The fundamental assumption of ﬁnancial mathematics theory is the notion of no-arbitrage within markets. This means that no investor can create a profit without any risk or initial investment. If this were to occur, by deﬁnition there exists an arbitrage opportunity, the exploiting of which would create a profitable opportunity. In reality if an arbitrage opportunity existed, market conditions are quickly adjusted to eliminate it. For example, if two institutions are oﬀering diﬀerent exchange rates on the same currency, although a proﬁt could be made by an individual in the short-term (by buying from one and selling to the other), the exchange rates would soon be adjusted to match the demand of the individual institution’s currency. To visualise the No-Arbitrage Principle in a simple market model, first consider a portfolio model displaying a range of investments.
A call option with an exercise price λ is a contract giving the holder the right, but no obligation, to purchase a share of stock for price λ at time t. The option carries a premium, and so it is possible to invest in a portfolio (x, y, z) of x shares of stock, y bonds and z call options.
V (t) denotes the value of the portfolio at time t
S(t) denotes the price of a share at time t
A(t) denotes the price of a bond at time t
C(t) denotes the value of the call option at its exercise time t
Consider a call option with exercise price λ equivalent to the initial price of one share S(0). If the share price increases, then the value of the option at time t, C(t), is equivalent to its payoﬀ S(t) – λ. If the share price decreases, then exercising the option would result in a loss and thus the option has no value; C(t) = 0. Note that the quantities S(0) and A(0) are known to investors, as they are the initial price of a share and a bond. A(t) is also known as the bond has a ﬁxed rate of return. S(t) is unknown, hence using the One-Step Binomial Model, it can be modelled as a discrete random variable that can take only two values, with respective probabilities p and 1 – p, depending on whether the share price increases or decreases.
Ultimately, the task is to price C(0), the option premium, such that it is consistent with the No-Arbitrage Principle. This article demonstrates the steps needed to calculate this price. The subsequent article will prove that this price satisﬁes the No-Arbitrage Principle.
First consider a portfolio by taking a position in x shares and y bonds. Let this equal the payoﬀ of the call option. This is known as replicating the option.
Solving for x and y:
Note x > 0 and y < 0. Therefore, to replicate the option, an investor must buy x shares of stock and borrow | y | amount of money. Using these values of x and y, compute the initial value of our portfolio at time t = 0. This is known as pricing the option.
The claim is that under No-Arbitrage, the initial value of the portfolio must equal the price of the call option, C(0), otherwise an arbitrage opportunity will occur. This will be proven by contradiction in the subsequent article.
By Ayush Bose
Sector Head: Aaron Hobb