22.2 Portfolio Strategies

The portfolio of an investor at time $ t$, i.e., the market value of the single equities (contracts) in his portfolio at time $ t$, is dependent on the development of the price $ \{$$S$$ _s;s<t\}$, $S$$ _s= (S^1_s,\ldots,S^d_s)^\top \in {\mathbb{R}}^d$ up to time $ t$, that is, on the information that is available at that particular time point. Given this it is obvious that his strategy, i.e., the development of his portfolio's value over time, should also be modelled as a $ {\cal{F}}_t$ adapted $ d$-dimensional stochastic process $ \phi_t$. In doing so $ \phi^i_t(\omega)$ represents how much is the security $ i$ in his portfolio at time $ t$ in state $ \omega$, where negative values indicate a short sell of the corresponding contract.

Definition 22.4  
Assume the following market model: $ {\cal M} =
(\Omega,{\cal{F}},{\P}, {\cal{F}}_t,$   $S$$ _t)$. A $ d$-dimensional stochastic process $ \phi_t$ adapted on the filtration $ {\cal{F}}_t$ is called a portfolio strategy. The stochastic process $ V(\phi_t)$ with $ V(\phi_t)\stackrel{\mathrm{def}}{=}\sum\limits_{i=1}^d
\phi^i_t S_t^i$ is called the value of the strategy $ \phi.$

Example 22.3   In the Black-Scholes model two financial instruments are traded on the market: a risky security $ S$ (stock) and a riskless security $ B$ (zero bond). As in Chapter 5, the stock price $ S_t$ is assumed to follow a geometric Brownian motion, so that the following stochastic differential equation is satisfied:

$\displaystyle dS_t = S_t(\mu dt + \sigma dW_t)$ (22.2)

The price of the zero bond $ B_t$ satisfies the differential equation:

$\displaystyle dB_t = rB_t dt $

with a constant $ r$. Without loss of generality it can be assumed that $ B_0=1$, which leads to $ B_t = {\rm e}^{rt}$.

The corresponding market model is thus $ \mathcal{M}_{BS} =
(\Omega,{\cal{F}},{\P},{\cal{F}}_t,$   $S$$ _t),$ where $S$$ _t
\stackrel{\mathrm{def}}{=}(S_t,B_t)^\top \in {\mathbb{R}}^2.$

The two-dimensional stochastic process $ \phi_t = (a_t,b_t)^\top $ now describes a portfolio strategy in which $ a_t(\omega)$ gives the number of stocks and $ b_t(\omega)$ gives the number of bonds in the portfolio at time $ t$ in state $ \omega$. The value of the portfolio at time $ t$ is then a random variable

$\displaystyle V(\phi_t) = a_tS_t + b_t B_t.$

A particularly important portfolio strategy is that once it is implemented it does not result in any cash flows over time, i.e., when the portfolio is re-balanced no payments are necessary. This means that eventual income (through selling securities, receiving dividends, etc.) is exactly offset by required payments (through buying additional securities, transaction costs, etc.) This is referred to as a self-financing strategy. One gets the impression that the change in value of the portfolio only occurs as the price of the participating securities changes.

Definition 22.5  
Let $ {\cal M} =
(\Omega,{\cal{F}},{\P}, {\cal{F}}_t,$$S$$ _t)$ be a market model and $ \phi$ a portfolio strategy with the value $ V(\phi_t)$. Then $ \phi$ is called
  1. self-financing, when $ dV(\phi_t) = \sum\limits_{i=1}^d \phi^i_t dS^i_t$ holds for $ {\P}$-almost sure,
  2. admissible, when $ V(\phi_t) \geq 0$ holds for $ {\P}$-almost sure.

In the following the Black-Scholes model will be considered. The subsequent specification shows that arbitrage is not possible in such a market: There is no admissible self-financing strategy with a starting value of $ V($$&phis#phi;$$ _0)=0$, whose end value $ V($$&phis#phi;$$ _T)$ is positive with a positive probability.

Lemma 22.1  
In the Black-Scholes model $ \mathcal{M}_{BS} =
(\Omega,{\cal{F}},{\P},{\cal{F}}_t, {\text{\rm\boldmath $\mathit S$}}_t), {\text{\rm\boldmath $\mathit S$}}_t=(S_t,B_t)^\top ,$ the portfolio strategy $ \phi_t = (a_t,b_t)^\top $ is exactly self-financing when the discounted process $ \tilde{V}_t$ with $ \tilde{V}_t = {\rm e}^{-rt}V_t$ satisfies the stochastic differential equation

$\displaystyle d\tilde{V}_t = a_td \tilde{S}_t, $

where $ \tilde{S}_t = {\rm e}^{-rt}S_t$ describes the discounted stock price.

The explicit specification of the corresponding strategy can be left out when it is clear from the context and $ V_t$ can be written as $ V(a,b)_t$. With the help of the Girsanov theorem a $ {\P}$ equivalent measure $ {\rm Q}$ can be constructed, under which the process of the discounted stock prices is a martingale. Using (A.2) one obtains

$\displaystyle d \tilde{S}_t = \tilde{S}_t\{(\mu-r)dt + \sigma dW_t\} .$ (22.3)

By setting

$\displaystyle X_t\stackrel{\mathrm{def}}{=}-\frac{\mu-r}{\sigma}$

the Novikov condition (see Remark A.2) is obviously fulfilled. Therefore, for $ {\rm Q}$ with
$\displaystyle \frac{d{\rm Q}}{d{\P}} = \xi_T$ $\displaystyle =$ $\displaystyle \exp (\int_0^T X_u dW_u-\frac{1}{2}\int_0^T X_u^2du )$  
  $\displaystyle =$ $\displaystyle \exp \{-\frac{\mu-r}{\sigma}W_T
-\frac{1}{2}(\frac{\mu-r}{\sigma})^2T \}$  

$ W^*_t\stackrel{\mathrm{def}}{=}W_t +\frac{\mu-r}{\sigma}t$ is a $ {\rm Q}$-Brownian Motion according to the Girsanov theorem. Because of (A.3) and using the definition of $ W^*_t$ it holds that

$\displaystyle d\tilde{S}_t = \tilde{S}_t \sigma dW^*_t .$ (22.4)

According to Itô's lemma this becomes

$\displaystyle \tilde{S}_t =
\tilde{S}_0 \exp \left(\int_0^t \sigma dW^*_u
- \frac{1}{2}\int_0^t \sigma^2du \right) $

and solves the stochastic differential equation. Since $ \sigma$ is constant, for all $ t$ the Novikov condition holds

$\displaystyle {\mathop{\text{\rm\sf E}}}\left[\exp\left(\int_0^t \sigma^2du \right)\right] < \infty. $

According to Remark A.2

$\displaystyle \exp\left(\int_0^t \sigma dW^*_u-\frac{1}{2}\int_0^t \sigma^2du
\right),$

that is $ \tilde{S}_t$, is also a $ {\rm Q}$-martingale.

$ {\rm Q}$ represents with respect to $ \widetilde S_t$ a $ {\P}$ equivalent martingale measure. It can be shown that given this form, it can be uniquely determined.

From the Definition of $ W^*_t$ and with the help of (A.2) one obtains

$\displaystyle dS_t = S_t(rdt + \sigma dW^*_t),$

i.e., under the measure $ {\rm Q}$ the expected value of the risky securities is equivalent to the certain value of the riskless bonds. Because of this the martingale measure $ {\rm Q}$ is also called the risk neutral probability measure and contrary to this $ {\P}$ is called the objective or physical probability measure of the Black-Scholes markets.

As a result of the $ {\rm Q}$-martingale properties of $ \widetilde S_t$, due to Lemma A.1, the discounted value of a self-financing strategy $ \widetilde V_t$ is itself a local $ {\rm Q}$-martingale. Thus the value of every admissible self-financing strategy that is a non-negative local $ {\rm Q}$-martingale is a $ {\rm Q}$-super martingale. Consequently it holds that: If the starting value of an admissible self-financing strategy is equal to zero, then its value at all later time points $ t$ must also be equal to zero. Thus in using an admissible self-financing strategy, there is no riskless profit to be made: The Black-Scholes market is free of arbitrage.

The following theorem represents the most important tool used to value European options with the help of the Black-Scholes model. It secures the existence of an admissible self-financing strategy that duplicates the option, thus the value of which can be calculated using martingale theory.

Theorem 22.6  
Assume that the Black-Scholes model $ \mathcal{M}_{BS}$ is given. The function $ X$ describes the value of an European option at the time to maturity $ T$ and is
$ {\rm Q}$-integrable.
a)
Then an admissible self-financing strategy $ (a_t,b_t)^\top $ exists, which duplicates $ X$ and whose value $ V_t$ for all $ t$ is given by

$\displaystyle V_t = {\mathop{\text{\rm\sf E}}}_{{\rm Q}}[{\rm e}^{-r(T-t)}X \mid {\cal{F}}_t] .$ (22.5)

b)
If the value $ V_t$ in a) is dependent on $ t$ and $ S_t$ and is written as a function $ V_t = F(t,S_t)$ with a smooth function $ F$, then it holds for the corresponding strategy that

$\displaystyle a_t = \frac{\partial F(t,S_t)}{\partial S_t}. $

Proof:
  1. One defines $ V_t$ by (A.5), where the function is defined follows from the $ {\rm Q}$-integrability of $ X$. Due to

    $\displaystyle \widetilde V_t = {\rm e}^{-rt}V_t = {\mathop{\text{\rm\sf E}}}_{{\rm Q}}[{\rm e}^{-rT}X \mid {\cal{F}}_t] $

    one identifies $ \widetilde V_t$ as $ {\rm Q}$-martingale. One should notice that $ {\rm e}^{-rT}X$, exactly like $ X$, is only dependent on the state at date $ T$ and thus it can be classified as a random variable on $ (\Omega,{\cal{F}}_t,{\rm Q})$.

    $ {\cal{F}}_t$ represents at the same time the natural filtration for the process $ W^*$, which, as was seen above, is also a $ {\rm Q}$-martingale. Therefore, according to Theorem A.5 there exists using the martingale representation a process $ H_t$ adapted on $ {\cal{F}}_t$ with $ \int_0^T H^2_t\sigma^2dt < \infty$ $ {\rm Q}$-almost sure, so that for all $ t$ it holds that:

    $\displaystyle \widetilde V_t = \widetilde V_0 + \int_0^t H_sdW^*_s
= V_0 + \int_0^t H_sdW^*_s .$

    Thus one sets:

    $\displaystyle a_t \stackrel{\mathrm{def}}{=}\frac{H_t}{\sigma \cdot \widetilde ...
...quad ,\quad
b_t \stackrel{\mathrm{def}}{=}\widetilde V_t - a_t \widetilde S_t .$

    Then after a simple calculation it holds that:

    $\displaystyle a_tS_t + b_tB_t = V_t $

    and $ (a_t,b_t)^\top $ is a $ X$ duplicating strategy. Furthermore, with (A.4) it holds for all $ t$:

    $\displaystyle a_t d\widetilde S_t = a_t \widetilde S_t \sigma dW^*_t
= H_t dW^*_t = d\widetilde V_t ,$

    i.e., $ (a_t,b_t)^\top $ is according to Lemma A.1 self-financing. Due to the non-negativity of $ X$ and the definition of $ V_t$, $ (a_t,b_t)^\top $ is also admissible.
  2. For $ V_t = F(t,S_t)$ it holds using Itô's lemma:
    $\displaystyle d\widetilde V_t$ $\displaystyle =$ $\displaystyle d\{{\rm e}^{-rt}F(t,S_t)\}$  
      $\displaystyle =$ $\displaystyle \frac{\partial\{{\rm e}^{-rt}F(t,S_t)\}}{\partial S_t}dS_t + A(t,S_t)dt$  
      $\displaystyle =$ $\displaystyle \frac{\partial F(t,S_t)}{\partial S_t} {\rm e}^{-rt}S_t(rdt+\sigma dW^*_t)
+ A(t,S_t)dt$  
      $\displaystyle =$ $\displaystyle \frac{\partial F(t,S_t)}{\partial S_t} \widetilde S_t \sigma dW^*_t
+ \widetilde A(t,S_t)dt$  
      $\displaystyle =$ $\displaystyle \frac{\partial F(t,S_t)}{\partial S_t} d\widetilde S_t
+ \widetilde A(t,S_t)dt.$  

    Since not only $ \widetilde V_t$ but also $ \widetilde S_t$ are $ {\rm Q}$-martingales, the drift term $ \widetilde A(t,S_t)$ must disappear. According to part a) of the theorem the corresponding strategy is self-financing and thus using Lemma A.1 the claim follows.
$ {\Box}$

Remark 22.4   With the relationships of the preceding theorems $ V_t$ is called the fair price for option $ X$ at date $ t$, because at this price, according to the previous argumentations, there is no arbitrage possible for either the buyer or the seller of the option. Equation (A.5) is called the risk neutral valuation formula, since it gives the fair price of the option as the (conditional) expectation of the (discounted) option value at maturity with respect to the risk neutral measure of the Black-Scholes model.

The result obtained from the last theorem was already formulated in Chapter 6 as equation (6.24).

Corollary 22.1  
The relationships of the preceding theorems hold. If the value $ X$ of the European option at date $ T$ is a function $ X=f(S_T)$ dependent on the stock price $ S_T$, then it holds that $ V_t =
F(t,S_t),$ where $ F$ for $ x\in [0,\infty[$ and $ t\in[0,T]$ is defined by:

$\displaystyle F(t,x) = {\rm e}^{-r(T-t)} \int_{-\infty}^{+\infty} f \left(x{\rm...
...T-t)+\sigma y \sqrt{T-t}}\right) \frac{{\rm e}^{-\frac{y^2}2}}{\sqrt{2\pi}}dy~.$ (22.6)

Proof:
With respect to $ {\rm Q}$, $ S_t$ contains the drift $ r$ and thus it holds that

$\displaystyle S_t = S_0 \exp\{(r-\frac{\sigma^2}{2})t + \sigma W^*_t\} .$

Thus $ S_T$ can be written in the following form:

$\displaystyle S_T = S_t(S_TS_t^{-1})
= S_t \exp\{(r-\frac{\sigma^2}2)(T-t) + \sigma(W^*_T-W^*_t)\} .$

Since $ S_t$ is measurable with respect to $ {\cal{F}}_t$ and $ W^*_T-W^*_t$ is independent of $ {\cal{F}}_t$, one obtains
$\displaystyle V_t$ $\displaystyle =$ $\displaystyle {\mathop{\text{\rm\sf E}}}_{{\rm Q}}[{\rm e}^{-r(T-t)}f(S_T) \mid {\cal{F}}_t]$  
  $\displaystyle =$ $\displaystyle {\mathop{\text{\rm\sf E}}}_{{\rm Q}}\Big[{\rm e}^{-r(T-t)}
f(S_t {\rm e}^{(r-\frac{\sigma^2}2)(T-t) + \sigma(W^*_T-W^*_t)})\mid {\cal{F}}_t\Big]$  
  $\displaystyle =$ $\displaystyle {\mathop{\text{\rm\sf E}}}_{{\rm Q}}\Big[{\rm e}^{-r(T-t)}
f(x {\rm e}^{(r-\frac{\sigma^2}2)(T-t)+\sigma(W^*_T-W^*_t)})\Big]_{x=S_t}$  

From this it can be calculated that $ V_t = F(t,S_t)$. $ {\Box}$

Example 22.4   Consider a European call $ X=\max\{0, S_T-K\}$. Using (A.6) the value at date $ t$ is exactly the value given by the Black-Scholes formula in Chapter 6.

$\displaystyle C(t,S_t)\stackrel{\mathrm{def}}{=}V_t = S_T\Phi(d_1) - K{\rm e}^{-r(T-t)}\Phi(d_2)$

with

$\displaystyle d_1\stackrel{\mathrm{def}}{=}\frac{\ln(\frac{S_T}K)+(r+\frac{\sig...
...rm{def}}{=}\frac{\ln(\frac{S_T}K)+(r-\frac{\sigma^2}2)(T-t)}{\sigma\sqrt{T-t}}.$