Real options valuation: KLM airlines flight options
The object of this paper is to deliver a real option valuation of an option on an air ticket from KLM Airlines. The paper consists of several parts: (i) a file explaining the problem, the modeling choices and the solutions (ii) a table with calculations and; (iii) the conclusion.
The value of an options contract relies on a variety of different variables. In addition to the value of the underlying asset itself, options are extremely complex to value. There are many pricing models in current use, though all basically incorporate the concepts of moneyness, rational pricing, put-call parity, option-time value.
Relevant Definitions:
Moneyness is a measure of the degree to which a derivative is likely to have positive monetary value at its expiration.
Rational pricing is the assumption in economics that asset prices will reflect the arbitrage-free price of the asset as any deviation from this price will be "arbitraged away."
Put-call parity is the relationship between the price of a call option and a put option -- both with the identical strike price and expiry.
Option-Time Value indicates that the value of an option consists of two components, its intrinsic value and its time value. Time value is simply the difference on option value and intrinsic value.
The most common options models are:
The Black-Scholes and the Black model
The Binomial options pricing model
The Monte Carlo option model
The Finite difference methods for option pricing
Other modeling approaches include:
The Heston model
The Heath-Jarrow-Morton framework
The Variance Gamma Model (see variance gamma process)
Black-Scholes Model
For this paper, the author has chosen the Black -- Scholes model, which is a mathematical description of financial markets and derivative investment instruments. The model centers on partial differential equations whose solution, the Black -- Scholes formula, is commonly used in the pricing of options.
The model was developed by Fischer Black and Myron Scholes in their joint1973 paper, "The Pricing of Options and Corporate Liabilities."
The model is commonly used for stock options, as follows:
S, be the price of the stock.
V (S, t), the price of a derivative as a function of time and stock price.
C (S, t) the price of a European call option and P (S, t) the price of a European put option.
K, the strike of the option.
r, the annualized risk-free interest rate, continuously compounded.
, the drift rate of S, annualized.
, the volatility of the stock's returns; this is the square root of the quadratic variation of the stock's log price process.
t, a time in years; we generally use: now=0, expiry=T.
, the value of a portfolio.
R, the accumulated profit or loss following a delta-hedging trading strategy.
Lastly, we will use N (x) which denotes the standard normal cumulative distribution function,
N'(x) which denotes the standard normal probability density function,
This model of the market for a particular equity is based on the following assumptions:
One can borrow and lend cash at a known constant risk-free interest rate.
A stock price follows a geometric Brownian motion with constant drift and volatility.
There are no transaction costs, taxes or bid-ask spread.
The underlying security does not pay a dividend.
All securities are infinitely divisible.
There are no restrictions on short selling.
There is no arbitrage opportunity.
From these conditions in the market for an equity, the authors demonstrated that "it is possible to create a hedged position, consisting of a long position in the stock and a short position in calls on the same stock, whose value will not depend on the price of the stock."
The Particulars of This Case
On Nov. 12, 2010, KLM Airlines, launched a new feature on its web site, according to a corporate news release, entitled, Take an option on a flight.
The news release states, in relevant part,
"While searching for tickets on our website it is possible to save flight information. At a click of your mouse you can save any flight schedule offered to your computer.
However, the fare of a saved flight schedule can change. That is why we now offer the possibility to temporarily fix the price of the ticket. For this, you pay EUR 10 to EUR 15.
By taking an option on a ticket, we save the ticket and fare for 14 days - depending on the departure date - for you. The price is guaranteed for that period. So you can decide about your purchase in your own time.
Taking an option does not oblige you to anything, and is available for almost all fare types.
Securing a ticket for a limited time is very simple. When visiting our website, at step 2 when selecting the flight...
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