Continued from part 1.

**Expected Value**

Suppose there’s a competition I can enter for free where I have a 50% chance of winning $1,000,000, and a 50% chance of receiving nothing. Obviously I’m going to enter as many times as possible. Suppose I enter 1,000 times: what will I expect to win?

On average I’m going to win $500,000 per entry ($1,000,000 x 50%), so I’d expect to win around $500,000,000 from 1000 entries.

We say that the ‘expected value’ of playing the game is $500,000. It’s what we ‘expect’ to win on average if we play many times.

Similarly if the game has a 50% chance of winning $1,000,000, a 20% chance of winning $200,000 and a 30% chance of winning $-500,000 we can say the ‘expected value’ of playing is:

50% x 1,000,000 + 20% x 200,000 + 30% x -500,000 = $390,000

And if I play 1000 times I expect to win around $390,000. Notice that to do this we are multiplying the percentage probability of each value by the amount we make in each case, and then adding it all up.

So to value our option contract using an expected value calculation we would:

- Work out the percentage probability of each value of the stock price in a year’s time
- Work out how much profit we make in each case (from the payoff graph)
- Multiply the percentage probability of each value by the amount we make and add them up to get an ‘expected value’ for the contract.

This can then be considered as the basis of the price of the option, although there’s also the time value of money to consider, which I will discuss later.

**Continuity**

In the discussion above I implied that to calculate the expected value of our option we would consider a series of distinct values of the price of the stock at maturity. In fact the Black Scholes method assumes that the stock price can take ANY value continuously between minus infinity and plus infinity at maturity. This actually makes the maths easier, although possibly more difficult for non-mathematicians to understand. Nevertheless the basic concepts remain the same.

**Time Value of Money**

Another concept that you need to understand is the time value of money. If I receive a cash flow of $1,000,000 in one year’s time then that is worth LESS to me than if I receive $1,000,000 today. This is intuitively obvious if you consider that if I receive $1,000,000 today I can buy a government bond and earn interest on it for a year. At the end of the year I’ll have more than $1,000,000.

The difference between the two values clearly depends on the interest rate that I can get over the year. If I can get an interest rate of i% then at the end of the year my $1,000,000 will be worth $1,000,000 x (1+i%). Equally if I am due to receive $1,000,000 x (1+i%) in a year’s time, I can say that that is worth $1,000,000 to me today. If I receive $1,000,000 in a year’s time that is worth $1,000,000 / (1+i%) to me today.

In general if I am going to receive $y in a year’s time I can say that that is worth y / (1+i%) to me today. In this case we call the 1/(1+i%) a ‘discount factor’. It’s what we ‘discount’ (multiply) the future cash flow by to get its current value.

Note that here i% is the interest rate I can get for investing in something with very little or no risk for a whole year. A different rate may apply if my cash flow is due in six months time (and clearly I’ll only be getting six months of interest at the different rate, so the calculation changes as well). A different discount factor will apply.

To avoid some of the difficulties of interest rates over different periods the Black-Scholes method assumes initially that interest rates are constant. It also uses what are called ‘continuously compounded interest rates’. I will explain these in more detail in another article. All you need to know for the purposes of this article is that if r is our continuously compounded interest rate for a period of T years then the discount factor will be .

**Black-Scholes**

The easiest way to understand the Black-Scholes formula intuitively is to consider what happens if we exercise the option. This has two parts:

i) Value of the Cash to Buy the Option

Firstly, if the option is exercised we pay the strike price (30 in our example). We pay the strike price only if the underlying stock price is above the strike at maturity. So to work out the expected value of this we need the probability simply that the stock price is above the strike at maturity. Let’s call this probability N(d2), and the strike price K. Then the expected value of this is just KN(d2). What ‘N()’ means, and the value of d2, will be discussed later.

KN(d2) is the value of the cash flow at maturity. As discussed above, to get the value of this cash flow today we need to discount it, and the discount factor is .

So the value of the cash to buy the option today is KN(d2).

ii) Value of the Stock Received if any

Secondly, if the option is exercised we get a unit of the stock. This is clearly worth whatever the stock price is in the market at maturity. But this again only happens if the underlying stock price is above the strike at maturity.

It turns out that the expected value of this valued as at today is proportional to S, the stock price today, and can be written as SN(d1). That is to say, SN(d1) is the expected value of something that is equal to the final stock price if the final stock price is above the strike, and equal to zero if the final stock price is below the strike.