MFM 5032 VAGLBs

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MFM 5032VAGLBs

• Lecture 4
• Gary Hatfield

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Agenda

• Questions
• Quiz
• Follow up from previous quiz, the value of ME
  fees
• Economic Scenario generation
• Calibration and model selection
• Fitting the funds
• Convergence
• How to test
• Ways to improve
• Hedging
• How delta hedging works
• Risks of delta hedging
• Tradeoffs of hedging more Greeks
• Practical considerations/Risk management etc.

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Valuation of the ME fees

• Consider a Variable Annuity with no rider.
• Along a given TR path, the ME charges taken at
  time t1 are given by
• ME(t1) AV_bcw(t1)me
• Actuarially adjusted, this becomes
• totME(t1)S(t) AV_bcw(t1)me
• The discounted value along that path is
• PV_totME(t1) P(0,t1)S(t) AV_bcw(t1)me

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Valuation of the ME fees

• The present value of all the ME charges (along
  that path) is

If there are no withdrawals, the risk-neutral
expected value of AV_bcw(t1) is
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Valuation of the ME fees

• Hence

• We conclude that the value of the ME primarily
  depends primarily on
• The size of the fee
• The survivorship
• iexp matters, but not as significant
• Interest rates do not matter
• Volatility does not matter

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Valuation of the ME fees

• In the presence of a rider, the previous
  conclusions are not strictly true
• Interest rates and volatility impact the
  likelihood of withdrawals driving the AV to zero
• Withdrawals can greatly reduce the value of the
  ME fees

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Economic Scenario Generation

• In order to perform a Monte Carlo Valuation, we
  need a way to generate economic scenarios that
  are distributed according to the risk-neutral
  density
• So far, we have looked at generating scenarios
  where the model is GBM with terms structure of
  volatility and interest rates.

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Economic Scenario Generation

• Three topics merit further discussion
• Calibration
• Model Selection
• Fitting the funds
• All three are interrelated

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ESG

• If we are given the yield curve and the
  volatility term structure, everything follows
  nicely
• But in general, we will not have complete market
  information
• As described earlier, the yield curve is not
  usually prescribed at all maturities
• Getting put option prices even more problematic

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ESG

• Put option sparseness
• Exchange traded options tend to have maturies
  less than two years (and are generally only
  liquid out to about one year)
• OTC options are traded on indices such as the
  SP500, but even then, liquidity rapidly
  vanishes. Any indications past 10 years are
  considered unreliable
• Hence, the insurer is essentially market making
  in long dated vol.

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ESG

• The result is that the insurer must take a view
  on what to use for long term volatility and
  forward interest rates
• And the reserve side, the prices calculated will
  not be consistent with traded out of the money
  options - GBM with term structure does not
  account for the observed skew in option prices

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ESG

• However, at this point, we need to ask ourselves
  what we are trying to accomplish
• Ideally, we could construct a model that recovers
  the prices of all traded options and which in
  turn interpolate/extrapolates to price of our
  VAGLB
• But this is generally accomplished with rather
  excessive complexity while not appreciably
  improving the quality of the extrapolation to
  long maturities
• gt the greater question is this What traded
  instruments do you need to calibrate to
  (calibrate means the model will correctly
  reprice) ?

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ESG calibration

• What traded instruments do you need to calibrate
  to?
• Philosophical, not a mathematical question
• Important questions
• What instruments are you going to hedge with?
• What traded instruments have similar option
  characteristics as the liability?
• Does increased complexity increase the
  correctness of the price?
• Which Greeks do you plan to hedge?

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ESG fund fitting

• If the AV were invested entirely in an SP500
  index fund (or an ETF), then life is simple
  (relatively speaking) just calibrate to SP500
  options (which are the most liquid options our
  there)
• In most cases however, the policyholders will
  select among many different investment options
• Typically, the investment funds themselves will
  not have liquid options on them

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ESG fund fitting

• However, one can attempt to hedge options on an
  illiquid basket via options on and futures and a
  smaller set of liquid indices
• Liquid indices SP500, NASDAQ, EuroStoxx 50,
  DJIA, Russell 2000, etc. (not all are as liquid
  as might be desired)
• In addition, a good mapping often requires
  mapping to at least one bond index

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ESG fund fitting

• Let the actual funds be F1, F2, , FM and the
  hedgable indices be I1, I2, , IN with MgtN
• Assume the policyholder invests in the funds Fi
  with weights fi , then we seek weights wk such
  that, from a return perspective

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ESG fund fitting

• By looking at time series returns, we can use
  regression to choose our weightings so that the
  sum of the squares of the residuals e(t) is
  minimized
• This is easily done using linear algebra or
  standard regression packages

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ESG fund fitting

• Considerations
• Relevance of time series versus sufficient data
  points
• Use time-weighted residuals?
• Some funds may not map well, but goal is for
  aggregate AV to map well
• If you map to highly correlated indices, then
  co-integration can be a problem

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ESG fund fitting

• Once the mapping is determined, you only need to
  generate risk neutral scenarios for the set of
  hedged indices
• If the noise term is significant (R2 ltlt1), then
  consider generating returns for it as well
  (otherwise, you understate the volatility of the
  funds)

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ESG fund generation

• Suppose the AV is successfully mapped onto N
  hedgeable indices for which one can obtain a
  suitable volatility term structure
• In order to perform Monte Carlo simulation, one
  needs correlation assumptions between the indices
• Unlike volatilities, getting market implied
  correlations is problematic usually must use
  historical
• Must test sensitivity to correlation assumptions

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ESG-Fund Generation

• Let ?(t) be the N x N variance-covariance matrix
  for each period (?i,j si sj )
• How do we generate the desired returns?
• Use the Cholesky decomposition of ?(t) (at each
  time step) i.e.
• Let ?(t) L(t)Lt(t), where L(t) is lower
  triangular and let w(t) be a vector of
  uncorrelated white noise, then Lt(t)w(t) gives
  the properly correlated random returns.

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Convergence

• When valuing an option, we seek to compute the
  expected value µ of the discounted cash flows
  under the risk-neutral probability distribution
• When using Monte Carlo, we take a finite sample
  of N draws from this distribution and compute the
  sample mean x
• x is an estimator of the true mean µ because x
  as N -gt8, x-gt µ
• x is unbiased as its expected value is µ.
• The variance of x converges to zero as N becomes
  large, but the smaller the variance of x is, the
  better it is as an estimator of µ

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Convergence

• How do we know whether or not x is giving us a
  decent estimate? That is to say, how do we know
  whether or not N is large enough?
• If the underlying distribution is normal (or
  close to normal), then we could look at the
  sample standard deviation s and assume that x
  N(µ,s2/Sqrt(N))

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Convergence

• But in many situations, the underlying
  distribution is not very normal looking
• For example, the payoff of a put option is very
  asymmetrical with a high probability of zero
  payoff
• The above assumption is still true
  asymptotically, provided N is large enough
• But that is what we are tying to figure out is N
  large enough?

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Convergence

• Only way to tell for sure is to run several
  experiments of N trials
• Calculate the sample variance of the estimator
  itself
• Brute force feel, but so it goes.

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Convergence

• Tricks to improve convergence
• Adjust the white noise to have a mean of
  exactly zero
• Adjust the white noise to have a variance of
  exactly 1
• Use anti-thetic sampling
• Use control variates
• Use importance sampling
• See Glasserman

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Delta Hedging -review

• First, the sensitivity of the option to various
  parameters is calculated (i.e. get the Greeks)
• For options values by Monte Carlo, requires the
  valuation to be rerun with parameters slightly
  increased or decreased
• For example, if S is the underlying, and ?(S) is
  the option price (for some option), define ?
  ?(S?S) and ?- ?(S-?S). Then

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Review of Delta Hedging

• In delta hedging, one attempts to match the first
  order sensitivity of the option to the
  underlying. That is, you match the delta (go
  figure)
• Because of the gamma, this technique loses a
  little bit of money every time the market moves
  (assuming you are short gamma i.e. you sold an
  option)

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Gamma Bleed
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Gamma a bleed and volatility

• Over a short period of time, the PL of a delta
  hedged position is approximately
• PL Theta(t) (?t) ½Gamma(t)(?x)2
• PL time decay gamma bleed
• If ?x sqrt(?t) s, then PL 0
• Over time,
• PL time value - ½ Gamma ?(?x)2
• time value - ½ Gamma Variance

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What happens when the realized volatility is much
higher (or lower) than the implied?
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What happens when the realized volatility is much
higher (or lower) than the implied?
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Delta hedging summary

• Gamma bleed is how the volatility is realized
• When the actual volatility matches what is
  assumed in the price, the gamma loss matches off
  with the time decay
• When the actual volatility is higher (lower), the
  hedging cost increases (decreases) but with a
  higher dispersion

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Risks of delta hedging

• You need to know what the volatility will be in
  order to get it right (or at least, you have to
  know how high it could go)
• If the underlying process behaves very
  differently from lognormal, all bets are off
  (jumps can kill you)
• Trading frequency can be more of an art than a
  science
• Intra day volatility can be higher than daily
• But can miss gap downs if not monitored
  frequently enough

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Managing more Greeks

• For long-dated option, interest rate risk not
  trivial
• If you dont hedge the rho, you may regret it
• Key rate sensitivity matters a lot for things
  like GLWB, so hedging interest rate risk requires
  hedging at more than one point on the curve

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Managing more Greeks

• If you hedge Vega (or gamma)
• You no longer fear increases in actual volatility
• You will pay the market price of that volatility
• You will still be subject to roll risk and
  counterparties willing to hedge long-term
  volatility are few and far between
• Managing Vega may (will) require a more robust
  valuation model

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Risk Management Considerations

• Hedge program requires use of derivatives
• Derivatives by nature involve high levels of
  leverage a small cash outlay can put the
  company at risk for millions
• Many of the largest derivatives losses in history
  were due to a single rogue trader
• Hence, controls and oversight are very important

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Risk Management

• Make sure that hedge programs are clearly defined
  in writing
• What is being hedged?
• How is it being hedged?
• What are the risk tolerances?
• When do trades take place?
• Who has authority to trade?

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Risk Management

• Need independent verification that terms of
  strategy are being followed
• Oversight needs to be frequent enough so that
  mistakes and/or misdeeds can be caught and
  corrected in a timely manner
• Traders need to have little or no control over
  calculations
• Incentives should align with desired outcomes

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Summary

• Value of ME fees
• Economic Scenario Generation
• Convergence
• Hedging and Risk Management