Naked, Covered and Stop-Loss strategies

We now turn to the above example, investigating what kind of [naive] strategies the financial institution can follow. Apparently, the institution can adopt what is called a naked position, which just means doing nothing. When the call expires, there are two cases:

• The price exceeds the strike price, $S_{T}>K$. Then, the institution know that the call will be exercised, and they will have to buy shares at $S_{T}$ in order to sell them immediately at $K$. They will realize a loss of $N\left( S_{T}-K\right)$, with present value $e^{-rT}N\left( S_{T}-K\right)$. This loss can well be substantially higher than the $\pounds 300k$ they received; for instance, if $S_{T}=\pounds 60$, the present value of the loss is $\pounds 981k$, with a net loss of $\pounds 681k$.

• The price is below the strike price, $S_{T}<K$. Then, the call will not be exercised and the profit for the institution is $\pounds 300k$.

The present value payoffs of this strategy are as expected the ones provided by just selling a call, as shown in the first part of figure 8.1

Figure 8.1: Payoffs of the Naked and Covered positions of the example discussed in section 8.1. The values of profits and losses are in thousands.

On the other hand, the institution can adopt a covered position. In this case they buy $100k$ shares as soon as they write the call option contract. For these shares they will have to pay $\pounds 4.9m$, the institution starts with a loss of $\pounds 4.6m$. Now the two cases above become:

• If $S_{T}>K$, then they deliver the shares that they already own. They are going to receive $\pounds 5m$, with present value $\pounds 4.90477m$. In addition they have already sustained an initial loss of $\pounds 4.6m$, leaving them with a fixed payoff with present value $\pounds 304.77k$.

• If $S_{T}<K$, the option will not be exercised. The institution will loose due to the difference $K-S_{T}$, which can be substantial. For example, if the stock price is $S_{T}=\$40$, then the present value of the loss will be $\pounds 981k$. The present value of the income is as before $%% \pounds 304.77k$, leaving a net loss of $\pounds 676k$.

The institution has in fact replicated a put option, the present value payoffs of this strategy are shown in the second part of figure 8.1

In fact, the above methodology gives rise to an important relationship, called the put-call parity, formally given as

$$C\left( S_{0},X,r,T,\sigma \right) +e^{-rT}K=P\left( S_{0},X,r,T,\sigma \right) +S_{0}$$.

which we are going to discuss a bit later in the lecture.

Apparently, the above strategies do not offer any satisfactory hedge. In both cases, the writer of the option has the chance to face extreme losses. If a perfect hedge were to be constructed, it should offer a fixed payoff and not one that varies across moneyness levels at expiration.

If the BS assumptions were correct, and the model generating the stock price was a geometric Brownian motion, then the price of one call is calculated to be $\pounds 2.402$; a perfect hedge would therefore ensure a payment of $%% \pounds 240k$, no matter what the terminal price is [this comes of course from the fact that the market is complete and the option can be replicated].

Another strategy that one would employ, would be buying the stock as soon as it reaches the strike price of $\pounds 50$ and sell it as soon as it drops below the strike price. In principle this would ensure that one is naked when $S_{t}<K$ and covered when $S_{t}>K$, for all $t\in \left[ 0,T\right]$. The problem of this approach, called a stop-loss strategy, arises from the nature of the Brownian motion. If the price reaches the strike price, say from below, then one cannot tell if it will continue to rise [and therefore buy the stock] or if it will decline again [and therefore do nothing]. It is obvious that one has to choose some value $\varepsilon$, and employ a strategy buying at $K+\varepsilon$ and selling at $%% K-\varepsilon$. As it is, this strategy apparently creates losses. On the other hand, if one tries to set $\varepsilon \rightarrow 0$, it is easily shown that the number of trades required will tend to infinity, making this approach not feasible.