Download Using Mean Reversion Skew andJoint Measure

Using Mean Reversion Skew and
Joint Measure Calibration to
Model Interest Rates over
Long Time Horizons
Alexander Sokol*
Head of Quant Research, CompatibL
*Includes material from a working paper by Hull, Sokol, and White
available from SSRN at http://ssrn.com/abstract=2403067.
Barrie & Hibbert Seminar, Edinburgh
May 12, 2014
Alexander Sokol (CompatibL)
Using Mean Reversion Skew and Joint Measure Calibration to Model Interest Rates over Long Time Horizons
1 / 49
1. Introduction
1.
2.
3.
4.
5.
6.
Introduction
Joint Measure Model
Calibrating the Market Price of Risk
Mean Reversion Skew
The Effect on FX Scenarios
Summary
Introduction
Alexander Sokol (CompatibL)
Using Mean Reversion Skew and Joint Measure Calibration to Model Interest Rates over Long Time Horizons
2 / 49
1. Introduction
The recent rise in importance of quantile-based risk
measures at long time horizons
• Valuation of long dated hybrid derivatives (up to 30y)
• Model is constructed of the specific instrument and calibrated to related
liquid vanilla instruments - the model works mostly as an interpolation
mechanism between market prices of calibration instruments
• Exotic trade is hedged at least until the last liquid point, giving the model risk
to the market as a whole (as long as the market uses similar models)
• XVA, PFE, limits, liquidity, insurance, pensions ...
• We do not calibrate to "European CVA" in order to calculate "Bermudan
CVA" - the model is no longer just an interpolation
• Market sensitivities of XVA are not hedged or not hedged completely - we
can no longer give model risk to anyone
• In the presence of wrong way risk, modeling of high quantiles is especially
important, because this is where the counterparty is most likely to default
Alexander Sokol (CompatibL)
Using Mean Reversion Skew and Joint Measure Calibration to Model Interest Rates over Long Time Horizons
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1. Introduction
The importance of measure choice in long term
simulation
• We live in the real world and think in terms of real world probabilities.
• When a risk manager wants to know the probability of 1bn cash outflow
between 10y and 11y from now, they want to know the real probability,
not the risk neutral probability.
• How different are these two probabilities? This is an example from the
interest rate world.
• Market price of risk is the reason why long term yields are higher than short
term yields.
• In risk neutral measure, we advance toward the forward.
• If we simulate in risk neutral measure, our expected short rate in 30y could
be 8% while in real world measure it could be 4 % (average short rate)
• Would a risk manager consider acceptable an error of 4% in interest rate
level when modeling cashflows of an interest rate portfolio?
Alexander Sokol (CompatibL)
Using Mean Reversion Skew and Joint Measure Calibration to Model Interest Rates over Long Time Horizons
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2. Calibration in Real World Measure
1.
2.
3.
4.
5.
6.
Introduction
Joint Measure Model
Calibrating the Market Price of Risk
Mean Reversion Skew
The Effect on FX Scenarios
Summary
Alexander Sokol (CompatibL)
Using Mean Reversion Skew and Joint Measure Calibration to Model Interest Rates over Long Time Horizons
5 / 49
2. Calibration in Real World Measure
Let’s start from the basics
• What is the reason we are using risk neutral measure?
• Prices of contingent claims are affected by both risk factor probabilities
and the excess return investors demand for taking risk.
• If we know both the probability and the excess return over the risk free rate,
we can price contingent claims
• Conversely, if we know the prices of contingent claims, we can back out the
product of probability and the extra discount factor due to the excess return,
but only in combination with each other
• We cannot derive them separately
• Because this combined probability would have been the real world
probability if the investors did not ask for additional return for taking risk,
it came to be called "risk neutral probability".
Alexander Sokol (CompatibL)
Using Mean Reversion Skew and Joint Measure Calibration to Model Interest Rates over Long Time Horizons
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2. Calibration in Real World Measure
When do we need the real world probability
• We do not need it for valuation or XVA because these are prices (under
certain assumptions).
• We do not need it for market risk but only because of the short time
horizon - at short timescales the volatility is much higher than the drift, so
the drift is a small correction.
• However, once we encounter the need to compute a quantile at a long
time horizon, we are no longer able to do this in the risk neutral measure.
Alexander Sokol (CompatibL)
Using Mean Reversion Skew and Joint Measure Calibration to Model Interest Rates over Long Time Horizons
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2. Calibration in Real World Measure
Specific calculations which cannot be done in risk
neutral measure
1
MPFE - maximum potential future exposure is a popular measure of
counterparty risk used in limit management and in evaluating the need
for collateral.
2
Liquidity - maximum cash outflows in a given time interval at a given
confidence level is used to manage liquidity.
3
Insurance - modeling insurance reserves requires real world, not risk
neutral probabilities.
4
Pensions - Pension funds are required to model the probability of a
shortfall in reserves.
Generally any risk measure based on a quantile for a long time horizon
requires real world measure modeling.
5
• Market risk does not fall under this definition because of its short time
horizon. For the time horizons of 1d or 10d, the difference between the
measures is minor.
Alexander Sokol (CompatibL)
Using Mean Reversion Skew and Joint Measure Calibration to Model Interest Rates over Long Time Horizons
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2. Calibration in Real World Measure
Properties of the real world model
• A simple argument based on the Girsanov’s theorem shows that volatility
is the same in both measures. When modeling in real world measure, vol
calibration can be historical or market implied.
• Of course the spot (today’s) values of risk factors are also the same in
both measures.
• The main problem with real world calibration arises when modeling long
term expectations of risk factors.
• There is no guarantee that at any given point of time in the future, however
distant, the interest rate will be exactly at its long term average level. Even if
there is an equilibrium distribution of rates within the model, the real world
rate will fluctuate around its long term average.
• We cannot simply draw our rate samples from this equilibrium distribution,
because for intermediate horizons we know that the rate will not yet reach
equilibrium.
• In risk neutral measure, we can calibrate these future expectations to the
market prices of long term swaps and bonds, which are highly liquid.
• To do the same in real world measure, we need to know the market
price of risk.
Alexander Sokol (CompatibL)
Using Mean Reversion Skew and Joint Measure Calibration to Model Interest Rates over Long Time Horizons
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2. Calibration in Real World Measure
Can we find a non-subjective calibration method for
the market price of risk
• Joint measure model
• We need a model which expressly incorporates the market price of risk and
can produce expectations in both measures - a joint measure model
• This will permit us to include market implied long term swap rates and bond
yields into the calibration procedure to determine the expected values of risk
factors at long time horizons.
• Market implied data plus one additional non-price input
• The market price of risk cannot be calibrated to market prices alone.
• This means we need one more calibration input, in addition to market prices,
to fix the market price of risk
• How should we choose this additional input?
Alexander Sokol (CompatibL)
Using Mean Reversion Skew and Joint Measure Calibration to Model Interest Rates over Long Time Horizons 10 / 49
3. Joint Measure Model
1.
2.
3.
4.
5.
6.
Introduction
Joint Measure Model
Calibrating the Market Price of Risk
Mean Reversion Skew
The Effect on FX Scenarios
Summary
Constructing a single forward looking model which can calculate both Qmeasure and P-measure distributions (the joint measure model).
Alexander Sokol (CompatibL)
Using Mean Reversion Skew and Joint Measure Calibration to Model Interest Rates over Long Time Horizons 11 / 49
3. Joint Measure Model
Backward vs. forward looking calibration
• We will distinguish a model calibrated to the historical data (backward
looking) and a model calibrated to the market’s expectations of the future
evolution of risk factors (forward looking).
• We can use market implied inputs (including swap rates and treasury
yields) to calibrate the long term drift in real world measure, if we have
one additional calibration input which is not a price.
• This one input can be calibrated to the historical data or to a forecast. I
will provide one example of each, but first we need to construct the
model.
Alexander Sokol (CompatibL)
Using Mean Reversion Skew and Joint Measure Calibration to Model Interest Rates over Long Time Horizons 12 / 49
3. Joint Measure Model
Outline of the approach
• When constructing a risk neutral model, one begins from the change of
measure
• When constructing a real world model, the traditional approach is not to
change the measure at all and continue working in real world
probabilities.
• Because the real world model cannot be calibrated to market implied prices
(only the historical or forecast based calibration methods are possible), it
cannot produce the same valuation as the risk neutral model.
• What we need instead is a joint measure model - the model which can
compute both risk neutral and real world probabilities.
• The key requirement to the joint measure model is that its "risk neutral
side" does not depend on non-price inputs used to determine the market
price of risk, and matches the prices obtained in a standard risk neutral
model.
• The joint measure model is always an extension of the corresponding risk
neutral model.
• Pricing and XVA do not require using the "real world side" of the joint
measure model.
Alexander Sokol (CompatibL)
Using Mean Reversion Skew and Joint Measure Calibration to Model Interest Rates over Long Time Horizons 13 / 49
3. Joint Measure Model
Outline of the approach
• To implement this approach, we will specify the SDEs for the model’s risk
factors in both measures.
• The two equations work with the same state variables, but result in
different stochastic process in each measure.
• Because the model’s state variable(s) are the same, we can then take
market prices from the risk neutral side of the model for all combinations
of state variables, and assign them to the same state variables in the real
world side.
• This closes the loop and creates a single model which is capable of
calculating both risk neutral and real world probability distributions of
prices.
• Note that the prices themselves are always risk neutral, but their
distribution can be obtained in both measures.
• In risk neutral measure for valuation and XVA
• In real world measure for PFE and liquidity
Alexander Sokol (CompatibL)
Using Mean Reversion Skew and Joint Measure Calibration to Model Interest Rates over Long Time Horizons 14 / 49
3. Joint Measure Model
Model selection
• This approach can be applied to a variety of interest rate models,
including short rate models (one and two factor) and yield curve models.
• Note that for each risk driver of the model, we will have a separate
market price of risk.
• When considering model choice, we should keep in mind that the market
price of risk is difficult to calibrate.
• Increasing model complexity could cause parameter estimation problems for
the real world side of the model.
• In this presentation, we will focus on one and two factor SDE for the short
rate.
• For one factor SDE, this includes such models as HW1F, CEV and CIR++.
• For two factor SDE, this includes HW2F and G2++.
• Including smile/skew and extensions to stochastic volatility are possible and
straightforward to implement.
Alexander Sokol (CompatibL)
Using Mean Reversion Skew and Joint Measure Calibration to Model Interest Rates over Long Time Horizons 15 / 49
3. Joint Measure Model
Introducing the market price of risk into a short rate
model
• We will illustrate the approach using a one factor short rate SDE.
• When all interest rates are driven by the a single risk factor as is the case
in a single factor short rate model, a riskless portfolio can be constructed
from bonds of different maturity.
• We will have as many of these combinations as the model has stochastic
shocks, e.g. in case of two factor SDE we will only be able to construct a
riskless portfolio from a security and two bonds, not one.
• Because for a single stochastic factor everything is 100% correlated with
a single risk driver, we can follow the exact same argument as one does
in the textbook derivation of risk neutral valuation for derivatives
contingent on the price of a single stock.
• The result of the risk aversion is that in real world measure the short rate
has additional drift term proportional to the volatility:
dr (RW ) = dr (RN) + λσdt
• We are now ready to specify the joint measure model.
Alexander Sokol (CompatibL)
Using Mean Reversion Skew and Joint Measure Calibration to Model Interest Rates over Long Time Horizons 16 / 49
3. Joint Measure Model
One factor joint measure SDE for the short rate
• In risk neutral measure, the short rate follows this process:
dr = µ(r , t)dt + σ(r , t)dz
This is the "risk neutral side" of the joint measure model.
• In the real world measure, we have the same volatility but there is an
additional drift term:
dr = (µ(r , t) + λ(r , t)σ(r , t))dt + σ(r , t)dz
This is the "real world side" of the joint measure model.
• The parameter λ(r , t) is the market price of risk, which in the most
general form of the model should be an additional stochastic variable.
• By convention, λ < 0 if the investors are risk averse.
• Because the two "sides" of the joint measure model are referencing the
same state variable (the short rate r ), we can compute market prices as
a function of r using the risk neutral "side" and then assign them to the
same value of r on the in real world "side".
Alexander Sokol (CompatibL)
Using Mean Reversion Skew and Joint Measure Calibration to Model Interest Rates over Long Time Horizons 17 / 49
3. Joint Measure Model
Two factor joint measure SDE for the short rate
• In a two factor SDE, each of the two stochastic risk drivers dz1,2 has its
own market price of risk λ1,2 .
• We will use symmetric (G2++) definition of the two factor model:
dx1,2 = µ1,2 (x1 , x2 , t)dt + σ1,2 (x1 , x2 , t)dz1,2
r = φ(t) + x1 + x2
This is the "risk neutral side" of the two factor joint measure model.
• In the real world measure, we have the same volatility but there is an
additional drift term for each x1,2 .
dx1,2 = (µ1,2 (x1 , x2 , t) + Λ1,2 (x1 , x2 , t))dt + σ1,2 (x1 , x2 , t)dz1,2
r = φ(t) + x1 + x2
Λ1,2 (x1 , x2 , t) = λ1,2 (x1 , x2 , t)σ1,2 (x1 , x2 , t)
This is the "real world side" of the two factor joint measure model.
Alexander Sokol (CompatibL)
Using Mean Reversion Skew and Joint Measure Calibration to Model Interest Rates over Long Time Horizons 18 / 49
3. Joint Measure Model
State variables for the two factor SDE
• The method used to obtain real world distribution of risk neutral prices
requires a slight modification for the two factor SDE because we now
have two state variables.
• The short rate no longer represents the full model state.
• This means we can no longer assign prices from one model to the other
model based on the short rate alone.
• Instead, we can use the pair consisting of the short rate and one long
rate with maturity T which satisfies the following equation:
1/a1 T 1/a2
where a1,2 is the reversion speed of x1,2 .
• With the values of mean reversion a1,2 typically found in two factor model
calibration, the 10y rate is usually a good choice for the state variable.
Alexander Sokol (CompatibL)
Using Mean Reversion Skew and Joint Measure Calibration to Model Interest Rates over Long Time Horizons 19 / 49
4. Calibrating the Market Price of Risk
1.
2.
3.
4.
5.
6.
Introduction
Joint Measure Model
Calibrating the Market Price of Risk
Mean Reversion Skew
The Effect on FX Scenarios
Summary
The market price of risk cannot be calibrated to market implied prices alone.
We will discuss two possible additional inputs permitting to calibrate λ, one producing the historical average value of λ and the other producing λ conditional
on the level of the short rate.
Alexander Sokol (CompatibL)
Using Mean Reversion Skew and Joint Measure Calibration to Model Interest Rates over Long Time Horizons 20 / 49
4. Calibrating the Market Price of Risk
Estimating λ from the shape of the yield curve
• The procedure of estimating the market price of risk from the slope of the
short end of the curve is well known (Stanton, Cox and Pedersen,
Ahmad and Wilmott) and can be summarized as follows.
• We know that the yield curve is upward sloping most of the time.
• This means that in risk neutral measure the short rate has upward drift.
• In risk neutral measure, we "advance toward the forward".
• Of course, in real world measure the drift averaged over a sufficiently
long time horizon must be zero.
• Unlike stocks or FX, the interest rates move within a stable range.
• The average yield curve slope at t → 0 then makes it possible to
estimate λ.
• Note that only the long term average slope can be used. At any given time,
some of the slope may arise because the traders expect the rates to move in
a certain direction.
• For example, when the yield curve is downward sloping, it does not mean
that the investors suddenly became risk seeking. It is more likely that they
simply expect the rates to fall.
Alexander Sokol (CompatibL)
Using Mean Reversion Skew and Joint Measure Calibration to Model Interest Rates over Long Time Horizons 21 / 49
4. Calibrating the Market Price of Risk
Estimates from the short end of the curve
• For the estimate of λ based on the slope of the short end of the yield
curve (maturities less than 1y were considered in all of the publications
listed on the previous slide), most authors agree on the value around
λ ∼ −1 for major currencies.
• If the normal vol is about 1%, the difference between risk neutral and real
world rate will grow ∼ 1% per year, or 30% of difference (in absolute terms)
between risk neutral and real world short rate accumulating over a 30y
horizon.
• Even if somewhat reduced by mean reversion, this is an absurd number
which shows that the value of λ ∼ −1 cannot be used over long time
horizons.
Alexander Sokol (CompatibL)
Using Mean Reversion Skew and Joint Measure Calibration to Model Interest Rates over Long Time Horizons 22 / 49
4. Calibrating the Market Price of Risk
Estimates from the regression of swap rates of
different maturity
• We propose to estimate λ from the regression of bond prices and interest
rates with maturities on the same timescale as in our model - up to 30y
horizon.
• Then the estimate is guaranteed to be reasonable, as this calibration
procedure ensures that λ generates historically observed and therefore
resonable differentials of rates of different maturity.
• The calibration procedure must take into account convexity and mean
reversion.
• Simple estimates of λ can be obtained by analytical approximation.
• Let’s look at the data to confirm that our calibration inputs do not have
excessive amount of noise.
Alexander Sokol (CompatibL)
Using Mean Reversion Skew and Joint Measure Calibration to Model Interest Rates over Long Time Horizons 23 / 49
4. Calibrating the Market Price of Risk
Regression of 30y vs. 10y swap rate
• X axis is 10y swap rate
• Y axis is 30y swap rate
• Best fit for x > 2% is y = 0.92x + 0.85%
Alexander Sokol (CompatibL)
Using Mean Reversion Skew and Joint Measure Calibration to Model Interest Rates over Long Time Horizons 24 / 49
4. Calibrating the Market Price of Risk
Regression of 30y vs. 5y swap rate
• X axis is 5y swap rate
• Y axis is 30y swap rate
• Best fit for x > 1% is y = 0.8x + 1.85%
Alexander Sokol (CompatibL)
Using Mean Reversion Skew and Joint Measure Calibration to Model Interest Rates over Long Time Horizons 25 / 49
4. Calibrating the Market Price of Risk
Regression of 10y vs. 2y swap rate
• X axis is 2y swap rate
• Y axis is 10y swap rate
• Best fit for x > 1% is y = 0.8x + 2.1%
Alexander Sokol (CompatibL)
Using Mean Reversion Skew and Joint Measure Calibration to Model Interest Rates over Long Time Horizons 26 / 49
4. Calibrating the Market Price of Risk
Could this be convexity?
• Same chart but with maximum possible convexity of all calibrations
• Also includes the line with slope 1.
• While convexity has to be taken into account, most of the difference
between rates of different maturity is due to the market price of risk
Alexander Sokol (CompatibL)
Using Mean Reversion Skew and Joint Measure Calibration to Model Interest Rates over Long Time Horizons 27 / 49
4. Calibrating the Market Price of Risk
Regression of OECD long vs. short rate in 40+
countries since 1960s
• X axis is OECD short rate, usually similar to LIBOR or CP
• Y axis is OECD long rate, usually similar to 10y treasury.
• The data is averaged within each decade, across all countries
Alexander Sokol (CompatibL)
Using Mean Reversion Skew and Joint Measure Calibration to Model Interest Rates over Long Time Horizons 28 / 49
4. Calibrating the Market Price of Risk
Dependence of the estimated market price of risk on
bond maturity using data from different time periods
• Starts at the value close to the consensus estimate from the papers
which use the short end yield curve slope to estimate
• After 5y is already close to its long term equilibrium value
Alexander Sokol (CompatibL)
Using Mean Reversion Skew and Joint Measure Calibration to Model Interest Rates over Long Time Horizons 29 / 49
4. Calibrating the Market Price of Risk
Backward-looking and forward-looking methods of
estimating λ
• The availability of clean fit to the regression of rates of different maturity
makes it possible to estimate the average λ from the first principles,
based on the historical data.
• This average can be calibrated directly to the linear regression of the swap
rates or treasury yields of different maturities around the center of the
distribution.
• We may also be interested in the estimate of the market price of risk
contingent on today’s market conditions.
• For example, if today the rates are near zero, the excess drift may not be the
same as the historical average.
• Under this approach, λ can be obtained from a nonlinear regression of the
rates of different maturity, conditional on the level of the rates today.
• We can think of the difference between these two methods the same way
as we think about the difference between historical and market implied
vols - there are backward-looking and forward-looking estimates which
are different, and both have a role in modeling.
Alexander Sokol (CompatibL)
Using Mean Reversion Skew and Joint Measure Calibration to Model Interest Rates over Long Time Horizons 30 / 49
4. Calibrating the Market Price of Risk
Projected interest rates in both measures
• Unlike the earlier estimates of λ ∼ −1 from the short end of the yield
curve, our calibration procedure produces much lower values of
λ ∼ −0.2 from swap rates or bond yields with maturities comparable with
the simulation horizon.
• These values of λ generate reasonable long term real world rate
distributions (shown below).
Alexander Sokol (CompatibL)
Using Mean Reversion Skew and Joint Measure Calibration to Model Interest Rates over Long Time Horizons 31 / 49
5. Nonlinear Reversion Speed
1.
2.
3.
4.
5.
6.
Introduction
Joint Measure Model
Calibrating the Market Price of Risk
Mean Reversion Skew
The Effect on FX Scenarios
Summary
We will show that the historical data is consistent with reversion speed which
is rapidly accelerating in a high rate regime (rate > 10 %).
Alexander Sokol (CompatibL)
Using Mean Reversion Skew and Joint Measure Calibration to Model Interest Rates over Long Time Horizons 32 / 49
5. Nonlinear Reversion Speed
Historical from OECD Stats website
• Available at no cost from stats.oecd.org
• Covers time period from 1950 to present
• Data from 40+ countries including some non-members of OECD
Alexander Sokol (CompatibL)
Using Mean Reversion Skew and Joint Measure Calibration to Model Interest Rates over Long Time Horizons 33 / 49
5. Nonlinear Reversion Speed
Historical time series for two very different sample
currencies
• USD - near zero short rate today
• In a near zero interest rate regime today
• Had previous periods of very low rates (e.g. after the dotcom crisis)
• AUD - normal levels of interest rates today
• Central bank is comfortable with high interest rates
• Short rate never below ∼ 4% for the past 40 years
• Not only for USD but also for AUD the rates exited the > 10% territory
much more rapidly than one would expect given the typical reversion
speed of 5% − 10% assumed in interest rate modeling
Alexander Sokol (CompatibL)
Using Mean Reversion Skew and Joint Measure Calibration to Model Interest Rates over Long Time Horizons 34 / 49
5. Nonlinear Reversion Speed
Cross-sectional analysis
• How can we avoid the effects of noise in estimating short term drift from
the historical data?
• We can use cross sectional analysis (averaging across currencies)
• This immediately leads to a philosophical question - how unique is a
given currency at a given point in time?
• Historical data supports the view that in most cases not so unique
• Currency crises related to high interest rate regimes are frequently
encountered in the historical record and follow the same set of patterns
Alexander Sokol (CompatibL)
Using Mean Reversion Skew and Joint Measure Calibration to Model Interest Rates over Long Time Horizons 35 / 49
5. Nonlinear Reversion Speed
Cross-sectional estimation of average realized
reversion
• Select a time lag
• Take all dates and currencies for which the short rate is available
• Group the observations into 1% buckets by rate level
• Take short rate observation with the time lag (at T+5y)
• Taking the average of the rate change between T and T+5y within each
bucket (across all currencies and dates for which the rate was within the
bucket) gives us the historical realized reversion.
Alexander Sokol (CompatibL)
Using Mean Reversion Skew and Joint Measure Calibration to Model Interest Rates over Long Time Horizons 36 / 49
5. Nonlinear Reversion Speed
Realized reversion of the short rate with 5y lag
• Starts linear for low and medium rates, then accelerates rapidly
• Best linear fit parameters in the low rate limit: reversion speed a = 6%,
target rate θ = 4%
Alexander Sokol (CompatibL)
Using Mean Reversion Skew and Joint Measure Calibration to Model Interest Rates over Long Time Horizons 37 / 49
5. Nonlinear Reversion Speed
Same calculation but with 1y lag
• More noise in the data because of the higher random shocks to drift ratio
• Despite the noise, the same effect is seen using 1y lag, with similar best
fit parameters for low rates: reversion speed a = 6%, target rate θ = 4%
• Repeating the analysis in different time periods also yields similar results
Alexander Sokol (CompatibL)
Using Mean Reversion Skew and Joint Measure Calibration to Model Interest Rates over Long Time Horizons 38 / 49
5. Nonlinear Reversion Speed
Comparing short, long, and immediate rates - similar
behavior for all three
• Short rate (e.g. LIBOR)
• Long rate (e.g. 10y bond yield)
• Immediate rate (e.g. OIS)
Alexander Sokol (CompatibL)
Using Mean Reversion Skew and Joint Measure Calibration to Model Interest Rates over Long Time Horizons 39 / 49
5. Nonlinear Reversion Speed
Model for the IR component with mean reversion
dependent on rate level
• The proposed approach will be demonstrated on the specific example
(one factor short rate SDE), however it can be applied to other model
types
• We start from SDE of the following form
dr (t) = µ(t, r )dt + σ(t, r )dw
where µ(t, r ) is the drift term which plays the role of mean reversion
• First, µ(t, r ) is calibrated such that the reversion is accelerating in the
high rate environment. The calibration target is historical record of high
interest rates, across currencies in similar economies
• As usual, σ(t, r ) can be calibrated to the option skew
Alexander Sokol (CompatibL)
Using Mean Reversion Skew and Joint Measure Calibration to Model Interest Rates over Long Time Horizons 40 / 49
5. Nonlinear Reversion Speed
Parameterization of µ(t, r )
• There are several choices of parameterization of µ(t, r (t)) with
accelerating reversion, all resulting in similar model properties
• Historical data is consistent with flat mean reversion speed (linear drift)
for low or moderate rates, and begins significantly deviating from it when
rates reach 8-10%.
• One possible choice which fits the observed data well is:
(
b(r − rt )2 for r > rt
µ(r , t) = −a(r − θ(t)) −
0 for r < rt
• One advantage of this parameterization is that near the money
calibration remains unaffected (as long as θ(t) rt for any t), and
analytical expansion in b becomes possible. The value of rt with this
approach appears to be around 8%.
• This can be easily extended to G2++ or other two factor models by
making reversion accelerate for each factor, based on the fit to short and
long rates
Alexander Sokol (CompatibL)
Using Mean Reversion Skew and Joint Measure Calibration to Model Interest Rates over Long Time Horizons 41 / 49
5. Nonlinear Reversion Speed
The effect of nonlinear reversion parameters
• In conventional IR models, reversion is linear: b = 0. Adding the
quadratic terms reduces the probability of extended periods of high rates.
• Keeping reversion linear near the target rate level θ(t) preserves the
usual volatility decay factor exp(−a(T − t)) in the limit of small vol.
• There is no reason to think that the reversion behavior of high- and
low-rate regimes is similar. This means that b should be calibrated only
for the high rates, irrespectively of what we assume about the low rate
regime.
• Models modified by making mean reversion nonlinear and accelerating
for high rates can be referred to as "+R" models, for example HW+R,
CEV+R, CIR++R etc.
Alexander Sokol (CompatibL)
Using Mean Reversion Skew and Joint Measure Calibration to Model Interest Rates over Long Time Horizons 42 / 49
5. Nonlinear Reversion Speed
Historical length of time in high rate regime
• From cross-sectional OECD rate history data, we can measure the
fraction of time the short rate was above a certain level historically
• In the historical time record, the rate which exceeds 10% comes back
down relatively quickly (except in case of hyperinflation which we will not
consider here)
Alexander Sokol (CompatibL)
Using Mean Reversion Skew and Joint Measure Calibration to Model Interest Rates over Long Time Horizons 43 / 49
5. Nonlinear Reversion Speed
Can explain this behavior with mean reversion skew?
• Same historical data as on previous chart
• Now also includes model data without mean reversion skew and with two
choices of mean reversion skew for 8% onset
Alexander Sokol (CompatibL)
Using Mean Reversion Skew and Joint Measure Calibration to Model Interest Rates over Long Time Horizons 44 / 49
6. The Effect on FX Scenarios
1.
2.
3.
4.
5.
6.
Introduction
Joint Measure Model
Calibrating the Market Price of Risk
Mean Reversion Skew
The Effect on FX Scenarios
Summary
The unrealstic quantiles for FX can be traced to the ability of interest rates to
become very high and stay high for an extended period of time under traditional
interest rate models
Alexander Sokol (CompatibL)
Using Mean Reversion Skew and Joint Measure Calibration to Model Interest Rates over Long Time Horizons 45 / 49
6. The Effect on FX Scenarios
FX drift in the risk neutral measure is the interest rate
differential
• FX drift in risk neutral measure is equal to the short rate differential
between the currencies:
dS(t)/S(t) = (rd − rf )dt + ...
where rd is the domestic short rate, and rf is the foreign short rate.
• In real world measure, the drift is also influenced by the interest rate
differential over the long term even though it can deviate from from it for
extended time periods (violation of UIP or carry trade effect).
Alexander Sokol (CompatibL)
Using Mean Reversion Skew and Joint Measure Calibration to Model Interest Rates over Long Time Horizons 46 / 49
6. The Effect on FX Scenarios
An example of the effect of extreme FX rate variation
on CVA
• This is an example where 95% quantile PFE of USD 10bn, while the
notional is USD 1bn.
• Examining this scenario shows that it represents USD collapse to 0.127
EUR, making the EUR cashflows we pay very expensive.
Alexander Sokol (CompatibL)
Using Mean Reversion Skew and Joint Measure Calibration to Model Interest Rates over Long Time Horizons 47 / 49
6. The Effect on FX Scenarios
Details of the scenario which predicted this FX rate
variation
• The interest rate differential results in FX drift
• In this simulation scenario, the normal model we use for the USD has the
short rate reaching 8%
• The normal model we use for EUR has short rate which first goes below
-2.5%, then it hovers near zero for a while and settles at 1.5%.
• Over 30 years, this results in drift pull which causes USD collapse
relative to EUR
• This effect is greatly reduced by introducing nonlinear reversion
Alexander Sokol (CompatibL)
Using Mean Reversion Skew and Joint Measure Calibration to Model Interest Rates over Long Time Horizons 48 / 49
7. Summary
1.
2.
3.
4.
5.
6.
Introduction
Joint Measure Model
Calibrating the Market Price of Risk
Mean Reversion Skew
The Effect on FX Scenarios
Summary
Alexander Sokol (CompatibL)
Using Mean Reversion Skew and Joint Measure Calibration to Model Interest Rates over Long Time Horizons 49 / 49
7. Summary
Joint measure model
• A joint measure model is proposed for calculating PFE, liquidity
measures, and other long term real world quantiles of risk neutral prices.
• The model is calibrated to market implied data with one additional input
to determine the real world probabilities - the regression of historical
swap rates or treasury yields of different maturities.
• The risk neutral side of the model depends only on market implied inputs.
• Our results indicate that the market price of risk in swaps and bonds
decreases rapidly with maturity - effectively a "term structure" of the
market price of risk.
• It is reasonable to expect that the market price of risk also depends on
the rate level (in other words, a "skew" of the market price of risk), and is
driven by its own stochastic factor.
Alexander Sokol (CompatibL)
Using Mean Reversion Skew and Joint Measure Calibration to Model Interest Rates over Long Time Horizons 50 / 49
7. Summary
Model parameterization
• The full set of parameters of the stochastic process driving λ(r , t) cannot
be obtained from the observed data.
• For this reason, we need to find the minimally suitable set of parameters
which are consistent with market observations.
• We found that λ = const is unable to fit both the short end and the long
end of the yield curve, and also yields absurd simulation results at long
time horizons.
• Assuming an effective term structure of λ(t) makes it possible to fit the
data across all maturities, and yields reasonable simulation results.
• The observed term structure of λ may be explained using a blend of
short term and long term yield curve risk drivers each with its own λ,
however the model itself does not depend on the precise way in which
the observed term structure of λ comes about.
• Note that the model can still have a single risk driver even if an effective term
structure of λ is assumed.
Alexander Sokol (CompatibL)
Using Mean Reversion Skew and Joint Measure Calibration to Model Interest Rates over Long Time Horizons 51 / 49
7. Summary
Calibration of the market price of risk
• We propose two calibration methods, one for the average historical λ,
and the other for λ conditional on today’s rate level.
• The proposed methods are based on regression of swap rates or bond
yields of different maturities, and do not rely on any pre-existing
assumptions about the form of the term structure and skew of λ.
• It is important that the calibration procedure takes into account rate
convexity when estimating λ.
• Unlike the previous estimates of λ from the short end of the yield curve
which produce unreasonable simulation results when applied over long
time horizons, the proposed calibration generates realistic real world
distributions at 30y horizon.
Alexander Sokol (CompatibL)
Using Mean Reversion Skew and Joint Measure Calibration to Model Interest Rates over Long Time Horizons 52 / 49
7. Summary
Unrealistic levels of interest rates at high quantiles
• Standard interest rate models predict that once the interest rate reaches
a high level (> 10 % ), it can stay at that level for an extended period of
time
• In historical data, the time periods of very high rates are short. In 5y after
reaching a peak, the rate typically drops by more than 50%.
• Unrealistic interest rate predictions also cause extreme variation of the
FX rate
• When one rate is high and the other is near zero or negative, the drift over a
long simulation horizon causes FX to change much more than it would
through its own idiosyncratic volatility
Alexander Sokol (CompatibL)
Using Mean Reversion Skew and Joint Measure Calibration to Model Interest Rates over Long Time Horizons 53 / 49
7. Summary
Proposed risk factor projection model with
accelerating reversion in high rate regime (+R models)
• The proposed model has nonlinear mean reversion which rapidly
accelerates in the high rate regime.
• It is most instructive to think about this not as a higher speed reversion to a
constant mean but a drift term which captures the rapid rate of departure
from a high interest rate regime which was observed historically.
• The reversion term is estimated from the historically observed speed of
departure from the high interest rate regime.
• Once non-calibrated parameters are determined, the calibration is done
in the usual way.
Alexander Sokol (CompatibL)
Using Mean Reversion Skew and Joint Measure Calibration to Model Interest Rates over Long Time Horizons 54 / 49