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Capital asset pricing model at Wiki

The efficient frontier was first defined by Harry Markowitz in his groundbreaking (1952) paper that launched portfolio theory. That theory considers a universe of risky investments and explores what might be an optimal portfolio based upon those possible investments.

Consider an interval of time. It starts today. It can be any length, but a one-year interval is typically assumed. Today’s values for all the risky investments in the universe are known. Their accumulated values (reflecting price changes, coupon payments, dividends, stock splits, etc.) at the end of the horizon are random. As random quantities, we may assign them expected returns and volatilities. We may also assign a correlation to each pair of returns. We can use these inputs to calculate the expected return and volatility of any portfolio that can be constructed using the instruments that comprise the universe.

The notion of “optimal” portfolio can be defined in one of two ways:

1. For any level of volatility, consider all the portfolios which have that volatility. From among them all, select the one which has the highest expected return.

2. For any expected return, consider all the portfolios which have that expected return. From among them all, select the one which has the lowest volatility.

Each definition produces a set of optimal portfolios. Definition (1) produces an optimal portfolio for each possible level of risk. Definition (2) produces an optimal portfolio for each expected return. Actually, the two definitions are equivalent. The set of optimal portfolios obtained using one definition is exactly the same set which is obtained from the other. That set of optimal portfolios is called the efficient frontier. This is illustrated in Exhibit 1

Efficient Frontier Exhibit 1

The green region corresponds to the achievable risk-return space. For every point in that region, there will be at least one portfolio that can be constructed and has the risk and return corresponding to that point. The efficient frontier is the gold curve that runs along the top of the achievable region. Portfolios on the efficient frontier are optimal in both the sense that they offer maximal expected return for some given level of risk and minimal risk for some given level of expected return.

In Exhibit 1, the green region corresponds to the achievable risk-return space. For every point in that region, there will be at least one portfolio constructible from the investments in the universe that has the risk and return corresponding to that point. The yellow region is the unachievable risk-return space. No portfolios can be constructed corresponding to the points in this region.

The gold curve running along the top of the achievable region is the efficient frontier. The portfolios that correspond to points on that curve are optimal according to both definitions (1) and (2) above.

Typically, the portfolios which comprise the efficient frontier are the ones which are most highly diversified. Less diversified portfolios tend to be closer to the middle of the achievable region.

The efficient frontier

Efficient Frontier. The hyperbola is sometimes referred to as the ‘Markowitz Bullet’

Every possible asset combination can be plotted in risk-return space, and the collection of all such possible portfolios defines a region in this space. The line along the upper edge of this region is known as the efficient frontier (sometimes “the Markowitz frontier”). Combinations along this line represent portfolios (explicitly excluding the risk-free alternative) for which there is lowest risk for a given level of return. Conversely, for a given amount of risk, the portfolio lying on the efficient frontier represents the combination offering the best possible return. Mathematically the Efficient Frontier is the intersection of the Set of Portfolios with Minimum Variance (MVS) and the Set of Portfolios with Maximum Return. Formally, the efficient frontier is the set of maximal elements with respect to the partial order of product order on risk and return, the set of portfolios for which one cannot improve both risk and return.

The efficient frontier is illustrated above, with return μp on the y-axis, and risk σp on the x-axis; an alternative illustration from the diagram in the CAPM article is at right.

The efficient frontier will be convex – this is because the risk-return characteristics of a portfolio change in a non-linear fashion as its component weightings are changed. (As described above, portfolio risk is a function of the correlation of the component assets, and thus changes in a non-linear fashion as the weighting of component assets changes.) The efficient frontier is a parabola (hyperbola) when expected return is plotted against variance (standard deviation).

The region above the frontier is unachievable by holding risky assets alone. No portfolios can be constructed corresponding to the points in this region. Points below the frontier are suboptimal. A rational investor will hold a portfolio only on the frontier.

An Asian option (or average value option) is a special type of option contract. For Asian options the payoff is determined by the average underlying price over some pre-set period of time. This is different to the case of the usual European option andAmerican option, where the payoff of the option contract depends on the price of the underlying instrument at maturity.

One advantage of Asian options is that these reduce the risk of market manipulation of the underlying instrument at maturity^[1].

Payout of Asian call options with arithmetic average

We describe the payout of some Asian call options.

The continuous case gives the payout  where T is the time to maturity, S is the price and K is the strike price.

For the case of discrete monitoring (with monitoring at the times ) we have the payout 

There exist Asian options using geometric average, as well as arithmetic average.

A protective options strategy that is implemented after a long position in a stock has experienced substantial gains. It is created by purchasing an out of the money put option while simultaneously writing an out of the money call option. 

Investopedia explains Collar
1. The purchase of an out-of-the money put option is what protects the underlying shares from a large downward move and locks in the profit. The price paid to buy the puts is lowered by amount of premium that is collect by selling the out of the money call. The ultimate goal of this position is that the underlying stock continues to rise until the written strike is reached.2. An example is a circuit breaker which is meant to prevent extreme losses (or gains) once an index reaches a certain level. 

Collars can protect you against massive losses, but they also prevent massive gains.

Bearish Calendar Spread video

20 Bullish Calendar Spread Option Strategies

The date on which something was introduced or started, such as a mutual fund.