Study notes 4 - lognormal distribution

Random variable xL that is continuously distributed in interval is said to have lognormal distribution described by probability density function fL(xL) if variable xN, that is defined as xN=lnxL, has normal distribution described by probability density function fN(xN) in interval -∞ to ∞.

Denote the mean and variance of normally distributed variable as μ and σ2 respectively, then
mean of XL will be

Study notes 3 - Assumptions made by Black Scholes theory

The following is quoted from site http://hilltop.bradley.edu/~arr/bsm/pg04.html

The Black and Scholes Option Pricing Model didn't appear overnight, in fact, Fisher Black started out working to create a valuation model for stock warrants. This work involved calculating a derivative to measure how the discount rate of a warrant varies with time and stock price. The result of this calculation held a striking resemblance to a well-known heat transfer equation. Soon after this discovery, Myron Scholes joined Black and the result of their work is a startlingly accurate option pricing model. Black and Scholes can't take all credit for their work, in fact their model is actually an improved version of a previous model developed by A. James Boness in his Ph.D. dissertation at the University of Chicago. Black and Scholes' improvements on the Boness model come in the form of a proof that the risk-free interest rate is the correct discount factor, and with the absence of assumptions regarding investor's risk preferences.

In order to understand the model itself, we divide it into two parts. The first part, SN(d1), derives the expected benefit from acquiring a stock outright. This is found by multiplying stock price [S] by the change in the call premium with respect to a change in the underlying stock price [N(d1)]. The second part of the model, Ke(-rt)N(d2), gives the present value of paying the exercise price on the expiration day. The fair market value of the call option is then calculated by taking the difference between these two parts.

Assumptions of the Black and Scholes Model:

1) The stock pays no dividends during the option's life

Most companies pay dividends to their share holders, so this might seem a serious limitation to the model considering the observation that higher dividend yields elicit lower call premiums. A common way of adjusting the model for this situation is to subtract the discounted value of a future dividend from the stock price.

2) European exercise terms are used

European exercise terms dictate that the option can only be exercised on the expiration date. American exercise term allow the option to be exercised at any time during the life of the option, making american options more valuable due to their greater flexibility. This limitation is not a major concern because very few calls are ever exercised before the last few days of their life. This is true because when you exercise a call early, you forfeit the remaining time value on the call and collect the intrinsic value. Towards the end of the life of a call, the remaining time value is very small, but the intrinsic value is the same.

3) Markets are efficient

This assumption suggests that people cannot consistently predict the direction of the market or an individual stock. The market operates continuously with share prices following a continuous Itô process. To understand what a continuous Itô process is, you must first know that a Markov process is "one where the observation in time period t depends only on the preceding observation." An Itô process is simply a Markov process in continuous time. If you were to draw a continuous process you would do so without picking the pen up from the piece of paper.

4) No commissions are charged

Usually market participants do have to pay a commission to buy or sell options. Even floor traders pay some kind of fee, but it is usually very small. The fees that Individual investor's pay is more substantial and can often distort the output of the model.

5) Interest rates remain constant and known

The Black and Scholes model uses the risk-free rate to represent this constant and known rate. In reality there is no such thing as the risk-free rate, but the discount rate on U.S. Government Treasury Bills with 30 days left until maturity is usually used to represent it. During periods of rapidly changing interest rates, these 30 day rates are often subject to change, thereby violating one of the assumptions of the model.

6) Returns are lognormally distributed

This assumption suggests, returns on the underlying stock are normally distributed, which is reasonable for most assets that offer options.

Study notes 2 - time value of money

This continues from my last post. I will write down some notes on Black-Scholes theory.

1. time value of money.
Basically this is about interest and it represents the 'opportunity cost'. If you choose not to invest money in options, you can receive interest with relatively low or no risk.

Bearing this in mind, it will be different to receive some amount of money today as compared to receive equal amount a year later. Assuming an annual interest of i%, the amount of $Y to be received a year later will be deemed equivalent to the amount $Y/(1+i%) received today. Over here, we have introduced the concept of 'discount factor' (1/(1+i%)) to help to define the effect of time value of the money.

In Black-Scholes theory, with a few important mathematical assumptions made, the discount factor is calculated to be exp(-rT). It is more accurately called 'continuously compound interest'. Although it looked obscured, it does ring a bell for those with engineering background, exponential decay with respect to a period of time T at a factor of r.

Forget about mathematics, let's climb on the giant's shoulders first.

2. when you deposit your money with continuously compound interest r at t0, then at time (t0+t), your money plus interest will be exp(rt).

Study notes 1 - terminology and concepts about options

I am reading the book Design.Patterns.and.Derivatives.Pricing by Mark S. Joshi. Since I am from Computer science and electronic engineering background, I have difficulty in understanding some of the financial terminologies.

I had some basic financial accounting knowledge from university general electives and some stock trading experience, so I venture to do some self studying through internet searching.

On chapter 1, a simple Monte Carlo model. A simple Monte Carlo simulation requires five parameters input, (expiry, strike, spot, vol, r, and NumberOfPaths).

The wikipedia page (http://en.wikipedia.org/wiki/Call_option) provides a very nice introduction about call options.

1. What is an option?
A call option is a contract formed between 'caller' and 'writer'. The caller predicts that the stock price (spot price) will rise beyond an agreed limit (i.e., the strike price) on a defined future date (Expiry date), thus he pays a premium to the writer to enter into a contract which entitles him(the 'caller') the right to exercise the option by purchasing the underlying stock from the writer at the strike price on this defined future date.

The caller has the right to exercise the option at his discretion, and if he chooses to do so, the writer must agree to sell the stock.

In essence, the buyer is paying for a chance to buy a stock at certain price (hopefully a discounted price), rather than to buy the real stock.

In the above explanation, stock is used just for illustrative purpose; in real world, the underlying financial instrument could be different.

2. how does an option differs from warrant?

To my understanding, from a mathematics point of view, the key difference is that warrants are dilutive, which means the company has to issue new shares when the warrants are exercised. Options is only about changing ownership of the underlying financial instruments.

3. how much does the buyer earn or lose?
=>> if spot price is higher than the strike price, and yes, that is what the buyer (caller) expected, his earnings amounts to
S=P-(Q+R)
(S) Trader A's total earnings .
(P) Sale of stock at spot price
(Q) Amount paid to purchase the stock at strike price upon exerise
(R) Contract commissions (the premium paid)

==> or, if the spot price is lower than the strike price, obviously, it does not make sense for the buyer(caller) to exercise the option (paying higher than market price to purchase the stock), thus his total loss will be equal to -R.

4. what is 'in-the-money' and payoff?
When the spot price is higher than the strike price, the option is said to be 'in-the-money', which means the option has monetary value to the buyer(caller).

The earnings for the buyer(caller) resulting from exercising the option is called 'payoff'.
When the option is 'in-the-money', the payoff is (spot price - strike price). Otherwise, the payoff is zero.

5. what is over-the-counter instrument?
This is a bit side track. Warrants are often called over-the-counter instruments, which means it is normally traded between financial institutions without exchange facilities, as opposed to exchange trading (like you trade stock in HKSE or SGX).

6. what are European call option and American call option?

A European call option allows the holder to exercise the option (i.e., to buy) only on the option expiration date. An American call option allows exercise at any time during the life of the option.

7. Call option vs put option

What we have discussed above is termed call option; in layman terms, if the contract is modified to entitle the buyer to sell the underlying stock at certain price, then the option is termed put option. Here of course the buyer is taking a 'short' position towards the underlying instrument.