The Greeks

In the last few lectures we analyzed the way that derivative contracts should be priced, and examined the assumptions and properties that would lead to arbitrage-free pricing. In this lecture we turn to the use of derivatives [mainly European calls and puts] as instruments to hedge against adverse movement of the market. First we will define the sources that generate changes in the value of the portfolio, and we will examine which of these sources represent some kind of risk that can be hedged. The Greeks of the portfolio will be introduced. Afterwards we will discuss alternative methods of dynamically hedging a given position. All graphs in the first parts are based on the following

Example 44 (Hedging, the setting)   Suppose that an institution has sold for $\pounds 300k$ a European call option on $N=100k$ shares of a [non-dividend paying] stock, or $C=\pounds 3$ the price of each call. Say that at the time of the contract [now] the stock price is $S_{0}=S=\pounds 49$, the strike price is $K=\pounds 50$, the interest rate is $r=5\%$ p.a. [continuously compounded], the stock return volatility is $\sigma =20\%$ p.a., the time to maturity is $T=20$ weeks [or $%% T=0.3846$ years], and the expected return on the stock is $\mu =13\%$ p.a.

At this point, it is important to remember the differential equation that any portfolio or derivative contract, with value $\Pi$, has to satisfy, if one makes the assumption that the underlying asset follows a geometric Brownian motion, namely

$$\frac{\partial \Pi }{\partial t}+rS\frac{\partial \Pi }{\partial ... ...1}{%% 2}\sigma ^{2}S^{2}\frac{\partial ^{2}\Pi }{\partial S^{2}}=r\Pi \text{.}$$

Later on we will define $\Delta _{\Pi }=\frac{\partial \Pi }{\partial S}$, $%% \Theta _{\Pi }=\frac{\partial \Pi }{\partial t}$ and $\Gamma _{\Pi }=\frac{%% \partial ^{2}\Pi }{\partial S^{2}}$, implying that

$$\Theta _{\Pi }+rS\Delta _{\Pi }+\frac{1}{2}\sigma ^{2}S^{2}\Gamma _{\Pi }=r\Pi \text{.}$$

In addition, a Taylor's expansion of the portfolio value would give that

$$\Delta \Pi = \frac{\partial \Pi }{\partial t}\Delta t+\frac{\partial \Pi }{%% ... ...artial S^{2}}%% \left( \Delta S\right) ^{2}+o\left( \Delta t\right) \text{, or}$$

$$\Delta \Pi \approx \Theta _{\Pi }\times \Delta t+\Delta _{\Pi }\times \Delta S+\frac{1}{2}\Gamma _{\Pi }\times \left( \Delta S\right) ^{2}$$