Derivatives and Structured Products in Portfolio Management Prof. Massimo Guidolin

20263– Advanced Tools for Risk Management and Pricing Spring 2016

CAREFIN Centre for Applied Research in Finance

1

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? CAREFIN Centre for Applied Research in Finance

2

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) CAREFIN Centre for Applied Research in Finance

3

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 CAREFIN Centre for Applied Research in Finance

4

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 CAREFIN Centre for Applied Research in Finance

5

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… CAREFIN Centre for Applied Research in Finance

6

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 CAREFIN Centre for Applied Research in Finance

7

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

CAREFIN Centre for Applied Research in Finance

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) 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 CAREFIN Centre for Applied Research in Finance

9

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 CAREFIN Centre for Applied Research in Finance

10

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 Ot=g(St,Vt) and it may be very complex o Pricing is performed on the basis of a convenient «pricing kernel»… CAREFIN Centre for Applied Research in Finance

11

A model of «completable markets»

 Consider a simple «off-the-shelf» model with stochastic volatility à la Heston (1993): Risk premium on diffusive risk

Price return

Diffusive shocks

Volatility shocks

Change in variance 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 Ot=g(St,Vt) and it may be very complex o Pricing is performed on the basis of a convenient «pricing kernel»… CAREFIN Centre for Applied Research in Finance

12

A model of «completable markets»

 The investor has initial wealth W₀ and she solves a standard problem of expected utility maximization: % in risk index

% in structured product

Coefficient of relative risk aversion

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 W0

 As an application of Merton’s (1971) principle of optimal stochastic control, we can derive optimal exposures to risk factors: CAREFIN Centre for Applied Research in Finance

13

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 gS provides an exposure to price shocks with a diffuse nature; a derivative with non-zero gV 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 gV ≠ 0  We obtain

Mean-variance, static component

Hedging component

CAREFIN Centre for Applied Research in Finance

14

The optimal demand of structured products  The optimal demand of SSP is inversely proportonal to gV/Ot, which measures the exposure to volatility per dollar invested o When gV/Ot 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 CAREFIN Centre for Applied Research in Finance

15

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 Ot 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 CAREFIN Centre for Applied Research in Finance

16

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 CAREFIN Centre for Applied Research in Finance

17

An example: asymmetric straddles Derivative Payoff

1.20

Obvious asiymmetry

1.00 0.80 0.60 0.40 0.20 0.00

-0.20

1.6

1.7

1.8

1.9

2

2.1

Optimal Portfolio Returns

2.2

2.3

Derivative wth clearprotective purposes (with some limit to extent) of capital

Portfolio Return

4% 3% 2% 1% 0%

-1% -2% -3% -4%

-5% -26%

-20%

-14%

-8% -2% 4% Risky Asset Return

10%

16%

CAREFIN Centre for Applied Research in Finance

18

An example: asymmetric straddles

 Such products are normally traded in the sense that variant is represented by the symmetric double win

o http://www.investimenti.unicredit.it/tlab2/it_IT/quotazioni/Doubl eWin/infoutili.jsp?idNode=9173#

 But this is not the point of our exercise: a pension fund may ask for any payoff Ot=g(St,Vt) 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 σ ρ ή r ξ ϒ T √( V) at t=0 √( V) at t=h τ t V at t=0 V at t+h V at t-h

CAREFIN Centre for Applied Research in Finance

0.0169 5 0.25 -0.4 2 0.05 4 4 5 0.15 0.15 0.1 0 0.0225 0.0225 0.0225

19

An example: asymmetric straddles Coefficient of Relative Risk Aversion

200%

Note:  = 4

140%

80% 20%

-40%

-100% -160%

Stocks

-220%

0

2

φt*

4

Asymmetric straddle ψt*

6

8 1-φt*-ψt*

10

Cash

 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 CAREFIN Centre for Applied Research in Finance

20

An example: asymmetric straddles Volatility Risk Premium Nota:  = 4

80% 60% 40% 20% 0%

-20%

-6

-4

φt*

-2

ψt*

0

Structured product 2 1-φt*-ψt*

4

6

 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 CAREFIN Centre for Applied Research in Finance

21

An example: asymmetric straddles Mean reversion in volatility

60%

Note:  = 4, ξ =4

50% 40% 30% 20%

Structured product

10% 0%

0

2 φt*

4 ψt*

6 1-φt*-ψt*

8

 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 CAREFIN Centre for Applied Research in Finance

22

An example: asymmetric straddles Investment Horizon

60%

Nota:  = 4, ξ =4

50% 40% 30% 20%

Structured product

10% 0%

0

5

φt*

10

ψt*

15

1-φt*-ψt*

20

 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 CAREFIN Centre for Applied Research in Finance

23

The risk-adjusted value of the derivative

Because any derivative with gV ≠ 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 gV ≠ 0 will complete the markets,, its economic value does not specifically depend on its payoff CAREFIN Centre for Applied Research in Finance

24

An example: asymmetric straddles Coefficient of Relative Risk Aversion Note:  = 4

35% 30% 25% 20% 15% 10% 5% 0%

0

2

4

6

8

10

 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 CAREFIN Centre for Applied Research in Finance

25

An example: asymmetric straddles Volatility Risk Premium Nota:  = 4, ξ =4

7% 6% 5% 4% 3% 2% 1% 0%

-6

-4

-2

0

2

4

6

 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! CAREFIN Centre for Applied Research in Finance

26

An example: asymmetric straddles Investment Horizon

4.6%

Note:  = 4

4.2% 3.8% 3.4% 3.0% 2.6%

0

5

10

15

20

 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 CAREFIN Centre for Applied Research in Finance

27

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

CAREFIN Centre for Applied Research in Finance

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 c

Fonte: R. Frascà, in I prodotti strutturati nel private banking (a cura di M. Camelia e B. Zanaboni) CAREFIN Centre for Applied Research in Finance

29

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. 0.13 CAREFIN Centre for Applied Research in Finance

30

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 CAREFIN Centre for Applied Research in Finance

31

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 CAREFIN Centre for Applied Research in Finance

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  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) CAREFIN Centre for Applied Research in Finance

33

Appendix

 The pricing kernel πt 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 giS and giV 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:

CAREFIN Centre for Applied Research in Finance

34