Unified Financial Analysis Risk & Finance Lab

© Brammertz Consulting, 2009 1Date: 20.04.23

Unified Financial AnalysisRisk & Finance Lab

Chapter 4: Market Risk Factors

Willi Brammertz / Ioannis Akkizidis


Agenda

> Market Risk factors overview

> Some important terms

> Market risk factors in different analysis modes


Risk factors

> Contract evens are state contingent!


Double role of market risk factors

> Market risk factors determine expected cash flows

> Market risk factors used for discounting (interest rates, FX, stock indices, ....)


Main categories

1. Yield curves

1. Term structures

2. Product rates

2. FX rates

3. Stock prices and indices

4. Commodity prices and indices


Agenda





Arbitrage free

> There is no free lunch

> There is no possibility to make systematic gain without taking risk

(1 + r1)(1 + r1,2) = (1 + r2)2


Risk neutral and risk free

> Risk free

> Risk does not exist

> This is generally not the case

> A risk free world can be constructed in some cases (call/put)

> Risk neutrality

> Risk does not matter (in decisions)

> Risk has a zero prize

> In a risk free world, risk neutrality applies


> A coin of unit 1 is tossed until head appears

> The payoff is 2^(i-1) if the first i-1 tosses were tails

> How much are people ready to pay for the game?

> How much should the fair pay-off be?

St. Petersburg paradox

= ∞


Another example from financeWhy do smiles exist?

>Implied volatilities tweak probabilities (σ, EQ) in order to get observed prices using risk free rates (same function as spreads)

Forward Strike

ImpliedVolatility


Another exampleForward rates and forecasts

> Forward rates are based on arbitrage considerations

> Example: (1 + r1)(1 + r1,2) = (1 + r2)2

> Predominant upward slope of yield curves embed upward bias

> Possible explanation: Mismatch between liquidity preferences and long term un-certainty

> “Risk free” yield curves contain some risk premium

> Arbitrage free models <> unbiased forecast


Keynes on risk

It is safe to say that enterprise which depends on hopes stretching into the future benefits the community as a whole. But individual initiatives will only be adequate when reasonable calculation is supplemented and supported by animal spirits, so that the thought of ultimate loss which often overtakes pioneers, as experienced undoubtedly tells us and them, is put aside as a healthy man puts aside the expectation of death. (The Theory of Money …. J.M Keynes)


Conclusions on risk

> The world is not risk neutral

> People prefer low but more likely gains to high but very unlikely pay-offs

> If people take risks, they want to be paid for it

> Nevertheless the bulk of finance (theory) is within a risk-neutral setting

> Risk neutrality is primarily reflected within market evolutions models

> Risk neutrality is achieved by arbitrary “tweaking” of parameters within models

> Risk neutrality is convenient: Easy mathematical solutions


Agenda





Different types of market modelling

15

Risk neutrality assumptionArbitrage free models

Risk is assumedReal world models


Static models (type I)

A yield curve and its derivations

> Zero coupons

> Discount factors

> Forward rates

is sufficient in a static risk-free world with linear instruments

> Border cases

> BS model for options

> Parametric VaR


Yield curves

> Interest rates and yield curves take the center stage of finance

> Yield curves solve the time problem of finance

> Yield curves are a (very useful) abstraction

> Only bond prices observable

> Yield curves represent par bonds

> Yield curves: most complex market objects

> Spreads to correct for specific risk


Some important yield curve properties

> Zero and coupon bearing rates

> Forward rates

(1 + r1)(1 + r1,2) = (1 + r2)2

> Forward and short rates

> Yield curves and discount factors


Stress tests – stressed models (type II)

Same like type I but

> Initial position calculated with actual market conditions

> Stress applied on actual market conditions

> Position recalculated with stressed conditions

> Stressed VaR

> Stress initial conditions

> Yield curves, FX-rates

> Volatilities

> Apply standard VaR techniques


Type IIILinear and non-linear instruments

Complex non-linear instruments need a distribution of the underlying dynamic risk factors paths for valuation

Simple instruments can be solved analytically


Stochastic arbitrage free models (type III)Stocks


Stochastic arbitrage free models (type III)Yield curves: General considerations

> Positivity

> Mean reversion

> No-arbitrage

> Volatility term structure

> Tractability


The Libor Market Model

> Arbitrage free term structure model

> Calibrated to

> Cap/Floor volatility term structure (LFM)

> Swaption term structure (LSM)

> Thanks to calibration to volatility term structures widely used in the interest rate option field

> Consistency with BS formulas for caps/floors and swaptions

> Although the two versions (LFM/LSM) are not compatible among each other

> Most widely accepted model thanks to consistency with BS


From type III to type VAnother important property

> Forward rates are not economic forecasted rates

> Forward rates are biased if taken for forecasting


Risk neutral and real world modelling

Markets

Counter-parties BehaviorContracts

Real world models

Arbitrage free models


Real world models (Type V)

Real world models are the models a typical economist would construct

> What-If

> Historical

> Monte Carlo Scenarios (e.g. Ornstein Uhlenbeck)

> Mean

> Variance

> Mean reversion

Taking all economic knowledge into account (Mi, GDP, Unemployment etc.)


Real world problems: Spreads

•Spreads are mini-yield curves (same mathematical treatment)•Spreads to model risk aversion (non risk neutral behavior)


Real world problems: Product rates

Product rates are eminently important for the profitability of the financial sector.

Product rates are interest rates, that do not follow interest movement in a 1:1 fashion. For example saving and current account rates.


Parallel valuation techniques

Arbitrage free models and real world models must coexist (example : determination of value within dynamic simulation)

Needs strict separation of cash flow generation and discounting!

For a „real“ real world case see chapter 17.4


Synopsis of valuation techniques

Unified Financial Analysis Risk & Finance Lab

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