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...
This amount comes on top of the price of the ticket.
Then, for up to 14 days, you have time to think about your purchase. If you decide not to buy the ticket, then the option will expire and the reservation will be cancelled automatically.
After you have taken an option, you will receive a confirmation e-mail from us. It contains details of the flight and the option. Plus a link which you can use to purchase the ticket.
Through the link in the e-mail, you will visit our website where you can pay for the ticket and finalize your purchase."
Contextual Facts/Financial Background on KLM for FY 2010-11
Positive third quarter operating results demonstrated the recovery of the firm from previous financial troubles, according to a corporate news release earlier this year.
Revenues up 13.9% to 5.92 billion euros
Operating result of 81 million euros, an improvement of 326 million euros year-on-year
Free cash flow of 72 million euros
Nine months to December 31, 2010
Revenues up 14.5% to 18.29 billion euros
Operating result of 525 million euros, an improvement of 1.3 billion euros
Net income of 980 million euros
Operating cash flow close to 1 billion euros
KLM Royal Dutch Airlines was founded in 1919, making it the world's eldest airline. In the Netherlands, KLM comprises the core of the KLM Group which further includes KLM cityhopper, transavia.com en Martinair. KLM serves 130 destinations using a modern fleet of 158 aircraft and employs more than 33,000 people around the planet.
KLM OPTIONS
On March 14, we examined the prices of KLM tickets from New York City to London Heathrow. According to the KLM web page, the prices were from $210, each-way, based on roundtrip purchase. Taxes & fees of up to $250 incl. 9/11 Security Fee of a maximum $10 per roundtrip not included in the pricing. Purchasing a 15 euro option on these flights would not be worth the expense -- as these options are merely a marketing gimmick, rather than a true financial risk management tool.
For an option to be a legitimate financial risk management tool, it must have the following features as stated above:
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.
Analysis Based on Black-Scholes Framework
What is more, unlike a stock price, which follows a geometric Brownian motion with constant drift and volatility, airline ticket costs over a two-week period are not volatile and are rather constant.
There are also transaction costs, taxes, for airline tickets, which do not impact the price of equities.
A simple mathematical model, A + B +C = D, thus, demonstrates the lack of moneyness of these airline options:
Formula for Pricing for Airline Tickets, Round Trip, New York To London (W/Option)
A + B + C = D
Price $420
Taxes $250
Option 15 Euros
Total Cost: $685
Less Cost of Option: $670
CONCLUSION: KLM OPTION VALUATION: INTRINSICALLY WORTHLESS
Clearly, based on the simple mathematical formula above the option actually increases the financial risk, i.e., the cost, of the airline ticket transaction. KLM's options are a mere marketing gimmick -- not true options that can hedge risk in a volatile market. The assumptions of Black-Scholes, and all of the other, aforementioned models, for options pricing, do not work for this kind of option, as (a) the option has no intrinsic value or moneyness; (b) there are taxes included in the price of the product; and (c) there are other variables which make Black-Sholes forecasting difficult for this kind of "option."
Primary references
1. Black, Fischer; Myron Scholes (1973). "The Pricing of Options and Corporate Liabilities." Journal of Political Economy 81 (3): 637 -- 654. doi:10.1086/260062. [1] (Black and Scholes' original paper.)
2. Merton, Robert C. (1973). "Theory of Rational Option Pricing." Bell Journal of Economics and Management Science (The RAND Corporation) 4 (1): 141 -- 183. doi:10.2307/3003143. http://jstor.org/stable/3003143. [2]
3. Hull, John C. (1997). Options, Futures,…
