The previous article defined the idea of arbitrage and subsequently the assumption of no arbitrage within financial markets. A portfolio consisting of shares of stock, bonds and call options was considered, where the share price was modelled as a discrete random variable using the One-Step Binomial Model. The No-Arbitrage Principle states the initial value of the portfolio must equal the price of the call option *C(0)* under this assumption, i.e.:

This article will prove by method of contradiction that this unique price of *C(0) *satisfies all market assumptions; by assuming a negation of the statement above and showing it reaches a statement that contradicts the No-Arbitrage Principle. The negated statement is then taken as false, implying that the original statement is true.

First assume that *C(0) – yA(0) > xS(0). *As *y < 0* (see previous article), is equivalent to the statement

*C(0) + |y*|*A(0) > xS(0).* Hence, at time *t = 0*, an investor should:

- Issue and sell one option for
*C(0).* - Borrow |
*y*|*A(0)*amount of money. - Purchase
*x*shares of stock for*xS(0).*

The cash amount remaining after these transactions is equal to *C(0) + |y*|*A(0) − xS(0),* which is positive according to the assumption. If the investor invests this amount in a risk-free security, the resulting portfolio has initial value *V(0) = 0. *At time *t = t*, the investor should then:

- Settle the option by paying a if the share price increases to
*S(0) +**a,*or pay nothing if the share price decreases. In general, the cost of the option to the investor is*C(t).* - Repay the loan he borrowed with interest, which will cost |
*y*|*A(t).* - Sell the stock, obtaining
*x(S(0)*+ a) if the share price increases, or*x(S(0) −**a)*if the share price decreases. In general, his cash return from the stock will be*xS(t)*.

The cash balance after these transactions is *xS(t) − *|*y*|*A(t) *− *C(t)*. Regardless of whether the stock increases or not, this quantity equals zero, by comparison with the equation *C(t) = xS(t) + yA(t).* However, the investor is now left with the initial risk-free investment *C(0) + |y*|*A(0) − xS(0), *with added interest. Thus, a portfolio *V *with initial value *V(0) = 0* has resulted in a profitable opportunity, violating the No-Arbitrage Principle. Hence a contradiction is reached, implying that our original assumption that *C(0) – yA(0) > xS(0) *is false.

Now assume that *C(0) – yA(0) < xS(0). *Again, this is equivalent to the statement *C(0) + |y*|*A(0) < xS(0).*

Hence, at time *t = 0*, an investor should:

- Buy one option for
*C(0)*. - Buy |
*y*| bonds for |*y*|*A(0)*. - Short sell
*x*shares of stock for*xS(0)*.

The cash amount remaining after these transactions is equal to *xS(0) − *|*y*|*A(0) − C(0), *which is positive according to the assumption. If an investor invests this amount into a risk-free security, the resulting portfolio has initial value *V(0) = 0. *At time *t = t*, the investor should then:

- Exercise the option if the share price increases to
*S(0) +**a,*receiving a in return, and do nothing if the share price decreases. In general, the income to the investor is*C(t)*. - Sell the bonds for |
*y*|*A(t).* - Close the short position by buying the stock back, paying
*x(S(0) +*a) if the share price increases, or*x(S(0) −**a)*if the share price decreases. In general, the investor must pay*xS(t)*to re-purchase the stock.

The cash balance after these transactions is *C(t) + *|*y*|*A(t) – xS(t).* Again, regardless of whether the stock increases or not, this quantity equals zero. However, the investor is now left with the initial risk-free investment *xS(0) − *|*y*|*A(0) − C(0)*, with added interest. Thus, a portfolio *V *with initial value *V(0) = 0* has resulted in a profitable opportunity, violating the No-Arbitrage Principle. Hence a contradiction is reached, implying that our original assumption that *C(0) – yA(0) < xS(0) *is false.

It follows from the axiom of trichotomy that *C(0) – yA(0) = xS(0)*. Hence *C(0) = xS(0) + yA(0), *completing the proof.

In a simplified One-Step model such as this, when considering a discrete random variable, it is relatively straightforward to deduce the option premium. However, different mathematical methods are needed to solve the option price when the situation is modelled in continuous time. For example, the famous Black-Scholes equation uses ideas from measure theory, stochastic calculus and partial differential calculus to model the price evolution of a call option. Its complexity is a reason why option *trading* is more popular than option pricing among investors. Despite this, the mathematics used to derive the formula is rigorous and worthy in its own right.

By Ayush Bose

Sector Head: Aaron Hobb