Ten
Modern Week Risk Management
with Options and Futures
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Copyright Bruce N. Lehmann 2000-2011
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The Potential Benefits of Risk
Management
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Risk management can reduce the PV of
taxes, bankruptcy costs, and the costs of
financial distress
Risk management can facilitate the
exploitation of investment opportunities
Risk management can increase the firm s
debt capacity
Risk management reduces the costs to
stakeholders, large shareholders, and
managers of bearing firm specific risk
Copyright Bruce N. Lehmann 2000-2011
!
A Risk Measure Should:
Facilitate the assessment of the uncertainty
regarding a portfolio s gains and losses
Quantify its contribution to overall portfolio
risk, not its risk viewed in isolation
Provide guidance regarding the role of an
asset or liability in hedging different risks
Be amenable to empirical validation
Have well understood strengths and
weaknesses
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Copyright Bruce N. Lehmann 2000-2011
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Portfolio Risk, Position Size, and
Risk Measures
The returns of a portfolio Rp are given by:
+ Rp = wi Ri
+ i.e., wi is the weight given to security i
Formally, the risk of generating the
portfolio return is additive
+ Risk[Rp] = Risk[wi Ri]
The risk arising from changes in wi that do
not affect market prices (i.e., small wi) is:
+ Risk[Rp] = wi Marginal Risk[Ri]
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Copyright Bruce N. Lehmann 2000-2011
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The Virtues and Defects of
Additive Marginal Risk Measures
Virtues
+ Additivity
+ Scale independence
Limitation
+ Measures impact of marginal portfolio changes
- i.e., effect of small position changes on portfolio risk
+ Does not answer question of how large changes
affect portfolio risk
+ There are empirically relevant market settings
in which this defect is serious
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Copyright Bruce N. Lehmann 2000-2011
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Portfolio Risk Measure 1:
Portfolio Standard Deviation
Portfolio return variance is:
+ = i =1 j=1 w i w j ij where ij is the
covariance between the returns of assets i and j
2
p
N
N
A little bit of calculus shows that:
+ Marginal Risk[Ri] = p/wi = pip; ip=ip/p2
- Perhaps better to separate upside and downside betas
Required inputs
+ Portfolio standard deviation
+ Betas of asset returns on portfolio return
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- Estimates or model-based, especially for options
Copyright Bruce N. Lehmann 2000-2011
!
Downside Risk and Portfolio
Standard Deviation and Beta
Benefits of risk management related to
downside risk
+ Standard deviation and beta treat unusual up
and down changes in value symmetrically
Ex: two portfolios with same mean return
and vol but different up and downside risks
+ Red: $100 MM of normally distributed stock
with volatility of 30% per year
+ Green: $76.89 MM in riskless asset plus call
option struck at current stock price
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Copyright Bruce N.
- options do not have normally distributed returns Lehmann 2000-2011!
Distribution of Return on Stock
vs. Bill + Call Portfolio
+ Red: normally distributed stock return
+ Green: Call option plus riskless asset
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- Same mean and standard deviation Copyright Bruce N. Lehmann 2000-2011!
of return
Portfolio Risk Measure 2:
Value-at-Risk (VaR)
VaR(%,N): % chance portfolio loss over
next N days will be worse than VaR
+ i.e., th quantile of portfolio return distribution
+ Prob[Loss < VaR] =
+ Basel Accords mandate bank = 10 and N = 10
VaR can only be calculated for a few
distributions like the normal distribution
+ VaR = z x p + (Market value expected value)
- N[zz]=; z~N(0,1) Marginal risk linear in pp
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+ No incremental information relative to Bruce N. Lehmann 2000-2011!
p2
Copyright
Portfolio Risk Measure 2:
Value-at-Risk (VaR), Continued
Required inputs
+ Same as p under normality
+ VaRp(%,N)/wi can be evaluated
numerically for other distributions
- sometimes estimated from historical data, simulated,
or chosen by judgement
Related risk measures more convenient
+ Expected downside loss = E{min[Rp,X]}
+ Semivariance = E{[(Rp-X)-]2}
+ VaR dominant partly because of Basel
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Copyright Bruce N. Lehmann 2000-2011
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Value at Risk for the Green and
Red Portfolios
VaR calculation requires choice of horizon
and probability level
Consider horizon of one year and 5%
quantile
+ Stock portfolio has a one-year 5% VaR of
$41.93 MM
+ Portfolio holding the risk-free asset and option
has a one-year 5% VaR of $15 MM
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Measured downside risk of stock is 2.8
times that of call portfolio
Copyright option Bruce N. Lehmann 2000-2011
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The Role of Risks with Nonlinear
Payoffs in Risk Management
Stock and bond portfolios have linear payoffs
+ Portfolios weighted average of returns
+ Return models linear in underlying risk factors
Options have nonlinear payoffs
+ Non-normal returns even if underlying
returns are normal
Normal VaR not appropriate for options
or other assets with nonlinear payoffs
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+ Dynamic trading strategies like delta hedging
problematic too if payoffs are nonlinear Bruce N. Lehmann 2000-2011!
Copyright
Option Betas on the Underlying
Asset in a Black-Scholes World
Call option equivalent to levered investment
in underlying asset in Black-Scholes world
Option beta is levered underlying asset
beta
+ call
S
S N(d1 )
=
asset =
asset
C
C
- i.e., beta of bond component of call price is zero
+ Like levered beta in weighted average cost of
capital calculation
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Option beta rises and falls as S rises and
falls
Copyright Bruce N. Lehmann 2000-2011
!
Portfolio Risk Management with
Additive Marginal Risk Measures
Examine portfolio and identify replicating
portfolios for assets in portfolio such as:
+ Matched duration bond portfolio
+ Delta-neutral hedged option portfolio
Treat portfolio risk as risk of replicating
portfolios for the assets in the portfolio
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+ Replicating portfolio weights treated as risk
exposures to changes in underlying asset prices
+ Poor risk measures if replicating portfolio
weights change rapidly as prices change
Copyright Bruce N. Lehmann 2000-2011
!
Delta-VaR
Delta-VaR uses linear return approximation
+ Bonds and yields: Pbond BPV x yield
+ Options and underlying: C [CS]ert + S
Delta-VaR is the VaR of the replicating
portfolios of the assets in the portfolio
+ Normal Delta-VaR treats changes in underlying
asset values as normally distributed
+ Replicating portfolio weights treated as risk
exposures to changes in underlying asset prices
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Copyright Bruce N. Lehmann 2000-2011
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The Protective Put and Wealth
Insurance
An investor with wealth W wants to:
+ invest in a portfolio of risky stocks
+ insure that wealth cannot fall too much
+ can be accomplished by purchasing the
portfolio and put options on the portfolio
The wealth management problem
+ W = NS x [S0 + P(S0 ,X,T)]; NS = # of shares
+ k x W > NS x [ST + max[XST,0)]
- k x W is the desired floor on wealth
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+ X = [k x W]/NS is desired strike price
Copyright Bruce N. Lehmann 2000-2011
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The Protective Put of the 1980s:
Portfolio Insurance
Suppose no options that trade or that have
the desired strike are available in the market
Why not use the replicating portfolio?
+ Short NS x [1 N(d1)] shares
+ Buy T period risk free bonds with a face value
of NS x {P(S0 ,X,T) [1 N(d1)] x S0}
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Portfolio insurance replaces a static hedge
via actual puts with a trading strategy in the
underlying asset and riskless bonds that
replicates options when the model is true
Copyright Bruce N. Lehmann 2000-2011
!
Delta-Gamma VaR
Delta-VaR uses linear return approximation
VaR reflects concern about downside risk
Convexity is implicitly a major concern
Delta-Gamma VaR adds a convexity
adjustment for large price changes via:
+ V = a + S + (S)2
+ Under normality, volatility is given by:
2(V) = 22(S) + 2[2(S)]2
!
Model risk and problems with distributional
assumptions remain
Copyright Bruce N. Lehmann 2000-2011
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Nick Leeson and the Limitations
of the Delta-VaR Risk Measure
Nick Leeson formed an approximately deltaneutral portfolio of the stock and straddles
+ Long Nikkei stock index futures
+ Short Nikkei straddles
- i.e., sold Nikkei puts and calls with the same strikes
Put/call parity and straddle deltas
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+ S0 + P(X,S0,T) = C(X,S0,T) + PV(X)
+ P(S,X,T) = C(X,S0,T) + PV(X) S0
+ -[C(X,S0,T)+P(S,X,T)] = S0-2C(X,S0,T)-PV(X)
+ Straddle delta depends only on Copyrightdelta Lehmann 2000-2011!
call Bruce N.
Nick Leeson and the Limitations
of Delta-VaR, Continued
Black-Scholes call option prices and deltas
+ Suppose S = 18,759, X = $19,000, = 19.8%,
T = , r = 4%
+ Black-Scholes call option price = 536.33
+ Black-Scholes call option delta =
+ [S0-2C(X,S0,T)-PV(X)] = [S0]-2[C(X,S0,T)]
= 1 2 x = 0!
Black-Scholes Delta-VaR = 0
+ Need Delta-Gamma VaR and Delta-Gamma
hedge from last week
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Copyright Bruce N. Lehmann 2000-2011
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