Deriving the Black-Scholes Partial Differential Equation

Abstract

Stochastic calculus forms a basis for many different models in financial mathematics. One such model is the Black-Scholes Options Price model which gives an appropriate price for a call option in a stock market under certain conditions. Deriving the partial differential equation for this model takes pieces from all sorts of stochastic processes such as Brownian motion, the Wiener Process, and Ito’s Lemma. This equation and its solution have revolutionized the field of financial mathematics and have even birth an entire field all about derivatives.

Introduction

The Black-Scholes model is one of the most famous financial mathematical models. From the partial differential equation that exists in the model, we can deduce a formula for calculating the price of a European call option. For the less financially literate, a European call option is a contract between a buyer and a seller of a stock. The contract tells the seller to keep that stock on sale for the exact price that it is at, called the strike price, but to only sell it to the buyer.

The contract also tells the seller that they can buy the stock for the listed price at the end of the period, or they can let it expire, and the seller can relist the stock. Either way, the buyer of the stock will pay the seller some amount of money to hold the stock. The main purpose of the Black-Scholes formula is to determine this price.

Before the discovery of the Black-Scholes formula in 1973, there were a few notable attempts at trying to find the price of a call for a stock.

The first of these was the Bachelier Model^[2] in the early 1900s. He derived that the call price c was equal to:

√
(S − X)N(d1) + σTn(d1)

Where:

• S = Stock Price
• X = Strike Price
• T = Time to expiration in years
• σ = Volatility of asset price
• n(x) = the standard normal density function

This result had a few flaws, one of which was that it took into account negative stock prices, which never happen. It also did not allow for any drift in the asset price, which would not allow one to account for interest rates and risk aversion^[2].

A new model was developed in 1965 called the Samuelson Model^[2] that addressed these concerns. It took into account a positive stock price and allowed for a positive drift in the asset price. He derived that the call price c was equal to:

Se(ρ−ω)TN(d1) − Xe−ωTN(d2)

Where:

• ρ = the average rate of growth of the asset price
• ω = the average rate of growth of the call price

We will see later on that Samuelson's equation was very close to what Black and Scholes derived in 1973.

What Black and Scholes actually derived was the solution to the partial differential equation that we will discuss. It wasn't until later that a man named Robert C. Merton came along and showed how to derive a differential equation which has the Black-Scholes formula as a solution. In 1997, Merton and Scholes received the Nobel Prize in economics. The reason given was "for a new method to determine the value of derivatives. Robert C. Merton and Myron S. Scholes have, in collaboration with the late Fischer Black, developed a pioneering formula for the valuation of stock options. Their methodology has paved the way for economic valuations in many areas. It has also generated new types of financial instruments and facilitated more efficient risk management in society." but their work has done so much more than that^[4].

Definitions

Before deriving the Black-Scholes partial differential equation, there are a few key theorems to observe.

2.1 Geometric Brownian Motion

Brownian Motion is the term that describes the motion of particles through a body of liquid or gas. These particles collide into each other repeatedly, constantly changing their direction. There are strong parallels between Brownian Motion and stock prices. Both have rapid, random changes constantly happening over time. We can use Brownian Motion to model what happens in a stock market and, ultimately, the Black-Scholes equation^[6].

2.1.1 The Wiener Process

The Wiener Process is a stochastic process that describes the change in a value over time in a one-dimensional plane. The Wiener Process has the following properties^[6]:

1. W0 = 0.
2. Wt is continuous along t.
3. Wt has independent increments. This means that for every choice of t0 < t1 < ... < tn, the incremental random variables Wt1 − Wt0, Wt2 − Wt1, ..., Wtn − Wtn−1 are jointly independent.
4. Wt has distribution Wt − Ws ∼ N(0, t − s). This means that no matter what increment of time you look at, the distribution of values in that time period will have a mean of 0 and a variance of t−s. This also shows that as t grows larger, the variance between values also increases.

We can see an example of this Wiener process repeated 25 times. As expected, W0 = 0, Wt is continuous, Wt has independent increments, and the variance between values increases as t increases.

2.1.2 Brownian Motion

We can use the Wiener Process to construct a stochastic differential equation for finding the change in a stock price over time. This is called Arithmetic Brownian Motion^[8]:

$\mathrm{dS}$
=
µ
dt
+
σ
dWt

This stochastic differential equation has two components: µdt, which is a drift constant µ times a change in time. This constant will give you the expected return of the stock given a very small change in time. The next component, σdWt, is a variance constant σ times a change in a Wiener Process dWt.

This stochastic differential equation gives a fairly accurate representation of example stock prices. This equation, however, has one flaw in it. Stock prices can never be negative. This can be fixed by multiplying both terms by our stock price S, which gives us the equation for Geometric Brownian Motion^[6]:

$\mathrm{dS}$
=
µS
dt
+
σSdWt

As we can now see, we no longer have a negative stock price. There are also other beneficial effects with this equation; If the company's stock price after a certain time is 0, say from bankruptcy, then the change in the stock price would also be 0.

2.3 Delta-Hedge Portfolio

In order to start deriving the Black-Scholes equation, we must first have something to start working from. We start from a portfolio, or a collection of assets, that we will define as Vt. This portfolio consists of two things, stocks and cash. We say we have φ units of stocks and ψ units of cash. Then,

Vt = φtSt + ψtCt

where

Ct = rCtdt^[2]

Deriving Black-Scholes

There are seven assumptions made by Black and Scholes before deriving their equation^[1]:

• The short-term interest rate is known and constant through time.
• The stock price follows a random walk in continuous time with a variance rate proportional to the square of the stock price. Thus, the distribution of possible stock prices at the end of any finite interval is log-normal. The variance rate of the return on the stock is constant.
• The stock pays no dividends or other distributions.
• The option is European.
• There are no transaction costs in buying or selling the stock or the option.
• It is possible to borrow any fraction of the price of a security to buy it or to hold it, at the short-term interest rate.
• There are no penalties for short selling.

To begin our derivation, we start with our portfolio defined earlier

Vt = φtSt + ψtCt,

and our goal is to apply it to Ito’s Lemma. (1)

(2)

In order to do this, we must first set Vt = f. This would mean that

(3)

because it’s already written in a differentiated form of t. Also,

(4)

thus

= 0 (5)

Plugging in (3), (4), and (5) into (2) we get

df = [ψtrCt + µStφt]dt + σStφtdWt (6)

Now we solve for our original equation by inputting

(7)

if we subtract over and divide by Ct we’re left with

(8)

and because Vt = f we have

(9)

plugging (4) and (9) back into (6) we get

(10)

which simplifies to

(11)

We can now set (11) equal to (2) and we immediately see that the last terms cancel each other out and we are left with

(12)

canceling out the dt’s and distributing over the rdt we get

(13)

canceling out like terms we have

(14)

and this

(15)

is the Black-Scholes partial differential equation.

Applications

While this partial differential equation is very useful in the field of finance, in order to actually perform some real-life examples, we need the solution to this partial differential equation. The solution to the equation is:

$C$
=
N
(
d1
)
St
−
N
(
d2
)
Ke
−
r
(
T
−
t
)

where

Where

C= The call price

St= The spot price

K= The strike price

N(x)= is the cumulative distribution function

r= risk-free rate of return

σ= the volatility of returns of the asset

T − t= the time to maturity of the call^[1]

Let’s use this formula to try and calculate the price of a call. For this example, let's use the market SPXL. SPXL trades in European-style options, so the Black-Scholes can work for this^[9]. A simple Google search will tell you that the stock price is close to around $60, and let’s say that the option we want to buy has a strike price of $58. This option would be called "in the money" if we owned this option because the stock price is $60 but the strike price is $58, so we could strike our call for $58 and then turn around and sell it for $60. Unfortunately, we don't own that stock, but we can calculate the price it would take to buy that call for.

Let’s say that this call will be held for 30 days as well. Now, according to Yahoo! Finance, the yield rate for SPXL is about 91% right now. This just leaves the volatility to discern, and then we can begin calculating. The volatility of a stock is very difficult to calculate with reverse-engineering Black-Scholes with recent call prices. Some people actually trade off the fact that it is very difficult to discern the volatility of the market. According to alphaquery.com, the volatility for SPXL is 36% or 0.36. Now we can input these numbers into our d1 and d2 to get

This leaves us with

d1 = 2.8

d2 = 2.44

Now, for calculating our cumulative distribution function, we want to find the area under the standard normal curve for segments d1 and d2 standard deviations away from the mean. Since our d values are so high, we can estimate them as 0.99. Plugging them into our Black-Scholes formula, we get

$C$
=
.
99
(
60
)
−
.
99
(
58
)
(
e
−
.
91
)

Which gives us a call price of about $36. We can see that this is a fair call price even though it's almost half of the price of the stock. The rate of return of the market is very high, and the stock we are buying is already in the money.

Discussion and Conclusion

Another thing that the Black-Scholes formula can calculate is the price of a put. A put is, in essence, insurance for your stock. It’s a contract between a buyer of a stock and a seller of a stock where the buyer purchases the stock from the seller but gives a little more money extra for insurance. Between the time of purchase and the expiration date, the buyer can tell the seller that they want to sell the stock back, and the seller must oblige.

In order to calculate the price of a put option, we must use something called put-call parity. When done, this turns our formula for the price of a call into:

$P$
=
N
(
−
d2
)
Ke
−
r
(
T
−
t
)
−
N
(
−
d1
)
St

where P is the price of the put^[2].

There have been some interesting articles written about the Black-Scholes equation. One article by Marek Kolman derived the Black-Scholes partial differential equation under Arithmetic Brownian Motion instead of Geometric Brownian Motion. This model has some very interesting properties. It allows you to calculate the price of a call when r=0. r can equal 0 in the stock market, so this is a good model to have^[5]. Another interesting article by Grossinho et al. attempts to construct a model for calculating the price of an American call based on the Black-Scholes Model. An American call is the same as a European call except you don’t decide whether you let the option expire or not strictly at the end of the period; you can strike at any time. This is extremely important as most stocks are traded in American options. Having this flexibility is important to a lot of stock traders, and they are willing to pay a premium for the choice. Thus, American options are generally more expensive than European^[7].

I feel further advancements could be made with the Black-Scholes equation. I feel like the Black-Scholes equation could be made less restrictive by removing some of the assumptions tied to it. The previous paragraph discussed the removal of some of these assumptions such as "The option is European". One assumption that I feel is paramount to removing is the one that states "There are no transaction costs in buying or selling the stocks or options". With how popular trading options is becoming, companies such as Robinhood and Acorns have popped up and charge fees to trade on them, violating that assumption.

References

1. Fischer Black and Myron Scholes. The pricing of options and corporate liabilities. The Journal of Political Economy, 81(3):637–654, 1973.
2. Espen Gaarder Haug. The Complete Guide To Option Pricing Formulas. McGraw-Hill, 1997.
3. Kiyosi Ito. On stochastic differential equations. Memoirs of the American Mathematical Society, (4), 1951.
4. Robert A. Jarrow. In honor of the nobel laureates robert c. merton and myron s. scholes: A partial differential equation that changed the world. The Journal of Economic Perspectives, 13(4):229–248, 1999.
5. Marek Kolman. Black-scholes model under arithmetic brownian motion. 2013.
6. Shan Lu. Power laws in complex graphs: parsimonious generative models, similarity testing algorithms, and the origins. Doctoral Dissertations, 2018.
7. Yaser Faghan Kord Maria do Rosario Grossinho and Daniel Sevcovic. Pricing american call options by the black-scholes equation with a nonlinear volatility function. 2018.
8. Sheldon M. Ross. Introduction to Probability Models. Academic Press, 2014.
9. Mark Wolfinger. Spx options vs. spy options, 2019.