Cont_IR_Models_Two

# Cont_IR_Models_Two - Con\$nuous Time Interest Rate ...

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Unformatted text preview: Con\$nuous Time Interest Rate  Models: Part Two  Pricing European Deriva\$ves  •  If a security has a payoﬀ VT at \$me T, then its  price at \$me 0 (today) is:  T V0 = E exp − ∫ 0 r( t ) dt ⋅ VT ( ) •  Some\$mes these prices can be computed  € analy\$cally using special techniques. In this  course, we will focus on compu\$ng prices  using laJces and simula\$on.  Monte‐Carlo Simula\$on  •  ct : cashﬂow at \$me t of the security to be  priced.  •  Time steps: t = Δ, 2Δ, 3Δ,…  •  Choose a short‐rate model (e.g. one deﬁned  by a stochas\$c diﬀeren\$al equa\$on).  •  Simulate the values:   {r(t ), 0 ≤ t ≤ NΔ} •  Discount factor for present valuing cf at \$me t:  v = exp(− ∫ r( s) ds) € t t 0 Monte‐Carlo Simula\$on  •  The value of the security at \$me zero is:  N C0 = E ∑ c nΔ v nΔ n =1 •  Note: This assumes the expecta\$on is taken  with respect to the risk‐neutral probabili\$es.  € •  The above expecta\$on can be computed  analy\$cally only in very special circumstances.   Monte‐Carlo Simula\$on  •  Simulate S paths for r(Δ), r(2Δ),…,r(NΔ).  Denote the ith simulated path by ri(Δ), ri(2Δ) …,ri(NΔ), i=1,…,S.  •  Es\$mate C0 by:   1S N ˆ C0 = ∑ ∑ c i ( nΔ )v i ( nΔ ) S i=1 n =1 •  Where ci(nΔ) is the cashﬂow at \$me nΔ based  € on the ith simulated interest rate path and:  v i ( nΔ ) = exp(−( ri (0) + ri (Δ ) + + ri ( nΔ ))Δ ) Monte‐Carlo Simula\$on  •  Rendleman‐Bar+er Model (Wt is a standard  Brownian mo\$on):  r( t ) = r(0) exp − ( σ2 2 t + σ ⋅ Wt ) •  Discre\$za\$on:  € r( nΔ ) = r(0) ⋅ exp− σ ⋅ nΔ + σ Z , Z ~ N (0, Δ ) i.i.d. 2 ∑ j j n 2 j =1 r( nΔ ) = r(( n − 1)Δ ) ⋅ exp − ( σ2 2 ⋅ Δ + σ ⋅ Zn ) € Monte‐Carlo Simula\$on  •  Vasicek Model (Wt is a standard Brownian  mo\$on):  dr( t ) = α (β − r( t )) dt + σ ⋅ dW t •  Discre\$za\$on (r0 given):  € € r( nΔ ) = r(( n − 1)Δ ) + α (β − r(( n − 1)Δ ))Δ + σ ⋅ Z n , Z n ~ N (0, Δ ) i.i.d. Stochas\$c Diﬀeren\$al Equa\$ons  dXt = b(Xt,t)dt + a(Xt,t)dWt X0 = x Means that, for small Δt, the change in X is (approximately) normally distributed with mean b(Xt,t)Δt and variance a2(Xt,t) Δt Xt+Δt - Xt ≈N(b(Xt,t)Δt, a2(Xt,t) Δt) This leads to a simple simulation algorithm. Pick a value for Δt (small, if you want an accurate simulation). Set X0=x and Xt+Δt = Xt + b(Xt,t)Δt + a(Xt,t) (Δt)1/2 wt Where wt is a sequence of i.i.d. N(0,1) variables. Mortgage Backed Securi\$es  •  c : Monthly mortgage payment from  underlying collateral pool of mortgages.  •  N : Length (in months) of the mortgages.  •  r0 : Fixed monthly mortgage rate.  1 α k = × balance on mortgage at (end) of month k (before payment) c € Mortgage Backed Securi\$es  •  wk : frac\$on of remaining mortgages that will  prepay (en\$re balance) at the end of month k.  •  fk : frac\$on of remaining mortgages at the  beginning of month k.  f1 = 1, f 2 = 1 − w1, f 3 = f 2 (1 − w 2 ) f k = (1 − w1 ) (1 − w k −1 ) € Mortgage Backed Securi\$es  •  Cashﬂow at the end of month k:  c k = c ⋅ f k ( w k ⋅ α k + (1 − w k )) •  Simple prepayment model:   w k = b1 + b2 arctan(rk b3 + b4 ) € •  One of b2 or b3 less than zero, so that wk decreases with rk.  € Mortgage Backed Securi\$es  w k = b1 + b2 arctan( rk b3 + b4 ) b1 = 0.01, b2 = −0.005, b3 = 10, b4 = 0.5 0.009  Prepayment Rate € w 0.008  0.007  0.006  0.005  0.004  0.003  0.002  0.001  0  0  0.007  0.014  0.021  0.028  0.035  0.042  0.049  0.056  0.063  0.07  0.077  0.084  0.091  0.098  0.105  0.112  0.119  0.126  0.133  0.14  0.147  0.154  0.161  0.168  0.175  0.182  0.189  0.196  r Monte Carlo for MBS  •  Pseudo‐Code:  For i=1 to S   Set ri(0)=r(0)   Set pv(i)=0   For j=1 to N     Generate ri(jΔ) from ri((j‐1)Δ)     Compute fi(jΔ), αi(jΔ), wi(jΔ), vi(jΔ)     Compute ci(jΔ)     Set pv(j) = pv(j) + ci(jΔ)vi(jΔ)   End  End  1 Return   S ∑ pv ( j) S j =1 € Example: Cancellable Swaps  •  •  •  •  •  Consider a swap with payment dates t1,…,tN.  Pay ﬁxed, receive ﬂoa\$ng.  Payments in arrears.  Denote rj=r(tj), and let L be the swap no7onal.  At \$me tj receive:  L( rj −1 − k ) € Cancellable Swaps  •  Have a cancellable op\$on of Bermudan type:   –  European op\$ons can only be exercised at the maturity  date.  –  American op\$ons can be exercised at any \$me up to and  including the maturity date.  –  Bermudan op\$ons can be exercised on a sequence of  predeﬁned dates.  •  At each payment date tj we have the op\$on to cancel  the swap (i.e. to stop all future payments).  •  Intui\$vely, the swap should be cancelled when interest  rates are declining.  •  How can we determine the op\$mal cancella\$on \$me  using Monte‐Carlo simula\$on?  Cancellable Swaps  •  As with bond op\$ons, we should compare the  exercise value (intrinsic value) with the  con7nua7on value (value of wai7ng).  •  Let Vj be the value of the swap at \$me tj.  •  Value at maturity of the swap:  VN = L ⋅ ( rN −1 − k ) •  The con\$nua\$on value at \$me tj is denoted by   Cj and is:     − r ΔV € Cj = E e ( j j +1 rj ) Cancellable Swaps  •  Recall that when using the binomial laJce to  price a bond op\$on, we had:   C j ,n 1 1 = [q j,nV j +1,n +1 + (1 − q j,n )V j +1,n ] = E 1 + r V j +1 rj = rj,n 1 + r j ,n j ,n € •  The challenge in using Monte‐Carlo is that rather  than having two possible future states, we have  inﬁnitely many possible future values of rj+1 given  rj.   –  Rather than a sum, the condi\$onal expecta\$on is an  integral  Cancellable Swaps  •  The integral deﬁning the condi\$onal  expecta\$on:  Cj = E e ( − r j ΔV j +1 rj )      is generally not analy\$cally tractable.  Numerical methods are needed to  € approximate it.  Cancellable Swaps  •  How can we use Monte‐Carlo simula\$on to price  the cancellable swap?  •  Can we es\$mate Cj by using a set of S paths?  ri,1, ri,2 , ..., ri,N ri,n = ri ( t n ) •  Here ri(tn) denotes the value of the short rate at  \$me tn on the ith simulated path, i=1,…,S.  € Cancellable Swaps  •  As with the binomial tree, we can do a “backward  recursion”, but we can’t simply use the value of  the swap Vi,j+1 on the current path i to compute  Ci,j as there is only one such future value on this  simulated path.  •  This would be like trying to compute the  condi\$onal expecta\$on with a sample size of one.  •  We are trying to avoid doing a “nested  simula\$on”.  Cancellable Swaps  •  Least Squares Monte‐Carlo method.  •  A sort of backward induc\$on.  •  At \$me tj we have:   –  S current values for the short rate (one for each  simulated path):  r1, j , r2, j , ..., rS, j –  S values for the instrument at the next \$me    V1, j +1,V2, j +1, ...,VS , j +1 € –  Recall that at the maturity date:   VN = f ( rN −1 ) € Cancellable Swaps  •  At \$me tj we have S observa\$ons of the pairs  (rj,Vj+1):  {(r1, j ,V1, j +1),(r2, j ,V2, j +1),...,(rS, j ,VS, j +1)} •  Use these to approximate, using regression,  how the response Vj+1 depends on the factor  €j.  r •  For example, we could es\$mate the regression  equa\$on:  e − rj Δ V j +1 = β 0, j + β1, j rj + β 2, j rj2 + ε Cancellable Swap  •  Es\$mate the con\$nua\$on value Cj by:  ˆ ˆ ˆ ˆ C j = β 0, j + β1, j rj + β 2, j rj2 •  Does this work?  •  We considered a quadra\$c func\$on for the  € approxima\$on.  •  It can be shown that by allowing more and  more terms in the model for Cj we can get  very close to the true con\$nua\$on value:  Cj = E e [ − r j ΔV j +1 rj = lim ∑ β p rjp k →∞ p =1 k Cancellable Swaps  •  Summary of the Least Squares Monte‐Carlo  Algorithm.  •  Step One: Generate S paths for the short rate  according to the chosen model.  •  Step Two: For each path, compute  Vi,N = L ⋅ ( ri,N −1 − k ) –  Set  t i* = t N –  As we move backward in \$me, we will update the  t es\$mated op\$mal exercise \$me    i* € € Cancellable Swaps  •  Step Three: Going backward from j = N‐1 to 1  ˆˆˆ –  Obtain the regression coeﬃcients                      by  β 0, j , β1, j , β 2, j es\$ma\$ng the following equa\$on using least  −r Δ e V j +1 = β 0, j + β1, j rj + β 2, j rj2 + ε squares:  –  Es\$mate the con\$nua\$on value on the ith path  € using the equa\$on:  j € ˆ ˆ ˆ ˆ Ci, j = β 0, j + β1, j ri, j + β 2, j ri,2j ˆ <0 t i* = t j Ci, j –  If              ,  cancel the swap and set  •  Otherwise, set:      € € Vi, j = L ⋅ ( ri, j −1 − k ) + Vi, j +1e € − rj Δ Cancellable Swaps  •  Step Four: Return the following es\$mator for  the value of the callable swap at \$me 0:  ˆ = 1 ∑ ∑ L ⋅ exp(−( r + + r )Δ ) ⋅ ( r − k ) V0 i, 0 i, j i, j S i=1 j =1 * S t i −1 € ...
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