Options

Credit card companies in the United States offered sub-prime credit cards usually with lower credit limits and charged high fees and interest rates sometimes as high as 30% or more. With slowdown in economic growth in the United States in 2002, the default rates for sub-prime credit card holders increased, compelling sub-prime credit card issuers to reduce or cease operations.

In 2007, many new vendors emerged making the market more competitive, forcing issuers to make the cards more attractive to consumers which resulted in the interest rates being available as low as 9.9% but even then in some cases it goes as high as 24%.

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Binomial option pricing model is an options valuation method developed by Cox, et al, in 1979. The binomial option pricing model uses an iterative procedure, allowing for the specification of nodes, or points in time, during the time span between the valuation date and the option’s expiration date.

Like the Black-Scholes model, this model also assumes a perfectly efficient market. The binomial model takes a risk-neutral approach to valuation. It assumes that underlying security prices can only either increase or decrease with time until the option expires worthless.

Advantages of Binomial model:

• Useful for valuing American options which allow the owner to exercise the option at any point in time until expiration.
• The model is simple mathematically when compared to the Black-Scholes model, and is relatively easy to build and implement with a computer spreadsheet.
• In this model it is possible to check at every point in an option’s life for the possibility of early exercise.
• The Binomial options pricing model approach is widely used as it is able to handle a variety of conditions for which other models cannot easily be applied. This is largely because the BOPM models the underlying instrument over time – as opposed to at a particular point.
• This model is also used to value Bermudan options which can be exercised at various points.
• This model is considered to be more accurate, particularly for longer-dated options, and options on securities with dividend payments.

Limitations of Binomial model:

• The main limitation of the binomial model is its relatively slow speed. Even with the power of computers available today this is not a practical solution for calculation of thousands of prices in a short span of time.

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The Black-Scholes model is used to calculate a theoretical call price, ignoring dividends paid during the life of the option, using the five key determinants of an option’s price viz., stock price, strike price, volatility, time to expiration, and short-term risk free interest rate.

The original formula for calculating the theoretical option price is as follows:

Where:

The variables are:

OP       = theoretical option price
S          = stock price

X         = strike price
t           = time remaining until expiration, expressed as a percent of a year
r           = current continuously compounded risk-free interest rate
v          = annual volatility of stock price

ln         = natural logarithm
N(x)     = standard normal cumulative distribution function
e           = the exponential function

Assumptions underlying the above formula:

• It is possible to short sell the underlying stock.
• There are no arbitrage opportunities.
• Trading in the stock is continuous.
• There are no transaction costs or taxes.
• All securities are perfectly divisible (e.g. it is possible to buy 1/100th of a share).
• It is possible to borrow and lend cash at a constant risk-free interest rate.

Except the volatility factor all the other parameters used in this model viz., strike price, time remaining till expiration, the risk-free interest rate, and the current underlying price are objective and are observable. Hence we can conclude that there is a direct relationship between the option price and the volatility. By observing the option price and pegging the other parameters in this formula it is possible to arrive at the volatility that is implied by the market. By applying such derived volatility implied by the market over the other strike prices and expiry we can test the validity of the Black-Scholes option pricing model. It would be observed that the implied volatility tends to be higher for lower strike prices, and lower for higher strike prices.

It is interesting to note that currencies tend to have more symmetrical curves, with implied volatility lowest at-the-money, and higher volatilities for deep in-the-money and out-of-the-money strike prices, while commodities have higher implied volatility for higher strike prices and lower implied volatility for lower strike prices, exactly the opposite of equities.

Lognormal distribution

The Black-Scholes pricing model assumes a lognormal distribution. A lognormal distribution is skewed to the right, which means it has a longer tail towards it right as compared with a normal distribution that is bell-shaped. The lognormal distribution allows for a stock price distribution of between zero and infinity and has an upward bias. This is because while a stock price can only drop 100%, it can rise by more than 100%.

Advantages of Black-Scholes model:

• It enables one to calculate a very large number of option prices in a very short time.

Limitations of Black-Scholes model:

• Black-Scholes model cannot be used to accurately price options with an American-style exercise as it calculates the option price at expiration only. Early exercise as in the case of American option cannot be priced correctly using this model which is a major limitation of this model.
• All exchange traded equity options (ETO) have American-style exercise as against the European options which can only be exercised at expiration. That means this model cannot be used for pricing most ERO options. The exception to this is an American call on a non-dividend paying asset as the call is always worth the same as its European equivalent since there is never any advantage in exercising early.

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