Derivatives and Structured Products in Portfolio Managementdidattica.unibocconi.it/mypage/dwload.php?nomefile=Lec_19... · Derivatives and Structured Products in Portfolio Management

1CAREFINCentre for Applied Research in Finance

Derivatives and Structured Products in Portfolio Management

Prof. Massimo Guidolin

20263– Advanced Tools for Risk Management and Pricing

Spring 2017


Motivation

The concept of efficient portfolio management may in principlehelp re-define the traditional shyness of asset managers in usingderivatives 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 whatconditions) by improving the risk-return trade-off?

o This means an increase in expected «risk-adjusted» performance② Does such a contribution also (or especially) hold also ex-

post?③ What is the link between the economic value of derivatives

and the benefits of treating volatility as a separate, additional asset class?

The theory and practice of modern asset pricing models offerprecise ideas on the economic value of derivatives in ptf. mgmt


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 TradedCommodities) and ETN (Exchange Traded Notes) when these implystrong 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 warrantso 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 regulationsboth when issued/structured (primary market) and when they are subsequently traded (secondary market)


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


Preliminary Concepts: Let’s get rid of myths… Structured product does not mean expensive

o Obviously, structuring services need to be paid for, especially whenthe 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 SSPs② The increase in risk-adjusted performance that a SSP makesavailable, 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


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 riskpositionso In fact, with dynamic leverage, this may exceed the issuance level

SSPs are often difficult to price and they require expert advisory, atleast in support to internal teams

The liquidity of SSPs is often supplied by the issuers themselvesand hence their own credit risk is interacted with liquidity risk

The cost of structuring may be reduced trough auctionmechanisms, i.e., placing issuers in competiton with one another; hence understanding such mechanisms is important…

The SSPs enjoy of sophisticated replication strategies, but after allthey are just securitized loan contracts without margin accountso In this sense, excessive attention by textbooks may make them look

like basket products, which they are not…


Generalities on the Economic Value of SSPs

There is pervasive evidence of aninverse correlation betweenmarket indices and volatilityo Volatility has become an impor-

tant asset class

o SSPs allow both to take positionsdirectly in volatility when this isthe 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 diversifiableo In fact, what matters will be the «completability» of markets through

trading of derivatives

SSPs represent excellent «wrappers» of long and short strategieson the volatility of market indices


The key result

The «asset pricing» literature has adopted a variety of empiricaland mathematical techniques on which factors and market forcesthat ought to be priced in equilibrium, to assess whether they drive a non-zero equilibrium risk premiumo 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

Derivatives generate economic value in portfolio managemntbecause they uniquely allow one to take positions of correct signand magnitude vs. the risk factors (diffusion, volatility, and jumps)


The key result

o Also the (negative) correlation between the diffusive and stochasticvolatility components plays a key role, although it fails to representper 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. weightseven though the objective of all models is to determine exposures

Derivatives generate economic value in portfolio managemntbecause they uniquely allow one to take positions of correct signand magnitude vs. the risk factors (diffusion, volatility, and jumps)


A model of «completable markets»

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 valueo 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

Markets are completable when introducing an appropriate and finite number of derivative securities makes markets complete


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 constantrisk free rate r and a risky asset, identified with a market index

o The investor may also include derivatives in her portfolioo Such derivatives yield an exposure to the risks B and Z that differs

across stocks and bonds because their payoff is (potentially) nonlinearo In fact, such a nonlinearity may provide market completion

o The derivative is the function Ot=g(St,Vt) and it may be very complexo Pricing is performed on the basis of a convenient «pricing kernel»…


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 constantrisk free rate r and a risky asset, identified with a market index

o The investor may also include derivatives in her portfolioDerivativesyield an exposure to the risks B and Z that differs across stocks and bonds because their payoff is (potentially) nonlinearo In fact, such a nonlinearity may provide market completion

o The derivative is the function Ot=g(St,Vt) and it may be very complexo Pricing is performed on the basis of a convenient «pricing kernel»…

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


A model of «completable markets» The investor has initial wealth W₀ and she solves a standard

problem of expected utility maximization:

o The derivative may carry a maturity below the investment horizonand 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 leadto closed-form solutions

o Under constraints, a CRRA utility function ensures that wealth willnot go negative and that portfolio allocations will not depend on W0

As an application of Merton’s (1971) principle of optimalstochastic control, we can derive optimal exposures to risk factors:

% in risk index

% in structured product

Coefficient of relative riskaversion


The optimal demand of structured products

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

The optimal weights in the risky index and the SSP have two com-ponents: one static, mean-variance, and a dynamic hedging one

Mean-variance, staticcomponent

Hedging component



The optimal demand of SSP is inversely proportonal to gV/Ot, which measures the exposure to volatility per dollar investedo 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 premiumo Wwhn ξ < 0, it is normal to find a negative demand of the derivative,

which yields a negative contribution to the risk-return trade-offo 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



But also when ξ = 0, an investor with γ ≠ 1 could anyway invest in the SSP because of the second termo 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


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 wino 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 optionsand 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 overalloptimal portfolioo Optimal ptf. is computed from the parameters reported below


An example: asymmetric straddles

-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.6 1.7 1.8 1.9 2 2.1 2.2 2.3

Derivative Payoff

-5%-4%-3%-2%-1%0%1%2%3%4%

-26% -20% -14% -8% -2% 4% 10% 16%

Portf

olio R

etur

n

Risky Asset Return

Optimal Portfolio Returns

Obvious asiymmetry

Derivative wthclearprotective

purposes (with some limit to extent) of

capital


An example: asymmetric straddles Such products are normally traded in the sense that variant is

represented by the symmetric double wino 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 mayask for any payoff Ot=g(St,Vt) to be structured and thenlisted, if deemed useful

The remaining resultsshown here are based onthe parameters on the side

Comparative staticsexercise follow

υ 0.0169k 5σ 0.25ρ -0.4ή 2r 0.05ξ 4ϒ 4T 5√( V) at t=0 0.15√( V) at t=h 0.15τ 0.1t 0V at t=0 0.0225V at t+h 0.0225V at t-h 0.0225

variance

Base case parameters

long run mean of volatilityrate of mean reversion

volatility of volcorrelation coefficient

premium- diffusive price risk-risk free rate

premium - volatility risk -risk aversion

investment horizon stock market volatility

time to expiration



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

-220%

-160%

-100%

-40%

20%

80%

140%

200%

0 2 4 6 8 10

Coefficient of Relative Risk Aversion

φt* ψt* 1-φt*-ψt*

Note: = 4

Stocks

Asymmetric straddle

Cash



The demans for the SSP reaches zero exactly in correpondence of ξ= 0, a sign that it is static mean-variance demand to dominateo 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

-20%

0%

20%

40%

60%

80%

-6 -4 -2 0 2 4 6

Volatility Risk Premium


Structured product

Nota: = 4


0%

10%

20%

30%

40%

50%

60%

0 2 4 6 8

Mean reversion in volatility


Structured product


The optimal demand of the SSP slightly increases as the meanreversion rate for long-run variance increases

When this occurs the variance of variance increases and there a larger demand for protection from risk

Note: = 4, ξ =4


0%

10%

20%

30%

40%

50%

60%

0 5 10 15 20

Investment Horizon


Structured product


The optimal demand of the SSP does not seem to depend on the investment horizon of the asset managero The demand of derivatives does not derive in any way from speculationo Recall that portfolio rebalancing occurs in continuous timeo We have used a SSP with short maturity but this has no large effects

Nota: = 4, ξ =4


The risk-adjusted value of the derivative

If we compare the maximized objective function with and withoutSSP, we can compute the certainty equivalent (i.e., risk-adjusted) that an asset manager should be ready to pay in order to haveaccess to ptf./hedging strategies based on the derivative:

Because any derivative with gV ≠ 0 will complete the markets,, itseconomic value does not specifically depend on its payoff

Because any derivative with gV ≠ 0 will complete the markets, itseconomic value does not specifically depend on its payoff



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 paymuch more, up to 30% per annum

0%

5%

10%

15%

20%

25%

30%

35%

0 2 4 6 8 10

Coefficient of Relative Risk AversionNote: = 4



The sign and size of the premium on volatility risk play a first-order 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!

0%

1%

2%

3%

4%

5%

6%

7%

-6 -4 -2 0 2 4 6

Volatility Risk Premium Nota: = 4, ξ =4


2.6%

3.0%

3.4%

3.8%

4.2%

4.6%

0 5 10 15 20

Investment Horizon


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

Note: = 4


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-returnprofiles: there are no limits to how much flexibility may be used


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-returnprofiles: 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)


The ex-post economic value of derivatives

There is also a growing empirical literature that has investigatedthe performance improvement that may be realized ex-post from inserting derivatives (both plain vanilla and structured) in equityand bond portfolios

The results are generally reassuring: derivatives markedlyimprove realized performance in recursive exerciseso Driessen and Maenhout (2007) show how realized performance

improvements mostly derive from the demand of derivatives comingfrom the myopic portfolio component

o Faias and Santa Clara (2011) have simulated in real time the risk-adjusted 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

When researchers have experimented with backtesting exercises, the outcome has been that the presence of derivatives considerablyimproves performance, especially shriking the variance and tails


The ex-post economic value of derivatives: a simple case

The performance backtesting between 2013 and 2014 of a simpleportfolio of certificates with a payoff equal to the one in slide 28 originated the following results

Visibly, the structured product does not have to produce anyexceptional performances: however it stabilizes ptf. value

Structured product

Fonte: R. Frascà, in I prodotti strutturati nel

private banking (a cura di M. Camelia e B. Zanaboni)


What is left to do? Extensions

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 marketso Econometrics applied to US data suggests that the jump risk

premium exceeds the premium paid for diffusive risks and this maycreate 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 methodso Up to this point the structured product has been exogenously fixed

There is a lot left to do to make the calculation/estimation of the economic value of derivatives and SSPs operational


What is left to do? Extensions

To give an explicit role to downside and drawdown constraintstypical 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 furtherto expected utility increase, such as VaR, tail risk, maxiumdrawdown 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 investmentproblems as in Hsuku (2007)

There is a lot left to do to make the calculation/estimation of the economic value of derivatives and SSPs operational


Appendix The pricing kernel πt mentioned above has structure:

This parametric formulation has the advantage of including twoparameters (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 gi

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 coefficientH(τ) is definedas:

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Derivatives and Structured Products in Portfolio Managementdidattica.unibocconi.it/mypage/dwload.php?nomefile=Lec_19... · Derivatives and Structured Products in Portfolio Management