Market Risk

Market risk considers the following set of risks to a portfolio:

• Equity risk
• Interest rate risk
• currency risk
• Commodity risk
• margining risk

Fundamental Review of the Trading Book is the regulation that require banks to have certain about of capital (for cushion in case of statistics)

Terminology §

• Value at Risk (VaR): investment

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Continuing from Portfolio Theory (Risk) def. Time series. Some of the time series like risk changes can be continuous, but we want to model discrete time-step changes. So we turn the continuous time $t$ into:

$${\tau }_t:=\underset{\text{ index }}{\underbrace{t}}\cdot \underset{\text{ quanta of time }}{\underbrace{\Delta t}}$$

• $\Delta t$ is usually $\frac{1}{365}$ or $\frac{1}{250}$(trading days per year), i.e. timestep of one day. def. Loss Operator. Instead of mapping $g$ from risk factor to portfolio value (as in a portfolio), we’re interested in how risk factor change maps to loss. Thus we define the Loss Operator $l$:

$$l_{[t]}(x):=-(g({\tau }_{t+1},z_t+x)-g({\tau }_t,z_t))$$

• $g:({\tau }_t,Z_t)\to V_t$ but as opposed to $f$ is a discrete time-step function
• $z_t$ is the risk factor vector
• $x$ is the risk factor change at time $t$

Delta Approximation §

We can approximate the loss operator with a first-order Taylor Approximation: def. First-Order (=Delta) Approximation of loss operator: