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Financial Risk Forecasting © 2011,2017 Jon Danielsson, page 1 of 45
Introduction Bonds Options Duration VaR Delta VaR Summary
Financial Risk Forecasting
Chapter 6
Analytical value-at-risk for options
and bonds
Jon Danielsson ©2017London School of Economics
To accompanyFinancial Risk Forecasting
www.financialriskforecasting.com
Published by Wiley 2011
Version 3.0, August 2017
Financial Risk Forecasting © 2011,2017 Jon Danielsson, page 2 of 45
Introduction Bonds Options Duration VaR Delta VaR Summary
Financial Risk Forecasting © 2011,2017 Jon Danielsson, page 3 of 45
Introduction Bonds Options Duration VaR Delta VaR Summary
The focus of this chapter
• Calculate VaR for options and bonds
• Not possible with methods from Chapters 4 and 5
• We start by using analytical methods, deriving VaRmathematically
• Monte Carlo methods are discussed in Chapter 7
• Preferred for most applications
Financial Risk Forecasting © 2011,2017 Jon Danielsson, page 4 of 45
Introduction Bonds Options Duration VaR Delta VaR Summary
VaR for options and bonds
• Chapters 4 and 5 showed how a VaR can be obtained anasset distribution
• That is not possible for assets such as bonds and options,as their intrinsic value changes with passing of time
• e.g. the price of bond converges to fixed value as time
to maturity elapses, so inherent risk decreases over time
• Value of bonds and options is non–linearly related to theunderlying asset
Financial Risk Forecasting © 2011,2017 Jon Danielsson, page 5 of 45
Introduction Bonds Options Duration VaR Delta VaR Summary
Organization
• The first two sections of these slides introduce theproblem of the nonlinear relationship between theunderlying asset and a bond and option
• The last two sections show how one can use mathematicalapproximations to obtain a closed form solution
• Generally, such methods are not recommended
• And is better to use the simulation methods in the nextchapter
Financial Risk Forecasting © 2011,2017 Jon Danielsson, page 6 of 45
Introduction Bonds Options Duration VaR Delta VaR Summary
Notation
T Delivery time/maturityr Annual interest rateσr Volatility of daily interest rate incrementsσa Annual volatility of an underlying assetσd Daily volatility of an underlying assetτ Cash flow
D∗ Modified durationC Convexity∆ Option deltaΓ Option gamma
g(·) Generic function name for pricing equationϑ Portfolio value
Financial Risk Forecasting © 2011,2017 Jon Danielsson, page 7 of 45
Introduction Bonds Options Duration VaR Delta VaR Summary
Bonds
Financial Risk Forecasting © 2011,2017 Jon Danielsson, page 8 of 45
Introduction Bonds Options Duration VaR Delta VaR Summary
Bond pricing
• A bond is a fixed income instrument
• Typially with regular payments
• Bond price is given by present value of future cash flows
T∑
t=1
τt
(1 + rt)t
• Where {τt}Tt=1 includes the coupon and principalpayments
• And rt is the interest rate in each period
Financial Risk Forecasting © 2011,2017 Jon Danielsson, page 9 of 45
Introduction Bonds Options Duration VaR Delta VaR Summary
Bond risk asymmetry
• Bond has face value $1000, maturity of 50 years andannual coupon of $30
• Yield curve is flat, annual interest rates at 3%
• So its current price is equal to the par value
• Now consider parallel shifts in the yield curve to 1% or 5%
Interest rate Price Change in price1% $1784 $7843% $10005% $635 -$365
Financial Risk Forecasting © 2011,2017 Jon Danielsson, page 10 of 45
Introduction Bonds Options Duration VaR Delta VaR Summary
Bond risk asymmetry
Interest rate
1.0% 3.0% 5.0% 7.0%
$400
$600
$800
$1000
$1200
$1400
$1600
$1800
$1000
$1784
$635
Financial Risk Forecasting © 2011,2017 Jon Danielsson, page 11 of 45
Introduction Bonds Options Duration VaR Delta VaR Summary
Bond risk
• Change from 3% to 1% makes bond price increase by$784
• Change from 3% to 5% makes it fall by $365
Financial Risk Forecasting © 2011,2017 Jon Danielsson, page 12 of 45
Introduction Bonds Options Duration VaR Delta VaR Summary
Options
Financial Risk Forecasting © 2011,2017 Jon Danielsson, page 13 of 45
Introduction Bonds Options Duration VaR Delta VaR Summary
Options
• An option gives its owner the right, but not theobligation, to call (buy) or put (sell) an underlying assetat a strike price on a fixed expiry date
• European options can only be exercised at expiration
• American options can be exercised at any point up to
expiration
• We will focus on European options, but the basic analysiscould be extended to many other variants
Financial Risk Forecasting © 2011,2017 Jon Danielsson, page 14 of 45
Introduction Bonds Options Duration VaR Delta VaR Summary
Black-Scholes equationPricing European options
• Black and Scholes (1973) developed an equation forpricing European options
• Refer to the Black-Scholes (BS) pricing function as g(·)• We use the following notation:
Pt Price of underlying asset at year tX Strike pricer Annual risk-free interest rate
T − t Time until expirationσa Annual volatilityΦ Standard normal distribution
Financial Risk Forecasting © 2011,2017 Jon Danielsson, page 15 of 45
Introduction Bonds Options Duration VaR Delta VaR Summary
• The BS function for an European option
putt= Xe−r(T−t) − Pt + callt
callt = PtΦ (d1)− Xe−r(T−t)Φ (d2)
where
d1 =log (Pt/X ) + (r + σ2
a/2) (T − t)
σa
√T − t
d2 =log (Pt/X ) + (r − σ2
a/2) (T − t)
σa
√T − t
= d1 − σa
√T − t
Financial Risk Forecasting © 2011,2017 Jon Danielsson, page 16 of 45
Introduction Bonds Options Duration VaR Delta VaR Summary
• Value of an option is affected by many underlying factors
• Standard BS assumptions:
• Flat nonrandom yield curve
• The underlying asset has continuous IID-normal returns
• Our objective is to map risk in the underlying asset ontoan option
• This can be done using the option Delta and Gamma
Financial Risk Forecasting © 2011,2017 Jon Danielsson, page 17 of 45
Introduction Bonds Options Duration VaR Delta VaR Summary
60 70 80 90 100 110
0
5
10
15
Stock price
Opt
ion
pric
e
12 months6 months1 months0 months
Financial Risk Forecasting © 2011,2017 Jon Danielsson, page 18 of 45
Introduction Bonds Options Duration VaR Delta VaR Summary
VaR for bonds
• There are several ways to approximate bond risk as afunction of risk in interest rates
• One way is to use Ito’s lemma, another to follow thederivation for options
• Here we only present the result, as a formal derivationwould just repeat the one given for options
Financial Risk Forecasting © 2011,2017 Jon Danielsson, page 19 of 45
Introduction Bonds Options Duration VaR Delta VaR Summary
Modified duration
• We define modified duration, D∗, as the negative firstderivative of the bond-pricing function, g ′(r), divided byprices:
D∗ = − 1
Pg ′(r)
• Modified duration measures price sensitivity of a bond tointerest rate movements
Financial Risk Forecasting © 2011,2017 Jon Danielsson, page 20 of 45
Introduction Bonds Options Duration VaR Delta VaR Summary
Duration-normal VaRTwo steps to calculate bond VaR
1. Identify the distribution of interest rate changes, dr
2. Map distribution onto bond prices
Financial Risk Forecasting © 2011,2017 Jon Danielsson, page 21 of 45
Introduction Bonds Options Duration VaR Delta VaR Summary
Duration-normal VaR
• We assume the distribution of interest rate changes isgiven by
rt − rt−1 = dr ∼ N(
0, σ2r
)
but we could use almost any distribution
• Regardless of whether we use Ito’s lemma or follow thederivation for options, we arrive at the duration-normalmethod to get bond VaR
• Here we find that bond returns are simply modfiedduration times interest rate changes so
RBondApproximately∼ N
(
0, (σrD∗)2
)
Financial Risk Forecasting © 2011,2017 Jon Danielsson, page 22 of 45
Introduction Bonds Options Duration VaR Delta VaR Summary
Duration-normal VaR
• Now the VaR follows directly:
VaRBond(p) ≈ D∗ × σr × Φ−1(p)× ϑ
Financial Risk Forecasting © 2011,2017 Jon Danielsson, page 23 of 45
Introduction Bonds Options Duration VaR Delta VaR Summary
Accuracy of duration-normal VaR
• The accuracy of these approximations depends onmagnitude of duration and the VaR time horizon
• Main sources of error are assumptions of linearity and flatyield curve
• We now explore these issues graphically
Financial Risk Forecasting © 2011,2017 Jon Danielsson, page 24 of 45
Introduction Bonds Options Duration VaR Delta VaR Summary
Bond prices and durationAccuracy of duration approximation for T=1
Interest rate
$960
$980
$1000
$1020
$1040
2% 4% 6% 8% 10%
Bond priceDurationapproximation
Financial Risk Forecasting © 2011,2017 Jon Danielsson, page 25 of 45
Introduction Bonds Options Duration VaR Delta VaR Summary
Bond prices and durationAccuracy of duration approximation for T=50
Interest rate
$500
$1000
$1500
$2000
$2500
2% 4% 6% 8% 10%
Bond priceDurationapproximation
Financial Risk Forecasting © 2011,2017 Jon Danielsson, page 26 of 45
Introduction Bonds Options Duration VaR Delta VaR Summary
Bond prices and durationAccuracy of duration approximation for T=1 and T=50
• The graphs compare bond prices and durationapproximation for two maturities, T = 1 and T = 50
• It is clear that duration approximation is quite accuratefor short-dated bonds, but very poor for long-dated ones
• We conclude that maturity is a key factor when it comesto accuracy of VaR calculations using duration-normalmethods
Financial Risk Forecasting © 2011,2017 Jon Danielsson, page 27 of 45
Introduction Bonds Options Duration VaR Delta VaR Summary
Error in duration-normal VaRVarious volatilities of interest rate changes
0 10 20 30 40 50 60
0.6
0.7
0.8
0.9
1.0
Maturity
VaR
(true
)V
aR(d
urat
ion)
σr=0.1%σr=0.5%σr=1.0%σr=2.0%
Financial Risk Forecasting © 2011,2017 Jon Danielsson, page 28 of 45
Introduction Bonds Options Duration VaR Delta VaR Summary
Error in duration-normal VaRHigher volatility of interest rate changes leads to larger error
• The graph on the previous slide shows how the accuracyof duration-normal VaR is affected by interest rate changevolatility
• Duration-normal VaR is compared with VaR (true), whichis calculated with a Monte Carlo simulation
• Looking at maturities from 1 year to 60 years andvolatility from 0.1% to 2.0%, we see that the error induration-normal VaR increases as volatility of interest ratechanges increases
Financial Risk Forecasting © 2011,2017 Jon Danielsson, page 29 of 45
Introduction Bonds Options Duration VaR Delta VaR Summary
Accuracy of duration-normal VaR
• Based on these observations, we conclude thatduration-normal VaR approximation is best forshort-dated bonds and low volatilities
• Quality declines sharply with increased volatility andlonger maturities
Financial Risk Forecasting © 2011,2017 Jon Danielsson, page 30 of 45
Introduction Bonds Options Duration VaR Delta VaR Summary
Convexity and VaR
• Straightforward to improve duration approximation byadding second-order term, thereby allowing for convexity
• However, even after incorporating convexity there is oftenconsiderable bias in VaR calculations
• Adding higher order terms increases mathematicalcomplexity, especially if we have a portfolio of bonds
• For these reasons, Monte Carlo methods are generallypreferred
Financial Risk Forecasting © 2011,2017 Jon Danielsson, page 31 of 45
Introduction Bonds Options Duration VaR Delta VaR Summary
Delta
• First-order sensitivity of an option with respect to theunderlying price is called delta, defined as:
∆ =∂g(P)
∂P=
{
Φ (d1) > 0 call
Φ (d1)− 1 < 0 put
• Delta is equal to ±1 for deep-in-the-money options(depending on whether it is call or put), close to ±0.5 forat-the-money options and 0 for deep out-of-the-moneyoptions
Financial Risk Forecasting © 2011,2017 Jon Danielsson, page 32 of 45
Introduction Bonds Options Duration VaR Delta VaR Summary
• A small change in P changes the option price byapproximately ∆, but the approximation gets graduallyworse as the deviation of P becomes larger
• We can graph the price of a call option for a range ofstrike prices and two different maturities to gauge theaccuracy of the delta approximation
• We let X = 100, r = 0.01 and σ = 0.2 and comparematurities of one and six months
Financial Risk Forecasting © 2011,2017 Jon Danielsson, page 33 of 45
Introduction Bonds Options Duration VaR Delta VaR Summary
Accuracy of Delta approximationOne month
Stock price
Opt
ion
pric
e
$90 $100 $110
$0
$2
$4
$6
$8
$10Payoff at expirationPayoff one month to expirationDelta
Financial Risk Forecasting © 2011,2017 Jon Danielsson, page 34 of 45
Introduction Bonds Options Duration VaR Delta VaR Summary
Accuracy of Delta approximationSix months
Stock price
Opt
ion
pric
e
$90 $100 $110
$0
$2
$4
$6
$8
$10Payoff at expirationPayoff six monthsto expirationDelta
Financial Risk Forecasting © 2011,2017 Jon Danielsson, page 35 of 45
Introduction Bonds Options Duration VaR Delta VaR Summary
Gamma
• Second-order sensitivity of an option with respect to theunderlying price is called gamma, defined as:
Γ =∂2g(P)
∂P2= e−r(T−t) Φ (d1)
Ptσa
√
(T − t)
• Gamma is highest when an option is a little out of themoney and dropping as the underlying price moves awayfrom the strike price
• We can see this by adding a plot of gamma to theprevious graph of option price with one month to expiry
• Not surprising since the price plot increasingly becomes
a straight line for deep in-the-money and
out-of-the-money options
Financial Risk Forecasting © 2011,2017 Jon Danielsson, page 36 of 45
Introduction Bonds Options Duration VaR Delta VaR Summary
Gamma for the one month option
Stock price
Opt
ion
pric
e
$0
$5
$10
$15
$20
$80 $90 $100 $110 $120
0.010
0.012
0.014
0.016
0.018
0.020
Gam
ma
Financial Risk Forecasting © 2011,2017 Jon Danielsson, page 37 of 45
Introduction Bonds Options Duration VaR Delta VaR Summary
Numerical example
• Consider an option that expires in six months (T = 0.5)with strike price X = 90, price P = 100 and 20% volatility
• Let r = 5% be the risk-free rate of return
• The call delta is 0.8395 and the put delta is −0.1605
• The gamma is 0.01724
Financial Risk Forecasting © 2011,2017 Jon Danielsson, page 38 of 45
Introduction Bonds Options Duration VaR Delta VaR Summary
Delta-normal VaR
• We can use delta to approximate changes in the optionprice as a function of changes in the price of theunderlying
• Denote daily change in stock prices as:
dP = Pt − Pt−1
• The price change dP implies that the option price willchange approximately by
dg = gt − gt−1 ≈ ∆dP = ∆(Pt − Pt−1)
where ∆ is the option delta at time t − 1; and g is eitherthe price of a call or put
Financial Risk Forecasting © 2011,2017 Jon Danielsson, page 39 of 45
Introduction Bonds Options Duration VaR Delta VaR Summary
• Simple returns on the underlying are
Rt =Pt − Pt−1
Pt−1
and following the BS assumptions, they are IID-normalwith daily volatility σd :
Rt ∼ N(
0, σ2d
)
• The derivation of VaR for options parallels the one forsimple returns in Chapter 5
Financial Risk Forecasting © 2011,2017 Jon Danielsson, page 40 of 45
Introduction Bonds Options Duration VaR Delta VaR Summary
Delta-normal VaRDerivation of VaR for options
• Denote Varo(p) as the VaR of an option, where p isprobability:
p =Pr (gt − gt−1 ≤ −VaRo(p))
=Pr (∆ (Pt − Pt−1) ≤ −VaRo(p))
=Pr (∆Pt−1Rt ≤ −VaRo(p))
=Pr
(
Rt
σd
≤ − 1
∆
VaRo(p)
Pt−1σd
)
Financial Risk Forecasting © 2011,2017 Jon Danielsson, page 41 of 45
Introduction Bonds Options Duration VaR Delta VaR Summary
Delta-normal VaRDerivation of VaR for options
• Now it follows that the VaR for holding an option on oneunit of the asset is:
VaRo(p) ≈ −|∆| × σd × Φ−1R(p)× Pt−1
• This means that the option VaR is simply δ multiplied bythe VaR of the underlying, VaRu:
VaRo(p) ≈ |∆|VaRu(p)
• We need absolute value because we may have put or calloptions and VaR is always positive
Financial Risk Forecasting © 2011,2017 Jon Danielsson, page 42 of 45
Introduction Bonds Options Duration VaR Delta VaR Summary
Quality of Delta-normal VaR
• The quality of this approximation depends on the extentof nonlinearities
• Better for shorter VaR horizons
• For risk management purposes, poor approximation ofdelta to the true option price for large changes in theprice of the underlying is clearly a cause of concern
Financial Risk Forecasting © 2011,2017 Jon Danielsson, page 43 of 45
Introduction Bonds Options Duration VaR Delta VaR Summary
Delta and Gamma
• We can also approximate the option price by thesecond-order expansion, Γ
• Since dP is normal, (dP)2 is chi-squared
• The same issues apply here as for bonds: Adding higherorders increases complexity a lot, without eliminating bias
Financial Risk Forecasting © 2011,2017 Jon Danielsson, page 44 of 45
Introduction Bonds Options Duration VaR Delta VaR Summary
Summary
• We have seen that forecasting VaR for options and bondsis much more complicated than for basic assets like stocksand foreign exchange
• The mathematical complexity in this chapter is not high,but the approximations have low accuracy
• To obtain higher accuracy the mathematics become muchmore complicated, especially for portfolios
• This is why the Monte Carlo approaches in Chapter 7 arepreferred in most practical applications