrictions.) The model has won wide acclaim due to its algebraic simplicity and suitability for empirical applications.

Generally speaking, Markowitz’s work on portfolio theory may be regarded as having established financial micro analysis as a respectable research area in economic analysis.

William F. Sharpe

With the formulation of the so-called Capital Asset Pricing Model, or CAPM, which used Markowitz’s model as a “positive” (explanatory) theory, the step was taken from micro analysis to market analysis of price formation for financial assets. In the mid-1960s, several researchers – independently of one another – contributed to this development. William Sharpe’s pioneering achievement in this field was contained in his essay entitled, Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk (1964).

The basis of the CAPM is that an individual investor can choose exposure to risk through a combination of lending-borrowing and a suitably composed (optimal) portfolio of risky securities. According to the CAPM, the composition of this optimal risk portfolio depends on the investor’s assessment of the future prospects of different securities, and not on the investors’ own attitudes towards risk. The latter is reflected solely in the choice of a combination of a risk portfolio and risk-free investment (for instance treasury bills) or borrowing. In the case of an investor who does not have any special information, i.e., better information than other investors, there is no reason to hold a different portfolio of shares than other investors, i.e., a so-called market portfolio of shares.

What is known as the “beta value” of a specific share indicates its marginal contribution to the risk of the entire market portfolio of risky securities. Shares with a beta coefficient greater than 1 have an above-average effect on the risk of the aggregate portfolio, whereas shares with a beta coefficient of less than 1 have a lower than average effect on the risk of the aggregate portfolio. According to the CAPM, in an efficient capital market, the risk premium and thus also the expected return on an asset, will vary in direct proportion to the beta value. These relations are generated by equilibrium price formation on efficient capital markets.

An important result is that the expected return on an asset is determined by the beta coefficient on the asset, which also measures the covariance between the return on the asset and the return on the market portfolio. The CAPM shows that risks can be shifted to the capital market, where risks can be bought, sold and evaluated. In this way, the prices of risky assets are adjusted so that portfolio decisions become consistent.

The CAPM is considered the backbone of modern price theory for financial markets. It is also widely used in empirical analysis, so that the abundance of financial statistical data can be utilized systematically and efficiently. Moreover, the model is applied extensively in practical research and has thus become an important basis for decision-making in different areas. This is related to the fact that such studies require information about firms’ costs of capital, where the risk premium is an essential component. Risk premiums which are specific to an industry can thus be determined using information on the beta value of the industry in question.

Important examples of areas where the CAPM and its beta coefficients are used routinely, include calculations of costs of capital associated with investment and takeover decisions (in order to arrive at a discount factor); estimates of costs of capital as a basis for pricing in regulated public utilities; and judicial inquiries related to court decisions regarding compensation to expropriated firms whose shares are not listed on the stock market. The CAPM is also applied in comparative analyses of the success of different investors.

Along with Markowitz’ portfolio model, the CAPM has also become the framework in textbooks on financial economics throughout the world.

Transaction risk insurance use rises as part of M&A deals

The article referenced below showcases some particularly  interesting and innovative developments in commercial insurance; quoting from this article,

“M&A insurance has changed the way deal professionals allocate risk, using insurance as a tool to bridge the gap on one of the most fundamental issues in any M&A transaction: the potential post-closing erosion of value, either of the consideration received by the seller or the business acquired by the buyer.”

Transaction insurance to cover mergers and acquisitions deal risk continues to rise, according to a new report from Aon, which found “extraordinary growth” in the use of the coverage between 2016 and 2018.

Important insights from portfolio and capital market theory

The portfolio and capital market theory topics covered during the last couple of Finance 4335 class meetings (and to be completed today) 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:

  • Portfolio Theory (covered on October 11-16)
      1. Mean-variance efficiency
      2. Portfolio Mean-Variance calculations
      3. Minimum variance portfolio (n = 2 case)
      4. Efficient frontier (n = 2 case under various correlation assumptions)
  • Capital Market Theory (covered October 16 and today)
    1. Efficient frontiers with multiple number (“large” n) of risky assets (aka the “general” case)
    2. 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.
    3. Introduction of a risk-free asset simplifies the portfolio selection problem since the efficient frontier becomes a straight line rather than an ellipse in E({r_p}), {\sigma _p} 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.
    4. 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 (E({r_j}) - {r_f}) from investing in a risky rather than risk-free asset, divided by the risk taken on by the investor ({\sigma _{j,M}}) 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: E({r_j}) = {r_f} + {\beta _j}(E({r_M}) - {r_f}).

     

Moral Hazard

During yesterday’s class meeting, we discussed the concept of moral hazard. In finance, the moral hazard problem is commonly referred to as the “agency” problem. Many, if not most real-world contracts involve two parties – a “principal” and an “agent”. Contracts formed by principals and agents also usually have two key features: 1) the principal delegates some decision-making authority to the agent and 2) the principal and agent decide upon the extent to which they share risk.

The principal has good reason to be concerned whether the agent is likely to take actions that may not be in her best interests. Consequently, the principal has strong incentives to monitor the agent’s actions. However, since it is costly to closely monitor and enforce contracts, some actions can be “hidden” from the principal in the sense that she is not willing to expend the resources necessary to discover them since the costs of discovery may exceed the benefits of obtaining this information. Thus, moral hazard is often described as a problem of “hidden action”.

Since it is not economically feasible to perfectly monitor all of the agent’s actions, the principal needs to be concerned about whether the agent’s incentives line up, or are compatible with the principal’s objectives. This concern quickly becomes reflected in the contract terms defining the formal relationship between the principal and the agent. A contract is said to be incentive compatible if it causes principal and agent incentives to coincide. In other words, actions taken by the agent usually also benefit the principal. In practice, contracts typically scale agent compensation to the benefit received by the principal. Thus, in insurance markets, insurers are not willing to offer full coverage contracts; instead, they offer partial insurance coverage which exposes policyholders to some of the risk that they wish to transfer. In turn, partial coverage reinforces incentives for policyholders to prevent/mitigate loss.

Similarly, in a completely different setting, consider the principal/agent relationship which exists between the owner and manager of a business. If the manager’s effort level is high, then the owner may earn higher profits compared with when the manager’s effort level is low. However, if managerial pay consists of a fixed salary and lacks any form of incentive compensation (e.g., bonuses based upon meeting or beating specific earnings targets), then the manager may be inclined to not exert extra effort, which results in less corporate profit. Thus, compensation contracts can be made more incentive compatible by including performance-based pay in addition to a fixed salary. This way, the owner and manager are both better off because incentives are better aligned.