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Risk free rate
The risk free rate (rfr) is the rate on government securities. The effect of the rfr on option prices is not as clear-cut as one would expect. As the economy expands, rates tend to increase, but so does the expected rate of share price increases, because dividends increase. It is also known that the present value of future cash flows also decreases as rates increase.
These two effects tend to reduce the prices of put options, i.e. the value of put options decreases as the rfr increases. However, it has been shown that the value of call options increase as the rfr increases, as the former effect tends to dominate the latter effect.
Dividends have the effect of reducing the share price on the ex-dividend date. This is positive for puts and negative for calls. The size of the expected dividend is important, and the value of call options is therefore negatively related to the size of the expected dividend. The opposite applies to put options.
Of these factors, the only one that is not observable is volatility, i.e. the extent of variance in the underlying asset price. This is estimated (calculated) from data in the immediate past.
It will be clear that as volatility increases, so does the chance that the share will do well or badly. The investor in a share will not be affected because these two outcomes offset one another over time. However, in the case of an option holder the situation is different:
• The call option holder benefits as prices increase and has limited downsize risk if prices fall.
• The put option holder benefits as prices decrease and has limited downsize risk if prices rise.
Thus, both puts and calls increase in value as volatility increases.
The Black-Scholes valuation model is as follows (European call option):
Pc = price of European call option
S0 = price of the underlying asset currently
E = exercise price of the option
e = base of the natural logarithm, or the exponential function
r = risk-free rate per annum with maturity at expiration date
N(d) = value of the cumulative normal distribution evaluated at d1 and d2
t = time to expiry in years (short-term = fraction of a year)
d1 = [ln(S0/E) + (r + a2/2)t] / ojt
d2 = d1 - o4~t
ln = natural logarithm (Naperian constant = 2.718)
a2 = variance (of price of underlying asset on annual basis)
a = standard deviation (of price of underlying asset on annual basis).
In the case of a European put option, the price formula changes to:
The one parameter of the model that cannot be directly observed is the price volatility of the underlying asset (standard deviation). It is a measure of the uncertainty in respect of returns on the asset. According to research, typically, volatility tends to be in the range of 20-40% pa. This can be estimated from the history of the assets. An alternative approach is implied volatility, which is the volatility implied by the option price observed in the market.
Implied volatilities are used to gauge the opinion of market participants about the volatility of a particular underlying asset. Implied volatilities are derived from actively traded options and are used to make comparisons of option prices.
The Black-Scholes option pricing model is not the Midas formula, because it rests on a number of simplifying assumptions such as the underlying asset pays no interest or dividends during its life, the risk-free rate is fixed for the life of the option, the financial markets are efficient and transactions costs are zero, etc. However, it is very useful in the case of certain options (see section on binomial model after the following section). Next we present an example.
Example of black-scholes option pricing
The underlying asset is a non-dividend-paying share of company XYZ the current share price of which is LCC 100. The option is a European call, its exercise price is LCC 100 and it has a year to expiry. The risk-free rate is 6.0% pa, historical volatility is 30% and the standard deviation of the share's returns is 0.1 per year. Thus:
S0 = LCC 100 E = LCC 100
r = 0.06
t = 1 a2 = 0.01 a = 0.1
d1 = [ln(S0/E) + (r + a2/2)t] / a4~t
= [ln(100/100) + (0.06 + 0.005)1] / 0.1 VÏ = 0.065 / 0.1
From the cumulative normal distribution table48 one can establish the value of N(d1):
N(d1) = N(0.65) = 0.7422. Similarly we find the value of N(d2):
d2 = d1 - o4~t
= 0.65 - 0.1 = 0.55
N(d2) = (0.55) = 0.7088 (from table).
We are now able to complete the model:
Pc = N(d1)S0 - E(e-rt)N(d2)
= (0.7422 x LCC 100) - (LCC 100 x 2.718-0.06x1 x 0.7088) = LCC 74.22 - (LCC 100 x 0.94177 x 0.7088)
= LCC 74.22 - LCC 66.75 = LCC 7.47.
The Black-Scholes model is regarded as a good valuation model for certain options, particularly for European options on commodities. However, it is regarded as less accurate for dividend paying options and particularly so if the option is of the American variety. Also, it tends to undervalue deep-in-the-money options. Another problem is the assumption of log normality of future asset prices.
Where the Black-Scholes is regarded as weak, the binomial model is used. This model involves the construction of a binomial tree, i.e. a diagram representing different possible paths that may be followed by the underlying asset over the life of the option.
In addition to these two valuation models, there are:
• Monte Carlo simulation.
• Finite difference methods (implicit finite difference method and explicit finite difference method).
In the derivative markets reference is often made to the Greek letters, known as the "Greeks". The "Greeks" measure different dimensions of risk in option positions as follows:49
The delta is the rate of change of the option price with respect to the price of the underlying asset. Theta
The theta of a portfolio of derivatives is the rate of change of the portfolio value with respect to the passage of time (ceteris paribus - when all else remains the same). It is often referred to as the time decay of the portfolio.
The gamma of a portfolio of derivatives on an underlying asset is the rate of change of the portfolio's delta with respect to the price of the underlying asset.
The Vega of a portfolio of derivatives is the rate of change of the value of the portfolio with respect to the volatility of the underlying asset.
The rho of a portfolio of derivatives is the rate of change of the portfolio value with respect to the interest rate.
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