Mini Chapter Three

Intuition behind Black Scholes

Black Scholes option pricing equations for a European Call ( C ) and European Put (P)

$C = S \times N (d_{1}) - X e^{- r t} \times N (d_{2})$	(1)
$P = X e^{- r t} \times N (- d_{2}) - S \times N (- d_{1})$	(2)
Where,
$d_{1} = \frac{{(\log ((\frac{s}{x})) + ((r + \frac{σ^{2}}{2})) \times t)}}{σ \sqrt t}$	(3)
$d_{2} = \frac{{(\log ((\frac{s}{x})) + ((r - \frac{σ^{2}}{2})) \times t)}}{σ \sqrt t}$	(4)

To define the terms used in the equation:

S is current spot price

- X is the strike price

- t is time to expiration of the option, expressed in years

- r is the risk free rate of interest

- Xe^-rt is the present value of the strike

- $σ^{2}$ is the variance of the underlying

- N is the cumulative standard normal distribution function

An intuitive explanation of the equation – consider just the call option premium as in equation (1) expressed as the difference between expected value of the underlying at time t, denoted by S multiplied by the conditional probability N(d₁) and the payment made to exercise the option Xe^-rt weighed by the probability N(d₂). To delve deeper we would need to understand N(d₁) and N(d₂) terms in the equation. Starting with the simpler N(d₂) below.

N(d₂) is the probability of the underlying’s value being above the strike at expiry. In other words it is the probability of the option being exercised and the strike price being paid. Mathematical expression of d₂ can be further understood with intuition and some basics of statistics. We go back to the assumptions of the Black Scholes model where returns of the underlying asset have a normal distribution but importantly each return outcome is composed of a constant drift and a stochastic random component that lends it a Geometric Brownian motion.

In the world of finance where assets and their returns are correlated the mean return is better denoted as being geometric instead of arithmetic; the former always being lower than the latter. A standard normal distribution of returns therefore would have a geometric mean expressed as its arithmetic mean eroded by half the variance or $r - \frac{σ^{2}}{2}$ or μ.

r - \frac{σ^{2}}{2}

or μ.

Since N(d₂) is the cumulative probability of S_t > X, you can think of d₂ as being equivalent to a Z score or the unit standard deviation by which the required rate of return for S to reach X deviates from the mean return.

Graph 6a - Normal distribution of returns with mean

r - \frac{σ^{2}}{2}

Graph 6a – Normal distribution of returns with mean

r - \frac{σ^{2}}{2}

Source: Pandemonium.

N(Z) or normal distribution of Z would give us the area under the normal probability distribution curve to the left of Z or cumulative probability of all return outcomes below Z. Going by the symmetry of a normal distribution (refer to the graph above and the shaded areas that are equal on both sides) N(-Z) would give us the cumulative probability for returns to the right of Z i.e. returns corresponding to Sa > X. Continuing with the notation above:

d_{2 =} - Ζ = - \frac{{(\log_{e} ((\frac{X}{S})) - ((r - \frac{σ^{2}}{2}))})}{σ}

\frac{{(\log_{e} ((\frac{S}{X})) + ((r - \frac{σ^{2}}{2}))})}{σ}

= \frac{{(\log_{e} ((\frac{S}{X})) + ((r - \frac{σ^{2}}{2}))})}{σ}

Graph 6b – Log Normal

Source: Pandemonium.

Quick refresher on the characteristics of a lognormal distribution

- It’s a right tailed continuous probability distribution where the distribution’s steepness and peak on the left relative to the tail on the right is a function of the volatility of the asset’s returns and the frequency with which it’s compounded.

- Another way to define – a variable whose natural logarithm generates a normal distribution is itself lognormally distributed.

- As for a notation if variable X is lognormally distributed then its log transformation would generate a variable Y that’s normally distributed.

- Y = log_e (X) or inversely X = e^Y ,where it’s clear that while Y can attain negative values, X being an exponential function (continuous compounding of corresponding rates of return) would always be positive, also a key reason why the lognormally distributed X would denote asset price values that in theory cannot go negative.

In the world of statistics, the maximum likelihood estimation is used to compute parameters that generate a probability density function that best fits the observed data set. The same can be done to arrive at a log normal distribution but it’s important to keep in mind that the µ and parameters in its equation are the mean and standard deviation of its log-transformed (normal) distribution. Unlike the normal distribution, the mean, median and mode are attributed to different levels of the lognormal variable. Relevant notations below (mathematical derivation has been kept out of scope but readily available on the web):

e^{μ - \frac{σ^{2}}{2}}

Lognormal probability density function:

\frac{1}{X σ √ 2 π} \times e^{- \frac{1}{2} {(\frac{\log (X) - μ}{σ})}^{2}}

Median:

e^{μ}

Mean:

e^{μ - \frac{σ^{2}}{2}}

Mode:

e^{μ - σ^{2}}

Using the above notations the expected value/mean/ATMF value of the underlying can be obtained as:

E[X_t] = X_ATMF = X₀ e^rt, where,

X is the underlying

r is the risk free rate of return

t denotes the option expiry period

It’s intuitive then – given the continuous compounding of returns the mean value of the lognormal variable X would be well beyond the peak likelihood value of X; in statistical parlance likelihood is referred to as point probability while probability by default is cumulative/area under the curve.

Mathematically this is expressed as

Ζ = \frac{(X - μ)}{σ}

where

x

is the required rate of growth such that

Χ = S e^{x} o r \log_{e} ((\frac{X}{S})) = x

Expanding the notation then,

Ζ = \frac{{(\log_{e} ((\frac{X}{S})) - ((r - \frac{σ^{2}}{2}))})}{σ}

d_{2 =} - Ζ = - \frac{{(\log_{e} ((\frac{X}{S})) - ((r - \frac{σ^{2}}{2}))})}{σ}

= \frac{{(\log_{e} ((\frac{S}{X})) + ((r - \frac{σ^{2}}{2}))})}{σ}

Note that this expression would need to be scaled for the option expiry expressed in years to exactly match equation 4 above.

N(d₁) is the conditional probability factor by which the discounted expected value of the underlying exceeds its current value. The ‘conditional’ element comes from the condition of the stock price being higher than the strike at option expiry. SN(d₁) therefore incorporates both the probability of exercise and the cumulative expected value the stock attains conditional/contingent on the stock price exceeding the strike at maturity. This can also be expressed as:

{P_{1} \times S_{1} + P_{2} \times S_{2} + P_{3} \times S_{3} + .... + P_{n} \times S_{n}}

where observations 1 to n are all greater than X with P denoting their respective probabilities.

Also note that

{P_{1} + P_{2} + P_{3} > .... + P_{n}} = Ν ((d_{2}))

- A simple example on probability can explain this further – consider a stock price that can only take 10 different discrete values (say 10, 20, 30, 40…100) at expiry with equal probability for each outcome. Expected value of this stock at expiry would therefore be the probability weighted sum of all outcomes i.e. 55. This can also be understood as the unconditional future expected value of the stock. However, since a call option exercise depends on the stock price being above a strike, now consider a strike value of 60. The conditional expected value of the stock above 60 would be:

10% x (70 + 80 + 90 + 100) = 34

SN(d₁) would therefore be equal to 34 ignoring the effects of discounting.

Black Scholes Assumptions – Black Scholes is one of the mathematical models which estimates the probability distribution of a given option’s payoff to derive the price of that option. The BS model makes a few assumptions regarding the distribution of prices of the underlying which may or may not hold in the real world but serves as a good benchmark to align market pricing based on different volatility/market conditions. Assumptions (most references explained above) include:

- - Asset returns are normally distributed that derive a log normal distribution of its prices allowing for prices being greater than 0

- - Underlying asset returns follow a geometric brownian motion

- - Continuous hedge market availability with no transaction costs

- - Underlying asset does not generate interim cash flows – this is important to think about as a dividend paying stock trading ex-dividend just ahead of that cash flow would not allow for continuous hedging

- - BS also assumes that Volatility of the underlying and the risk free rate remain constant and predictable during the life of the option.

Other pricing models used by academicians and finance practioners such as Binomial models, GARCH, Auto-regressive, Local Vol, Jump Diffusion, Monte Carlo et al. also attempt to estimate probability paths of the underlying’s prices. To keep the content concise and relevant from a daily need-to-know perspective I would refrain from discussing these any further.

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Mini Chapter Three

Intuition behind Black Scholes

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Varda Pandey