Portfolio Theory (Risk)

def. Risky Portfolio. A portfolio contains a risk factors (multivariate random variable) $\overrightarrow{Z_t}$, and has a total value $V_t$, changing along indexed time $t$. The measurable function $f$ is the map from risk factors and time:

$$(Z_t,t)\stackrel{f}{\mapsto }{V}_t$$

Consider $Z_t,V_t$ a Stochastic Process $\{Z_t{\}}_{t=0,1,\dots },\{V_t{\}}_{t=0,1,\dots }$ with adapted filtration $\{{\mathcal{F}}_t{\}}_t$. We then consider the changes to this porfolio:

$$\begin{array}{rl}\text{Change }{X}_{t+1}& :=Z_{t+1}-Z_t\\ \text{Loss }{L}_{t+1}& :=-[\underset{\text{ future value }}{\underbrace{f(t+1,z_t+X_{t+1})}}-\underset{\text{ current value }}{\underbrace{f(t,z_t)}}]\end{array}$$

where $z_t$ is the realized value of $Z_t$ at time $t$ information ${\mathcal{F}}_t$ def. First-order Approximation of Risky Portfolio wrt to risk factors in vector $Z_t$ is:

$$\stackrel{\Delta }{L_{t+1}}:=-\left(\underset{\text{ time derivative}}{\underbrace{f_t^{(1)}{|}_{t,z_t}}}+\sum _{\forall i}\underset{\text{ risk factor derivative }}{\underbrace{f_{z_i}^{(1)}{|}_{t,z_t}{X}_{t+1,i}}}\right)$$

The goal is then to:

1. Model the distribution of $X$, the risk factors
2. Analytically solve the distribution of $L$ or linear approximation $\stackrel{\Delta }{L}$ Example 1. Stock portfolio with underlyers $\{S_{t,i}{\}}_{t\in \mathbb{N}}$ with each ${\lambda }_i$ positions, then:

• Risk factors: $Z_{t,i}=e^{S_{t,i}}$, log prices per custom
• Loss: $L_{t+1}=-{\sum }_{\forall i}{\lambda }_i\cdot S_{t,i}\cdot (e^{X_{t+1,i}}-1)$
• Linear Loss: $L_{t+1}^{\Delta }=-V_t{\sum }_{\forall i}{w}_{t,i}{X}_{t+1,i}$
  • where weights $w_{t,i}:=\frac{{\lambda }_i{S}_{t,i}}{V_t}$ Modeling $X$ as mean vector $\mu$ and covariance matrix $\Sigma$:

$$\begin{array}{r}\mathbb{E}[\stackrel{\Delta }{L_{t+1}}]=-V_t\underset{\text{ dot }}{\underbrace{w^{\top }\mu }}\\ \text{Var}[\stackrel{\Delta }{L_{t+1}}]=V_t^2{w}^{\top }\Sigma w\end{array}$$

Easy! Example 2. European Call Option. Recall the Black Scholes European Option Pricing Formula. We take the rate $r$, volatility $\sigma$ and underlyer $S_t$ as the risk factors, pack them into vector $Z_t=(\ln S_t,r,\sigma {)}^{\top }$. The option price is denoted $C$. Then: Modeling risk factor changes:

$$X_{t+1}=\left[\begin{array}{c}\ln \frac{S_{t+1}}{S_t}\\ r_{t+1}-r_t\\ {\sigma }_{t+1}-{\sigma }_t\end{array}\right]$$

Thus deriving linearized loss:

$$\stackrel{\Delta }{L_{t+1}}=-(\underset{\text{ "Theta" }}{\underbrace{C_t^{(1)}}}+\underset{\text{ "Delta" }}{\underbrace{C_{S_t}^{(1)}}}{S}_t{X}_{t+1,1}+\underset{\text{ "Rho" }}{\underbrace{C_r^{(1)}}}{X}_{t+1,2}+\underset{\text{ "Vega" }}{\underbrace{C_{\sigma }^{(1)}}}{X}_{t+1,3})$$

where each partial derivative has a special name. Example 3. Loan Portfolio with default risk. We consider giving out a bunch of loans to different people at time $t$, and maturity $t+1$:

• Principal $k_i$
• PV of principal = Exposure $e_i:=\frac{k_i}{1+r}$
• Defaults or not: RV $Y_i$ is $1$ if default, $0$ is solvent
  • ! Default doesn’t mean they don’t pay you, instead they pay you $(1-{\delta }_i)k_i$.
  • ${\delta }_i$ is the “loss ratio” given default
  • Probability of default $p_i$
  • This is the risk factor $Z=(Y_1,Y_2,\dots {)}^{\top }$ For simplifying notation, given default the expected PV, book value is:

$$\mathbb{E}\left[{\text{Loss}}_i|Y_i=1\right]:={\delta }_i{p}_i{e}_i$$

• This is not the value of the loan, since (as we see later) risk-neutrality means there must be a premium on the current price due to default risk Now we have expectation of value, and value:

$$\begin{array}{rl}\mathbb{E}[V_t]& =\sum _{\forall i}(1-{\delta }_i{p}_i)e_i\\ V_{t+1}& =\sum _{\forall i}(\underset{\text{ if solvent }}{\underbrace{(1-Y_i)k_i}}+\underset{\text{ if default }}{\underbrace{Y_i(1-{\delta }_i)k_i}})=\sum _{\forall i}{k}_i(1-{\delta }_i{Y}_i)\end{array}$$

Now, we use expectation for $V_t$ and realized value for $V_{t+1}$, because the former must be calculated at time $t$, and the only thing we know is expectation; the latter is calculated at $t+1$ which means $Y_i$s are all known, thus the realized value is known. Finally:

$$\begin{array}{rl}{L}_{t+1}& =V_t-\frac{V_{t+1}}{1+r}\\ & =\sum _{\forall i}(1-{\delta }_i{p}_i)e_i-\sum _{\forall i}\underset{=e_i}{\underbrace{\frac{k_i}{1+r}}}(1-{\delta }_i{Y}_i)\\ & =\sum _{\forall i}({\delta }_i{e}_i{Y}_i-{\delta }_i{e}_i{p}_i)\end{array}$$

Distribution of Loss from Distribution of Underlyer §

$$\text{Loss }{L}_{t+1}:=-[\underset{\text{ future value }}{\underbrace{f(t+1,z_t+X_{t+1})}}-\underset{\text{ current value }}{\underbrace{f(t,z_t)}}]$$

Recall the definition of loss above. To determine this $L_{t+1}$’s distribution we must

1. Model RV $X_{t+1}$’s distribution
   • Usually historical data is used
2. Model RV (=mapping function) $f(t+1,z_t+X_{t+1})$’s distribution from it
   • Valuation models, like the BSM are used In general we have three methods to do this:

1. Analytical Method §

• Choose $X_{t+1}$ and $f$ such that $L_{t+1}$ is analytically tractable
  • Assume $\stackrel{\Delta }{L_{t+1}}$ is a good enough approximation
  • Assume $X_{t+1}$ is a Normal Distribution Thus we will have a general form of:

$$\stackrel{\Delta }{L_{t+1}}=-\left(c_t+\frac{\partial b}{\partial t}{X}_{t+1}\right)$$

comparing to the example 1 above, the constant term $c_t=0$, $b$ are the weights

2. Historical Simulation §

3. Monte Carlo Simulation §