Derivatives and Structured Products in Portfolio Management

Transcription

1 Derivatives and Structured Products in Portfolio Management Prof. Massimo Guidolin Advanced Tools for Risk Management and Pricing Spring

2 Motivation The theory and practice of modern asset pricing models offer precise ideas on the economic value of derivatives in ptf. mgmt The concept of efficient portfolio management may in principle help re-define the traditional shyness of asset managers in using derivatives and their asset & liability applications In this lecture, we work on 3 research questions: 1 Is it possible that derivatives may create on an ex-ante basis economic value in ptf. management (and so under what conditions) by improving the risk-return trade-off? o This means an increase in expected «risk-adjusted» performance 2 Does such a contribution also (or especially) hold also expost? 3 What is the link between the economic value of derivatives and the benefits of treating volatility as a separate, additional asset class? 2

3 Definitions and Preliminary Concepts Even though a literature exists that has examined the effects of introducing individual derivatives in ptfs, we shall discuss the expected utility increase of securitized structured products (SSP) o SSPs are often option portfolios themselves Morever, SSPs include in principle also ETF (Exchange Traded Funds), structured mutual funds and especially ETC (Exchance Traded Commodities) and ETN (Exchange Traded Notes) when these imply strong structuring elements E.g., when they are of a «reverse» type (== short) and leveraged However, the baseline case is represented by the investment certificates and covered warrants o Such options may be both «plain vanilla» (European and American style) and exotics In particular, Asian and barrier options o In Italy, SSPs are financial constracts that are subject to regulations both when issued/structured (primary market) and when they are subsequently traded (secondary market) 3

4 Preliminary Concepts: Let s get rid of myths Structured product does not mean risky ( delta > 1) or speculative o In fact, among the most popular SSPs are (totally or conditionally) equity-protected certificates and «reverse» ETFs and ETCs Yet, SSPs may be used to take on additional risks in addition to classical, diffusive risk Structured product does not mean «complex» o SSPs exist that are characterized by very simple and intuitive payoffs For instance, leveraged certificates o The complexity of a SSP would derive mostly from a precise payoff need at maturity (or dynamically, from the need to make payments) o SSPs satisfy needs, they do not create them Structured product does not mean «illiquid» o Certificates are listed on the Milan Stock Exchange (Sedex) or on the Euro TLX; ETFs and ETCs on ETFplus; ETNs btw. ETFplus and Sedex They benefit from market making obligations imposed when issued They are surely more illiquid than the majority of corporate bonds 4

5 Preliminary Concepts: Let s get rid of myths Structured product does not mean expensive o Obviously, structuring services need to be paid for, especially when the SSPs are created to satisfy complex and unique needs o Yet the comparison costs need to be performed not with zero costs (or very low costs as in the case of govvies), but with: 1 The cost that ought to be borne to directly purchase the derivatives that should be used to replicate the payoffs of the SSPs 2 The increase in risk-adjusted performance that a SSP makes available, that as we shall see may be considerable Moreover, a few categories of SSPs, i.e., ETFs, ETCs, and ETNs are well known to imply rather modest costs Structured product does not mean extreme credit risk o Not a risk higher than stocks and bonds from same issuers! Structured product does not mean «opaque» o In principle, one is not considering to delegate parts of ptf. management but to insert relatively liquid, listed securities in it 5

6 What is true in some of the myths? SSPs may be very efficient to take leveraged (short or long) positions and in this case they represent tools to take large risk positions o In fact, with dynamic leverage, this may exceed the issuance level SSPs are often difficult to price and they require expert advisory, at least in support to internal teams The liquidity of SSPs is often supplied by the issuers themselves and hence their own credit risk is interacted with liquidity risk The cost of structuring may be reduced trough auction mechanisms, i.e., placing issuers in competiton with one another; hence understanding such mechanisms is important The SSPs enjoy of sophisticated replication strategies, but after all they are just securitized loan contracts without margin accounts o In this sense, excessive attention by textbooks may make them look like basket products, which they are not 6

7 Generalities on the Economic Value of SSPs SSPs represent excellent «wrappers» of long and short strategies on the volatility of market indices There is pervasive evidence of an inverse correlation between market indices and volatility o Volatility has become an important asset class o SSPs allow both to take positions directly in volatility when this is the underlying asset, or indirectly, as a function of the structure of their payoffs The economic value of SSPs will depend on the degree of market completeness, i.e., of the fact that all sources of risk be diversifiable o In fact, what matters will be the «completability» of markets through trading of derivatives 7

8 The key result Derivatives generate economic value in portfolio managemnt because they uniquely allow one to take positions of correct sign and magnitude vs. the risk factors (diffusion, volatility, and jumps) The «asset pricing» literature has adopted a variety of empirical and mathematical techniques on which factors and market forces that ought to be priced in equilibrium, to assess whether they drive a non-zero equilibrium risk premium o The idea is that the factors with zero risk premium simply increase the variance and therefore they do not belong to the efficient frontier, or equivalently, these just need to be «escaped» (neutralized) This becomes a pure risk management issue, not relevant here o In essence, such factors are: (A) The diffusive, continuous component of a price (or wealth) process (B) The randomness of the volatility of such a process (C) The potential presence of jumps This is instead the discontinuous component of the process 8

9 The key result Derivatives generate economic value in portfolio managemnt because they uniquely allow one to take positions of correct sign and magnitude vs. the risk factors (diffusion, volatility, and jumps) o Also the (negative) correlation between the diffusive and stochastic volatility components plays a key role, although it fails to represent per se an additional risk factor o Other factors have been occasionally isolated and discussed in the literature, but their role (especially their premia) are less evident For instance, the skewness and kurtosis: yet these derive from stochastic volatility and the presence of jumps Alternatively, the presence of stochastic regimes in the «intensity» of the drift process, of stochastic volatility, etc. The presence of jumps in stochastic volatility The presence of stochastic size and «intensity» of jumps The presence of co-jumps when different assets are modelled o I will introduce models that map such expositions in the ptf. weights even though the objective of all models is to determine exposures 9

10 A model of «completable markets» Markets are completable when introducing an appropriate and finite number of derivative securities makes markets complete When markets are complete then: 1 All state-contigent pay-off profiles (that is, that depend on the state of the world underlying) may be replicated (i.e., created) through an appropriate portfolio o An asset manager can creates all payoffs «he wantes» 2 If multiple SSPs exist that lead to market completion, they will all yield identical (and positive) economic value o This is because such a value derives from completion, i.e., the possibility that a SSPs gives to make some payoff profiles possible If markets are incomplete and remain so in spiete of the SSPs, then the economic value of different derivatives may be different It is possible (necessary?) to investigate which structuring profiles maximize the improvement in risk-adjusted performance 10

11 A model of «completable markets» Consider a simple «off-the-shelf» model with stochastic volatility à la Heston (1993): o The primitive securities are a riskess bond that pays the constant risk free rate r and a risky asset, identified with a market index o The investor may also include derivatives in her portfolio o Such derivatives yield an exposure to the risks B and Z that differs across stocks and bonds because their payoff is (potentially) nonlinear o In fact, such a nonlinearity may provide market completion o The derivative is the function O t =g(s t,v t ) and it may be very complex o Pricing is performed on the basis of a convenient «pricing kernel» 11

12 A model of «completable markets» Consider a simple «off-the-shelf» model with stochastic volatility à la Heston (1993): Price return Change in variance Risk premium on diffusive risk Diffusive shocks Volatility shocks Vol mean reversion Long-run variance Vol of vol Correlation btw. diffusive & volatility shocks o The primitive securities are a riskess bond that pays the constant risk free rate r and a risky asset, identified with a market index o The investor may also include derivatives in her portfolioderivatives yield an exposure to the risks B and Z that differs across stocks and bonds because their payoff is (potentially) nonlinear o In fact, such a nonlinearity may provide market completion o The derivative is the function O t =g(s t,v t ) and it may be very complex o Pricing is performed on the basis of a convenient «pricing kernel» 12

13 A model of «completable markets» The investor has initial wealth W₀ and she solves a standard problem of expected utility maximization: % in risk index Coefficient of relative risk aversion % in structured product o The derivative may carry a maturity below the investment horizon and in this case the position in it is to be dynamically «rolled over» over time (in a continuous manner) o The objective function may be different but the algebra may not lead to closed-form solutions o Under constraints, a CRRA utility function ensures that wealth will not go negative and that portfolio allocations will not depend on W 0 As an application of Merton s (1971) principle of optimal stochastic control, we can derive optimal exposures to risk factors: 13

14 The optimal demand of structured products The optimal weights in the risky index and the SSP have two components: one static, mean-variance, and a dynamic hedging one At this point, these exposures can be uniquely transformed in portfolio weights by using the definitions: o A derivative with non-zero g S provides an exposure to price shocks with a diffuse nature; a derivative with non-zero g V provides an exposure to the supplementary volatility risk, Z This simple linear transformation will always be possible if and only if markets are complete because g V 0 We obtain Mean-variance, static component Hedging component 14

15 The optimal demand of structured products The optimal demand of SSP is inversely proportonal to g V /O t, which measures the exposure to volatility per dollar invested o When g V /O t is large, it takes a small amount of the derivative to obtain the desidered exposure to volatility risk The myopic component of the demands of the SSP carries a sign that depends on ξ, the volatility risk premium o Wwhn ξ < 0, it is normal to find a negative demand of the derivative, which yields a negative contribution to the risk-return trade-off o This is not a major problem: many SSPs also exist in «reverse» style Moreover, such a component grows as ρ declines, i.e., in an increasing way as volatility provides hedging of the diffusive risk 15

16 The optimal demand of structured products But also when ξ = 0, an investor with γ 1 could anyway invest in the SSP because of the second term o In particular, when γ > 1, the investor likes to take a short exposure in volatility to insure herself against uncertainty making H(T-t) < 0 The specific nature of the structured product enters through O t and its derivative and hence it depends as it may be obvious from the SSP under examination The second term of the demand for the risky index provides and adjustment for the fact that the SSP generally has a non-zero delta In the absence of derivatives, the demand for the risky index would be simply be η/γ 16

17 An example: asymmetric straddles Let s consisder a typical SSP normally marketed by financial institutions as an investment certificate with partial capital protection, an (asymmetric) double win o It is a long bet on volatility that also plays a function of capital protection in the left-hand tail The payoff function has structure: where p( ) and c( ) are the prices of Europan put and call options and we set 1 = 4 and 2 = 1 o In essence, a portion of invested wealth is devoted to exploit the volatility in the tails of the distribution of the risky index The two pictures that follow show the payoff of this SSP as a function of the underlying index and the payoff of the overall optimal portfolio o Optimal ptf. is computed from the parameters reported below 17

18 An example: asymmetric straddles Derivative Payoff Obvious asiymmetry Derivative wth clearprotective purposes (with some limit to extent) of capital Portfolio Return Optimal Portfolio Returns 4% 3% 2% 1% 0% -1% -2% -3% -4% -5% -26% -20% -14% -8% -2% 4% 10% 16% Risky Asset Return 18

19 An example: asymmetric straddles Such products are normally traded in the sense that variant is represented by the symmetric double win o ewin/infoutili.jsp?idnode=9173# But this is not the point of our exercise: a pension fund may ask for any payoff O t =g(s t,v t ) to be structured and then listed, if deemed useful The remaining results shown here are based on the parameters on the side Comparative statics exercise follow long run mean of volatility rate of mean reversion volatility of vol correlation coefficient premium- diffusive price riskrisk free rate premium - volatility risk - risk aversion investment horizon stock market volatility time to expiration variance Base case parameters υ k 5 σ 0.25 ρ -0.4 ή 2 r 0.05 ξ 4 ϒ 4 T 5 ( V) at t= ( V) at t=h 0.15 τ 0.1 t 0 V at t= V at t+h V at t-h

20 An example: asymmetric straddles Coefficient of Relative Risk Aversion 200% Note: = 4 140% 80% 20% -40% -100% -160% Asymmetric straddle Stocks -220% φt* ψt* 1-φt*-ψt* For 3, the weights are realistic: less than 20% is invested in the derivative, while between 20 and 50% of wealth is invested in risky assets (for instance, a stock index) o Because we have set ξ = 4, the demand of derivatives is always positive Cash 20

21 An example: asymmetric straddles 80% Volatility Risk Premium Nota: = 4 60% 40% 20% 0% -20% Structured product φt* ψt* 1-φt*-ψt* The demans for the SSP reaches zero exactly in correpondence of ξ = 0, a sign that it is static mean-variance demand to dominate o The increase in the demand of the SSP is basically replaced on a one-to-one basis by the demand of the underlying risky index 21

22 An example: asymmetric straddles 60% Mean reversion in volatility Note: = 4, ξ =4 50% 40% 30% 20% 10% 0% Structured product φt* ψt* 1-φt*-ψt* The optimal demand of the SSP slightly increases as the mean reversion rate for long-run variance increases When this occurs the variance of variance increases and there a larger demand for protection from risk 22

23 An example: asymmetric straddles 60% Investment Horizon Nota: = 4, ξ =4 50% 40% 30% 20% 10% Structured product 0% φt* ψt* 1-φt*-ψt* The optimal demand of the SSP does not seem to depend on the investment horizon of the asset manager o The demand of derivatives does not derive in any way from speculation o Recall that portfolio rebalancing occurs in continuous time o We have used a SSP with short maturity but this has no large effects 23

24 The risk-adjusted value of the derivative Because any derivative with g V 0 will complete the markets, its economic value does not specifically depend on its payoff If we compare the maximized objective function with and without SSP, we can compute the certainty equivalent (i.e., risk-adjusted) that an asset manager should be ready to pay in order to have access to ptf./hedging strategies based on the derivative: Because any derivative with g V 0 will complete the markets,, its economic value does not specifically depend on its payoff 24

25 An example: asymmetric straddles 35% Coefficient of Relative Risk Aversion Note: = 4 30% 25% 20% 15% 10% 5% 0% For 3, an asset manager is ready to pay at least 200 bps per year to have access to a portfolio strategy that includes derivatives An aggressive ptf manager with 1, would be ready to pay much more, up to 30% per annum 25

26 An example: asymmetric straddles 7% Volatility Risk Premium Nota: = 4, ξ =4 6% 5% 4% 3% 2% 1% 0% The sign and size of the premium on volatility risk play a firstorder role: because static demand is crucial, it takes ξ 0 Here the news from the empirical literature are as good as odd: most papers report ξ 0 but there is a debate on its sign! 26

27 An example: asymmetric straddles 4.6% Investment Horizon Note: = 4 4.2% 3.8% 3.4% 3.0% 2.6% Investors with a longer horizon assign slightly less value to derivatives, because their variance risk declines However, the 200 bps found keeps representing a significant and remarkable lower bound in economic terms 27

28 How complex can the structured product be? To an institutional investor, the objective of inserting derivatives in her portfolio choice consists of «tailoring» the resulting risk-return profiles: there are no limits to how much flexibility may be used 28

29 How complex can the structured product be? To an institutional investor, the objective of inserting derivatives in her portfolio choice consists of «tailoring» the resulting risk-return profiles: there are no limits to how much flexibility may be used c Fonte: R. Frascà, in I prodotti strutturati nel private banking (a cura di M. Camelia e B. Zanaboni) 29

30 The ex-post economic value of derivatives When researchers have experimented with backtesting exercises, the outcome has been that the presence of derivatives considerably improves performance, especially shriking the variance and tails There is also a growing empirical literature that has investigated the performance improvement that may be realized ex-post from inserting derivatives (both plain vanilla and structured) in equity and bond portfolios The results are generally reassuring: derivatives markedly improve realized performance in recursive exercises o Driessen and Maenhout (2007) show how realized performance improvements mostly derive from the demand of derivatives coming from the myopic portfolio component o Faias and Santa Clara (2011) have simulated in real time the riskadjusted returns obtainable from investing in cash, the S&P 500 index and four plain vanilla, 1-minth options o They find Sharpe ratio increases 0.50 monthly vs

31 The ex-post economic value of derivatives: a simple case The performance backtesting between 2013 and 2014 of a simple portfolio of certificates with a payoff equal to the one in slide 28 originated the following results Structured product Fonte: R. Frascà, in I prodotti strutturati nel private banking (a cura di M. Camelia e B. Zanaboni) Visibly, the structured product does not have to produce any exceptional performances: however it stabilizes ptf. value 31

32 What is left to do? Extensions There is a lot left to do to make the calculation/estimation of the economic value of derivatives and SSPs operational As in Liu and Pan (2003), it would be interesting to extend these exercises to assessing the role of SSPs as tools to separate jump risks from diffusive risks in complete markets o Econometrics applied to US data suggests that the jump risk premium exceeds the premium paid for diffusive risks and this may create a demand for deep OTM put options To research, as in Branger and Breuer (2008) if SSP s (say, certifiates) can keep a role alo when they are used in a portfolio which already contains plain vanulla options (or options on VIX!) o They find a positive answer because only the highly non-linear payoff of complex SSPs may provide market completion To optimize/endogenize the structures under test (in incomplete markets) as in Haugh and Lo (2001), using numerical methods o Up to this point the structured product has been exogenously fixed 32

33 What is left to do? Extensions There is a lot left to do to make the calculation/estimation of the economic value of derivatives and SSPs operational To give an explicit role to downside and drawdown constraints typical of ALM and pension funds, as in Cui, Oldenkamp, e Vellekoop (2013) who use CRRA with displacement o But we know already from Ingersoll (1986) that the qualitative nature of the problem does not change when the downside constraint imposed has a proportional nature To employ evaluation criteria of performance different and further to expected utility increase, such as VaR, tail risk, maxium drawdown etc. (unfortunately Sharpe ratio remain popular) o Cui, Oldenkamp, and Vellekoop (2013) find that CER under CRRA utility and other criteria tend to provide similar results To study problems in which one faces cash outflows over time, e.g., exploting the similarity with consumption and investment problems as in Hsuku (2007) 33

34 Appendix The pricing kernel π t mentioned above has structure: This parametric formulation has the advantage of including two parameters (η e ξ) to separately price both risk factors Ab application of Itô s lemma to the price equation yields the following SDE: o g i S and g i V measure the reactivity of the price of the i-th SSP to infinitesimal changes in the price of the risky ptf. and of variance the coefficient H(τ) is defined as: 34