The Black-Scholes Equation WITHOUT Tears (or Probability!)

By Doron ZEILBERGER

The Fair price of a European Call Option (Black-Scholes Equation)

The Crown jewel of mathematical finance is

The Black-Scholes Equation([BS])

Suppose that a certain stock costs now S_{0} dollars, its volatility is σ, and the interest rate is r. The fair price of an option to buy that stock, for the price of K dollars, at time T, is

S_{0} Φ (\frac{log \frac{S_{0}}{K} + (r + \frac{1}{2}σ^{2})T}{σ\sqrt{T}} ) − K e^{−rT} Φ (\frac{log \frac{S_{0}}{K} + (r − \frac{1}{2}σ^{2})T}{σ\sqrt{T}} ) ,  (BS)

where

Φ(z) := ∫_{−∞}^{z} \frac{1}{\sqrt{2π}} e^{−x^{2}/2}dx .

The purpose of the present, purely expository, classroom note is to derive it from scratch, without any probability theory, following the beautiful Cox-Ross-Rubinstein[CRR] discrete-time model, but in a way that should be understood by the proverbial smart middle-school student. The only prerequisite is the ability to solve two linear equations with two unknowns. I was greatly helped by Alison Etheridge’s wonderful book[E].

Let S_{T} be the price of the stock at time T. Note that if S_{T} < K, it would be stupid for the buyer to exercise the option, since he can just buy the stock in the free market (rather than pay more for it), and he definitely lost the money spent buying the option. But if S_{T} > K, then he would exercise it, and by reselling it, would make S_{T} − K dollars. If S_{T} − K (adjusted) is larger than the price that he paid for the option then he came out ahead. Regarding the seller, he can “take a chance”, pocket what he got from the buyer, and do nothing. If S_{T} < K he lucked-out, but if S_{T} > K then he would have to buy the stock for S_{T} dollars, and sell it to the buyer, at a loss, for K dollars, that is equivalent to paying the buyer S_{T} − K dollars. If this (adjusted) amount is less than what he got for the option, then he still came out ahead, but if it is more, then he lost.

But doing nothing is only one option, (no pun intended!).

For the sake of simplicity, we will only consider the case where the interest rate, r, is zero. The extension to general r is straightforward.

Let’s Bet!

A stock today costs 12 dollars. It is known that tomorrow it is either going to cost 6 dollars or 24 dollars.

The seller offers the buyer, if he pays him 8 dollars.

• 3 dollars if the stock would go down;

• 15 dollars if the stock would go up. Schematically:

12 / \

6 [3] 24 [15] Let’s assume that the seller does not buy any stock. Then this deal is a gamble whether the stock would go up or down, and the actual prices, 6 and 24, are irrelevant. If the stock would go down the buyer would lose 5 dollars (and the seller would make 5 dollars), but if the stock would go up, the buyer would make 7 dollars (and the seller would lose 7 dollars). If the buyer estimates that the probability that the stock would go down is q, his expected gain is q(−5) + (1 − q)7 = 7 − 12q, so he would agree to the deal if q < \frac{7}{12}, if he is rational. But the fair price has nothing to do with what the buyer (or seller) think about the future of the stock. It is determined by the AXIOM of OPTION PRICING: If you want to have a chance to make money, you should be willing to risk losing money! In other words: IT SHOULD BE IMPOSSIBLE TO HAVE A STRATEGY THAT WOULD LET YOU MAKE MONEY (AT LEAST SOMETIMES) AND NEVER LOSE MONEY It turns out that 8 dollars is too high of a price for the above deal. Here is how to figure out the fair price of the above deal. We assume that the seller of this bet has no money of his own, and will get in jail if he can not fulfill his obligations to the buyer. So he has to design a mixed portfolio, only using the money that he would get from the buyer of the bet. Buy a units of stock and b dollars in cash (b can be negative, so he can borrow). The value of this portfolio today is 12a + b (since one unit of stock costs 12 dollars today) The value of this portfolio tomorrow is

• if the price of the stock went down to 6 dollars: 6 a + b

• if the price of the stock went up to 24 dollars: 24 a + b.

In order to meet his obligations (and not get arrested), he needs to make sure that

6a + b ≥ 3 , 24a + b ≥ 15 .

But if any of the inequalities is strict then he made money!, (at least sometimes). And that is not allowed, so to barely meet his obligations he has to solve the system of two linear equations in the two unknowns, a and b:

{6a + b = 3 , 24a + b = 15 } ,

whose solution is a = \frac{2}{3}, b = −1. The cost of this portfolio today is

12a + b = 12 · \frac{2}{3} + (−1) = 7 .

So the price must be at most 7 dollars. What if it is less? Then the buyer can borrow \frac{2}{3} units of stock, and lend 1 dollar, and buy the deal, and come out ahead, forbidden! So the fair, arbitrage-free, price of the bet is exactly seven dollars. Note that the mixed portfolio ‘buy \frac{2}{3} units of stock and borrow 1 dollar’, the so-called hedging portfolio, is only of theoretical interest. If the seller of the above deal would follow it, he is guaranteed to never make any money, so in real life he would use a different portfolio, based on his belief of the future market, and take a risk of losing money. But the price of seven dollars guarantees that neither buyer nor seller would be able to have a possibility of making money, without ever losing it, and hence is considered the fair price. The General Single Period Binary Case A stock today costs S_{0} dollars. It is known that tomorrow it is either going to go down to S_{d} dollars or go up to S_{u} dollars. The seller offers the buyer, the following deal.

• C_{d} dollars if the stock would go down;

• C_{u} dollars if the stock would go up. Schematically:

S_{0} / \

S_{d} [C_{d}] S_{u} [C_{u}] Here is how to figure out the fair price of that deal.

We assume that the seller has no money, and will get in jail if he can not fulfill his obligations. All he has is the money that he got from the buyer. How much money should the buyer give the seller for the above deal? The seller has to design a mixed portfolio. Buy a units of stock and b dollars in cash (b can be negative, so he can borrow). The value of this portfolio today is a · S_{0} + b (since one unit of stock costs S_{0} dollars today). The value of this portfolio tomorrow is

• if the price of the stock went down to S_{d} dollars: a · S_{d} + b

• if the price of the stock went up to S_{u} dollars: a · S_{u} + b. In order to meet his obligations (and not get arrested), we need

aS_{d} + b ≥ C_{d} , aS_{u} + b ≥ C_{u} .

But if any of the inequalities is strict then he would make money!, (at least sometimes), without any risk of losing money, and that is forbidden, so to barely meet his obligations he has to solve the system of two linear equations and two unknowns

aS_{d} + b = C_{d} , aS_{u} + b = C_{u} ,

whose solution is

a = \frac{C_{u} − C_{d}}{S_{u} − S_{d}} , b = \frac{C_{d}S_{u} − C_{u}S_{d}}{S_{u} − S_{d}} .

The cost of this portfolio today is

\frac{C_{u} − C_{d}}{S_{u} − S_{d}} · S_{0} + \frac{C_{d}S_{u} − C_{u}S_{d}}{S_{u} − S_{d}}

= \frac{(S_{0} − S_{d}) C_{u} + (S_{u} − S_{0}) C_{d}}{S_{u} − S_{d}} .

We just proved (using Algebra 1), the Fundamental Lemma of Option Pricing Suppose that the price of a certain stock today is S_{0} dollars, and it is known that tomorrow it is either going to cost dS_{0} (d < 1) or uS_{0} (u > 1). The fair price of a deal to be paid C_{d} dollars if the stock goes down and C_{u} dollars if the stock goes up is

\frac{(1 − d) C_{u} + (u − 1) C_{d}}{u − d} = \frac{1 − d}{u − d} C_{u} + \frac{u − 1}{u − d} C_{d} .

Let’s abbreviate

α := \frac{1 − d}{u − d} , β := \frac{u − 1}{u − d} .

Then the fair price is

αC_{u} + βC_{d} .

What if there are Two Days?

Suppose that today’s stock costs S_{0} dollars, and tomorrow it would go either down to S_{d} or up to S_{u}. If tomorrow it went down to S_{d}, then the day after tomorrow it may either go further down to S_{d,d}, or pick up, and go up to S_{d,u}. If tomorrow it went up to S_{u}, then the day after tomorrow it may drop to S_{u,d} or go further up to S_{u,u}. Assume that the seller offers the buyer the following conditional promises

• C_{d,d} dollars if after two days the price of the stock is S_{d,d}

• C_{d,u} dollars if after two days the price of the stock is S_{d,u}

• C_{u,d} dollars if after two days the price of the stock is S_{u,d}

• C_{u,u} dollars if after two days the price of the stock is S_{u,u}

Then to figure out the fair price, one uses backwards induction. First figure out the fair price tomorrow under the assumption that the stock went down, get a certain number (using the fundamental lemma), let’s call it C_{d}. Similarly for C_{u}, and then use the formula once again to get the fair price today.

Now let’s make the simplifying assumption that if the price of the stock today is S, then the next day it is either dS or uS, and d and u are always the same from day to day. Hence

S_{d,d} = S_{0}d^{2} , S_{d,u} = S_{u,d} = S_{0}ud , S_{u,u} = S_{0}u^{2} .

Also let’s assume that the payoff, at the end, only depends on the price of the stock, not on the history, so denoting C_{u,u} = C_{0}, C_{u,d} = C_{d,u} = C_{1}, C_{d,d} = C_{2}, it is immediate that the fair price, using the above formula is

α(αC_{0} + βC_{1}) + β(αC_{1} + βC_{2}) = α^{2}C_{0} + 2αβC_{1} + β^{2}C_{2} .

If there are three days, then the price is

α · (α^{2}C_{0} + 2αβC_{1} + β^{2}C_{2}) + β · (α^{2}C_{1} + 2αβC_{2} + β^{2}C_{3})
= α^{3}C_{0} + 3α^{2}βC_{1} + 3αβ^{2}C_{2} + β^{3}C_{3} .

After n days,

∑_{i=0}^{n} \binom{n}{i} α^{n−i}β^{i}C_{i} ,

that is easily proved by induction on n. Here C_{i} is the payoff if the price of the stock, after n days, is S_{0} d^{i}u^{n−i}. So far it was abstract gambling. If the seller promises the buyer to sell the stock, after n days, for K dollars then the value of the promise is

C_{i} = S_{0}d^{i}u^{n−i} − K , if S_{0}d^{i}u^{n−i} > K

C_{i} = 0 , if S_{0}d^{i}u^{n−i} < K .

We just derived the Discrete Black-Scholes Formula (Alias the Cox-Ross-Rubinstein Formula) The fair price of the option is

∑_{i;S_{0}d^{i}u^{n−i} > K} \binom{n}{i} α^{n−i}β^{i} (S_{0}d^{i}u^{n−i} − K) .

Defining (the partial sum of the binomial theorem)

B(n, m, p) := ∑_{i=0}^{m} \binom{n}{i}p^{i}(1 − p)^{n−i} ,

this can be expressed, more succinctly as

S_{0} · B(n, m, βd) − K · B(n, m, β) ,  (CCR)

where m is the solution of S_{0}d^{m}u^{n−m} = K, and β = \frac{u − 1}{u−d}. Taking the limit Going back to the ‘continuous’, Black-Scholes, scenario, if the expiration date is T, then chop it into n periods, where n is very very big. Each of the steps has duration

Δt = \frac{T}{n} .

For a stock of ‘volatility’ σ, it turns out that, for very large n, u = 1 + σ\sqrt{Δt} , d = 1 − σ\sqrt{Δt} .

Computing β := \frac{u−1}{u−d} = \frac{1}{2}, plugging into (CCR), and taking the limit n → ∞, using the fact, due to de Moivre (see [Z] for an elementary proof) that

B(n, np + \sqrt{np(1 − p)}x) → Φ(x) , with p = βd = d/2 for the first term, and with p = β = \frac{1}{2} for the second term, of (CCR) one immediately gets (using Maple) the Black-Scholes equation (BS) (with r = 0) displayed at the beginning of this article.

References

[BS] Fischer Black and Myron Scholes, The pricing of options and corporate liabilities, The journal of political economy (1973), 637-654.

[CCR] John C. Cox, Stephen A. Ross, and Mark Rubinstein, Option pricing: A simplified approach, Journal of financial Economics 7(3)(1979), 229-263.

[E] Alison Etheridge, “A course in Financial Calculus”, Cambridge University Press, 2002.

[Z] Doron Zeilberger, The Automatic Central Limit Theorems Generator (and Much More!), “Advances in Combinatorial Mathematics: Proceedings of the Waterloo Workshop in Computer Algebra 2008 in honor of Georgy P. Egorychev”, chapter 8, pp. 165-174, (I.Kotsireas, E.Zima, eds. Springer Verlag, 2009.)

http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/georgy.html

Doron Zeilberger, Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen Rd., Piscataway, NJ 08854-8019, USA.

Email: zeilberg at math dot rutgers dot edu .

Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger.

May 14, 2015