Black-Scholes model in valuing options

In this article, we will learn about the Black -Scholes model in valuing options. Most people know that their stocks can fall, but if asked to specify the odds, they would most likely blink in puzzlement. Imagine for a moment, that a student from your class showed symptoms of a contagious and dangerous disease. You would expect a warning or perhaps some advice on what precautions to take from the college authority.

But if a letter arrived from the college stating that you have an 18% chance of catching the sickness, a 12% chance of missing at least a week of college and a 2% risk of fatality. Wouldn’t you consider it odd and would want to know about the college’s doctor and the kind of medicine he practiced that gave such statistics. The Black-Scholes formula did this for Long-Term capital management in investing. It is much harder to calculate the odds in investing. Investing confronts us with risk and uncertainty.

Long -Term  Capital Management, a private fund responsible for the 1998 financial crisis, did something like this in a letter to its investors in 1995. In an attachment penned down by his academic stars, Merton, and schools, Long-Term did not merely concede the possibility of loss, it calculated the supposed odds of its occurring. The letter stated, “investors may experience a loss of 5% or more in about one month in five, and a loss of 10%  or more in about one month in ten.”

How could they have the odds?
They key for Long-Term was the volatility, in bond prices. By plugging in thousands of bond prices into the formula, they found the historic volatility i.e. how much the bonds fluctuated in the past and they used it to assess the future risk.

Black-Scholes formula was behind the huge success of the fund in its early years.

I believe the story very well tells the importance of the formula we are going to study.

The Black-Scholes Formula is a model that determines the price of European options. It was developed in 1973 by Fisher Black, Robert Merton and Myron Scholes. When applied to a stock option, the model incorporates the constant price variation of the stock, the time value of money, the option’s strike price and the time to the option’s expiry. The model is most widely used to find the implied volatility using market rice of an option.

The Black-Scholes Formula goes as follows,

Without dividends

With dividend yield(q)

The Black -Scholes formula is applied to both call and put options ,yielding and not yielding, dividends.

The terms used in the formula are as follows:

The result can be proved by two methods: PDE (Partial Differential Approach) and the martingale process (using the 5-step Binomial Method). The CM2 material has detailed proof.

Black Scholes PDE

Paul Samuelson, the first financial economist to win a Nobel Prize, noted, “ The essence of the Black-Scholes formula is that you know, with certainty, not what  the deal of the cards will be but what kind of universe is being sampled, which gives you the assumption of the lognormal process.”

Assumptions of the model 

  • The price of the underlying share follows a geometric Brownian motion. i.e. the share price changes continuously through time according to the stochastic differential equation:

dSt = St (µdt + σdZt) This is the same as the lognormal model.

  • To Black,  Scholes, and Merton, price changes in financial markets were random. No one could predict any particular change, but over a long period, they assumed that the distribution of all such prices would mirror the pattern of other random events like coin flips,  dice rolls or the heights of high school students. 
  • Merton assumed that volatility was so constant that prices would trade in continuous time ie. without any jumps. Merton’s markets were as smooth as well-brewed java, in which prices would indeed flow like cream. 
  • Unlimited short selling (that is, negative holdings) is allowed.
  • There are no taxes or transaction costs.
  • There are no risk-free arbitrage opportunities.
  • The risk-free rate of interest is constant, the same for all maturities and the same for borrowing or lending.
  • The underlying asset can be traded continuously and in infinitesimally small numbers of units.

The key general implication of the underlying assumptions is that the market in the

underlying share is complete: that is, all derivative securities have payoffs which can be replicated.

Criticisms of the model
Mitchell Kapor,  a friend and partner of Merton while publishing  Tiny  TROLL( a desktop graphics and statistics program)  wasn’t the only one who wondered if were also wondering the same. 

  • The volatility parameter σ  may not be constant over time.
  • The long-term drift parameter may not be constant over time. In particular, interest rates will impact the drift.
  • The distribution of security returns log(Su/ St ) has a taller peak in reality than that implied by the normal distribution. This is because there are more days of little or no movement in the share price.
  • The distribution of security returns log(Su/ St ) has fatter tails in reality than that implied by the normal distribution. This is because there are more extreme movements in security prices.
  • The sample paths of security prices are not continuous, but instead, appear to jump occasionally.

The Garman-Kohlhagen Model
Mark Garman and Steven Kohlhagen were the founders of the Garman‐Kohlhagen model. The Garman-Kohlhagen Formula is a variant on the Black-Scholes option pricing formula, applied to find the prices of currency options.

This model alleviates the restrictive assumption used in the Black-Scholes model that borrowing and lending is performed at the same risk free rate. In the foreign exchange market there is no reason that the risk free rate should be identical in each country. This model can cope with the presence of two risk-free rates.

The formula goes as follows:

where,
rd is the domestic risk-free rate
rf is the foreign risk-free rate

The risk-free foreign interest rate, in this case, can be thought of as a continuous dividend yield being paid on the foreign currency. Since an option holder does not receive any cash flows paid from the underlying instrument, this should be reflected in a lower option price in the case of a call or a higher price in the case of a put.

The Garman Kohlhagen model provides a solution by subtracting the present value of the continuous cash flow from the price of the underlying instrument. The model has the same assumptions and limitations as the Black -Scholes model.

Today every financial company, fund managers use Black-Scholes formula on computers for derivative pricing.

 

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