Stochastic Processes and Financial Mathematics
(part two)
Chapter 16 Application and extension of the Black-Scholes model
In this section we study the issue of how to make enough connections between the Black-Scholes model and reality that it can be used in the process of trading financial derivatives.
16.1 Transaction costs and parity relations
In our definition of the Black-Scholes market we assumed that it was possible to buy and sell, continuously, without any cost to doing so. In reality, there are costs incurred each time a stock is bought/sold, in the form of administrative cost and taxes.
For most contingent claims \(\Phi (S_T)\), the replicating portfolio \(h_t=(x_t,y_t)\) given by Theorem 15.3.1 changes continuously with time. This would mean continually incurring transaction costs, which is not desirable.
Our first idea for dealing with transaction costs comes form the (wishful) observation that it would be nice if we could replicate contingent claims with a constant portfolio \(h_t=(x,y)\) that did not vary with time.
That is, we don’t trade stock for cash, or cash for stock, during \((0,T)\). Note that the hedging portfolios \(h_t=(x_t,y_t)\) that we found in Theorem 15.3.1 are (typically) non-constant, since \(y_t=\frac {\p F}{\p s}\) will generally not be constant. Worse, Corollary 15.3.3 showed that these hedging portfolios were unique: there are no other self-financing portfolio strategies, based only on cash and stock, which replicate \(\Phi (S_T)\).
A possible way around this limitation is to allow ourselves to hold portfolios that include options, as well as just cash and stock. Let us illustrate this idea with European call/put options. First, we need some notation. Given a contingent claim \(\Phi (S_T)\) with exercise date \(T\) we write \(\Pi _t(\Phi )\) for the price of this contingent claim at time \(t\), and \(h^{\Phi }_t=(x^\Phi _t,y^\Phi _t)\) for its replicating portfolio.
In the case of European call/put options, the key to finding constant replicating portfolios is the ‘put-call parity relation’, which we’ll now introduce (although we have touched on the discrete time version of it in exercise 5.4). Let \(\Phi ^{call}\) and \(\Phi ^{put}\) denote the contingent claims corresponding respectively to European call and put options, both with strike price \(K\) and exercise date \(T\). Let \(\Phi ^{stock}\) and \(\Phi ^{cash}\) denote the contingent claims corresponding respectively to (the value at time \(T\) of) a single unit of stock, and a single unit of cash. Thus we have
\(\seteqnumber{0}{16.}{0}\)\begin{align*} \Phi ^{cash}(S_T)&=1\\ \Phi ^{stock}(S_T)&=S_T\\ \Phi ^{call}(S_T)&=\max (S_T-K,0)\\ \Phi ^{put}(S_T)&=\max (K-S_T,0). \end{align*} The put-call parity relation states that
\(\seteqnumber{0}{16.}{0}\)\begin{equation} \label {eq:put-call} \Phi ^{put}(S_T)=\Phi ^{call}(S_T)+K\Phi ^{cash}(S_T) -\Phi ^{stock}(S_T). \end{equation}
This can be verified from the definitions of the functions, and considering the two cases \(S_T\geq K\) and \(S_T\leq K\) separately. Doing so is left for you, in exercise 16.1. It follows (strictly speaking, using the result of exercise 15.7) that if we buy a portfolio, at time \(0\), consisting of
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• one European call option (with strike \(K\) and exercise \(T\)),
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• \(Ke^{-rT}\) units of cash,
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• minus one units of stock,
then at time \(T\) this portfolio will have the same payoff as a European put option (with strike \(K\) and exercise \(T\)).
We could rearrange (16.1) into the form \(\Phi ^{call}(S_T)=\ldots \) and carry out the same procedure for a call option. So, we learn that if we allow ourselves to hold portfolios containing options, we can hedge European call/put options using constant portfolios. It’s possible to hedge other types of contract too:
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Example 16.1.2 Consider a contract with contingent claim
\[ \Phi ^{straddle}(S_T) = |S_T-K| = \begin {cases} K-S_T & \text { if }S_T\leq K\\ S_T-K & \text { if }S_T>K. \end {cases} \]
This type of contingent claim is known as a straddle, with strike price \(K\) and exercise time \(T\). It is easy to see that the parity relation
\[\Phi ^{straddle}(S_T)=\Phi ^{put}(S_T)+\Phi ^{call}(S_T)\]
holds. Hence, we can hedge a straddle by holding a portfolio of one call option, plus one put option, with the same strike price \(K\) and exercise date \(T\).
There is a drawback here. This hedging strategy requires that we purchase calls and puts (whenever we like), based on the stock \(S_t\) with a strike \(K\) and a exercise dates \(T\) of our own choosing. For calls/puts and equally common types of derivative, this is, broadly speaking, possible. For exotic types of option, even if we can find a suitable relation between payoffs in the style of (16.1) the derivative markets are often less fluid and the hedging portfolio we wish to buy may simply not be available for sale.
A ‘next best’ approach, for general contingent claims (of possibly exotic options), is to try and approximate a general contingent claim \(\Phi (S_T)\) with a constant hedging portfolio consisting of cash, stock and a variety of call options with a variety of strike prices and exercise times. It turns out that this approximation is possible, at least in theory, with arbitrarily good precision for a large class of contingent claims. The formal argument, which we don’t include in this course, relies on results from analysis concerning piecewise linear approximation of functions. Unfortunately, in most cases a large number/variety of call options are needed, and this greatly increases the transaction cost of just buying the portfolio at time \(0\).
In short, there is no easy answer here. Transaction costs have to be incurred at some point, and there is no ‘automatic’ strategy that is best used to minimize them.