RISK NEUTRAL VERSUS REAL WORLD
VALUATION
Pieter de Boer
July 2009
Zanders
T +31 35 692 89 89
Postbus 221
Brinklaan 134
1400 AE Bussum
1404 GV Bussum
F +31 35 692 89 99
E info@zanders.eu
Risk neutral versus real world valuation
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Contents
1
2
Introduction ............................................................................................................3
1.1
Introduction ....................................................................................................... 3
1.2
Objective ........................................................................................................... 3
1.3
Approach ........................................................................................................... 4
1.4
Structure of this report ........................................................................................ 4
Risk neutral valuation ..............................................................................................5
2.1
What is risk neutral valuation? .............................................................................. 5
2.2
An intuitive justification of risk neutral valuation ...................................................... 5
2.3
Problems using real world valuation ....................................................................... 7
2.4
Risk neutral valuation in a continuous framework ..................................................... 7
2.5
From a real world to a risk neutral measure............................................................. 9
3
Guaranteed product ............................................................................................... 11
4
Black-Scholes-Hull-White model............................................................................. 12
5
4.1
Hull-White model .............................................................................................. 12
4.2
Black-Scholes model.......................................................................................... 14
4.3
Correlation between the interest rate and the stock price ......................................... 14
4.4
BSHW model .................................................................................................... 16
Results of the risk neutral valuation of the product ................................................. 19
5.1
6
7
Results ............................................................................................................ 19
The stochastic discount factor ................................................................................ 22
6.1
Theory on the stochastic discount factor ............................................................... 22
6.2
The stochastic discount factor in the BSHW model .................................................. 23
Results of the real world valuation of the product ................................................... 27
7.1
Results ............................................................................................................ 27
7.2
Backtest .......................................................................................................... 30
8
Summary and conclusions ...................................................................................... 34
9
References ............................................................................................................ 36
Appendix explanation used data .................................................................................. 37
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1
1.1
Introduction
Introduction
The importance of market consistent valuation (MCV) has risen in recent years throughout the
global financial industry (Gibbs & McNamara, 2007). This is due to the new regulatory landscape1
and because banks and insurers see the necessity of better understanding of the uncertainty in the
market value of their balance sheet. Part of these balance sheets on the liability side often contain
guaranteed contracts, which are difficult to value with standard valuation techniques due to the
optionality and nonlinearity embedded in the product. Valuation of such guaranteed contracts at
the present time uses the concept of risk neutral valuation, for example in Nielsen & Sandmann
(1995).
However, determining the uncertainty in the future value, for example needed for regulatory or
economic capital calculations, of such products is more difficult, because future outcomes are not
simulated correctly. For example, when using risk neutral simulations, stock prices are assumed to
grow with the risk free interest rate, which is not realistic.
Using real world simulations, future outcomes are correctly simulated, since the stock prices grow
at the actual expected return (the risk free rate combined with a risk premium). However, the
valuation of a product using a standard (risk neutral) discount factor is inconsistent, since the
returns are not risk neutral but the discount factor is.
Therefore, a combination of these two methods is needed to be able to correctly simulate future
outcomes, value the (guaranteed) products and determine the uncertainty in it‟s future value. Real
world simulations are needed to correctly simulate future values of the variables and a proper
(stochastic) discount factor is needed to value the products in the future correctly.
In this thesis the focus will be on the MCV of an equity-linked guaranteed contract. This product is
a combination of a savings product and investment product. It contains a guaranteed return, a part
of the 1-month Euribor interest rate, and the possibility of an extra investment return if the
underlying index performs above a contractually determined level.
1.2
Objective
The objective of this thesis is:
Combine risk neutral valuation and real world simulations to correctly measure the
uncertainty in future risk neutral value.
This objective is achieved by answering the following questions:
1
For example, Basel II, IFRS and Solvency II, which forces banks and insurers to report the values and risks of
products based on a market consistent valuation method.
Risk neutral versus real world valuation
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1. What is risk neutral valuation and why does valuation based on real world simulations not
result in a correct future value?
2. What is the risk neutral value of the equity linked guaranteed contract at the present time?
3. What is the uncertainty of this risk neutral value in the future, i.e. one year?
1.3
Approach
This thesis determines the risk neutral value of the product at the present time and measures the
uncertainty in the future risk neutral value. For the risk neutral valuation at the present time we
use a stochastic simulation model. This stochastic simulation model assumes a geometric Brownian
motion for the stock prices, just as in a Black-Scholes model (Black & Scholes, 1973), where the
volatility of the stock price is obtained by extracting the implied at the money (ATM) volatility of an
option on the stock.
Furthermore, a one factor Hull-White model (Hull & White, 1990) is used to simulate the interest
rate. This arbitrage free interest rate model is entirely determined by the current state of the
market. The parameters will be determined by the concept of calibration. This stochastic simulation
model is estimated in a risk neutral setting. The next step is to combine risk neutral valuation and
real world simulations to correctly measure the uncertainty in future risk neutral value. In order to
do so, real world simulations, by assuming realistic returns, are used instead of risk neutral
simulations, in the same setting as described above.
Since an appropriate discount factor is not directly available when real world simulations are used,
it‟s necessary to find a stochastic discount factor or “deflator” consistent with the real world
simulations. Using this stochastic discount factor, a valuation method equivalent2 to risk neutral
valuation can be used to determine the risk neutral value of the product at the present time and
measure the uncertainty in the future value.
1.4
Structure of this report
In the next chapter, the concept of risk neutral valuation is explained. Chapter 3 describes the
chosen guaranteed contract in detail. Chapter 4 describes the Black-Scholes-Hull-White (BSHW)
model used for the risk neutral valuation at the present time. The results of the BSHW-model are
presented in chapter 5. Chapter 6 describes the properties of deflators and their usefulness in
correctly measuring the uncertainty in future risk neutral value. This chapter describes the
incorporation of deflators in the BSHW model. Chapter 7 contains the outcomes of the model
described in the preceding chapter. Finally, chapter 9 gives a conclusion. The appendix discusses
the way the parameters are obtained.
2
I.e., not exactly the same, since the calculations are different, but the outcomes are identical for the numbers
of simulations approaching infinity.
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2
Risk neutral valuation
This chapter explains the methodology of risk neutral valuation. First, section 2.1 explains what
risk neutral valuation is, then a justification is given in section 2.2, illustrated by a simple example.
This example is also used to clarify why real world valuation cannot be used in a straightforward
way in section 2.3. Furthermore, a simple derivation of risk neutral valuation in a continuous
framework is given in section 2.4, as well as a short elaboration on changing from a real world to a
risk neutral probability measure in section 2.5.
2.1
What is risk neutral valuation?
Risk neutral valuation is a method frequently used in derivatives pricing. “It is the single most
important tool for the analysis of derivatives” (Hull J. , 2006). Common risk neutral models are the
Black-Scholes (BS) and the Hull and White (HW) models, which are used for valuing derivatives.
Both models are arbitrage-free models, based on the assumption that no profit can be made
without being exposed to risk. The HW-model is used for pricing interest rate options, where the
BS-model is used for pricing stock options.
The underlying assumption in the BS-model is that the option price and stock price depend on the
same underlying source of uncertainty. This enables the construction of a particular portfolio that
consists of part of the stock and the option which eliminates this source of uncertainty. This
portfolio is instantaneously riskless and has to earn the risk-free rate due to the exclusion of
arbitrage opportunities (Hull J. , 2006). Using the above portfolio, the BS differential equation can
be derived.
This differential equation is independent from all parameters that reflect a risk preference. The
solution to the differential equation is derived in a risk-free world and can be used in the real
world, which leads to the principle of risk neutral valuation. Risk neutral valuation, as described
above, can be used for valuation of a portfolio at the present time.
2.2
An intuitive justification of risk neutral valuation
An example can be used for an intuitive justification of risk neutral valuation. Consider for
simplicity a two-state binomial model (Rendleman & Bartter, 1979). In this case, the current stock
price is €100 and can either go up to €200, the ‟up state of the world‟, or down to €50, „the down
state of the world‟, within one year.
Stock price today
Stock price one year
200
100
50
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Assume a call option with strike price €100 is available in the market and the risk free rate is 10%.
It is now possible to create a riskless portfolio consisting of (buying) one share and shorting a
fraction (∆) of call options, with the price of the call option equal to C. If the stock price will go up,
the payout of the portfolio is €200 - ∆ call options (C). If the stock price goes down, the payout will
be €50.
Creating a riskless portfolio means having the same payout in both states of the world. Clearly, in
the down state of the world, the payout is €50. To have the same payout in the up state, 1.5 call
options should be sold, since the payout of the option in the up state of the world is €100. If
arbitrage opportunities3 are excluded, the return of the portfolio must equal the risk free interest
rate, i.e. 10%. Since the portfolio costs 100-1.5*C at the present time and returns 50 in one year,
the value of C can be calculated:
50 − 100 − 1.5 ∙ 𝐶
= 10%
100 − 1.5 ∙ 𝐶
(1)
Solving this equation for C yields a value of € 36.36, the price of the call option. Interesting to note
is that in the calculations so far, the investor‟s risk preference and the probabilities of the stock
going up or down are not used. Thus, the fact that the option prices are calculated without any
knowledge or assumptions about the risk preference of the investor, enables one to assume risk
neutrality in computing option prices. This assumption is only made for convenience, what can be
clarified by an expansion of the previous example.
Assuming risk neutrality of the investors means that investors do not require a risk premium for
any sort of investment. Therefore, the expected return on the stock must be equal to the risk free
interest rate, 10%. Therefore, it‟s possible to calculate the probability for the stock price to go up
(p) under a risk neutral probability measure:
p ∙ 200 + 1 − p ∙ 50 = 100 ∙ 1 + 0.1
(2)
This results in a risk neutral probability of 2/5 to reach the up state of the world. The risk neutral
probability can be used to determine the price of the call option, C. The value of the call option is
€100 if the stock price goes up and 0 otherwise:
2
3
100 +
0 = 40
5
5
(3)
The expected value of the call option at the end of the year is €40 and the discount factor equals
the risk free rate. This results in a value of the call option of €36.36, the same value as calculated
using no arbitrage arguments. This means that by assuming risk neutral investors, the same values
3
An arbitrage opportunity is an opportunity that guarantees a certain riskless positive excess return.
Risk neutral versus real world valuation
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are found. One major advantage of assuming risk neutrality of the investors is that the discount
factor is equal to the risk free rate. On the contrary, it‟s not easy to find the future discount factor
in the real world, which will be illustrated in the next section. Due to this advantage, among others,
risk neutrality of the investors is an assumption often used in option valuation.
2.3
Problems using real world valuation
Assume that the expected return on the stock in the real world is 15%, i.e. the market price of risk
is 5%. Now, it is possible to calculate the probability for the stock price to go up under a real world
probability measure:
p ∙ 200 + 1 − p ∙ 50 = 100 ∙ (1 + 0.15)
(4)
This results in a value of 0.433 for the real world probability of the stock price moving up. The
expected payoff of the option at the end of the year in the real world is then:
0.433 ∙ 100 + 0.567 ∙ 0 = 43.33
(5)
Discounting this value at 15% (the expected return of the stock) results in a value of the call
option at the present time of 37.68, which is known not to be the correct value in the (real world)
market. In the preceding section, this value was found to be 36.36, by assuming no risk free
arbitrage opportunities.
This example illustrates that the proper discount rate is higher than 15% due to uncertainty, but
it‟s not straightforward what this discount factor should be. It is now clear to see that risk neutral
valuation is more convenient than real world valuation, since both the expected return of all the
assets as well as the discount rate are all known in a risk neutral world. This in contrast with real
world valuation, where it‟s difficult to determine the proper discount factor.
Above it is illustrated that the real world probabilities can not be used for proper valuation of the
option. However, these real world probabilities are needed to be able to assess the risk in holding
the option. The above is merely an illustration of the difficulties that arise when using real world
probabilities for valuation of options.
2.4
Risk neutral valuation in a continuous framework
In the preceding sections, the model was set in a discrete environment. Its purpose was to clarify
the method of risk neutral valuation, but it does not claim to be set in a realistic world. However,
risk neutral valuation can also be used in a continuous, more realistic, framework. This section
gives a justification for the use of risk neutral valuation in a continuous setting. At the same time,
the Black-Scholes-Merton differential equation is (partly) derived.
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Consider a derivative, for example a call option C, whose underlying stock price, S, follows a
geometric Brownian motion process (which in fact is the same assumption as in the Black-Scholes
model):
dSt = μSt dt + ςSt dWtS
(6)
where Ws is a Wiener process and µ and σ, respectively the expected return on the stock and the
volatility of the stock price, are constants. Now Itô‟s lemma (Ito, 1951) can be employed to
equation 6 in order to express dC, i.e. the change in the value of the call option, as (Broadie &
Detemple, 2004):
dC =
∂C
∂C
1 ∂2 C 2 2
∂C
μS +
+
ς S dt +
Sς dW
2
∂S
∂t
2 ∂S
∂S
(7)
The next step is to create a riskless portfolio consisting of the stock and the derivative. Consider a
portfolio consisting of shorting one call option and buying ∆ shares. To create a riskless portfolio,
the appropriate amount of shares is equal to
∂C
∂S
(Hull J. , 2006). The change in the created riskless
portfolio, dV, can be expressed as a combination of equations 6 and 7:
dV = −dC +
∂C
∂C
∂C
1 ∂2 C 2 2
∂C
∂C
∂C
dS = −
μS +
+
ς S dt −
Sς dWs +
μSdt +
ςS dW
∂S
∂S
∂t
2 ∂S2
∂S
∂S
∂S
(8)
where the last two terms of the right-hand side equation cancel out with its first and fourth term.
The above equation is a riskless portfolio and due to the assumption of no arbitrage, it has to earn
the risk free interest rate, r. So, this leads to the following equality:
dV = rV dt
or
−
∂C
1 ∂2 C 2 2
+
ς S dt = r
∂t
2 ∂S 2
−C +
∂C
S
∂S
(9)
This equation can be rewritten as:
∂C
1 ∂2 C 2 2
∂C
+
ς S − rC +
rS = 0
∂t
2 ∂S 2
∂S
(10)
The above equation is the Black-Scholes-Merton (BSM) partial differential equation. The only
difference is that in this equation the derivative is specified as a call option, whereas in the BSM
partial differential equation no such specification is given. Solving this equation yields the value of
C.
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From equation 10 is noted that no element of investor risk preference enters the BSM equation. In
particular, the expected return on the stock, µ, is absent from the equation and since any form of
risk preference is absent, it can‟t affect the equation. This leads to the concept of risk neutral
valuation. Since risk preference is apparently not of any influence, risk neutrality can be assumed.
This means that in this model, the expected return as well as the discount factor can be set equal
to the risk free rate.
Assuming risk neutrality of the investors is a matter of convenience. Other assumptions about risk
preferences can be made as long as a proper discount factor is accompanying the assumption. The
solution to the Black-Scholes-Merton equation is valid in both cases, as well in the real world as in
a risk neutral world, but under a risk neutral measure it‟s easier to calculate. Risk neutral valuation
does not claim to be a realistic reflection of the real world, but it does appropriately value a
derivative.
2.5
From a real world to a risk neutral measure
When using risk neutral valuation, the assumption is made that the stock price increases with the
risk free interest rate. However, in equation 6, µ is the expected rate of return on the stock, which
does not necessarily equal the risk free rate, r, in the real world. Equation 6 is a stochastic
differential equation under the real world probability measure, but this equation can be rewritten to
an equation under the risk neutral probability measure, a requirement for using risk neutral
valuation.
For this purpose, assume a probability space (Ω, F, ℱ, ℙ), where Ω is the sample space, F is the
sigma field, ℙ is the real world probability measure and ℱ is the natural filtration Ft
0≤t≤T
(Bingham
& Kiesel, 2000). Suppose the stock price is a ℱ-adapted random process, the risk neutral
probability measure is ℚ and the real world probability measure is ℙ. Recall WtSP is a Wiener process
(or Brownian motion) under the real world probability measure ℙ, but now define
SQ
Wt
= WtSP +
μ−r
t
ς
SQ
WtSP = Wt
or
−
μ−r
t
ς
(11)
At this moment, it is possible to make use of Girsanov‟s theorem (Etheridge, 2002). It states that
WSQ is also a Brownian motion process, but now under a different probability measure, ℚ.
Furthermore, it states that that this process is a martingale. Girsanov‟s theorem states that this is
achieved by merely changing the drift of the stock price process. In this case, the change of drift
was the market price of risk, i.e.
μ−r
ς
, which makes sure that the drift of the new process equals the
risk free rate (since the expected return of the stock can be seen as the interest rate plus the
market price of risk).
It can be shown that by changing the drift in the above fashion, the process is changed to a risk
neutral process. By substituting the result of equation 11 into equation 6 this result is made
obvious and yields
Risk neutral versus real world valuation
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SQ
dSt = rt St dt + ςs St Wt
(12)
Thus, by rewriting the original stock price process and using Girsanov‟s theorem, it can be seen
that the drift of the stock price process under the risk neutral measure equals the risk free rate.
The above assumptions imply for the expectation of the stock price under a risk neutral probability
measure that
St = E ℚ e−r (T−t) ST | Ft
(13)
Thus, the stock price is discounted by the risk free interest rate. Similarly, for any derivative with
payoff CT , such as a call option, it implies that
Ct = E ℚ e−r (T−t) CT | Ft
(14)
equals the arbitrage-free value. This risk neutral value solves the BSM partial differential equation,
as shown in equation 10.
So far, it is shown how risk neutral valuation works and how it can be derived, as well as how to
rewrite a process under a real world probability measure to a risk neutral probability measure. With
explaining risk neutral valuation and what the difficulties are with real world simulation, the first of
the subquestions in the introduction is answered. Using this information, it is possible to determine
the risk neutral value of a product.
Risk neutral versus real world valuation
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3
Guaranteed product
In this thesis, the focus is on a fictitious product. This product can be described as an equity linked
guarantee product. Equity linked guarantees are not uncommon among insurance companies and
banks, since such a product is appealing to consumers, because it offers a combination of saving
and investing.
The specific product offers a minimum guaranteed return, which is equal to 50% of the 1-month
Euribor interest rate. Furthermore, the product contains the possibility to gain an additional
investment return. The extra investment return will be δ, a constant, of the return on the
Amsterdam Stock Exchange (AEX) index under the condition that the return is positive. This last
part is seen as a (part of a) call option on the AEX index.
In this thesis, the value of the product will be determined if it (δ) incorporates a quarter, half, or
three-quarter of the return of the call option on the AEX index. The attractive property of this
product is that it guarantees a certain interest rate and on top of that the return of a part of the
possible rise of the stock market is also incorporated in the product. A possible set of payouts of
the product is given in table 3.1.
δ = 1/2
Year 1
Year 2
Year 3
Year 4
Year 5
Interest rate
5%
4,5%
4%
4,5%
4%
Stock price change
-2%
7%
3%
-5%
1%
Total return
2,5%
5,75%
3,5%
2,25%
2,5%
Table 1 Example of possible payouts of the product
To be able to asses the current value and the future development of the value of the product, it is
necessary to investigate the development of the variables that influence the value of this product
the most. In this case, it is clear to that the product is sensitive to changes in the stock price and
the interest rate. These risks are examined throughout the thesis.
Risk neutral versus real world valuation
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4
Black-Scholes-Hull-White model
In this chapter, a stochastic simulation model is constructed to value the product described in the
previous chapter. As mentioned in chapter 3, the interest rate and the stock price were the main
determinants of the value of the product. Therefore, the purpose is to build a proper stochastic
model containing these variables. For the (stochastic) simulation of the interest rate, a one factor
HW model is used. The HW model is fitted to the current term structure for the risk neutral
parameters. In the BS model, a lognormal distribution is assumed for the stock prices. The one
factor HW model can be incorporated in the Black Scholes world, which results in the BSHW model.
The BSHW model is then used for the (risk neutral) valuation of the product. Throughout the
chapter the equations are assumed to be under a risk neutral probability.
4.1
Hull-White model
In using interest rate models, two kinds of models can be distinguished: a general equilibrium
model (for example the Vasicek or CIR model) and the no-arbitrage model (for example the Ho-Lee
model). Equilibrium models use assumptions about economic variables to estimate the interest rate
(Hull J. , 2006). No-arbitrage models exactly fit today‟s term structure, i.e. the current interest rate
term structure is fitted into the model in such a way that arbitrage opportunities for interest rate
derivatives are excluded.
In this thesis, the one-factor (no-arbitrage) HW model is used. Reason for this choice is that the
HW model incorporates mean-reverting features and, with proper calibration, fits the current
interest rate term structure without arbitrage opportunities (Rebonato, 2000). Furthermore, an
appealing future of the HW model is its analytical tractability (Hull & White, 1990).
Now, assume a probability space (Ω, F, ℱ, ℚ), where Ω is the sample space, ℚ is the risk neutral
probability measure, F is the sigma field and ℱ is the natural filtration Ft
0≤t≤T
as described in
chapter 2. Suppose the interest rate is also a ℱ-adapted random process. The specification of the
HW model for the process of the short rate under a risk neutral probability measure can be
expressed as:
drt = θ t − art dt + ςr dWtrQ
(15)
where a and σ are constants, WrQ is a Wiener process and θ(t) is a deterministic function, chosen
in such a way that it exactly fits the current term structure of the interest rates. For θ(t) to exactly
fit the current term structure, Hull and White (1990) provides the following solution
θ t =
∂ f M (0, t)
ςr 2
+ a f M 0, t +
1 − e−2at
∂t
2a
(16)
Risk neutral versus real world valuation
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Where f M 0, t is the market instantaneous forward rate at time 0 with maturity t. Now integrating
equation 15 yields
r t = r s e−a
t−s
t
+
e−a
t−u
s
= r s e−a
t−s
t
θ u du + ςr
+ α t − α s e−a
t−s
e−a
t−u
dWrQ u
s
t
+ ςr
(17)
e−a
t−u
dWrQ u
s
with
α t = f M o, t +
ςr 2
( 1 − e− at )2 )
2a2
(18)
Thus, it can be concluded that 𝑟 𝑡 conditional on the natural filtration Fs is normally distributed
with (Brigo & Mercurio, 2001):
E r t
Var r t
Fs = r s e−a (t−s) + α t − α s e−a (t−s)
Fs =
ςr 2
2a
1 − e− 2a
(19)
(20)
t−s
Now for purposes clarified later on in this chapter, it is possible to split up r(t) in a stochastic and a
deterministic part as:
(21)
r t = x t + α t
Hence, it is possible to rewrite equation (19) to
E r t
Fs = x s e−a (t−s) + α t
(22)
To be able to use the above stochastic differential equation for Monte Carlo simulations, equations
(19) and (20) have to be rewritten to a discrete process. This solution can be written as (Rebonato,
2000):
rt+∆t = rt e−a ∆t + α t + ∆t − α t e−a
∆t
+ εr ςr
1− e −2a
2a
∆t
εr ~ N 0,1
(23)
Using equation (23), the simulation process can be started, but it can be noted that two constants
are incorporated in equation(15). Of course, these two constants, the mean reversion speed a and
Risk neutral versus real world valuation
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the volatility ςr , could be chosen arbitrarily or based on historical data, but in order to achieve a no
arbitrage model, values have to be chosen in such a manner that the difference between the
market price of an (chosen) interest rate derivative and the HW price of the specific derivative is
minimized. This process is called calibration and is elaborated on in chapter 8.
4.2
Black-Scholes model
In 1973 Black and Scholes published a paper that turned out be a breakthrough in option
valuation. In this paper, the Black-Scholes model was presented and from this model a closed form
solution for the price of a European option could be derived. The BS model, possibly with proper
adjustments, is still used for many option valuation problems in the current financial industry. One
of the main assumptions made in the model is that the stock price follows a geometric Brownian
motion and stock price changes are log-normally distributed.
In this thesis, a stochastic simulation model is presented, which assumes the Black-Scholes world
(i.e., it is assumed that the stock price follows a geometric Brownian motion and the changes in
the stock price are log-normally distributed), but with stochastic interest rates. The stock price
process follows equation (12), with the constant risk free interest rate r replaced by the timevarying and stochastic interest rate rt and σ equal to the implied volatility of the current market
option prices. Using Itô‟s lemma, the process followed by the log of the stock price can be
expressed in a discrete environment as
∆ ln S t + ∆t =
rt −
ςs 2
2
∆t + ςs εs ∆t
εs ~ N 0,1
(24)
where εs a random sample from the standard normal distribution. Equation (24) can be rewritten
to a similar equation (Hull, 2006), but with a stochastic interest rate
T
S T = S 0 e0
r u du −
ςs2
T + ςs ε T
2
(25)
Using the above equation, it is possible to derive the value of an option, even when the payoff
structure is complicated. Since the chosen guaranteed product has no analytical closed form
solution, Monte Carlo simulation is needed, although it is computationally costly. It is important to
note that when an option does have a closed form solution, using Monte Carlo simulations will
generate equivalent values for the option. The answers from both calculation methods are
theoretically equal, but in practice, with a finite number of simulations, the values can differ
marginally.
4.3
Correlation between the interest rate and the stock price
The BSHW model also incorporates a correlation between the interest rate and the stock price (as
can be seen, for example, from the importance for stock markets of the decision of central banks
Risk neutral versus real world valuation
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on the interest rate). This means that an unexpected movement of the stock price or interest rate
influences the other. Thus, the Brownian motions for the stock price and interest rate are
correlated, with ρ equal to the correlation parameter:
dWrQ t dWsQ t = ρ dt
(26)
The correlation between the two processes needs to be incorporated in the model. It can be
captured by a Cholesky decomposition of the correlated random samples of both processes. The
Cholesky decomposition states that a symmetric and positive semi-definite matrix (the correlation
matrix possesses these properties) can be decomposed in a lower and upper triangular matrix.
Using the Cholesky decomposition, the correlation matrix, Σ, can be decomposed in the following
manner:
𝛴 = L LT
(27)
The correlation matrix is in this case a 2 x 2 matrix, since only the correlation between the interest
rate and the stock price is included. Therefore, we can decompose Σ in the following way:
Σ11
Σ21
Σ12
Σ22
=
1 ρ
ρ 1
=
l11
l21
0
l22
l11
0
l21
l22
(28)
Where the equality can be solved by:
l11 =
l21 =
l22 =
Σ11 = 1
Σ21
= ρ
l11
Σ22 − l221 =
(29)
1 − ρ2
By using the Cholesky decomposition, the matrix L is found. Multiplying this matrix L with the
generated random samples of both the stock price process and the interest rate process, results in
correlated disturbances. In this way, the correlation between both processes is captured, which is
necessary for a proper stochastic simulation model.
Risk neutral versus real world valuation
Page 16 of 38
4.4
BSHW model
The two correlated Brownian motions can now be rewritten to:
WrQ
WSQ
=
1
ρ
0
1 − ρ2
WrQ
WπQ
(30)
Where WrQ and WπQ are independent and the correlation between the interest rate and the stock
price is taken into account by the Cholesky decomposition. As can be seen from the results from
equation (30), one of the two equations is unaffected by the Cholesky decomposition. In this case,
the process for the interest rate is chosen to be the first process and the stock price the second
process. This choice is arbitrary, it could be the other way around, but that is of no effect on the
results, merely the way to get to the results. Now, the Cholesky decomposition can be used to
account for the correlation between the interest rate and the stock price and the system of two
equations can be written as:
dr = θ t − ar dt + ςr dWrQ
dSt = rt St dt + ςs St ρ dWrQ + ςs St
(31)
1 − ρ2 dWπQ
(32)
It is assumed that both the interest rate as the stock price are ℱ-adapted random processes, just
as was previously assumed. Since the interest rate process is unaffected by the Cholesky
decomposition, the conditional expectation and variance given in equation (19) and (20) are also
unaffected. However, the conditional expectation and the variance for the stock price are affected
by the decomposition.
On top of that, the results for the stock price in the first part of section 4.2 are not based on a
stochastic interest rate, but on a deterministic interest rate. This assumption is loosened later on
and a stochastic interest rate is accounted for in the BSHW model and this severely changes the
expectation and variance of the stock price.
Now, both processes are properly defined and so is the model. The interest rate process can be
simulated, but it is still necessary to calculate the expectation and variance of the stock price. It is
possible to integrate equation (32) (just as equation (15) and thus equation (15) was integrated)
to obtain (with time steps t an T):
t+∆t
S t + ∆t = S t exp
r u du −
t
∆t 2
ς + ςs ρ WrQ t + ∆t − WrQ t
2 s
+ ςs 1 − ρ2 WπQ t + ∆t − (WπQ t
(33)
Risk neutral versus real world valuation
Page 17 of 38
Where r(t) can be split up just as in equation(21), with x(t) satisfying the following property (Brigo
& Mercurio, 2001):
t+∆t
1 − e−a ∆t
ς
x t +
a
a
x u du =
t
t+∆t
1 − e−a (t+∆t−u) dWrQ u
(34)
t
Thus, equation 33 can also be written as:
𝑆 𝑡 + ∆𝑡 = 𝑆 𝑡 exp
1 − 𝑒 −𝑎 ∆𝑡
1
𝑥 𝑡 − 𝜎 2 ∆𝑡 + 𝜎𝜌 𝑊 𝑟𝑄 𝑡 + ∆𝑡 − 𝑊 𝑟𝑄 𝑡
𝑎
2
∗ exp 𝜎 1 − 𝜌2 𝑊 𝑠 𝑡 + ∆𝑡 − 𝑊 𝑠 𝑡
𝑡+∆𝑡
∗ exp
𝑓 𝑀 0, 𝑢 𝑑𝑢 +
𝑡
2
𝜎
2𝑎2
𝑡+∆𝑡
+
𝜎
𝑎
𝑡+∆𝑡
[ 1 − 𝑒 −𝑎
𝑡+∆𝑡−𝑢
] 𝑑𝑊 𝑟𝑄 𝑢
𝑡
1 − 𝑒 − 𝑎𝑢
2
(35)
𝑑𝑢
𝑡
Rewriting equation (35), it is possible to find a solution for the expectation of the log-stock price
under the risk neutral probability:
E ℚ [ ln
S T
1 − e−a ∆t
1
Fs =
x t − ςs 2 ∆t
S t
a
2
+ ln
(36)
f M 0,T
ς2r
2 −aT
1
+ 2 ∆t +
e
− e−at −
e−2aT − e−2at
M
f 0, t
2a
a
2a
where x(t) can be split up in:
x t = r t − f M 0, t −
ςr 2
( 1 − e− at )2
2a2
(37)
The variance for the stock price can also be found from equation (35) under the risk neutral
probability:
Varℚ [ ln
S T
S t
Fs =
ςr2
a2
∆t +
2
a
e−a ∆t −
1
2a
e−2a ∆t −
3
2a
+ ςs 2 ∆t +
2ρ ς s ς r
a
∆t −
1
a
1 − e−a ∆t
(38)
At this time, the expectation and variance of the interest rate and stock price are known. These can
be used to produce Monte Carlo simulations of the model, where the interest rate, stock price and
the correlation between them are taken into account.
However, for the stock price process an expectation and variance is given, but the distribution is
not.
Risk neutral versus real world valuation
Page 18 of 38
With the use of these Monte Carlo simulations, the risk neutral value and (for example) the Value
at Risk (VaR) at the present time can be determined for the product described in chapter 3. These
results will be described in chapter 5. To be able to calculate the uncertainty of the risk neutral
value in the future, some extra tools are required, which will be discussed in chapter 6.
Risk neutral versus real world valuation
Page 19 of 38
5
Results of the risk neutral valuation of the product
In this chapter, the results of the risk neutral valuation in the BSHW model will be presented. Also,
some measures for risk, for example VaR, will be presented. Results will be presented for different
values of δ, the extra investment return based on the performance of the AEX index. The date at
which the product is valued is chosen to be June 26, 2006.
5.1
Results
At the 26th of June 2006, the AEX-index denoted 427,26 and had an implied ATM (at the money)
volatility of 12,27% based on a maturity of one year. The mean reversion parameter in the HW
model was calibrated at 2,2% and the volatility in the HW model at 0,67%, which will be further
explained in chapter 8. Also, the zero interest rate curve is given as input for the model.
Using the results obtained in chapter 4, 10,000 simulations are run to simulate the (1-month)
interest rate and the stock price. In figure 5.1, the history and a forecast for the next 10 years,
including a lower- (LB) and an upper bound (UB), of the interest rate is shown. The expectation of
the interest follows the zero curve that was given as input, just as expected. The lower- and upper
bound for the interest rate shows that the HW model suggests that the interest rate can fluctuate
severely. The uncertainty in the interest rate becomes larger as the time (of the forecast period)
increases, which implies that the impact of the interest rate on an interest rate-linked product
increases along with the maturity.
1-month interest rate
97.5% UB. RNV
9%
Average estimate RNV
2,5% LB RNV
8%
7%
6%
5%
4%
3%
2%
1%
0%
1998
2000
2002
2004
2006
2008
2010
2012
2014
Figure 5.1 The history and forecast, including a confidence interval of the 1-month
interest rate.
Risk neutral versus real world valuation
Page 20 of 38
As can be seen from figure 5.1, the 2.5% lower bound approaches zero as time increases.
Normally, the interest rate does not become negative in reality. In the HW model, it is possible for
the interest rate to become negative, which is considered to be a drawback of the HW model
(Levin, 2004).
In figure 5.2, the history and a forecast for the next 10 years of the stock price, including a upperand lower bound are shown.
1800
AEX-index
2,5% LB RNV
97,5% UB RNV
Average estimate RNV
1600
1400
1200
1000
800
600
400
200
0
1995
1997
1999
2001
2003
2005
2007
2009
2011
2013
2015
2017
Figure 5.2 The history and forecast, including a upper- and lower bound, of the stock
price.
By running the simulations, 10.000 scenarios of possible payouts of the product are generated. In
this way, the average value and confidence interval boundaries can be extracted from the
simulations. As mentioned in chapter 3, δ was defined as part of the return of the call option on the
AEX index. In table 2, the results of the value of the product and a lower- and upper bound are
given for different values of δ.
δ
0,25
0,50
0,75
average
11,7
28,2
42,4
2.5% LB
-35,2
-30,2
-28,6
97.5% UB
100,2
131,0
160,6
Table 2 The average value of the product and confidence interval for different values of δ
As can be seen from table 2, as δ increases the value of the product increases as well. This makes
sense, since a larger part of the investment in a call option on the AEX-index, generates a higher
return on average than receiving the 1-month swap interest rate.
Risk neutral versus real world valuation
Page 21 of 38
Furthermore, the width between the upper- and lower bound is striking. This seems
counterintuitive, however, the product has a term of 30 years, which makes the current value more
volatile and that explains the width of the confidence interval.
In table 1, the values of the product at the 26th of June in 2006 are given. Comparing these values
to the values of the product at the 29th of August in 2008 shows that the market conditions in
August 2008 lead to a higher value of the product. These values are given below in table 3.
δ
0,25
0,50
0,75
average
30,0
57,2
84,0
5% C.I.
-26,0
-19,0
-14,2
95% C.I.
130,6
188,1
237,7
Table 3 The average value of the product for different values of δ at August 29, 2008
The higher average value of the product, when valued at the 29th of August in 2008, results from
the rise of the volatility of the stock price. The recent increase in the implied volatility can be
related to the „credit crunch‟.
Risk neutral versus real world valuation
Page 22 of 38
6
The stochastic discount factor
In this chapter, the use of and theory about the stochastic discount factor will be discussed. It is
explained how the stochastic discount factor (SDF) is a bridge between the real world and the risk
neutral world. Furthermore, a formula for the SDF will be derived in the BSHW model and it will be
justified how the SDF can be used for the valuation of the product described in chapter 3.
6.1
Theory on the stochastic discount factor
Assume once more a probability space (Ω, F, ℱ, ℙ), where Ω is the sample space, ℙ is the
probability measure and ℱ is the natural filtration Ft
0≤t≤T.
Suppose X is a ℱt -measurable random
variable, the risk neutral probability measure is ℚ and the real world probability measure is ℙ. L,
the Radon-Nikodym derivative of ℚ with respect to ℙ (Etheridge, 2002), is given by
L=
dℚ
dℙ
(39)
and
Lt = E ℙ L | Ft
(40)
For equivalent probability measures4 ℚ and ℙ, given the Radon-Nikodym derivative from equation
(39), the following equation holds for the random variable X (Duffie, 1996)
E ℚ X = E ℙ LX
(41)
and
E ℚ X | Ft = E ℙ X
LT
| Ft
Lt
(42)
It can be seen from the above equation that the expectation of X under the probability measure
ℚ is equal to the expectation of L times X under the probability measure ℙ.
Suppose Wtℚ is a bivariate ℚ-Brownian motion with the natural filtration that was given above as
Ft .
4
ℚ and ℙ are equivalent probability measures when it is provided that ℚ(A) > 0 if and only if ℙ(A)
> 0, for any event A (Duffie, 1996).
Risk neutral versus real world valuation
Page 23 of 38
Define:
t
Lt = exp −
0
θ′s dWsP −
1
2
t
0
θ′s θs ds
(43)
And assume that the following equation holds, with θt being ℱ-adapted
E exp(
1
2
T
0
θ′t θt dt ) < ∞
(44)
Where the probability measure ℙ is defined in such a way that L = Lt is the Radon-Nikodym
derivative of ℙ with respect to ℚ. Now it is possible to use the preceding to rewrite equations (41)
and (42) to catch the link between risk neutral pricing and pricing under a „real world‟ probability
measure
T
E ℚ exp −
t
rs ds X T | ℱt = E ℙ exp −
T
t
T
rs ds −
t
θ′s dWs −
1
2
T
t
θ′s θs ds X T | Ft
(45)
Combining the above equations and using Girsanov‟s theorem states that the process
Q
Wt = WtP +
t
0
θs ds
(46)
is a standard Brownian motion under the probability measure ℙ. A useful feature of this theorem is
that when changing the probability measure from risk neutral to real world, the volatility of the
random variable X is invariant to the process. In changing from a risk neutral to a real world
probability measure, it is essential to make WtP a standard Brownian motion. By Girsanov‟s
theorem, this is reduced to finding the correct θs in equation (46). With this knowledge, it is
possible to return to the BSHW model and find a proper SDF.
6.2
The stochastic discount factor in the BSHW model
Now processes for the interest rate and the stock price under the real world probability measure ℙ
are assumed. Returning to the stochastic differential equations (SDE) for the stock price and the
interest rate, bearing in mind the results of the Cholesky decomposition in chapter 4, it is possible
to obtain the SDE for the stock price and the interest rate under a real world probability measure.
Under this measure, it is necessary to account for the market price of risk for both the stock price
and the interest rate.
Risk neutral versus real world valuation
Page 24 of 38
The SDE for the interest rate and the stock price, both similar to the SDE under the risk neutral
probability measure in equations 31 and 32, can be written as:
drt = μr − art dt + ςr dWtrP
(47)
dSt = μt St dt + ςs St ρ dWrP
t + ς s St
(48)
1 − ρ2 dWtπP
Where μr is the real world expectation for the interest rate, based on historical data for the one
month interest rate and μt is the expected growth of the stock price, which is equal to the expected
growth rate under a risk neutral probability measure plus a constant market risk premium.
Now, according to the theory of section 5.1, to rewrite from a probability measure ℙ to a
probability measure ℚ, it is sufficient to find θs from equation 46. This leaves the following two
equations:
WtrQ = WtrP +
WtπQ
t
0
= WtπP +
θrs ds
t
0
(49)
θss ds
By choosing a proper value for θrs , the substitution of the first part of equation 49 into equation 31
should be equal to equation 47. By solving this inequality, θrs is found to be:
θrt =
μr − θ(t)
ςr
(50)
Something similar can be done to compute θst . With this knowledge, substituting the second part of
equation 49 into equation 32 and solving yields:
πQ
dWt
= dWtπP +
πs
ςs
1−
ρ2
dt −
ρ
1 − ρ2
rQ
dWt − dWtrP
(51)
Which results in:
θss
=
θrs
1
1 − ρ2
0
−ρ
1 − ρ2
1
πs
ςs
μr − θ(t)
ςr
(52)
Assuming that equation 44 holds, which is a requirement, the proper stochastic discount factor in
the BSHW model can be written as:
Risk neutral versus real world valuation
Page 25 of 38
T
SDF t, T = exp −
t
T
rs ds −
t
θss dWπℙ −
1
2
T
t
θss 2 ds −
T
t
θrs dWrℙ −
1
2
T
t
θrs 2 ds
(53)
In equation 53, the stochastic discount factor is stated. To be able to use Monte Carlo simulations
to value the guaranteed product under the real world probability measure, the expectation and
variance of the stock price and the interest rate under the real world probability measure need to
be found.
In this case, it is assumed that the stock price grows with μt , which equals the risk free interest
rate plus the market price of risk, as described in equation 51. Furthermore, it is assumed that the
market price of risk is a constant. This means that μt can be split up in the interest rate plus a
constant, πs , and so the expectation and variance can be calculated similarly to the calculations
under the risk neutral measure.
Also, for the interest rate a (negative) risk premium is added to the equation. This is done, since
the expectation of the interest rate is different under the real world probability measure. Under the
risk neutral probability measure, the interest rate process follows the initial forward curve, which is
not a realistic assumption, since under a „normal‟ initial forward curve, the interest rate increases
as the maturity increases.
At this point, the SDF, the interest rate and the stock price processes can all be simulated. Using
real world simulations, it can be shown that the value of the product at the present time is equal to
the value of the product using risk neutral simulations, i.e. equations 54 and 55 can be proved to
hold.
30
ℙ
V0 = E [
SDF 0, T ∗ CF T | F0 ]
(54)
SDF 1, T ∗ CF T | F1 ]
(55)
T=1
30
V1 = E ℙ [
T=2
Through the tower property of conditional expectations it must hold that
30
E ℙ [V1 |F0 ] = E ℙ [
SDF 1, T ∗ CF T | F0 ]
(56)
T=2
The above formulas state that the market value of the product can be seen as the sum of its
discounted cash flows. Since the interest rate and the stock price are known, the payout (and thus
the cash flow) of the product are known per month. The discount factor is known, both at t=0 and
at t=1, so the present as well as the value of the product in one year can be calculated using this
method.
Risk neutral versus real world valuation
Page 26 of 38
Only, using real world simulations, comments on the uncertainty in the future value of the product
are useful and valid, so also the conditional expectation is necessary in order to calculate the 1year VaR.
The 95% 1-year VaR is defined as that particular value, which in 95% of the simulations the
change in the value of the guaranteed product lies under. When calculating the VaR, it must be
taken into account that the value of the guaranteed product is expected to change. So, the change
in the value is defined as
V1 − V0 = E ℙ [SDF 0,1 ∗ CF 1 | F0 ]
+ Eℙ [
30
T=2 SDF
1, T ∗ CF T F1 − E ℙ [
30
T=2 SDF
1, T ∗ CF T | F0 ]
(57)
It holds that the first part is the expected cash flow in the first period and adding both expectations
can be seen as the uncertainty in the value due to a possible change in market conditions.
To be able to determine the VaR of the product, the interest rate process, stock price process and
the stochastic discount factor are simulated. With these instruments, the result of formula (57) can
be calculated for all simulations. By taking the 95th percentile of the simulation process for formula
(57), the 1-year 95% VaR of the guaranteed product can be extracted. In the next chapter, the
results will be presented.
However, a drawback of the model must be noted here. The conditional expectations in formulas
(55) and (57) assume that the parameters of the model are known at t=1 (..|F1 ). Obviously, the
(known) market conditions at t=0 will be different from the (unknown) market conditions at t=1.
This parameter risk is neglected in these calculations, which is a drawback of the model.
Risk neutral versus real world valuation
Page 27 of 38
7
Results of the real world valuation of the product
This chapter presents the results of the real world valuation by means of the BSHW model. Also,
some measures for risk, for example VaR, will be presented. Results will be presented for different
values of δ, the extra investment return based on the performance of the AEX index. The date at
which the product is valued is chosen to be the 26th of June in 2006.
7.1
Results
Just as in chapter 5, the AEX-index denoted 427,26 had an implied ATM (at the money) volatility of
12,27% based on a maturity of one year. The mean reversion parameter in the HW model was
calibrated at 2,2% and the volatility at 0,67%. Also, the zero interest rate curve is given as input
for the model. The average 1-month interest rate μr is chosen to be 4,27% based on historical
data. Furthermore, the risk premium, πs , is fixed at 3%.
Using the results obtained in chapter 6, 10,000 simulations are run to simulate the (1-month)
interest rate and the stock price. In figure 3, the history and a forecast for the next 10 years,
including a lower- and upper bound, of the interest rate are shown. Just as under the risk neutral
probability measure, the uncertainty in the interest rate becomes larger as the time (of the
forecast period) increases, which implies that the impact of the interest rate on an interest ratelinked product increases along with the maturity.
8%
7%
1-month interest rate RW
Average estimate RW
97.5% UB RW
2.5% LB RW
6%
5%
4%
3%
2%
1%
0%
1999
2001
2003
2005
2007
2009
2011
2013
2015
Figure 7.1 The history, forecast and its lower- and upper bound interval of the 1-month
interest rate.
Figure 7.1 depicts the development of the 1-month interest rate under the real world probability
measure compared to the development under the risk neutral probability measure. Clearly, under
the risk neutral probability measure the interest rate increases faster. The latter can be explained
by the fact that it follows the current zero curve very closely, as can also be seen from figure 7.1.
Risk neutral versus real world valuation
Page 28 of 38
The interest rate tends to the long-term average under the real world probability measure, but it
must be noted that this is a slow process. If a longer forecast period would have been chosen, this
would be clearer.
6.0%
1-mo nth inte re s t ra te RW
Ave ra g e e s tima te RW
Ave ra g e e s tima te RNV
The fo rw a rd z e ro curve
5.0%
4.0%
3.0%
2.0%
1.0%
0.0%
1999
2001
2003
2005
2007
2009
2011
2013
2015
Figure 7.2 The history, both forecasts and zero curve of the 1-month interest rate
In figure 7.2, the history, the zero curve and both forecasts are shown for the next 10 years.
Figure 7.3 shows the history, forecast and a lower- and upper bound of the AEX-index under both
probability measures. As can be seen, the average predicted value of the AEX-index has a smooth
course, but the width of the confidence interval shows that the predicted values of the index are in
fact rather volatile. As expected, under the real world probability measure the index increases
faster on average. This is due to the addition of the risk premium of 3% under this real world
measure.
Risk neutral versus real world valuation
Page 29 of 38
1000
800
AEX-index
RN 98% C.I.
RW 98% C.I.
RN average estimate
RW average estimate
600
400
200
0
Jan-04
The level of the AEX-index (412,84) at August 31, 2008 is used as a starting point for the analysis. The
current level lies outside the 98% confidence interval.
Jan-05
Jan-06
Jan-07
Jan-08
Jan-09
Jan-10
Jan-11
Figure 7.3 The history and forecast, including a lower- and upper bound of the stock
price
Table 4 shows the average value and lower- and upper bound for the guaranteed product under
investigation for different values of δ. On the left hand side, the average value and lower- and
upper bound are reproduced for calculations under the real world probability measure. On the right
hand side, the results from chapter 5 are shown again.
30/6/2006
Real World
δ
average
2,5% LB
0,25
13,3
-33,6
0,5
27,5
-30,5
0,75
42,7
-33,6
97.5% UB
100,7
127,2
153,5
δ
0,25
0,5
0,75
average
11,7
28,2
42,4
Risk Neutral
2,5% LB
-35,2
-30,2
-28,6
97.5% UB
100,2
131,0
160,6
Table 4 The average value of the product and the C.I. for different values of δ.
Clearly, the results for both calculation methods are very similar. This actually makes sense, since
the theory discussed in chapters 4 and 6 states that the average results should be the same. The
(minor) differences can first be explained by the fact that a different set of simulations is run for
both methods. Second, a discrete approximation for a part of the stochastic discount factor had to
be made in order to use it in the stochastic simulation model.
When taking these reasons into account, the differences between the estimated values under a risk
neutral and a real world probability measure can be surmountable. Also, the difference in the
estimated values of the product at the 29th of August 2008 are reasonable. These values are given
in table 5.
Risk neutral versus real world valuation
Page 30 of 38
29/8/2008
Real World
δ
average
2,5% LB
0,25
32,1
-23,9
0,5
54,4
-17,8
0,75
82,5
-17,3
97.5% UB
133,3
181,6
225,1
δ
0,25
0,5
0,75
average
30,0
57,2
84,0
Risk Neutral
2,5% LB
-26,0
-19,0
-14,2
97.5% UB
130,6
188,1
237,7
Table 5 The average value of the product for different values of δ at August 29, 2008.
Just as under the risk neutral probability measure, the average values are much higher due to the
rise in the implied volatility. Now, the present value of the product is calculated under a real world
probability measure. Using these real world simulations, the value of the product in one year and
the uncertainty in this value can be calculated.
Also, the 95% VaR of the product can be calculated. By taking the difference between the value of
the product at the present time and the expected market value at t=1, one can simulate the
change in the value of the product. This change should be corrected for the actual expected
change, since the time to maturity of the product has declined at t=1. The VaR is defined by the
difference between the average at t=1 minus its 95% boundary. The results are given in table 6
and 7:
Risk Neutral
Date
Expected market
value in 1 year
30/6/2006
28,1
34,5
29/8/2008
57,4
67,2
5% LB
Real World
VaR
Expected market
value in 1 year
20,2
6,4
28,4
35,6
21,2
7,2
44,1
9,8
55,7
66,4
45,0
10,7
95% UB
5% LB
95% UB
VaR
Table 6 The market values and the VaR of the guaranteed product
Whether the value of the product in one year is estimated correctly can be tested by using the
method of backtesting. This method is described in the next section.
7.2
Backtest
To examine the forecast capabilities of the model, the results can be tested by the method of
backtesting. Backtesting is a method where the model is used to calculate the present value of the
product on several different moments in the past and after that compares the results to the actual
(already known) outcomes.
The model under the real world probability measure is used to predict the value of the product in
one year. However, it is difficult to create enough observations and therefore, a rolling window of
one year is used to create more observations.
Since the dataset starts in May 2003, still 51 observations are available for the backtest. In all of
these 51 observations, it will be tested whether the actual value of the product lies outside the
90% confidence interval of the predicted value, generated in the model under the real world
probability measure. The results for the backtest are shown in figure 7.4.
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20
90% CI boundaries RW
90% CI boundaries RN
Realised market value
15
10
5
0
-5
-10
jun-04
dec-04
jun-05
dec-05
jun-06
dec-06
jun-07
dec-07
jun-08
Figure 7.4 Backtest results for the prediction of the value of the product in one year.
What can be concluded from the above figure, is that in particular the observations in the last year
of the dataset fall outside the predicted confidence interval. In total 15 of the 51 observations, lie
outside the predicted 90% confidence interval. These results can be mainly attributed to the rise in
the implied volatility due to the turbulent market conditions from May 2007 on, which can be seen
in figure 7.5.
30,0
Implied volatility AEX-index
22,5
15,0
7,5
0,0
jun-04
dec-04
jun-05
dec-05
jun-06
dec-06
jun-07
dec-07
jun-08
Figure 7.5 A graph of the implied volatility of the AEX-index
Whether the model passes the backtest can be calculated in a likelihood ratio testing framework
(Christoffersen, 1998). In this framework, suppose that It
T
t=1
is the indicator variable for the
interval forecast given by the model under the real world probability measure, which means that
whenever It =1 the actual value lies in the interval. The conditional coverage can be tested by
Risk neutral versus real world valuation
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comparing the null hypothesis that E It = p with the alternative hypothesis that E It ≠ p, given
independence.
The likelihoods under the null hypothesis and under the alternative hypothesis are given by:
L p; I1 , I2 , … , I52 = px 1 − p
n−x
L π; I1 , I2 , … , I52 = πx 1 − π
n−x
(58)
x
Where the maximum likelihood estimate of π is n , the number of values outside the interval
forecast divided by n, the total number of observations. Using these likelihoods, a likelihood ratio
test for the test of the conditional coverage can be formulated
LR cc = −2 ∗ log L p; I1 , I2 , … , I52 / L π ; I1 , I2 , … , I52
~ χ2 2
(59)
Where the test statistic is actually asymptotically Chi-Squared distributed with s(s-1) degrees of
freedom, with s=2 as the number of possible outcomes. Using this statistic it is difficult to take the
autocorrelation (due to the use of the rolling window) into account. So, the resulting conclusions
are less powerful. In this case, the LR-test statistic is 14,4, which is higher than the 5,99 from the
(5%) confidence level of the Chi-squared distribution, what justifies the conclusion that the model
is inaccurate.
However, the recent crisis is a very unexpected event. If this point would have been left out of
consideration, the backtest would have a totally different outcome. These results are presented in
figure 7.6.
20
90% CI boundaries RW
90% CI boundaries RN
Realised market value
15
10
5
0
-5
-10
jun-04
dec-04
jun-05
dec-05
jun-06
dec-06
Figure 7.6 Results of the backtest until may 2007
The LR test statistic for this dataset is 0,048, which would lead to not rejecting the model, as
opposed to a rejection taking the data from May 2007 until August 2008 into account.
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When these tests are performed for the model under a risk neutral probability measure, both tests
perform similar as in the model under the real world probability measure, see table 7.
Date
Until August 2008
Until May 2007
Real world
Risk neutral
1% critical value
5% critical value
14,4
0,05
23,6
1,67
9,21
5,99
Table 7 Results of the backtest for both models
So, when the data until May 2007 are used to backtest the model under a real world probability
measure, it is not rejected. This is similar to the model under the risk neutral probability measure.
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8
Summary and conclusions
The objective of this article was to determine the uncertainty in the future market value of the
product. To be able to determine this, a stochastic discount factor (SDF) was used. The SDF, also
called deflator, was needed for proper valuation using real world simulations. For the HWBS model
an SDF could be constructed, which was used to value the guaranteed product.
The most important conclusions that can be drawn from the results and the backtest are:

Valuation under the real world probability measure calculates the market value of a product with
the use of a stochastic discount factor. The main advantage of using real world simulations is
that the simulations can also be used for a „realistic‟ simulation of random variables.

Combining the real world simulations with a stochastic discount factor is very useful for banks
and insurers. They can use this method to estimate the current value of their products and,
more importantly, estimate the uncertainty in this value in one year in a consistent way. This
can be used in regulatory (e.g. Basel II or Solvency II) and economic capital calculations.

Capital calculations are typically based on a one year 99% VaR. When using real world
simulations and a standard discount factor, estimated average values are inaccurate, therefore,
resulting VaR calculations can be as well. When using risk neutral valuation to estimate the VaR,
only current market conditions are taken into account. Current market conditions are not
necessarily a good measure for future outcomes, which could also lead to inaccurate VaR
estimations.
However, some drawbacks of the model must be noted.

The model under the real world probability measure, using the SDF, did not pass the backtest.
The null hypothesis that the model correctly predicts the uncertainty in the future value is
rejected. The failure of the model in the backtest needs to be taken seriously. However, as
already mentioned, the market conditions in the last period of the sample, are quite unusual.
Whenever the dataset is cut off at May 2007, the model passes the backtest unlike the model
under a risk neutral probability measure. Of course, doing this would be a case of data mining,
but it does not alter the fact that the current market conditions are difficult to take into account.
It could be defined as an outlier, some theories state that the recent crisis is comparable to the
crisis in the twenties.

Two variables, the stock price and interest rate, are modelled stochastically. When more
variables are modelled stochastically, the SDF becomes more complicated. For banks and
insurers, who also model variables like exchange rates and volatility stochastically, several more
random variables enter the model. As the results have shown, the value of the product greatly
depends on this input and modelling this input as a random variable could help to improve the
Risk neutral versus real world valuation
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forecasting qualities of the model. However, this would make the model and the SDF more
complicated and less practical.
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9
References
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Broadie, M., & Detemple, J. B. (2004). Option pricing: valuation models and applications.
Management Science , 1145–1177.
Christoffersen, P. F. (1998). Evaluating interval forecasts. Washington: International Monetary
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Duffie, D. (1996). Dynamic asset pricing theory . Princeton University Press.
Etheridge, A. (2002). A course in financial calculus. Cambridge: Cambridge University Press.
Gibbs, S., & McNamara, E. (2007). Practical issues in ALM and stochastic modelling for actuaries.
Christchurch: Trowbridge Deloitte.
Hull, J. (2006). Options, Futures and other derivatives. New Jersey: Prentice Hall.
Hull, J., & White, A. (1990). Pricing interest-rate-derivative securities. The Review of Financial
Studies , 573-592.
Ito, K. (1951). On stochastic differential equations. Memoirs, American Mathematical Society , 151.
Levin, A. (2004). Interest rate model selection. New York: Andrew Davidson & Co., Inc. .
Nielsen, J., & Sandmann, K. (1995). Equity-linked life insurance: a model with stochastic interest
rates. Insurance: Mathematics and Economics , 225-253.
Rebonato, R. (2000). Interest-rate option models. Chichester: John Wiley & Sons.
Rendleman, R. J., & Bartter, B. J. (1979). Two-state option pricing. The journal of finance , 10931110.
Tamba, Y. (2005). Pricing a bermudan swaption with a short rate lattice method. Osaka: 21st
Century Center of Excellence Program (COE).
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Appendix explanation used data
This appendix discusses the assumptions that were made about the parameters needed for the
input in the model. Also, some estimation techniques for certain parameters needed in the model
will be elaborated on.
Used data
First, the estimation of the parameters in the HW model will be discussed. This estimation is
obtained by calibration. When calibrating the mean reversion speed and the volatility (Tamba,
2005), a specific interest rate derivative has to be chosen. Here, the instruments that are used for
calibration are at the money swaption premiums5 (Levin, 2004). The prices of the swaption
premiums are extracted for different maturities and expiry dates.
Then, in the HW model the swaption premiums can be calculated for a fixed mean reversion speed
and volatility. Of course, using several different swaption premiums (i.e. swaption premiums for
swaptions with different maturities etc), no exact fit can be found. So, a specific goodness of fit
measure has to be chosen, in this case the absolute difference between the prices of the swaption
premiums generated from the HW model and the actual (market priced) swaption premiums is
minimized.
In this model, the mean reversion parameter and volatility are chosen to be constant. This means
that there are two volatility parameters (Hull J. , 2006). Using the absolute difference between
both (above described) prices as the goodness of fit measure, the best fitting values of the mean
reversion speed and the volatility can be calculated with an optimization approach. Here, the
calibrated mean reversion speed equals 2,2% and the calibrated volatility 0,67%.
When backtesting the model in section 7.2, the calibration procedure was used every month. This
was necessary to exclude arbitrage opportunities at the start of every month used in the backtest
procedure.
In the model described in chapter 4 and extended in chapter 6, a constant volatility of the stock
price is assumed. The volatility can be estimated by the implied volatility, which is determined by
option prices observed in the market.
In an ordinary option pricing model, such as Black-Scholes, the price of an option can be written as
a function of the volatility. This suggests that the other parameters in the model are kept constant.
If an option price is observed in the market, with a known maturity and strike price (normally
chosen to be close to the present stock price), the same price can be obtained via the BlackScholes model by changing the volatility. The latter can be achieved by an iterative procedure. This
vested volatility is also known as the implied volatility (ATM and with a maturity of one year).
Furthermore, the risk premium is given as input for the model. A proper estimation for the risk
premium is very difficult. An estimation based on historical data is not very plausible, since the
5
All swaption premiums are extracted from Bloomberg.
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value of the risk premium is highly dependable on the chosen length of the historical data. A risk
premium of 3% for the AEX-index does not seem fairly unreasonable and is chosen as constant risk
premium.
Also, some adjustments have to be made for the interest rate. Under a risk neutral probability
measure, the interest rate follows the zero curve, which is not a realistic assumption. It is more
realistic to expect the interest rate to drift around some long-term average, which was defined to
be μr . This long-term average can be estimated using historical data of the 1-month Euribor
interest rate and equals 3,8% for our sample period.