A Quintessential Bet
May 6, 2005
An investor, with a capital of $1 million, wanted to bet on the S&P 500 stock index for a year. The S&P 500 can only go up or down each day from the starting price of the day.
The investor wants a very simple payout. Assuming that there are 252 trading days in a year the investor wants his payout at the end of the year to be the number of the up days divided by the total number of days. Here, the up days mean the trading days when S&P 500 went up.
Where, u = number of up days in a year with 252 trading days.
If, at the end of the day, the index ends exactly where it started off in the morning, then that day will be considered a down day. This means that if and only if the index goes up the investor gets a payout (as shown above) otherwise he gets nothing.
Thus if out of 252 days the S&P 500 index went up on 100 days then his payout will be $1,000,000*(100/252) i.e. $396,825; if the index went up on 25 days then his payoff will be $1,000,000*(25/252), i.e. $99,206.
The investor approaches a private bank Quantless and asks a hot shot private banker for a price of this contract. How much premium should he pay to buy such a contract?
The private banker immediately calls up the trader of his treasury desk and asks for a price of this contract. After two hours of cookies, coffee, red wine and even a discussion on Picasso, the trader called back the private banker and said it will take a day to figure out the price as the resident quant will have to perform some sophisticated modelling to get the price. So the private banker promised the investor a price within 24 hours.
Three days later the private banker called the investor back and said that the investor's notional amount of $1 million was too small for the bank to do this trade.
Now consider another trade. A street vendor who sells noodle on the street figures out that if it rains then her business for the day will be zero and on the other hand if it does not rain she will make a hundred dollars for the day. She also figures out that the monsoon season will last for 3 months.
So she goes to the very small next door bank QuantSmart and tells the manager that she wants to buy protection for 3 months such that at the end of 3 months she gets a payout as $9,000 (her assumed 3 months revenue) times the number of days it did rain divided by 90.
Where, T = number of days it rained in the 90 day period.
Thus, if out of 90 days it rained on 17 days, then the payout of the noodle lady (that the bank QuantSmart will pay to her) would be $9,000*(17/90), i.e. $1,700.
The manager, who was an ordinary graduate in Social Sciences and who never ever got an interview call from the likes of Quantless, furiously scribbled something on a piece of napkin on his desk and five minutes time gives the noodle lady a quote. She is so pleased that she takes out cash from the wallet and does the deal on the spot.
A year later the street vendor, the noodle lady, strikes gold and opens a million dollar account with the Manager of her bank QuantSmart.
The Private Banker, in the meantime quits Quantless, and starts his own wealth management and client advisory firm. His newsletter mostly talks about different kinds of red and white wine and the latest fashion shows in town. The resident quant of the bank, who famously modeled the investor’s trade with all the mathematics in the world but ultimately could not come up with the price was rumoured to be teaching financial engineering in an top Ivy League business school.
So dear readers, the questions are:
- What was the price that the Manager of the bank QuantSmart gave to the noodle lady and what should have been the price that the quant of the bank Quantless should have quoted the investor?
- If you have a million dollars to spare would you put it in the hands of some banker in Quantless or would you rather donate it to a charity?
Any comments and queries can
be sent through our
More on Quantitative Finance >>
back to top