B2. Application of option pricing theory in investment decisions Flashcards
Types of option.
An option gives the holder the right (but not the obligation) to buy or sell an asset at a pre-agreed price (however option price needs to be paid regardless of whether option is exercised or not). There are 2 types of option:
• Call option – right to buy (money is spent)
• Put option – right to sell (money is received).
Factors determining the value(price) of option.
The major factors determining the price of options are as follows: • Current asset price. • The exercise price. • Time to expiry of the option. • Interest rates. • Volatility of underlying item. .
Factors determining the value(price) of option.
Current asset price.
For a call option, the greater the price for the underlying item the greater the value of the option to the holder (option price $4, share price $5). For a put option the lower the share price the greater the value of the option to the holder. The price of the underlying item is the market prices for buying and selling the underlying item.
Factors determining the value(price) of option.
The exercise price.
For a call option the lower the exercise price the greater the value of the option. For a put option the greater the exercise price, the greater the value of the option.
Factors determining the value(price) of option.
Time to expiry of the option.
The longer the remaining period to expiry, the greater the probability that the underlying item will rise in value. Call options are worth more the longer the time to expiry (time value) because there is more time for the price of the underlying item to rise. Put options are worth more if the price of the underlying item falls over time.
Factors determining the value(price) of option.
Interest rates.
The seller of a call option will receive initially a premium and if the option is exercised the exercise price at the exercised date. If interest rate rises the present value of the exercise price will diminish and he will therefore ask for a higher premium to compensate for his risk. Higher interest rates=higher value of call option.
Factors determining the value(price) of option.
Volatility of underlying item.
Variability adds to the value of an option. The greater the volatility of the price of the underlying item the greater the probability of the option yielding profits.
The volatility
The volatility represents the standard deviation of day-to-day price changes in the underlying item, expressed as an annualized percentage.
The following steps can be used to calculate volatility of underlying item, using historical information:
Calculate daily return
P_i/P_0
where
Pi = current price and
Po = previous day’s price
Take the ‘In’ of the daily return using the calculator (x)
Square the result above to get, say, X2
Calculate the standard deviation as (Σ means average over n period)
=√(((∑x^2 )/n-((∑x)/n)^2 ) )
Then annualise the result using the number of trading days in a year.
daily volatility x √(trading days)
Application of Black-Scholes model.
Black-Scholes model is a model for determining the price of a call option. Writers of options need to establish a way of pricing them. This is important because there has to be a method of deciding what premium to charge to the buyers.
The market value of a call option (at time 0) can be calculated as:
C_0=P_a N(d_1 )-P_e N(d_2)e^(-rt)
N(dx) is the cumulative value from the normal distribution tables for the value dx
d_1=(In(P_a/P_e )+(r+0.5s^2 )t)/(s√t)
d_2=d_1-s√T
Where
Pa = current price of underlying asset (e.g. share price)
Pe = exercise price
r = risk free rate of interest
t = time until expiry of option in years
s = volatility of the share price (as measured by the standard deviation expressed as a decimal)
N(d) = equals the area under the normal curve up to d (see normal distribution tables)
e = 2.71828, the exponential constant (calculator)
In = the natural log (log to be base e)
Using the Black-Scholes model to value put options.
The put call parity equation is on the examination formula sheet:
p=c-P_a+P_e e^(-rt)
Steps:
Step 1: Value the corresponding call option using the Black-Scholes model.
Step 2: Then calculate the value the put option using the put call parity equation.
BSM underlying assumptions and limitations.
The model assumes that:
• The options are European calls, i.e. exercisable on a fixed date. BSOP model will undervalue American style options because it does not take into account time flexibility.
• There are no transaction costs or taxes.
• The investor can borrow at the risk-free rate.
• The risk-free rate of interest and the share’s volatility is constant over the life of the option and is known.
• The future share price volatility can be estimated by observing past share price volatility. Historical deviation is often a poor guide to expected deviation in the future (so based on judgment).
• The share price follows a random walk and that the possible share prices are based on a normal distribution.
• No dividends are payable before the option expiry date.
BSM application to American call options
One of the limitations of the Black-Scholes formula is that it assumes that the shares will not pay dividends before the option expires. If this holds true, then the model can also be used to value American call options. In fact, if no dividends are payable before the option expiry date, the American call option will be worth the same as a European call option.
BSM application to shares where dividends are payable before the expiry date.
The Black Scholes formula can be adapted to call options with dividends being paid before expiry by calculating a ‘dividend adjusted share price’:
• Simply deduct the present value of dividends to be paid (before the expiry of the option) from the current share price.
• Pa becomes Pa – PV (dividends) in the Black-Scholes formula.
• PV of dividend = De-rt, Where D = dividend
Limitations of DCF analysis and real options.
The conventional NPV method assumes that a project commences immediately and proceeds until it finishes, as originally predicted. Therefore, it assumes that a decision has to be made on a now or never basis, and once made, it cannot be changed. It does not recognise that most investment appraisal decisions are flexible and give managers a choice of what actions to undertake.
The real options method estimates a value for this flexibility and choice, which is present when managers are making a decision on whether or not to undertake a project. Real options build on net present value in situations where uncertainty exists and, for example:
• when the decision does not have to be made on a now or never basis, but can be delayed,
• when a decision can be changed once it has been made, or
• when there are opportunities to exploit in the future contingent on an initial project being undertaken.
Therefore, where an organisation has some flexibility in the decision that has been, or is going to be made, an option exists for the organisation to alter its decision at a future date and this choice has a value.
With conventional NPV, risks and uncertainties related to the project are accounted for in the cost of capital, through attaching probabilities to discrete outcomes and/or conducting sensitivity analysis or stress tests.
Options, on the other hand, view risks and uncertainties as opportunities, where upside outcomes can be exploited, but the organisation has the option to disregard any downside impact.
