I always believed that to really understand a subject, one needs to be able to explain said topic in the simplest manner possible. This is the key motivation for me to come up with the QuantSimplified series. This series seeks to achieve 2 objectives — one is to challenge me to really master some fundamental concepts in quantitative finance, and the second is to help aspiring quants pick up some intuitions on these concepts. But truth be told, this is by no means an easy undertaking, and hence it took me about 4 months to think of a second topic to write about after my first.
The topic of this post was inspired by a conversation that I had a couple of months back. The conversation was about quants and the role of quants in communicating complicated concepts to the general layperson. I was asked to explain ‘Ito’s Lemma’ simply, and how it was used in trading and option pricing.
I think the answer I gave back then was decent, but it could do better. So I spent some spare time in the next couple of months building a story that could draw links between the theory and the practice. And before we touch the lemma, we need to first discuss ‘Stochastic processes’.
“Stochastic” is a difficult word
I gotta admit, when I first started my quant finance journey, stochastic calculus has always been one of the more intimidating topics. To begin with, it sounds difficult — and is difficult to pronounce.
But its meaning is surprisingly simple; “stochasticity” means randomness. When it comes to randomness, it leads to a core topic — probability.
Consider the following random game: you are given a fair coin, and you are asked to toss it. At every toss, if you get a ‘head’, you assign a score of ‘1’ to your score sheet. Otherwise, if you get a ‘tail’, you assign a score of ‘-1’ instead. You are asked to play this for many rounds and you are to accumulate the points as you progress into the game.
To introduce some terms,
- Sample space: it is a set of possible outcomes of the random game. In our case, it’s ‘Head’ (‘H’) or ‘Tail’ (‘T’)
- Random variable: it is a numerical value that is assigned to an outcome. Here, the random variable takes a discrete value of either -1 or 1. The score that we earn is our random variable
If we only played one turn, our sample space will be limited to 2 outcomes, and our random variable will be restricted to 2 possible values. The more rounds we play, the sample space grows as the outcome will need to consider a sequence of outcomes (e.g. {‘H’, ‘T’, ‘H’, ‘H’}) and the random variable would also expand to take in more possible values.
If we record our cumulative score at each turn on a sheet, that series of score records can be said to follow a stochastic process. In essence, a stochastic process is a time series of random variables. To illustrate a stochastic process, let us simulate the game and plot our cumulative score after 10 rounds.
Here it is, and oof I do have really bad luck here! Let’s see how this looks after playing for a thousand turns.
That is better, I managed to get a positive score! But wait… ignoring the scale of the score itself, doesn’t this plot look like how a stock price would move?
Asset price dynamics follow a stochastic process
Just like our scores in the coin-tossing game, the price of an asset is a random variable, and its future price is uncertain. In quantitative finance, we model the asset prices in order to determine the value of its derivative instruments. But when it comes to modeling, there are a few problems to solve:
- What assumption(s) do we make on the distribution of the random variables?
- What are the value of the parameters of the distribution, after we solved problem 1?
The first problem is likely to be the most difficult one to solve. Unlike our coin-tossing game, in which we know the probability of each outcome, the true distribution of the asset price (or return) is unknown. We often need to make some sort of assumptions based on empirical observations. The most commonly used distribution is a normal distribution. When plotting historical asset returns in a histogram, we observed that the return distribution follows quite closely — but not exactly — a Gaussian distribution. There are other tweaks and variations to the Gaussian distribution, such as adding jumps, but those are for another day.
So how do we relate asset price to the asset return, if we were to model the asset return as a normal distribution? Here, we made use of a small math trick. Assuming that the asset return usually takes on a relatively small value, i.e. (<< 1), we can approximate the asset return as the logarithmic difference of asset price at time t and at time t-1, i.e.
With this approximation, we can now model the asset price as a “lognormal” distribution. In fact, a stochastic process based on a lognormal distribution is known as a “Geometric Brownian Motion”, and it takes a stochastic differential equation (SDE) as follows:
Risk-neutrality
Now that we have made the assumptions necessary to overcome problem 1, we need to deal with the second problem, to determine the mean and the variance of the lognormal distribution. We shall ignore the variance for now, and focus on the mean.
From the above equation, the stochastic process is made up of a deterministic drift and a stochastic component which is the product of the variance and the asset price and a Wiener process. A Wiener process W(T) follows a normal distribution with a mean of 0 and a variance of T. The deterministic drift describes the general direction of the price evolution.
The estimation of the drift is highly model dependent. Different models with highly differentiated assumptions would arrive at different values of the drift. It is highly unlikely that two fundamentally different models could give rise to the same drift. But more importantly, it is almost impossible to know if any model is spot on in its guess of the actual drift of the asset price, given that the true distribution of the asset price is not determinable.
For instance, investor A ran her model and is highly bullish on the asset and thinks that the drift would be at 5%. On the contrary, investor B’s model determined that the asset is going to depreciate, and thus the drift should be at -2%. There is no way one can know beforehand which model is correct. These models are working on ‘real-life’ probabilities, or what the industry calls the ‘P’-measure. If we cannot reconcile in the ‘P’-world, what could be an alternative?
For many assets, one can rely on risk-neutral pricing to determine the drift. Without going too much into the details, the key idea is that if we can hedge a portfolio perfectly, the expected value of the portfolio measured with respect to a reference asset would remain the same as the initial value of the portfolio w.r.t. that reference asset.
This reference asset has a unique name in quantitative finance, and it is known as the numeraire. The numeraire has to meet certain conditions; for one it must be a tradable asset, and it cannot have negative values.
As it turns out, to replicate and hedge a forward contract or an option contract, the numeraire is the money market account. Under this numeraire, the drift is simply the risk-free rate, denoted as ‘r’. Therefore this measure is known as the ‘risk-neutral measure’, and the industry calls it the ‘Q’-measure. Below shows the SDE of the asset price under the Q-measure.
Very quickly on the Ito’s Lemma: SDEs based on Geometric Brownian Motion cannot be easily solved as a Wiener process is nowhere differentiable and we can’t use the usual techniques that we apply on non-stochastic equations. Ito’s Lemma introduces a simple technique (based on Taylor’s series expansion) that allows us to solve such SDEs. This is crucial because we need the asset price dynamics for option pricing.
Option pricing = pricing of likelihood
The bulk of this article has been explaining what is a stochastic process and how we determine the parameters of the assumed distribution. We have not explained why we need to model the asset price as a stochastic process. The answer is rather simple — it is because of the non-linearity of the option contract payout.
Consider a forward contract. If one were to sell a forward contract on AAPL today (assume the share price is at $179 per lot), one could remain risk neutral by borrowing $179 from the money market to buy the share today to lock in today’s price. As the seller holds a short position on the money market, he would need to pay the cost of borrowing, let’s say 5.3% annualized. This additional cost needs to be added to the forward contract pricing. Likewise, if during the lifetime of the contract, there is an expected dividend payment of $0.24 per lot which would be paid to the seller who would hold the actual share, this dividend payment (or yield, q) would need to be deducted from the contract price. Because the payout of the forward contract is linear, i.e. for every dollar appreciation in the underlying stock the forward appreciates by a dollar, and likewise if the value of the underlying stock decrease by a dollar, we need not care about the variance of the asset price distribution, but only the mean. In fact, we could potentially derive the analytical solution of the forward price through intuition alone.
However, an option contract has a payout structure that favors the buyer over the seller. Take the call option as a case in point: a buyer of the call would gain a dollar for every dollar the underlying asset exceeds the pre-agreed price level (‘strike price’). On the other hand, if the underlying asset price falls below the strike, the buyer is not obliged to exercise the option and thus can simply let the option expires. This means that for underlying asset prices below the strike, the payout is zero.
Since the option payout structure disadvantages the seller, the seller would demand a premium on the option. To determine this premium amount, one needs to evaluate the probability of being exercised against — meaning the probability that the underlying price would exceed the strike price level. This is where the stochastic process comes in, to allow market makers to determine based on an assumed distribution the pricing of the option.
And this is how stochastic process and calculus are relevant in quantitative finance — it allows us to price derivatives that are non-linear. If one further expands on the equation below, one would arrive at the Black-Scholes option pricing equation. This also explains why there are cumulative distribution functions within the BS equation. We are truly working on determining the likelihood of being exercised against to price the premium.
The key is volatility
Now that we have covered the relevance of stochasticity in quantitative finance, we are now back with one last important point — volatility. We have not mentioned how one determines the variance (which is volatility-squared) to be used for the distribution. Unfortunately, there is no easy answer to this one.
Volatility is important, because the larger the volatility (i.e. the standard deviation), the greater the likelihood the underlying price exceeds the strike price. One would price the option premium higher under a more uncertain environment.
One common method used is to look at the market-implied volatility. If there is a liquid options market, demand and supply would help determine a price level that incorporates the market participants' expectations of volatility. By using the market option prices, one could work in reverse with the BS equation to determine the volatility used — therefore this is called the “implied” volatility. This implied volatility could then be used, and one can adjust based on one’s own expectation of the volatility.
The real world is not that straightforward
I think it is important that we appreciate the basic theory of option pricing, and how stochastic processes help make this possible. But there are still many issues in the real world that cannot be simplified.
Firstly, we made a bold assumption that the underlying asset price follows a lognormal distribution. We know for a fact that the asset return is actually not normal — it tends to be negatively skewed with fat tails. Our assumptions introduced some biases in our pricing models.
Secondly, although we could derive implied volatilities, what if there are no liquid option markets? How do we then determine volatility?
Lastly, what if the option cannot be hedged? For instance, volatility and weather are underlying assets that are intangible and cannot be bought or sold. How do we derive risk-neutral pricing for volatility and weather derivatives?
All these are crucial questions to ask because they are real and prominent in the industry. I hope this article has provided you with a sense of option pricing and the complicated math behind the theory so that you can ponder these difficult questions further.
Thank you for reading the second article of the series, and that you have gained some insights on the basic of option pricing in quantitative finance! For more primer on implied volatility, you may refer to my previous article “The many facets of Risk”. Please do comment to provide your feedback on how to make this series better. Thank you!
To help you under the concept discussed here better, here is a brain teaser:
Consider our coin-tossing game again. If the rule of the game is that after tossing the coins for 50 turns if your cumulative score is greater than 6 (i.e. you have tossed 6 more heads than you did tails), I will pay you $5 + a dollar for every point exceeding 6. Ignoring the efforts needed to toss the coin (50 times can be very tiring I know), what is the minimum amount that you should pay to play the game?
