From Utility Indifference Pricing to Risk Premium Pricing For a
Contingent Claim on a Non-Tradable Asset
Antoine Conze
Hiram Finance
[email protected]
November 2016
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Introduction
We consider a non-tradable asset whose value over time {Xt } is assumed to follow a geometric Brownian
motion
dXt
= αdt + σdWt
Xt
where α is the drift term, σ is the volatility term and {Wt } is a standard Brownian motion on a
probability space (Ω, F, P ) with P being the real world probability measure. We also consider a
contingent claim with payoff f (XT ) on maturity T , offered to investors for a price V0 .
A typical instance of such as situation is the case of a non public company undergoing an LBO,
where the asset is the non-tradable share of the company and f (XT ) is the payoff on a warrant package
that managers can buy in.
How should the price V0 be determined ? Because the underlying asset is not tradable, standard
hedging based option pricing theory does not apply. Two alternatives are:
• utility indifference pricing: representing investors preferences with a utility function and determining the contingent claim price from an indifference argument;
• risk premium pricing: applying the hedging based option pricing methodology, but with the
underlying asset’s drift modified with a risk premium to account for its non tradability.
Which of the two alternatives is preferable ? In the context of a management warrant package,
some authors (see for instance [1], [2]) apply the first method. On the other hand the second method
is usually easier to implement, if only because a flat risk premium yields Black & Scholes type pricing
formulas, and is often favored by financial advisors 1 .
The purpose of this note is to point out that the two approaches are in fact conceptually equivalent,
in the sense that the utility function risk aversion in the first method can be mapped into a risk
premium in the second method, albeit non flat, such that both methods yield the same price V0 .
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Utility Indifference Pricing
Assume the investor has initial wealth w0 and a utility function U (w), twice continuously-differentiable,
strictly increasing and strictly concave. Utility indifference pricing (see for instance [3] for an overview)
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Some financial advisors also apply a hedging based risk neutral option pricing methodology and then discount the
resulting price with a flat factor to account for the non tradability of the asset, but this seems to lack justification if the
same flat factor is applied regardless of the contingent claim payoff profile.
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states that the investor should be indifferent between not doing anything thus keeping her wealth
to its initial value2 w0 , or buying the contingent claim for a price V0 , yielding a terminal wealth
w0 − V0 + f (XT ). This translates to
U (w0 ) = E P [U (w0 − V0 + f (XT ))] .
(1)
The initial price V0 is then obtained by solving for (1).
Non-Linearity An important feature of utility indifference pricing is that the price is non-linear in
the number of contingent claims3 . Indeed, consider two contingent claims with almost surely different
payoffs f 1 (XT ) and f 2 (XT ) with prices respectively V01 and V02 , and θ ∈ [0, 1]. Then the price V0θ of
the contingent claim θf 1 (XT ) + (1 − θ)f 2 (XT ) is a strictly concave function of θ as can be seen from
E P U (w0 − (θV01 + (1 − θ)V02 ) + θf 1 (XT ) + (1 − θ)f 2 (XT ))
> θE P U (w0 − V01 + f 1 (XT )) + (1 − θ)E P U (w0 − V02 + f 2 (XT ))
= θU (w0 ) + (1 − θ)U (w0 ) = U (w0 )
h
i
= E P U (w0 − V0θ + θf 1 (XT ) + (1 − θ)f 2 (XT )) .
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Risk Premium Pricing
Risk Neutral Pricing If the asset was tradable standard hedging based risk neutral option pricing
theory would apply and the price of the contingent claim would be computed as the risk neutral
expectation
V0 = E Q [f (XT )]
where Q is the risk neutral probability measure under which the dynamics of the asset is
dXt
= σdWtQ
Xt
with {WtQ } a standard Brownian motion under Q. The measure Q is obtained from the RadonNikodym derivative
α
1 α 2
dQ
= exp − WT −
T .
(2)
dP
σ
2 σ
The price V0 can then be obtained as V0 = V (X0 , 0) where V is the solution to the backward
Kolmogorov partial differential equation
∂V
1
∂2V
(x, t) + σ 2 x2 2 (x, t) = 0
∂t
2
∂x
V (x, T ) = f (x).
Risk Premium Pricing Staying within the realm of option pricing methodology, a common practitioner approach to deal with the non tradability of the asset is to introduce a risk premium {λt } to
the measure Q defined by (2), adapted to the probability space (Ω, F, P ), such that
dXt
= −λt dt + σdWtQλ
Xt
with {WtQλ } a standard Brownian motion under the risk premium adjusted probability measure Qλ ,
and then to price the contingent as in the tradable case as the expectation under Qλ
V0 = E Qλ [f (XT )] .
2
3
To simplify the exposition we assume throughout this note that interest rates are zero.
Meaning non-linear in k when the continent claim payoff is kf (XT ).
2
(3)
The measure Qλ is obtained from the Radon-Nikodym derivative
dQλ
= exp −
dQ
Z
0
T
λt
1
dWtQ −
σ
2
Z
0
T
λt
σ
2
!
dt .
Assuming a risk premium process in the form of a function λt = λ(Xt , t) the backward Kolmogorov
partial differential equation is modified to
∂V
∂V
1
∂2V
(x, t) − λ(x, t)x
(x, t) + σ 2 x2 2 (x, t) = 0
∂t
∂x
2
∂x
V (x, T ) = f (x).
(4)
Linearity Given a risk premium process {λt }, the resulting price V0 is linear in the number of
contingent claims, as is readily seen from (3) or (4).
Non-Linear Risk Premium Pricing Once the problem has been setup as a partial differential
equation, the risk premium function can be made more general so as to depend on the value function
itself. For instance to account for an increased risk premium for a contingent claim on a larger notional
or with a larger sensitivity to the asset value, one may model the risk premium as
∂V
λt = λ Xt , t, V (Xt , t),
(Xt , t) .
∂x
The backward Kolmogorov partial differential equation is then modified to
∂V
∂V
1
∂2V
∂V
(x, t) − λ x, t, V (x, t),
(x, t) x
(x, t) + σ 2 x2 2 (x, t) = 0
∂t
∂x
∂x
2
∂x
V (x, T ) = f (x).
(5)
The partial differential equation becomes non linear, hence the price V0 = V (X0 , 0) is no longer linear
in the number of contingent claims.
Risk Premium as a Cost One can think of the risk premium λt as representing the “cost” per
unit of asset value of making the asset tradable. For a tradable asset incurring such a cost, risk neutral
pricing would apply but would have to factor in the cost applied to the hedge ratio Xt ∂V (Xt , t), and
∂x
the resulting partial differential equation would be exactly (5).
For instance if the asset was in fact tradable but with a thin repo market, the cost of repo would
have to be properly modeled and factored in when pricing a contingent claim (e.g. a long position in
a call option) whose hedge requires short selling the underlying. In this case λt would be the repo
cost, that is the difference between the risk free rate and the rate on the repo collateral. The repo
cost is often modeled as a constant, but with a very thin repo market one might want to model it as
being dependent on the number of shares to be shorted when hedging the contingent claim, that is as
a function
∂V
λt = λ
(Xt , t) .
∂x
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From Utility Indifference Pricing to Risk Premium Pricing
Let
yt = EtP [U (w0 − V0 + f (XT ))] .
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Then yt = y(Xt , t) where y is the solution to the backward Kolmogorov partial differential equation
∂y
∂y
1
∂2y
(x, t) + αx (x, t) + σ 2 x2 2 (x, t) = 0
∂t
∂x
2
∂x
y(x, T ) = U (w0 − V0 + f (x)).
(6)
Now let
V (x, t) = V0 − w0 + U −1 (y(x, t))
By construction we have V0 = V (X0 , 0), and
y(x, t) = U (w0 − V0 + V (x, t))
∂V
∂y
0
(x, t) = U (w0 − V0 + V (x, t))
(x, t)
∂t
∂t
∂y
∂V
0
(x, t) = U (w0 − V0 + V (x, t))
(x, t)
∂x
∂x
2
∂2V
∂V
∂2y
0
00
(x, t) = U (w0 − V0 + V (x, t)) 2 (x, t) + U (w0 − V0 + V (x, t))
(x, t)
∂x
∂x2
∂x
and from (6) we obtain
00
1
∂V
1
∂2V
U (w0 − V0 + V (x, t))
∂V
(x, t) + αx
(x, t) + σ 2 x2 2 (x, t) + σ 2 x2 0
∂t
∂x
2
2
U (w0 − V0 + V (x, t))
∂x
2
∂V
(x, t) = 0
∂x
(7)
V (x, T ) = f (x).
Utility Indifference Risk Premium Identifying (7) with (5), we conclude that utility indifference
pricing and risk premium pricing yield the same price when the risk premium is chosen as
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∂V
∂V
(Xt , t) = −α + σ 2 A(w0 − V0 + V (Xt , t))Xt
(Xt , t)
λt = λ Xt , t, V (Xt , t),
(8)
∂x
2
∂x
00
0
where A(w) = −U (w)/U (w) is the utility function coefficient of absolute risk aversion.
Equation (8) maps the utility function risk aversion into a utility indifference risk premium. Note
that the risk premium depends on V (Xt , t) (unless the absolute risk aversion is constant) and on
∂V (X , t), hence the price V = V (X , 0) is non-linear in the number of contingent claims as is
0
0
∂x t
expected.
Constant Absolute Risk Aversion
(CARA) the risk premium is
In the case A(w) = a of a constant absolute risk aversion
1
∂V
λt = −α + σ 2 aXt
(Xt , t)
(9)
2
∂x
It is worth noting that in this case the risk premium adjusted for the real world drift, λt + α, is
proportional to Xt ∂V (Xt , t), which represents the contingent claim price absolute sensitivity to the
∂x
underlying asset log return ln(Xt /X0 ). A contingent claim with larger log return absolute sensitivity
will result in a larger risk premium.
Constant Relative Risk Aversion
(CRRA) the risk premium is
In the case wA(w) = a of a constant relative risk aversion
Xt ∂V (Xt , t)
1
∂x
λt = −α + σ 2 a
2
w0 − V0 + V (Xt , t)
(10)
In this case the risk premium adjusted for the real world drift, λt +α, is proportional to Xt ∂V (Xt , t)/(w0 −
∂x
V0 +V (Xt , t)), which represents the contingent claim price relative (in terms of total wealth) sensitivity
to the underlying asset log return. A contingent claim with larger log return relative sensitivity will
result in a larger risk premium.
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References
[1] Detemple, J., & Sundaresan, S. (1999). Nontraded asset valuation with portfolio constraints: a
binomial approach. Review of Financial Studies, 12(4), 835-872
[2] Henderson, V. (2005). The impact of the market portfolio on the valuation, incentives and optimality of executive stock options. Quantitative Finance, 5(1), 35-47.
[3] Henderson, V., & Hobson, D. (2004). Utility indifference pricing-an overview. Volume on Indifference Pricing.
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