REAL OPTIONS Alain Bensoussan International Center for Decision and Risk analysis School of Management University of Texas at Dallas In Honor to Professor Wolfgang Runggaldier, July 17 2007 – p. 1/81
INTRODUCTION Real Options theory is an approach to mitigate risks of investment projects which is based on two ideas. The first one is hedging, borrowed from financial options, when market considerations can be introduced. The project risk must be correlated to the market risk ,in which case tradable assets can be used to hedge. The second idea is flexibility . There is flexibility in the process of decision making. In particular, one may scale down or up the project, one may stop it, one may change orientation. This flexibility allows to react properly when information is obtained on the uncertainties of the evolution. In Honor to Professor Wolfgang Runggaldier, July 17 2007 – p. 2/81
EXAMPLES option to defer natural-resource extraction, real estate staged investment long-development capital-intensive projects option to alter operating scale facility planning and construction in cyclical industries option to abandon new product introductions in uncertain markets option to switch goods subject to volatile demand In Honor to Professor Wolfgang Runggaldier, July 17 2007 – p. 3/81
TRADITIONAL NPV Consider a project which will bring a cash flow P ( t ) per unit of time. Its value is � T V = EP ( t ) exp − rtdt 0 According to the NPV rule, the project can be decided if its net present value V − I ≥ 0 , where I is the investment cost. The flexibility is not valued. The discount factor is the risk-free interest rate. In Honor to Professor Wolfgang Runggaldier, July 17 2007 – p. 4/81
REFERENCES A.K. DIXIT, R.S. PINDYCK Investment under Uncertainty , Princeton University Press, Princeton,New Jersey, 1994. L. TRIGEORGIS, Real Options ,MIT Press, 1996. T. COPELAND, V. ANTIKAROV, Real Options , Texere, N.Y. 2003 In Honor to Professor Wolfgang Runggaldier, July 17 2007 – p. 5/81
COMPLETE MARKET MODEL We assume continuous time. The randomness is characterized by n standard independent Wiener processes w j ( t ) . We denote by F t = σ ( w j ( s ) , j = 1 , · · · , n ; s ≤ t ) There are n "basic" assets on the market whose prices are denoted by Y i ( t ) whose evolution is governed by dY i ( t ) = Y i ( t )( α i ( t ) dt + σ ij ( t ) dw j ) where α i ( t ) , σ ij ( t ) are processes adapted to F t . In Honor to Professor Wolfgang Runggaldier, July 17 2007 – p. 6/81
The market is complete when the matrix σ ( t ) is invertible. In this case, the information obtained by observing the evolution of the prices of assets is sufficient to recover the underlying source of noise modelled by the Wiener processes. We assume that these basic assets do not carry coupons. In Honor to Professor Wolfgang Runggaldier, July 17 2007 – p. 7/81
MARKET RISK INDICATOR In addition to the random assets there is a riskless asset whose evolution is characterized by Y 0 ( t ) = exp rt Denoting by α ( t ) the vector with components α i ( t ) we consider the process θ ( t ) = σ − 1 ( t )( α ( t ) − r 1 I ) whose definition makes direct use of the invertibility of the matrix σ ( t ) . In Honor to Professor Wolfgang Runggaldier, July 17 2007 – p. 8/81
We next define Z ( t ) by the relation dZ ( t ) = − Z ( t ) θ ( t ) .dw ( t ) , Z (0) = 1 This process ( a martingale ) is an indicator of the market risk In Honor to Professor Wolfgang Runggaldier, July 17 2007 – p. 9/81
RISK PREMIUM AND CAPM This is justified by the formula α i dt = rdt − cov ( dY i ( t ) Y i ( t ) , dZ ( t ) Z ( t ) |F t ) (1) The expected return of each individual asset is the risk-free return plus a premium linked to the specific risk of the asset. Other Approaches: Risk-neutral probability, martingale, CCAPM In Honor to Professor Wolfgang Runggaldier, July 17 2007 – p. 10/81
TRADABLE ASSET We shall consider now an asset whose value P ( t ) evolves according to � dP = P ( α ( t ) dt + σ j dw j ) j This asset, different from the basic assets of the market, gets its random component from the randomness of the market, but with specific volatility, hence specific risk. In Honor to Professor Wolfgang Runggaldier, July 17 2007 – p. 11/81
We consider an expression similar to (1) , namely µdt = rdt − cov ( dP ( t ) P ( t ) , dZ ( t ) Z ( t ) |F t ) (2) which is called the risk-adjusted expected rate of return of P In Honor to Professor Wolfgang Runggaldier, July 17 2007 – p. 12/81
BASIC ASSUMPTION We assume that µ ( t ) − α ( t ) = δ ( t ) ≥ 0 (3) We shall then say that the asset is tradable . The quantity δ ( t ) will be interpreted as a dividend associated to the asset P ( t ) . The logic is that we can build a risk-free portfolio made of the asset P ( t ) and a short position on the basic assets of the market. Its value is P ( t ) − � i π i ( t ) Y i ( t ) . In Honor to Professor Wolfgang Runggaldier, July 17 2007 – p. 13/81
For this portfolio to be risk-free, we must choose π i ( t ) such that � P ( t ) σ j ( t ) = π i ( t ) Y i ( t ) σ ij ( t ) i which defines uniquely π i ( t ) , since the matrix σ is invertible. We then claim that its expected return at any time per unit of time should be equal to the risk-free return. In Honor to Professor Wolfgang Runggaldier, July 17 2007 – p. 14/81
Recalling that P ( t ) carries a dividend δ ( t ) we then have � P ( t )( α ( t ) + δ ( t )) − π i ( t ) α i ( t ) Y i ( t ) = i � r ( P ( t ) − π i ( t ) Y i ( t )) i Using (1) and recalling the definition of µ ( t ) (2), we obtain the relation (3). In Honor to Professor Wolfgang Runggaldier, July 17 2007 – p. 15/81
COUPONS In the case the basic assets Y i ( t ) carry a coupon δ i ( t ) per unit value and unit of time, then one must replace ij (( σ ∗ ) − 1 ) ij σ j ( t ) δ i ( t ) . It is then useful to r with r + � consider the case when P ( t ) is simply one of the basic assets Y k ( t ) . Formula (3) reduces to the relation defining θ k . In Honor to Professor Wolfgang Runggaldier, July 17 2007 – p. 16/81
VALUE OF A PROJECT We consider now that P ( t ) represents the output flow of a project, or the revenue of a company. We will assume α , σ ij deterministic constants. We also consider that P carries a dividend δ per unit-value and unit of time also constant to simplify. We have the relation α + δ = µ where µ , the risk-adjusted expected rate of return of P is given by (2). In Honor to Professor Wolfgang Runggaldier, July 17 2007 – p. 17/81
VALUATION EQUATION The project carries a flow of profits given by π ( P, t ) when the output is P at time t . Denote by V ( P, t ) the value of owning the project ( we can also think of a firm instead of a project and speak about the value of the firm). We consider the value of ownership of the project as a tradable asset and write its differential ∂ 2 V ∂P Pα + 1 dV = ( ∂V ∂t + ∂V ∂P 2 P 2 σ 2 ) dt + 2 ∂V � ∂P P σ j dw j j σ 2 = � σ 2 j j In Honor to Professor Wolfgang Runggaldier, July 17 2007 – p. 18/81
The risk-adjusted rate of return of the ownership of the project is then µ = r + ∂V P cov ( σ j dw j , dZ � � (( σ ∗ ) − 1 ) ij σ j δ i − ˜ V [ Z )] ∂P ij j Therefore µ − rV = ∂V V ˜ ∂P P ( µ − r ) We then write that the expected capital return on the ownership of the project plus the profit flow per unit value is equal to ˜ µ . In Honor to Professor Wolfgang Runggaldier, July 17 2007 – p. 19/81
This means ∂ 2 V ∂P Pα + 1 µ = π + ∂V ∂t + ∂V ∂P 2 P 2 σ 2 V ˜ 2 We see that α can be eliminated. There remains ∂ 2 V ∂P P ( r − δ ) + 1 ∂V ∂t + ∂V ∂P 2 P 2 σ 2 − rV + π = 0 (4) 2 In Honor to Professor Wolfgang Runggaldier, July 17 2007 – p. 20/81
SOLUTION OF THE VALUATION EQUATION Equation (4) is a partial differential equation whose space variable P lies in (0 , ∞ ) . We need boundary conditions. As far time is concerned, we will in most cases look for stationary solutions of equation (4) namely ∂ 2 V ∂P P ( r − δ ) + 1 ∂V ∂P 2 P 2 σ 2 − rV + π = 0 (5) 2 which is possible when coefficients are independent of time ( as we assumed), the profit flow is also independent of time and the horizon is infinite. For finite horizon, V ( P, T ) = 0 whose interpretation is clear. In Honor to Professor Wolfgang Runggaldier, July 17 2007 – p. 21/81
Concerning the variable P we need conditions at 0 and ∞ . For P = 0 we see formally on equation (5) that V (0) = π (0) r It is natural to assume that π (0) = 0 , so V (0) = 0 . This assumes implicitly that V does not have a singularity at 0 . Concerning the condition at infinity, we need to specify a growth condition. We shall require a growth condition similar to that of π ( P ) . Leaving aside a particular solution which will be valid for P large, the general solution is an exponential V ( P ) = exp βP In Honor to Professor Wolfgang Runggaldier, July 17 2007 – p. 22/81
β is a solution of 1 2 σ 2 β ( β − 1) + ( r − δ ) β − r = 0 The roots of this quadratic expression are β 1 , β 2 with � β 1 = 1 2 − r − δ [1 2 − r − δ σ 2 ] 2 + 2 r + σ 2 σ 2 � β 2 = 1 2 − r − δ [1 2 − r − δ σ 2 ] 2 + 2 r − σ 2 σ 2 and β 1 > 1 , β 2 < 0 In Honor to Professor Wolfgang Runggaldier, July 17 2007 – p. 23/81
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