4 Maximizing Growth Rate under the Threat of a Crash: An arbitrary number of crashes and other refinements
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Part Five advanced topics
Theorem 7 ‘Best portfolio under the threat of at most n crashes’
The optimal strategy in the log-utility case if at most n crashes of size k ∈ 0, k ∗ can occur is
given by the unique solution(s) of the differential equation(s)
1
π˙ˆ j (t) = ∗ 1 − πˆ j (t)k ∗
k
πˆ j (t) (µ − r) −
1
2
πˆ j (t)2 σ 2 + πˆ j −1 (t)2 σ 2
,
πˆ j (T ) = 0
with
1
µ−r
, j = 1, . . . , n, πˆ 0 (t) =
∗
k
σ2
0 ≤ πˆ j (.) <
where πˆ j (.) is the optimal portfolio process for the worst-case problem if at most j crashes
can occur.
Proof
We give an induction on n, the maximum number of crashes. For the case of n = 1 (and
also n = 0) all the claims follow from Theorem 3. Now that the above assertions are already
proved for n − 1 and that πˆ n−1 (.) is the corresponding equilibrium strategy (i.e. all future time
instant yield binding constraints simultaneously for the worst case problem), then, by induction,
vn−1 (t, x) the corresponding value function is given by
vn−1 (t, x) = log(x) + r (T − t) + log 1 − πˆ n−1 (t)k ∗
T
(µ − r) πˆ n−2 (s) − 12 πˆ n−2 (s)2 σ 2 ds
+
t
1
0.9
0.8
0.7
0.6
pi
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p*
pi_tilde
1_Crash_pi
2_Crash_pi
3_Crash_pi
0.5
0.4
0.3
0.2
0.1
0
0
2
4
6
8
10
Time
Figure 67.3 Optimal portfolios with and without crash possibility (large time horizon, at most three
crashes).
asset allocation under threat of a crash Chapter 67
(where this only holds for n − 1 > 0). The rest of the proof is now totally similar to that of
Theorem 3 with an obvious change of notation.
In Figure 67.3 we illustrate the n-crash-situation, showing the situation for the 3-crashsituation with the same data as used for producing Figure 67.2. Of course, the more crashes
possibly to come the less is the optimally invested fraction of wealth into the risky stock.
67.4.2
Changing Volatility After a Crash
It is a common phenomenon that after a crash has happened the volatility has the tendency to
increase. For our worst-case problem this has the consequence that the ‘starting value function’
v0 (t, x) has to be computed with a different value of σ . In particular, in the n-crash case
it might be necessary to calculate all value functions with different values of σ where they
are valid.
67.4.3
Further Possible Reﬁnements
There are still a lot of possible problems in the above setting which are worth considering and
which might be the subject of future research:
•
•
•
•
inclusion of the possibility for consumption;
explicit solution of the problem for a general utility function;
inclusion of liquidity constraints;
additional consideration of derivatives for portfolio insurance.
67.5
SUMMARY
It is particularly pleasing when a model produces results that tie in with intuition. We see that
in this model since it recommends reducing exposure to the risky investment as our ‘retirement’
approaches.
FURTHER READING
• See Korn (1997) for the maths of optimal portfolios.
• For further work in this area see Korn (2003, 2005), Korn & Menkens (2004a,b).
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CHAPTER 68
interest-rate modeling
without probabilities
In this Chapter. . .
•
the Epstein–Wilmott model, a non-Brownian motion
model for interest rates:
• how to value interest-rate products in a worst-case
scenario
• the Yield Envelope
• optimal static hedging
68.1
INTRODUCTION
The two main classical approaches to pricing and hedging ﬁxed income products may be
termed ‘yield-based’ and ‘stochastic.’ The former (see Chapter 13) assumes that interest rates
are constant for each product, which, of course, is inconsistent across products. These ideas
are used a great deal for the simpler, ‘linear’ products. The latter approach (see Chapters 30
and 35) assumes that interest rates are driven by a number of random factors. It is used for
‘non-linear’ contracts, contracts having some form of optionality. In the stochastic models an
equation for the short-term rate will give as an output the whole yield curve.
Both of these approaches can be criticized. The yield-based ideas are not suited to complex
products and the popular stochastic models are inaccurate. For the latter, it is extremely difﬁcult
to estimate parameters, and after estimating them, they are prone to change, making a mockery
of the underlying theory. One of the main problems is the assumption of a ﬁnite number of
factors. From such an assumption it follows that you can delta hedge any contract with this
same number of simpler contracts. For example, in a one-factor world you can hedge one part
of the yield curve with any other part, something which is clearly not possible in practice. Is
it acceptable to hedge a six-month option on a one-year bond with a ten-year bond? Although
practitioners use common sense to get around this (they would hedge the option with the
one-year bond), this common sense is not reﬂected in the modeling.
In this chapter I address the problem from a new perspective, by assuming as little as possible
about the process underlying the movement of interest rates. I will model a short-term interest
rate and price a portfolio of cashﬂows in a worst-case scenario, using the short rate as the
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Part Five advanced topics
rate for discounting. One of the key features of the model in this chapter is that delta hedging
plays no important role. The resulting problem is non linear and thus the value of a contract
then depends on what it is hedged with. This approach necessarily correctly prices traded
instruments; no ﬁtting is necessary. I also describe the Yield Envelope. This is a sophisticated
version of the yield curve. We ﬁnd a yield spread at maturities for which there are no traded
instruments.
68.2 WHAT DO I WANT FROM AN INTEREST RATE MODEL?
Here is my list of properties of my ideal interest rate model.
•
•
•
•
•
•
As few factors as possible, but able to model any realistic yield curve
Easy to price many products quickly
Insensitivity of results to hard-to-measure parameters, such as volatilities and correlations
Robustness in general
Sensible ﬁtting to data
Strategy for hedging
This list has been built up after conversations with many practitioners. In this chapter I
describe a model that delivers all of these and more. In fact, we won’t be seeing any mention
of volatilities or correlations, or delta hedging. The only hedging will be entirely static.
68.3 A NON-PROBABILISTIC MODEL FOR THE BEHAVIOR
OF THE SHORT-TERM INTEREST RATE
Motivated by a desire to model the behavior of the short-term interest rate, r, with as much
freedom as possible, I assume only the following constraints on its movement:
r− ≤ r ≤ r+
(68.1)
and
c− ≤
dr
≤ c+ .
dt
(68.2)
Equation (68.1) says that the interest rate cannot move outside the range bounded below by
the rate r − and above by the rate r + . Equation (68.2) puts similar constraints on the speed of
movement of r. The constraints can be time-dependent and, in the case of the speed constraints,
functions of the spot interest rate, r.
There is an obvious difference between the classical stochastic models of the spot rate, with
their Brownian motion evolution of r and locally unbounded growth, and the model I am now
presenting. I can justify this on several grounds: (i) We are perhaps trying to model a longterm behavior for which we are less concerned about the very short-term movements (a weak
justiﬁcation); (ii) The Brownian models can also be criticized. It is still an open question exactly
what the stochastic process is that underlies the evolution of ﬁnancial quantities; Brownian
motion is often chosen for its nice mathematical properties (a slightly better justiﬁcation);
interest-rate modeling without probabilities Chapter 68
(iii) A combination of the model here together with bands, jumps etc. discussed later will, in
practice, be indistinguishable from the real process (excellent justiﬁcation, if it’s true).
The worst-case scenario that we will be addressing is hard to criticize as long as it gives decent
prices. Why then don’t we simply present an interest rate version of the uncertain volatility
model of Chapter 52. This has been done by Lewicki & Avellaneda (1996) in a Heath, Jarrow &
Morton framework. It is not entirely satisfactory because of the usual problem: What we model
and what we trade are two different things. If we are going to use a delta-hedging argument,
then we have to assume that what we hedge with is perfectly correlated with our contract; this
can never be the case in the ﬁxed-income world. Different points on the yield curve may be
correlated but they are certainly not perfectly correlated for all time. In the model we present
here there is no delta hedging and we do not depend on any correlation between different parts
of the yield curve.
68.4
WORST-CASE SCENARIOS
AND A NON-LINEAR EQUATION
In this section we derive the equation governing the worst-case
price of a ﬁxed-income portfolio, ﬁrst presented by Epstein &
Wilmott (1997). Let V (r, t) be the value of our portfolio when
the short-term interest rate is r and the time is t. We consider
the change in the value of this portfolio during a time step dt.
Using Taylor’s theorem to expand the value of the portfolio
for small changes in its arguments, we ﬁnd that
V (r + dr, t + dt) = V (r, t) +
∂V
∂V
dr +
dt + · · · .
∂r
∂t
Note that there is no second r-derivative term because the process is not Brownian. We want
to investigate this change under our worst-case assumption. This change is given by
min(dV ) = min
dr
dr
∂V
∂V
dr +
dt .
∂r
∂t
Since the rate of change of r is bounded according to (68.2), we ﬁnd that
min(dV ) = min
dr
dr
∂V
∂V
dr +
dt
∂r
∂t
= c
∂V
∂r
∂V
∂V
+
∂r
∂t
dt
where
c(x) =
c+
c−
for x < 0
for x > 0.
We shall require that, in the worst case, our portfolio always earns the risk-free rate of
interest. This gives us
∂V
∂V
+c
∂t
∂r
∂V
− rV = 0.
∂r
(68.3)
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Part Five advanced topics
This is a ﬁrst-order non-linear hyperbolic partial differential equation. For an instrument
having a known payoff at maturity we will know the ﬁnal data V (r, T ). Also, if there is a cash
ﬂow K at time ti , then we have
V (r, ti− ) = V (r, ti+ ) + K.
Thus we solve backwards in time from T to the present, applying jump conditions when
necessary.
In addition to the worst-case scenario, we can ﬁnd the value of our portfolio in a best-case
scenario. This is equivalent to a worst-case scenario where we are short the portfolio. We
therefore have
−Vbest = (−V )worst .
68.4.1
Let’s See That Again in Slow Motion
Now we’ve done the math, let’s see that again with some numbers.
Suppose we hold a contract that, depending on the evolution of interest rates over the next
dt time step, will be worth 1.03, 1.02, 1.015, 1.01 or 1.02. This is illustrated in Figure 68.1.
The short-term interest rate is currently 8%. What is the worst that can happen? Clearly, the
worst is for the value of 1.01 to be realized, since that is the lowest of all the possible values
after the next time step. So we assume that’s what happens. In that case, what is the value of
the contract now? PVing the future value gives
?=
1.01
.
1 + 0.08 dt
If the time step is one day this is just 1.0097.
1.03
1.02
?
1.015
1.01
1.02
Figure 68.1 Five possible outcomes.
Example
Our ﬁrst ‘proper’ example will be the simplest possible: We will value a zero-coupon bond
in the worst and best cases according to the model. Results are shown in Figure 68.2 In this
example I have valued a zero-coupon bond in the two scenarios, best- and worst-cases, and
plotted the yield for different maturities. The spot rate is initially 6%, is allowed to grow or
decrease at 4% p.a. at most, and cannot go outside the range 3–20%. The important point to
interest-rate modeling without probabilities Chapter 68
20
18
16
14
Yield
12
10
8
6
4
2
0
0
2
4
6
8
10
T
Figure 68.2 Yields in worst- and best-case scenarios for a zero-coupon bond.
0.2
0
1
0.15
2
Maturity
0.1
4
6
0.05
8
20%
18%
15%
12%
8%
5%
3%
10%
0
10
Spot interest rate
Figure 68.3 Yields in worst-case scenario for a zero-coupon bond.
note about this ﬁgure is how wide the yield spread is. If the model always gave such extreme
results then it would be of no practical use. Fortunately we can reduce this dramatically by static
hedging, and we will see examples of this shortly. This example is shown again in Figures 68.3
and 68.4. In these are shown the yields in the two cases but now against both the spot rate
and maturity.
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Part Five advanced topics
0.25
0
0.2
1
2
Maturity
0.15
4
0.1
6
0.05
8
20%
18%
12%
8%
10%
5%
15%
0
10
3%
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Spot interest rate
Figure 68.4 Yields in best-case scenario for a zero-coupon bond.
68.5 EXAMPLES OF HEDGING: SPREADS FOR PRICES
Since our uncertain interest rate model solely places bounds on the short-term interest rate, it
is not surprising that the best we can do is to ﬁnd bounds for the value of a contract. We have
derived a partial differential equation for the value of a contract in a worst-case scenario, a
lower bound, and for the value in the best case, an upper bound. We therefore ﬁnd a spread
for the possible price of a contract. Consequently, long and short positions in a contract have
different values.
Finding a spread for prices is not necessarily a disadvantage of the model. After all, the
market itself has such a property (the bid-offer spread). In some sense, spreads are therefore a
more realistic result than a single price. However, it becomes a disadvantage when the spreads
are so large that the result becomes meaningless. We require a method to reduce large spreads
to more sensible levels; this is the process of static hedging.
Typically, we ﬁnd that the spread between the worst- and best-case values is larger than
the market spread. To reduce the spread, we hedge with market-traded instruments. Hedging is
shown schematically in Figure 68.5. At the top of this ﬁgure is our original, or ‘target’ contract.
Some of the cashﬂows are known amounts and some are ﬂoating. Below this are shown the
contracts that are available for hedging; to keep things simple I’ve shown them all as zerocoupon bonds. How many of each of these available hedging bonds should we buy or sell to
give our target cashﬂow the highest value in the worst-case scenario?
There is an optimal static hedge for which the worst-case value of the bond is as high as
possible, and another for which the best-case value of the bond is as low as possible. This
optimization technique was described in detail in another context in Chapter 60. To ﬁnd this
optimal static hedge in the worst-case scenario, we maximize the value of our zero-coupon
bond with respect to the hedge quantities of the hedging instruments. In the best-case scenario,
we minimize with respect to the hedge quantities.
We expect that the market price of a hedging instrument is contained in the spread of values
for the instrument generated by our model. If this were not the case, we could make an arbitrage
interest-rate modeling without probabilities Chapter 68
Original Contract:
Hedging Contracts:
Figure 68.5 A schematic diagram of the hedging problem.
proﬁt by selling (buying) the instrument at a price above (below) its maximum (minimum)
possible value, assuming that the interest rate moves within the constraints of our model.
Observe how the nonlinearity in the model means that the value of a portfolio depends
on what it is hedged with. This means that the ‘ﬁtting’ that we saw in Chapter 31 becomes
irrelevant. In fact, we are not concerned with the market prices of traded instruments except
in so far as we exploit these instruments for hedging. We never say that we ‘believe’ market
prices are ‘correct’ only that they tell us how much we must pay to get a particular cashﬂow.
68.5.1
Hedging with One Instrument
Consider a contract consisting of a set of cash ﬂows. We wish to
value this contract in a worst-case scenario. Suppose that there
exists a market-traded instrument, with known market price (a
zero-coupon bond, for instance). We hedge with this instrument
and price the resulting portfolio in a worst-case scenario. The
value of the overall portfolio is
VALUE(contract + hedging instrument).
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Part Five advanced topics
where ‘VALUE’ means the solution of the non-linear partial differential equation,
Equation (68.3), with relevant ﬁnal and jump conditions.
The cost of setting up this static hedge is equal to the current market value of the hedging
instrument. The marginal value of our hedged contract is therefore the value of the overall
portfolio minus the cost of the static hedge,
MARGINAL VALUE(hedged contract) = VALUE(contract + hedging instrument)
− cost of hedge.
Example 1
Let’s try to hedge a ﬁve-year zero-coupon bond with a one-year zero-coupon bond. The latter
has a market price of $0.90484, the spot rate is currently 10%, the interest rate range is 3–20%,
and the spot rate cannot change at a rate faster than 4% p.a., either up or down. In Figure 68.6
are shown the worst- and best-case values for the ﬁve-year bond as a function of the number
of one-year bonds with which it is hedged. First look at the lower curve, the worst-case value.
Note that it has a maximum. To get the most from the ﬁve-year bond in the worst case it must
be hedged with −2.15 of the one-year bond. If you hold the ﬁve-year bond short then to get
the best out of it you must hedge it with 2.29 of the one-year bond.
1.4
Value
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1.2
Best case
1
0.8
0.6
Worst case
0.4
0.2
−20
−15
−10
−5
0
0
5
10
15
20
Number of bonds
Figure 68.6 The value of the hedged ﬁve-year bond in the worst and best cases as a function of
the number of one-year bonds with which it is hedged.