Mini Chapter One
Option contracts
 Option contracts – are an arrangement entered between two parties where the buyer has the right but not the obligation to exercise a short or a long position in the underlying asset at a preagreed price (strike). The seller/writer of the option however has the obligation to either buy or sell the underlying asset as per the direction the option buyer chooses to exercise. Corresponding to the long versus short view on the underlying asset there are two types of options – Calls and Puts.
 Buying an option also limits one’s losses to the premium paid, hence the payoff is asymmetrical (i.e. levered) around the strike compared to an outright forward at the same strike.
 A long call position is a right to buy and a long put is a right to sell the underlying asset at their respective strikes. Graph 1 below shows the payoff (at maturity) for a buyer/seller of a call and a put option.
Source: Pandemonium.
 Options get exercised by their buyers in case of a positive payoff i.e. when they are ‘in the money’. In case, of a negative payoff i.e. ‘out of the money’ there isn’t any obligation to exercise it and the loss only amounts to the premium paid by the option buyer.
 Option types also differ depending upon the timing of their exercise: a) American options can be exercised any time up to and including the expiration date while b) European options may be exercised only at their expiration date. A longer running optionality to exercise in the American type versus the European counterpart makes the former more valuable than latter.
Stochastic Processes
Before jumping into the pricing of an option let’s first understand the process that an underlying’s price follows to reach a terminal value at the end of any given time period. This process of changes in price is stochastic in nature and results in a randomly distributed movement of the asset’s prices. This would be at the heart of the framework(s) for arriving at the probability distribution of the underlying prices. As we go along the use of intuition would be palpable enough to break down even the more intimidating concepts/equations to hopefully help cement a friendlier approach whenever/wherever you read these again.
Stochasticity can be for both discrete (i.e. underlying taking random values at certain discrete points in time) and continuous time periods (i.e. underlying taking just about any value within a range). Options theory refers to a continuous process but the real world is more discrete.
Random movement of the underlying variable generates a probability distribution for its various outcomes, and among the several types taking the form of a normal distribution that’s analysed by a specific mean and a variance. Let’s begin with the various types of stochastic processes and how they build on each other:
 Markov Property – to summarise the idea, think of only “present” to have any relevance as the present value of a variable contains all information to determine its future value i.e. the past values or the path the variable takes to get to its present value holds no significance. A popular disclaimer of investment products citing past performance to not guide future returns takes a leaf from Markov. A key corollary of the markov process is that normal distributions (known to be independent of each other as previous paths do not determine futures ones) across time periods can be additive i.e. the sum of two independent normal distributions would result in a standard normal distribution with mean and variance being the sum of the respective individual distributions. Do note that variance being a “squared” metric can be added but standard deviation being the squareroot of variance can’t be. Therefore, if change in the underlying variable in year 1 and 2 have independent respective normal distributions with mean = 0, variance = 1, the sum of these distributions would have a mean of 0 and variance 2 but a standard deviation of √2. That’s also why you’d see references to √T where T is the cumulative time period in question implying that the standard deviation for a distribution for time T is scaled by the square root of time.
 Wiener Processes – are defined by movement in a variable with the markov property that attains a normal distribution with mean = 0 and variance = 1 per year. Consider a variable X, the change/returns on which over a small time period Δt follows the Wiener process. The returns can be expressed as the equation below:
 ΔX = N (√Δt) where N is the normal distribution with a mean of 0 and variance of 1

 Further, following markov the change in X for any two different time intervals are independent of each other which lends ΔX a normal distribution with mean 0 and variance Δt or standard deviation of √Δt.

 A probability distribution for a longer time interval T (made up of smaller intervals Δt ranging from 1 to n) would be a sum of each of the distributions corresponding to the respective intervals with a mean of 0 and standard deviation of √T.

 Lastly, the Wiener process is also used interchangeably with the Brownian motion concept in Physics.
 Generalised Wiener Processes – was a further refinement to the wiener process to account for a constant “drift” in the movement of the underlying variable in addition to the random stochastic component weighed by a constant variance. To draw an analogy – basic wiener process assumed a drift of 0 and a variance of 1. You can think of drift to mean a known pattern of expected returns that can either be enhanced or depleted by the random movement in returns. Even in case of no volatility (no randomness) one can still expect the drift to create a linear trend of returns. For equation lovers here is the expression for change on the underlying variable X following generalised wiener:.
 The generalised wiener process dz for changes in X can be denoted as, dX = a dt + b dz
 Elaborating the above for small movements in time (Δt),
ΔX = a Δ t+b N (√Δt), where a is the constant drift and b is the constant that scales the wiener movement N (√Δt).
 Elaborating the above for small movements in time (Δt),
 Nonzero value of b adds the variability to the underlying’s movement i.e. b = zero would eliminate the randomness. In that case X would have a constant rate of movement “drift” with respect to time T expressed as, X_{1} = X_{0} + aT

 Wiener process above (dz as in the generalised equation) is normally distributed with mean 0 and standard deviation b
 Subsequently this creates a normal distribution for ΔX with mean aΔt and standard deviation b√Δt
 Ito Process – a further refinement of the generalised wiener process takes into account the variability of both a and b i.e. when both the drift and the variance of the random component are a function of the underlying X and time T. Stock prices tend to have constant return expectations. Thus changes or drift would change based on the value of the stock. Expected drift on a stock with return expectation of 5% would be 5$ if the stock was priced at 100 and would be 50$ if the stock was priced at $1000. Hence the ito process changes the above Generalized weiner process equation as below:
dX = a (X, t) dt + b (X, t) dz
Where now a and b themselves are a function of the value of the underlying and are subject to change w.r.t both changes in the underlying and the passage of time
 Again for a small time period Δt, ΔX = a (X, t) Δt + b(X, t) N(√Δt), which assumes that for small time periods a (X, t) and b (X, t) do not change.
 Applying this thought process to the process of changing stock prices of a nondividend paying stock would mean that if the expected return on a stock was μ then drift of the stock price change would be expected to be μS such that:
dS = μSdt in case of no uncertainty or zero volatility of return of the stock
 In reality we also need to account for the uncertainty. A reasonable assumption made for stock prices is that the variability of the percentage return of a stock price would be the same for small passage of time suggesting that the standard deviation of the change in stock price would also be proportional to the stock price. Thus adding the variability term to the above equation we would get the following equation for stock price behaviour
dS = μSdt + 𝞂Sdz
 Note that this is akin to the ito process equation above wherein a(X, t) – Drift is replaced by μS (function of S) and b(X, t) is replaced by 𝞂 S (again a function of S).
 As the drift term is directly proportional to the underlying stock price – μ can also be thought of as the average geometric mean return expected by the market with the variability term akin to a brownian motion as mentioned earlier – this equation also describes a Geometric Brownian Motion applied to stock prices.
 This is the most widely used equation of stock prices and the similarity with the Ito process then lends us to use further mathematical processes to analyse the probability distribution of the prices through time. Ito’s Lemma is a wellknown mathematical tool in stochastic calculus – derivation would be out of scope here – and widely applied in the world of economics and quantitative finance. Its application is used to describe the evolution of a stochastic process (this could be a process ‘f’ that’s a function of a stochastic variable like equity/FX prices) through time that’s largely governed by a drift and a random component. In other words, modelling the stochastic movement of financial assets is facilitated by Ito’s Lemma to generate probability distributions of the various outcomes of the underlying asset in a certain time frame.
We move next to the pricing of options and an intuitive explanation of the Black Scholes pricing model that’s based on a Brownian motion of the underlying and is a consequence of the application of the lemma.