Financial Products and Markets

1. Financial Products and Markets Lecture 7

2. Risk measurement • The key problem for the construction of a risk measurement system is then the joint distribution of the percentage changes of value r1, r2,…rn. • The simplest hypothesis is a multivariate normal distribution. The RiskMetrics™ approach is consistent with a model of “locally” normal distribution, consistent with a GARCH model.

3. Value-at-Risk • Define Xi = riciP(t,ti) the profit and loss on bucket i. The loss is then given by –Xi. A risk measure is a function (Xi). • Value-at-Risk: VaR(Xi) = q(–Xi) = inf(x: Prob(–Xi x) > ) • The function q(.) is the  level quantile of the distribution of losses (Xi).

4. VaR as “margin” • Value-at-Risk is the corresponding concept of “margin” in the futures market. • In futures markets, positions are marked-to-market every day, and for each position a margin (a cash deposit) is posted by both the buyer and the seller, to ensure enough capital is available to absorb the losses within a trading day. • Likewise, a VaR is the amount of capital allocated to a given risk to absorb losses within a holding period horizon (unwinding period).

5. VaR as “capital” • It is easy to see that VaR can also be seen as the amount of capital that must be allocated to a risk position to limit the probability of loss to a given confidence level. VaR(Xi) = q(–Xi) = inf(x: Prob(–Xi x) > ) = inf(x: Prob(x + Xi > 0) > ) = = inf(x: Prob(x + Xi  0) 1 – )

6. VaR and distribution • Call FX the distribution of Xi. Notice that • FX(–VaR(Xi)) = Prob(Xi –VaR(Xi)) = Prob(– Xi >VaR(Xi)) = Prob(– Xi > F–X –1()) = Prob(F–X (– Xi ) > ) = 1 –  • So, we may conclude Prob(Xi  –VaR(Xi)) = 1 – 

7. VaR in a parametric approach • pimarking-to-market of cash flow i ri, percentage daily change of i-th factor Xi, profits and losses piri • Example: ri has normal distribution with mean i and volatility i, Take  = 99% Prob(rii–i 2.33) = 1% If i = 0, Prob(Xi = ri pi–i pi 2.33) = 1% VaRi = i pi 2.33 = Maximum probable loss (1%)

8. VaR methodologies • Parametric: assume profit and losses to be (locally) normally distributed. • Monte Carlo: assumes the probability distribution to be known, but the pay-off is not linear (i.e options) • Historical simulation: no assumption about profit and losses distribution.

9. VaR methodologies • Parametric approach: assume a distribution conditionally normal (EWMA model ) and is based on volatility and correlation parameters • Monte Carlo simulation: risk factors scenarios are simulated from a given distributon, the position is revaluated and the empirical distribution of losses is computed • Historical simulation: risk factors scenarios are simulated from market history, the position is revaluated, and the empirical distribution of losses is computed.

10. Value-at-Risk criticisms • The issue of coherent risk measures (aximoatic approach to risk measures) • Alternative techniques (or complementary): expected shorfall, stress testing. • Liquidity risk

11. Coherent risk measures • In 1999 Artzner, Delbaen-Eber-Heath addressed the following problems • “Which features must a risk measure have to be considered well defined?” • Risk measure axioms: ·  Positive homogeneity: (X) = (X) · Translation invariance: (X + ) = (X) –  · Subadditivity: (X1+ X2) (X1) + (X2)

12. Flaws of VaR • Value-at-Risk is the quantile corresponding to a probability level. • Critiques: • VaR does not give any information on the shape of the distribution of losses in the tail • VaR of two businesses can be super-additive (merging two businesses, the VaR of the aggregated business may increase • In general, the problem of finding the optimal portfolio with VaR constraint is extremely complex.

13. Expected shortfall • Expected shortfall is the expected loss beyond the VaR level. Notice however that, like VaR, the measure is referred to the distribution of losses. • Expected shortfall is replacing VaR in many applications, and it is also substituting VaR in regulation (Base III). • Consider a position X, the extected shortfall is defined as ES = E(X: X VaR)

14. Elicitability • A new concept is elicitability, that means that there exists a function such that one can measure whether a measure is better then another. • In other words, a measure is elicitable if it results from the optimization of a function. For example, minimizing a quadratic function yields the mean, while minimizing the absolute distance yields the median. • Surprise: VaR is elicitable, while ES is not. • A new class of measures, both coherent and eligible? Expectiles!

15. Economic capital and regulation • Since the 80s the regulation has focussed on the concept of economic capital, defined as the distance between expected value of an investment and its VaR. • In Basel II and Basel III the banks are required to post capital in order to face unexpected losses. The capital is measured by VaR • In Solvency II and Basel IV VaR will be substituted by expected shortfall.

16. Non normality of returns • The assumption of normality of of returns is typically not borne out by the data. The reason is evidence of • Asimmetry • Leptokurtosis • Other casual evidence on non-normality • People make a living on that, so it must exist • If nornal distribution of retruns were normal the crash of 1987 would have a probability of 10–160, almost zero…

17. Why not normal? Options… • Assume to have a derivative sensitive to a single risk factor identified by the underlying asset S. • Using a Taylor series expansion up to the second order

18. Why non-normal? Leverage… • One possible reason for non normality, particularly for equity and corporate bonds, is leverage. • Take equity, of a firm whose asset value is V and debt is B. Limited liability implies that at maturity Equity = max(V(T) – B, 0) • Notice that if at some time t the call option (equity) is at the money, the return is not normal.

19. Why not normal? Volatility • Saying that a distribution is not normal amounts to saying that volatility is constant. • Non normality may mean that variance either • Does not exist • It is a stochastic variable

20. Dynamic volatility • The most usual approach to non normality amounts to assuming that the volatility changes in time. The famous example is represented by GARCH models ht =  +  shock2t-1 +  ht -1

21. Arch/Garch extensions • In standard Arch/Garch models it is assumed that conditional distribution is normal, i.e. H(.) is the normal distribution • In more advanced applications one may assume that H be nott normally distributed either. For example, it is assumed that it be Student-t or GED (generalised error distribution). Alternatively, one can assume non parametric conditonal distribution (semi-parametric Garch)

22. Volatility asymmetry • A flow of GARCH model is that the response of the return to an exogenous shock is the same no matter what the sign of the shock. • Possible solutions consist in • distinguishing the sign in the dynamic equation of volatility. Threshold-GARCH (TGARCH) ht =  +  shock2t-1 +  D shock2t-1 +  ht -1 D = 1 if shock is positive and zero otherwise. • modelling the log of volatility (EGARCH) log(ht ) =  + g (shockt-1 / ht -1 )+  log(ht -1 ) with g(x) = x + ( x - E(x )).

23. High frequency data • For some markets high frequency data is available (transaction data or tick-by-tick). • Pros: possibility to analyze the price dynamics on very small time intervals • Cons: data may be noisy because of microstructure of financial markets. • “Realised variance”: using intra-day statistics to represent variance, instead of the daily variation.

24. Subordinated stochastic processes • Consider the sequence of log-variation of prices in a given price interval. The cumulated return R = r1 + r2 +… ri + …+ rN is a variable that depends on the stoochastic processes a) log-returns ri. b) the number of transactions N. • R is a subordinated stochastic process and N is the subordinator. Clark (1973) shows that R is a fat-tail process. Volatility increases when the number of transactions increases, and it is then correlated with volumes.

25. Stochastic clock • The fact that the number of transactions induces non normality of returns suggest the possibility to use a variable that, changing the pace of time, could restore normality. • This variable is called stochastic clock. The technique of time change is nowadays one of the most used tools in mathematical finance.

26. Implied volatility • The volatility that in the Black and Scholes formula gives the option price observed in the market is called implied volatility. • Notice that the Black and Scholes model is based on the assumption that volatility is constant.

27. The Black and Scholes model • Volatility is constant, which is equivalent to saying that returns are normally distributed • The replicating portfolios are rebalanced without cost in continuous time, and derivatives can be exactly replicated (complete market) • Derivatives are not subject to counterpart risk.

28. Beyond Black & Scholes • Black & Scholes implies the same volatility for every derivative contract. • From the 1987 crash, this regularity is not supported by the data • The implied volatility varies across the strikes (smile effect) • The implied volatility varies across different maturities (volatility term structure) • The underlying is not log-normally distributed

29. Smile, please!

34. Trading strategies with options • Trade the skew: betting on a reduction of the skewness = flattening of the smile • Trade of the fourth moment: betting on a decrease of out and in the money options and increase of the at-the-money options. • Volatility surface: change of volatility across strike prices and maturities.