Advanced Topics in Corporate Finance - IMW

Advanced Topics in Corporate FinanceThomas DanglTU Wienthomas.dangl@tuwien.ac.athttp://www.imw.tuwien.ac.at/danglVGSF, 2010Continuous-Time Finance Toolbox

The Market Price of Risk (1)• Assume the dynamics of a single underlying process X(t) [which is notnecessarily traded] asdX(t) = µ(t, X(t))dt + σ(t, X(t))dW (t)X(0) = X 0where dW (t) is the increment to a Wiener process W (t). This is a Markovprocess.• The value of X will characterize the state of the firm (level of current cashflows,value of the unlevered productive assets, etc.)Continuous-Time Finance Toolbox 1

The Market Price of Risk (2)• There exists a money account B(t) with the dynamicsdB(t) = r(t)B(t)dt,B(0) = B 0 .We assume in what follows r(t) = r = constant. Many of the results, however,hold in the general case of stochastic interest rates.• Assume further that there exists a claim F = F (t, X(t)) contingent on theprocess X, where F denotes the market value of the claim.• It is assumed that F is twice differentiable in X and once in t.Continuous-Time Finance Toolbox 2

The Market Price of Risk (3)• Then Itô’s Lemma gives the dynamics of F asdF = ∂F∂tF (0) = F 0 .dt +∂F∂X dX + 1 2 ∗ ∂2 F∂X 2σ2 dt,• Please note: This is NOT a valuation equation. If X(t) is NON-traded, thereis NO WAY to derive a unique market price of the claim F from the knowledgeof X and its dynamics.Continuous-Time Finance Toolbox 3

The Market Price of Risk (4)• If there is more than one claim traded on a well-functioning market, i.e.F i (t, X(t)) contingent on the realizations of X, the F s can be priced relativeto one reference claim (under some regularity conditions).• From Itô’s Lemma we know:dF i ={ ∂Fi∂t + ∂F i∂X µ(t, X(t)) + 1 }2 σ2 (t, X(t)) ∂2 F i∂X 2dt + ∂F iσ(t, X(t))dW.∂XContinuous-Time Finance Toolbox 4

The Market Price of Risk (5)• In many cases, holding a claim is associated with the right to receive certainpayoffs.• Let D i (t) denote the accumulated payouts associated with holding claim F ifrom time 0 to time t.If δ i (t, X(t)) is the flow (again contingent on X), then we haveD i (t) =∫ t0δ i (τ, x(τ))dτ,or the flow differentialdD i (t) = δ i (t, X(t))dt.Continuous-Time Finance Toolbox 5

The Market Price of Risk (6)• Then the gain differential dG i can be defined asdG i = dF i + dD i ,where dF i is the capital gain differential and dD i is the payout differential.dG i ={ ∂Fi∂t + ∂F i∂X µ(t, X(t)) + 1 }2 σ2 (t, X(t)) ∂2 F i∂X 2 + δ i(t, X(t)) dt+ ∂F iσ(t, X(t))dW (t).∂XContinuous-Time Finance Toolbox 6

The Market Price of Risk (7)• Write the relative gain differential asdG iF i= α i (t, X(t))dt + σ i (t, X(t))dWwhereα i = 1 F i{ ∂Fi∂t + ∂F i∂X µ + 1 2 σ2∂2 F i∂X 2 + δ }σ i = 1 F i{ ∂Fi∂X σ }(arguments (t, X(t)) are dropped for brevity)Continuous-Time Finance Toolbox 7

The Market Price of Risk (8)• Let us take two different claims, F i and F j , and form an instantaneouslyriskless portfolio:V = h i F i + h j F j• Define the relative portfolio weights:u i = h iF iV , u j = h jF jV⇒ h i = u iVF i,h j = u jVF jContinuous-Time Finance Toolbox 8

The Market Price of Risk (9)• For valuation, the total gain differential must be considereddV = h i dG i + h j dG j{}dG i dG j= V u i + u jF i F j= V {u i α i + u j α j } dt + V {u i σ i + u j σ j } dW• Then form a riskless gain portfoliou i + u j = 1u i σ i + u j σ j = 0Continuous-Time Finance Toolbox 9

The Market Price of Risk (10)• This leads to portfolio weightsu i =u j =−σ jσ i − σ jσ iσ i − σ jThis only makes sense if σ i ≠ σ j !• The gain dynamics of this portfolio are thendV = V{ }αj σ i − α i σ jdt + 0dWσ i − σ jContinuous-Time Finance Toolbox 10

The Market Price of Risk (11)• Since this portfolio is instantaneously riskless, absence of arbitrage impliesdV = r(t)dt⇒ α jσ i − α i σ jσ i − σ j= r(t),α j σ i − α i σ j = r(σ i − σ j ),(α j − r)σ i − (α i − r)σ j = 0,α j − r− α i − rσ j σ i= 0,⇒ α i − r(t)σ iwhere λ(t) is the market price of risk.= α j − r(t)σ j= λ(t), ∀i, j,Continuous-Time Finance Toolbox 11

Valuation (1)• Now we substitute for α i , σ i :write the no-arbitrage condition as α i − r = λ(t)σ i :1F i{ ∂Fi∂t + ∂F i∂X µ i(t, X(t))+ 1 2 σ i(t, X(t)) ∂2 F i∂X 2 + δ(t, X(t)) }− r = 1 F iσ i (t, X(t))λ(t) ∂F i∂Xr(t)F i = ∂F i∂t + ∂F i∂X {µ i(t, X(t)) − λ(t)σ i (t, X(t))}+ 1 ∂ 2 F i2 ∂X 2σ2 (t, X(t)) + δ(t, X(t))F i (T, x) = φ(x)• Note: T can be an adapted random stopping time.Continuous-Time Finance Toolbox 12

Valuation, Special Cases (1)• Let us regard two special cases for the dynamics of X(t)◦ Brownian motiondX = µ dt + σ dW (µ, σ = constant),X(0) = X 0 .◦ Geometric Brownian motiondXX= µ dt + σ dW (µ, σ = constant)Continuous-Time Finance Toolbox 13

Valuation, Special Cases (2)• Correspondence between Brownian motion and Geometric Brownian motion:Suppose X(t) with dynamicsdX = µ dt + σ dW,X(0) = X 0 .⇒ X(t) = X 0 += X 0 +∫ t0∫ t0dX,µ dt +∫ t0= X 0 + µt + σW (t).σ dW,Continuous-Time Finance Toolbox 14

Valuation, Special Cases (3)• Definey(t) = exp{X(t) − 1 2 σ2 t}.⇒ dy = yµ dt + yσ dW,y 0 = e X 0i.e. y(t) is a Geometric Brownian motion, and E[y(t)] = y 0 e µt .• Note:E[y] = E[ {y 0 exp (µ − 1 }]2 σ2 )t + σW (t)We know that W (t) is normally distributed with mean 0 and variance t. Thus,we can integrate the density weighted value of y over the range (−∞, ∞) toget the above statement.Continuous-Time Finance Toolbox 15

Valuation, Special Cases (4)• Valuation Equationr(t)F = ∂F∂t + ∂F∂X {µ − λ(t)σ} + 1 2∂ 2 F∂X 2σ2 + δ(t, X(t)).• Note:◦ λ(t), r(t) can be any processes adapted to the information set F t⇒ may be path dependent,⇒ solution complicated and F ≠ F (X(t), t) but F = F (t, {X(τ)|0 ≤ τ ≤t}).◦ special functional form of δ(t, X(t)) might also cause problems when solvingthe equation (see Feynman-Kač below).Continuous-Time Finance Toolbox 16

Valuation, Special Cases (5)• Remember• Valuation equationdXX= µ dt + σ dW (µ, σ = constant)r(t)F = ∂F∂t + ∂F∂X X{µ − λ(t)σ} + 1 2∂ 2 F∂X 2X2 σ 2 + δ(t, X(t))Continuous-Time Finance Toolbox 17

Valuation, Special Cases (6)• Assume X(t) is itself a traded security, paying δ(t, X(t)) per time interval tothe security trader.◦ We know that◦ Then∂X∂t = 0,∂X∂X = 1,∂ 2 X∂X 2 = 0.α x = 1 {µ(t, X(t)) + δ(t, X(t))},Xσ x = 1 {σ(t, X(t))}.XContinuous-Time Finance Toolbox 18

Valuation, Special Cases (7)• no arbitrage → X must satisfyα X − rσ X= λ(t).You can also interpret this equation as: the dynamics of the traded asset Xdefine or reveal λ(t).• Now1X {µ(t, X(t)) + δ(t, X(t))} − r(t) = λ(t) 1 σ(t, X(t)),X⇒ r X = µ(t, X(t)) + δ(t, X(t)) − λ(t)σ(t, X(t)).Continuous-Time Finance Toolbox 19

Valuation, Special Cases (8)• If X is a Geometric Brownian motion:• Next step:r = µ + δ(t)X − λ(t)σµ − λ(t)σ = r(t) − δ(t)X◦ Assume λ is constant, i.e. λ(t) = λ.◦ Assume r is constant, i.e. r(t) = r.• ⇒ if µ and σ are constant, µ − λσ = ˆµ is constant.ˆµ is the risk-neutral drift of the underlying.• Now we consider some special setups in order to derive different modules thathelp us to determine general solutions of the valuation equation.Continuous-Time Finance Toolbox 20

Valuation, Solving the HJB-Equation (1)• Suppose F (x) is a claim that pays K whenever the process X first hits a lowerbarrier x < X 0 , i.e. δ(t, X(t)) = 0.Let X follow a simple Brownian motion. Then the valuation equation is:rF = ∂F∂t + ∂F∂x ˆµ + 1 ∂ 2 F2 ∂x 2 σ2 .• Since F is not explicitly time dependent,∂F∂t= 0,⇒ rF = ∂F∂x ˆµ + 1 2∂ 2 F∂x 2 σ2 .This is an ordinary second order differential equation which is homogenous inthe derivatives of F .Continuous-Time Finance Toolbox 21

Valuation, Solving the HJB-Equation (2)• Solution approach: Assume (ansatz)F = e βx ,⇒ ∂F∂x = βeβx ,∂ 2 F∂x 2 = β2 e βx .• Substitute into drift equation:re βx = βe βxˆµ + 1 2 β2 e βx σ 2 , ⇒ r − β ˆµ − 1 2 β2 σ 2} {{ }fundamental quadratic= 0.• The solutions are:β 1,2 = − ˆµ σ 2 ± √ ( ˆµσ 2 ) 2+ 2 r σ 2} {{ }> ˆµ σ 2 if r>0Continuous-Time Finance Toolbox 22

Valuation, Solving the HJB-Equation (3)• Since we know that r > 0 ⇒, β 1 > 0 and β 2 < 0.rΒContinuous-Time Finance Toolbox 23

Valuation, Solving the HJB-Equation (4)• The general solution is thenF (x) = Ae β 1x + Be β 2x• Needs two boundary conditions.• Assume a claim F pays K when x hits a lower barrier x:(i) lim x→x+ F (x) = K,(ii) lim x→∞ F (x) = 0.• The transversality condition (ii) is often called the ”exclusion of speculativebubbles”.Continuous-Time Finance Toolbox 24

Valuation, Solving the HJB-Equation (5)• From (ii) it follows A = 0 since β 1 > 0.• Condition (i) impliesF (x) = Be β 2x⇒⇒ F (x) = e β 2(x−x)} {{ } KD(x)B = Ke −β 2x• D(x) is a discount factor. Since β 2 < 0,⇒ D(x) ↓ 0 for x ↑ ∞,⇒ D(x) ↑ 1 for x ↓ x.Continuous-Time Finance Toolbox 25

Valuation, Solving the HJB-Equation (6)• Assume a claim that pays K when X hits an upper barrier ¯x.• The general solution is the same as above but boundary conditions change.(i) lim x→x− F (x) = K(ii) lim x→−∞ F (x) = 0• From (ii) it follows that B = 0 since β 2 < 0.• Condition (i) impliesF (¯x) = Ae β 1¯x⇒ A = Ke −β 1¯x .F (x) =e −β 1(¯x−x)} {{ } Kdiscount factorContinuous-Time Finance Toolbox 26

Valuation, Solving the HJB-Equation (7)• Now suppose that X follows a Geometric Brownian motion, F is a claim thatpays K if X hits a lower threshold x, again ∂F∂t= 0, F = F (x).• Valuation EquationrF = ∂F∂x xˆµ + 1 ∂ 2 F2 ∂x 2 x2 σ 2• Solution approach (ansatz)F (x) = x β⇒ ∂F∂x = βxβ−1∂ 2 F∂x 2 = β(β − 1)x β−2Continuous-Time Finance Toolbox 27

Valuation, Solving the HJB-Equation (8)• Substitution into the valuation equation yieldsrx β = β ˆµ xx β−1} {{ }x β+ 1 2 β(β − 1)σ2 x 2 x β−2} {{ }x β• Get the fundamental quadraticr − β ˆµ − 1 2 β(β − 1)σ2 = 0• Again we have β 1 > 0, β 2 < 0.Continuous-Time Finance Toolbox 28

Valuation, Solving the HJB-Equation (9)• Roots of the fundamental quadratic:β 1,2 = 1 2 − ˆµ σ 2 ± √ (12 − ˆµ σ 2 ) 2+ 2rσ 2Continuous-Time Finance Toolbox 29

Valuation, Solving the HJB-Equation (10)• If r > ˆµ, we have furthermore β 1 > 1:β 1 = 1 2 − ˆµ σ 2 + √ √√√√ ( 12 − ˆµ σ 2 ) 2+ 2rσ 2} {{ }∗∗ = 1 4 − ˆµ σ 2 + ( ˆµσ 2 ) 2+ 2rσ 2 > 1 4 − ˆµ σ 2 + ( ˆµσ 2 ) 2+ 2ˆµσ 2 = ( 12 + ˆµ σ 2 ) 2⇒ β 1 > 1 2 − ˆµ σ 2 + √ (12 − ˆµ σ 2 ) 2= 1 2 − ˆµ σ 2 + 1 2 + ˆµ σ 2 = 1Continuous-Time Finance Toolbox 30

Valuation, Solving the HJB-Equation (11)• If X(t) is the value of a traded asset that pays no dividend:⇒ ˆµ = r⇒ β 1,2 = 1 2 − r √1σ 2 ± 4 − r σ 2 + r2σ 4 + 2r} {{ σ 2}12 + r σβ 1 = 1β 2 = − 2rσ 2Continuous-Time Finance Toolbox 31

Valuation, Solving the HJB-Equation (12)• Back to the claim F , the general solution istBoundary conditions:(i) lim x→x+ F (x) = K(ii) lim x→∞ F (x) = 0 ⇒ A = 0• From (i) it follows thatF = Ax β 1+ Bx β 2F (x) = Bx β 2= K, ⇒ B = K( 1x) β2, F (x) =( β2xKx)} {{ }Dwhere D is the appropriate discount factor for a geometric Brownian motionabsorbed at a lower threshold.Continuous-Time Finance Toolbox 32

Valuation, Solving the HJB-Equation (13)• Note: The claim F is essentially an Arrow-Debreu security associated with theevent that the process hits x from above for the first time.• Analogously, if X is absorbed at an upper barrier ¯x the boundary conditionsare(i) lim x→¯x F (x) = K(ii) lim x→0+ F (x) = 0⇒ B = 0 ⇒ F (x) =( x¯x) β1K,} {{ }Dwhere D is the appropriate discount factor for a geometric Brownian motionabsorbed at an upper threshold.Continuous-Time Finance Toolbox 33

Feynman-Kač Representation (1)• Assume risk neutrality ⇒ asset pricing is done based on expectations withrespect to the objective probability measure P .• Take a cash flow δ(t, X(t)) which is contingent on the general diffusion X(t).• Calculate the expected value of the aggregated discounted cash flow up tosome (random stopping) time T , and assume the claim pays K at T .F P (t, X t ) = E P [ ∫ Tt] [ ]δ(τ, x(τ))e −r(τ−t) dτ|X(t) = x t + E P e −r(T −t) K= F P δ (t, X t ) + F P K(t, X t ).• First focus only on F P δ (t, X t), the cash flow-created part of the claim value,and decompose the integral into integration over [t, t + dt] and [t + dt, T ].Continuous-Time Finance Toolbox 34

Feynman-Kač Representation (2)F P δ (t, X t ) = δ(t, X t )dt + E P [ ∫ T= δ(t, X t )dt+e −rdt E P [E P [ ∫ Tt+dtt+dtδ(τ, x(τ))e −r(τ−t) dτ|X(t) = x t]δ(τ, x(τ))e −r(τ−t−dt) dτ|X(t + dt) = x t+dt]= δ(t, X t ) + e −rdt E P [F (t + dt, X t + dX)|X(t) = x t ]= δ(t, X t ) + e −rdt E P [F (t, X(t)) + dF (t, X(t))|X(t) = x t ]|X(t) = x t]Continuous-Time Finance Toolbox 35

Feynman-Kač Representation (3)• Two rearrangements:◦ Write e −rdt = 1 − rdt + o(dt)◦ E P [F + dF ] = E P [F ] + E P [dF ] = F + E[dF ]• Applying Itǒ’s Lemma givesdF ={ ∂F∂t + ∂F∂x µ(t, x(t)) + 1 ∂ 2 }F2 ∂x 2 σ2 (t, x(t)) dt + ∂F σ(t, x(t))dW∂xsince E(dW ) = 0,⇒ E(dF ) =( ∂F∂t + ∂F∂X µ + 1 2∂ 2 )F∂X 2σ2 dtContinuous-Time Finance Toolbox 36

Feynman-Kač Representation (4)• Put all parts together:F = δdt + (1 − rdt){F +( ∂F∂t + ∂F∂x µ + 1 ∂ 2 ) }F2 ∂x 2 σ2 dt• Ignore all terms of order o(dt) in the following equation:( ∂FF = δdt + F − rF dt +∂t + ∂F∂x µ + 1 ∂ 2 )F2 ∂x 2 σ2 dt• It follows the valuation equation for a claim that pays δ for λ = 0, i.e. forrisk-neutral agents:⇒ rF = ∂F∂t + ∂F∂x µ + 1 ∂ 2 F2 ∂x 2 σ2 + δThe value at T is determined by the boundary condition: F (T, X T ) = K.Continuous-Time Finance Toolbox 37

Feynman-Kač Representation (5)• Thus, there is a one-to-one correspondence between the two statements(i) F is given by the expectations over the stochastic integralF P (t, X t ) = E P [ ∫ Ttδe −r(τ−t) dτ|X(t) = x t]+E P [ e −r(T −t) K|X(t) = x t](ii) F is the solution of the partial differential equationrF = ∂F∂X µ + 1 ∂ 2 F2∂X + δF (T, X T ) = K(T, X T )Continuous-Time Finance Toolbox 38

Feynman-Kač Representation (6)• We see that the expected value of the stochastic integral over the discountedcash flows represents the valuation equation only if λ = 0 ⇔ risk-neutralagents.• Otherwise we have to change the probability measure in order to get thevaluation right!• i.e., we have to change the dynamics of the underlying:µ → (µ − λσ)P → Q ⇔[ ∫ ]TF (t, X t ) = E Q δe −r(τ−t) dτ|X(t) = x tt+E Q [ e −r(T −t) K(T, X(T ))|X(t) = x t]Continuous-Time Finance Toolbox 39

Feynman-Kač Representation (7)• or alternativelyF (t, X t ) =∫ Tt∫ ∞0δe −r(τ−t)dQdP dP dτ + ∫ ∞0e −r(T −τ) K dQdP dP |X(t) = x twhere dQdPis the Radon-Nykodym derivative which explicitly characterizes themeasure transformation.• Comment: Numerical methods like tree-approaches or Monte-Carlo simulationsuse the integral-representation to solve the corresponding PDE.Continuous-Time Finance Toolbox 40

Example• Assume◦ δ = 0 for all t and K(T, X(T )) = K = constant.◦ T is a random stopping time defined by the first hit of an upper threshold¯x.◦ X is a geometric Brownian motion.• In this case,F (t, X 0 ) = E Q [ e −rT K ] = E Q [ e −rT ] K =( X¯X) β1K⇒ E Q [ e −rT ] =( X¯X) β1Continuous-Time Finance Toolbox 41

Further Example (1)• Some more advanced Arrow-Debreu securities.• X = geometric Brownian motion◦ F pays ¯K if an upper barrier ¯x is hit first,◦ F pays K if a lower barrier x is hit first,◦ δ = 0• F = Ax β 1 + Bx β 2, two boundary conditions F (x) = K, F (¯x) = ¯K• A¯x β 1 + B¯x β 2 = ¯KAx β 1 + Bx β 2 = KContinuous-Time Finance Toolbox 42

Further Example (2)• This impliesF = ¯D ¯K + D K¯D = xβ 2x β 1 − x β 1x β 2x β 1¯xβ 2 − xβ 2¯xβ 1= E Q [ e −rT ∧ ¯x is hit first at T ]= ¯D(x; ¯x, x)D = xβ 2¯x β 1 − x β 1¯x β 2x β 2¯xβ 1 − xβ 1¯xβ 2= E Q [ e −rT ∧ x is hit first at T ]= ¯D(x; ¯x, x)Continuous-Time Finance Toolbox 43

Further Example (3)• Are these events mutually exclusive?• The discount factors have the following properties:lim x→ ¯x ¯D(x; ¯x, x) = 1, limx→ ¯x D(x; ¯x, x) = 0,lim x→ x ¯D(x; ¯x, x) = 0, limx→ x D(x; ¯x, x) = 1,(lim x→ 0 ¯D(x; ¯x, x) = x¯x) β1, lim x→ 0 D(x; ¯x, x) = 0,( β2lim¯x→∞ ¯D(x; ¯x, x) = 0,xlim¯x→∞ D(x; ¯x, x) = x).Continuous-Time Finance Toolbox 44

General Solution for δ ≠ 0 (1)• Now we search for a general solution of the valuation equation if δ ≠ 0 ⇒valuation equation is non-homogeneous• Consider the special case of a geometric Brownian motion and a claim that isnot explicitly time dependent, i.e. ∂F∂t = 0⇒ rF = ∂F∂x xˆµ + 1 2 x2 σ 2∂2 F+ δ(t, x(t))∂x2 • We know the general solution of the homogeneous differential equation¯F h = Ax β 1+ Bx β 2Continuous-Time Finance Toolbox 45

General Solution for δ ≠ 0 (2)• The general solution of the non-homogeneous differential equation is thegeneral solution of the homogeneous differential equation + a specialsolution of the non-homogeneous differential equation.• We know the general solution of the homogeneous differential equation¯F h = Ax β 1+ Bx β 2.• To compute a special solution of the non-homogeneous equation, we use theFeynman-Kač representation, and assume as a special case that the currentregime prevails forever (i.e., there is no absorbing upper or lower barrier, etc.),thus, T = ∞.∫∞¯F = E Q δ(τ, x(τ))e −r(τ−t) dτtContinuous-Time Finance Toolbox 46

General Solution for δ ≠ 0 (3)• General solution:F = F h + ¯F• Assume δ = c is constant.∫∞⇒ ¯F = E Q0ce −rτ dτ =[− c r e−rτ] ∞0= 0 − (− c r e0 ) = c r⇒ ¯F is the current value of the perpetual flow c.Continuous-Time Finance Toolbox 47

General Solution for δ ≠ 0 (4)• General solution:F = Ax β 1} {{ }∗+ Bx β 2} {{ } + c∗∗ }{{} r∗∗∗We know * and ** represent the value of the claim if the regime changes atsome upper or lower threshold. *** is the fundamental solution.• If F is really the claim to the perpetuity c, ⇒ A = B = 0, i.e. we excludespeculative bubbles!F = ¯F = c rContinuous-Time Finance Toolbox 48

Examples (1)• Consider a bond contract, a consol bond that pays a constant coupon c withoutcredit risk ⇒ ¯F = c r• If X(t) is a geometric Brownian motion that characterizes the state of thefirm and we know that the firm will default on its obligations if X hits x. Thevalue of the bond in bankruptcy is K ⇒ there is a regime shift at x.⇒ F = Ax β 1+ Bx β 2+ c r• Two boundary conditions(i) F (x) x→∞−→ c r(ii) F (x) x→x−→ K. . . probability of default vanishes.Continuous-Time Finance Toolbox 49

Examples (2)• From (i): A = 0 since β 1 > 1 . . . excluding speculative bubbles.F (x) = Bx β 2+ c r = K, ⇒ B = (K − c r ) ( 1x) β2.• Value of a risky bond:F (x) = (K − c ( β2x} {{ r } x) ) + c} {{ } }{{} r∗ ∗∗ ∗∗∗* is the differential payoff if the firm defaults. Claimholder loses the perpetuitycrbut receives K.** is the appropriate discount factor for this uncertain event.*** is the fundamental value, i.e. the value of the bond if the current regime,i.e. the firm is solvent, remains forever.Continuous-Time Finance Toolbox 50

Examples (3)• Assume δ = kx where k is constant. ⇒ x characterizes the cash flow of thefirm and F is a claim to a fixed fraction of this cash flow.[∫ ∞]¯F t = E Q kx(τ)e −r(τ−t) dτ|x(t) = x t==t∫ ∞E Q tt∫ ∞= k x tr − ˆµ⇒ ¯F kx(x) =r − ˆµt[kx(τ)e −r(τ−t)] dτkx t eˆµ(τ−t) e −r(τ−t)} {{ }e −(r−ˆµ)(τ−t)dτ(Fubini)⇒ gives sense only if ˆµ < r what we assumeContinuous-Time Finance Toolbox 51

Examples (4)• Then the general solution of the valuation equation isF (x) = Ax β 1+ Bx β 2+kxr − ˆµ .• Assume the firm runs into bankruptcy if x hits x and claimholders lose theright to participate in cash flows after the first hit of x, ⇒ F is an equityclaim.• Please note: Since in this simple modelwill never voluntarily default.kxr−ˆµ> 0 in any case, equityholders• Two boundary conditions:(i) lim x→∞ (F (x) − ¯F (x)) = 0, ⇒ no speculative bubbles,(ii) lim x→x F (x) = 0.Continuous-Time Finance Toolbox 52

Examples (5)• From (i) ⇒ A = 0 since β 1 > 0( β2• From (ii) ⇒ B = −kx 1r−ˆµ x)⇒ this is the value of the equity claim to afraction k of the firm’s cash flow.F (x) = −kx ( β2x}r −{{ˆµ}x)} {{ }∗ ∗∗+ kxr − ˆµ} {{ }∗∗∗* in the case of bankruptcy, claimholders lose their claim.** appropriate discount factor, the corresponding Arrow-Debreu security*** fundamental valueContinuous-Time Finance Toolbox 53

Regime Shifts (1)• Two cases(i) shifts at absorbing barriers,(ii) shifts at non-absorbing barriers.• ad (i): We already had two examples, risky bond and risky equity contract.• In general: Suppose a claim which is worth F (x) as long as x > x, when xhits x the first time the regime changes and the claim is worth F d (x).⇒ since F (x) = δdt + e −rdt E Q (F + dF ),we require lim F (x) = F d (x)x→xThis is the so-called value matching condition.Continuous-Time Finance Toolbox 54

Regime Shifts (2)• If there is a lump sum payment of K associated with the regime shift:lim F (x) = F d(x) − Kx→x• Example:Convertible debt: A firm has two different debt issues outstanding, a consolbond requiring a continuous coupon flow c 1 , and a convertible bond requiringa coupon c 2 . Assume at an upper level ¯x bondholders convert and after thata fraction of k of the firm shares are owned by former bondholders, i.e. afraction of k of the cashflows benefit the former bondholders.Continuous-Time Finance Toolbox 55

Regime Shifts (3)• What is the value of equity and debt?Let F 1 denote the pre-conversion valueLet F 2 denote the post-conversion value of the convertible debt contract.• One key assumption: risk neutral dynamics of x do not change!• Start with F 2 which is essentially equity:F 2 (x) =( kc1r − kx ) ( ) β22 x+ kxr − ˆµ x} {{ 2 r − ˆµ − kc 1r}D(x;¯x=∞,x 2 )The Pre-conversion Value is(F 1 (x) = K − c )2D(x; ¯x, xr1 ) +(F 2 (¯x) − c 2r)¯D(x; ¯x; x1 ) + c 2r .Continuous-Time Finance Toolbox 56

Regime Shifts (4)• ad (ii): Regime shifts at a non-absorbing barrierExample: Assume you hold a claim which pays x(t) whenever x(t) ≥ ˜x anda fixed coupon c whenever x < ˜x, i.e. x(t) can freely move back and forthacross the critical threshold ˜x (e.g. preferred stock).• Two boundary conditions have to be met:(a) value matching: lim x→˜x− F (x) = lim x→˜x+ F (x)(b) smooth pasting: lim x→˜x− F (x) ′ = lim x→˜x+ F (x) ′Both boundary conditions are required by the fact that F (x) has a Feynman-Kač representation of the form:F (x) = c(t)dt + e −rdt E Q (F + dF )Continuous-Time Finance Toolbox 57

Regime Shifts (5)• Implication of the boundary conditions:FxFx• Attention: This smooth-pasting condition is not a FOC!Continuous-Time Finance Toolbox 58

Regime Shifts (6)• Some authors even modeled the case where X(t) changes its dynamics at ˜x,i.e.◦ for X ≤ ˜x◦ for X > ˜xµ = µ 1 and σ = σ 1µ = µ 2 and σ = σ 2• Attention: SDE has no strong solution in this case.Continuous-Time Finance Toolbox 59

Regime Shifts (7)Now back to the example:• X(t) follows a geometric Brownian motion.• Assume there are no further regime shifts.• It follows◦ general solution for x < ˜x◦ general solution for x ≥ ˜xF 1 = A 1 x β 1+ B 1 x β 2+ c rF 2 = A 2 x β 1+ B 2 x β 2+xr − ˆµContinuous-Time Finance Toolbox 60

Regime Shifts (8)• Need four boundary conditions:(i) B 1 = 0 since lim x→0 (F 1 (x) − c r ) = 0(ii) A 2 = 0 since lim x→∞ (F 2 (x) −xr−ˆµ ) = 0(iii) Value-matching at ˜x(iv) Smooth-pasting at ˜x• (iii) and (iv) determine A 1 and B 2 .lim A 1x β 1+ cx→˜x− r = lim B 2x β 2+x→˜x+xr − ˆµlim β 1A 1 x β1−1 = lim β 2B 2 x β2−1 + 1x→˜x− x→˜x+ r − ˆµContinuous-Time Finance Toolbox 61

Optimal Decision Making (1)• Question: When should a firm default?• Assume a firm has debt outstanding that requires a continuous coupon serviceof c and the firm earns a cashflow of X.⇒ F (x) =( −xr − ˆµ r) + c ( ) β2x+ xx r − ˆµ − c r• If the firm defaults ⇒ The game is over!F d (x) = 0∀x⇒ x is an absorbing barrier!Continuous-Time Finance Toolbox 62

Optimal Decision Making (2)x• Note: Nowr−ˆµ − c rturns negative for small values of x ⇒ endogenous defaultis a valuable option!• We used already the so-called value-matching condition:For a given x,limx→x+ F (x) = 0Continuous-Time Finance Toolbox 63

Optimal Decision Making (3)• When should the firm default? The value of equity for different default triggers.x optxcrContinuous-Time Finance Toolbox 64

Optimal Decision Making (4)• x → x optis the choice of shareholders!• Want to climb to the highest ”value-level” ⇒ indifference, or FOC:∂F d= ∂F}{{} ∂x }{{} ∂x∗ ∗∗* free boundary** value functionThis is the so-called smooth-pasting condition (see the art of smooth-pasting).In the special case ∂F d∂x= 0 ⇒ optimality• Attention: There are two completely different frameworks in which smoothpastingconditions occur! Confusion!Continuous-Time Finance Toolbox 65

Optimal Decision Making (5)• If you have a ”transparent” barrier, the smooth-pasting condition is a necessarycondition to ensure the value-function to be consistent with expectationformation.• If you have a free boundary problem, i.e. the optimization of some interventionthreshold,then the smooth-pasting condition can be violated. A violation,however, means that the intervention strategy is not optimally chosen fromthe point of view of the security holder.Continuous-Time Finance Toolbox 66

Optimal Decision Making (6)• Back to the example: A firm can decide to default on its obligations tomaintain its coupon payments.E(x) =( −xr − ˆµ r) + c ( ) β2x+ xx r − ˆµ − c r∂E∂x | x=x =( ( −xβ 2r − ˆµ r) + c ( ) )β2x −1 1x x + 1 | x=x = 0r − ˆµ( −x⇒ β 2r − ˆµ + c )= − xr r − ˆµ( )β 2 − 1 x= c β 2 r − ˆµ rContinuous-Time Finance Toolbox 67

Optimal Decision Making (7)• After rearrangingxr − ˆµ} {{ }∗=β 2 c}β 2{{− 1} }{{} r∗∗∈[0,1]forˆµ

Optimal Decision Making (8)• What is the value of the corresponding debt contract?D(x) =(− c r( β2x x+ (1 − α)+r − ˆµ)x) c rcrDx opt1Α x rΜ xContinuous-Time Finance Toolbox 69

Optimal Decision Making (9)• The transition at bankruptcy is not smooth, since bondholders do not optimizex (equityholders do determine x)• This is essentially the Leland (1994) firm model in a cash flow formulation andwithout taxes.Continuous-Time Finance Toolbox 70

Optimal Decision Making (10)• We can think of many extensions - some of them are already done:◦ Taxes: Leland (1994)◦ Dynamic capital structure choice: Fischer, Heinkel, and Zechner (1989),Goldstein, Ju, and Leland (2001), Dangl and Zechner (2004)◦ Maturity structure of debt: Leland (1994b): ”exponential debt”, Leland andToft (1996): ”linear debt”◦ Optimal maturity structure of debt: Dangl and Zechner (2009)Continuous-Time Finance Toolbox 71