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 today’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. 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.

Problem Set 7 helpful hints

  1. The least risky combination of Security A and Security B in Problem 1 is found by calculating {w_A} = \displaystyle\frac{{\sigma _B^2 - {\sigma _{AB}}}}{{\sigma _A^2 + \sigma _B^2 - 2{\sigma _{AB}}}} and {w_B} = 1 - {w_A}.
  2. It will always be the case for 2 security portfolios that by following the minimum variance portfolio weighting scheme in the previous bullet point, such a portfolio must have zero variance if {\rho _{AB}} = 1 or -1.
  3. In part B of Problem 2, the Sharpe Ratio for security j is \displaystyle\frac{{E({r_j}) - {r_f}}}{{{\sigma _j}}}.

Capital Market Synopsis from last Friday’s Finance 4335 class meeting

The Capital Market Theory lecture note upon which last Friday’s Finance 4335 class discussion was based provides the following important insights:

  1. Borrowing and lending at the riskless rate of interest in combination with investing in (mean-variance efficient) risky portfolios makes it possible for investors to obtain superior risk-return tradeoffs 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[ {\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 so-called market portfolio. 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 portfolio; such portfolios are referred to as “lending” portfolios. When \alpha = 1, the investor is fully invested in the market portfolio. Finally, when \alpha> 1, the investor funds her investment in the market portfolio 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.

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.

3. 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 underpriced (overpriced). Investors will recognize this mispricing and bid up (down) the underpriced (overpriced) security until its expected return conforms to the CAPM equation.

4. 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 the macroeconomic factors that 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).

Extra credit opportunity in Finance 4335: “Brewing up Entrepreneurship: Microbrewers and the Fight for Distribution Rights”

Extra Credit Opportunity:  Baugh Center Free Enterprise Forum this Thursday, 4-5:30 pm in Foster 240: “Brewing up Entrepreneurship: Microbrewers and the Fight for Distribution Rights.”

The story involves a lawsuit by start-up microbrewers who were forced by legislation to give their distributor rights away. The distributors who made significant money reselling the rights to retailers had pushed this legislation. It is a landmark case that overturns laws dating back to the prohibition era.

I will offer extra credit if you attend this presentation and submit (via email sent to risk@garven.com) a 1-2 page executive summary of what you learned from the presentation. The executive summary is due by no later than 5 p.m. on Monday, March 27 (any time after then will be considered “late” and therefore not be eligible for extra credit). The extra credit will replace your lowest quiz grade (assuming the extra grade is higher).

U.S. Household Net Worth Reaches Record $92.8 Trillion

While $92.8 trillion is obviously a really big number, consider that in October 2015, Credit Suisse put total global wealth at $250 trillion in their annual global wealth report.

U.S. household net worth climbed to a record $92.8 trillion in the fourth quarter of 2016, as the end-of-year surge in stocks and a steady climb in home prices…

What are the “correct” answers for the following questions?

Today’s Wall Street Journal features a “Wealth Management” report in it’s Personal Finance section at https://www.wsj.com/news/types/journal-reports-wealth-management.

Here are four questions that are asked, and answered either “yes” or “no” by experts on each topic. I read all of these articles and find the “yes” arguments the most convincing. How about you?

1. Should College Students Be Required to Take a Course in Personal Finance?

https://www.wsj.com/articles/should-college-students-be-required-to-take-a-course-in-personal-finance-1489975500

2. Is It Prudent for Investors to Use Target-Date Funds for Retirement Savings?

https://www.wsj.com/articles/is-it-prudent-for-investors-to-use-target-date-funds-for-retirement-savings-1489975382

3. Should the U.S. Raise the Age for Mandatory IRA Withdrawals?

https://www.wsj.com/articles/should-the-u-s-raise-the-age-for-mandatory-ira-withdrawals-1489975261

4. Are We Better Off Curbing the Role of the Consumer Financial Protection Bureau?

https://www.wsj.com/articles/are-we-better-off-curbing-the-role-of-the-consumer-financial-protection-bureau-1489870870

Outline and PowerPoint from the March 17 meeting of Finance 4335

The topics covered last Friday (portfolio and capital market theory) definitely rank in the upper echelon of important finance topics; after all, the scientific foundation of these topics won Nobel economics prizes for Markowitz (portfolio theory) and Sharpe  (capital market theory) in 1990.  As a further help to students, here I post the outline that we followed in class that day; the details for the outline are to be found in the 28 page lecture note that I posted at  http://risk.garven.com/wp-content/uploads/2017/03/March-17-Finance-4335-Lecture-Portfolio-and-Capital-Market-Theory.pdf (this lecture note is basically a lightly edited version of the Portfolio Theory and Capital Market Theory lecture notes that are also available from the course website.

  • Review from March 3
    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
    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 is now 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 portfolio by finding the tangency between highest indifference curve and the efficient frontier.  The point of tangency 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}).

     

Finance 4335