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# Arvid Kjellberg- Jakub Lawik - Juan Mojica - Xiaodong Xu.

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Arvid Kjellberg- Jakub Lawik - Juan Mojica - Xiaodong Xu

 by Journal of Derivatives in Fall 1994  Where to use it?  if there is a function x = f(r) of the short rate r that follows a mean reverting arithmetic process  Our project:  Hull and White trinomial tree building procedure  Excel Implementation

 Short Rate (or instantaneous rate)  The interest rate charged (usually in some particular market) for short term loans.  Bonds, option & derivative prices can depend only on the process followed by r (in risk neutral world)  t - t+Δt investor earn on average r(t) Δt  Payoff:

 Short rate  And we define the price at time t of zero-coupon bond that pays off \$1at time T by:  If R (t,T) is the continuously compounded interest rate at time t for a term of T-t:  Combine these formulas above:  This equation enables the term structure of interest rates at any given time to be obtained from the value of r at that time and risk-neutral process for r. And we define the price at time t of zero-coupon bond that pays off \$1at time T by: This equation enables the term structure of interest rates at any given time to be obtained from the value of r at that time and risk-neutral process for r.

 How is related to the Hull/White model?  was further extended in the Hull-White model  Asumes short rate is normal distributed  Mean reverting process (under Q)  Drift in interest rate will disappear if  a : how fast the short rate will reach the long-term mean value  b: the long run equilibrium value towards which the interest rate reverts  Term structure can be determined as a function of r(t) once a, b and σ are chosen. r = θ = b/a

 How is related to the Hull/White model?  Ho-Lee model is a particular case of Hull & White model with a=0  Assumes a normally distributed short-term rate  SR drift depends on time  makes arbitrage-free with respect to observed prices  Does not incorporate mean reversion  Short rate dynamics:  σ (instantaneous SD)  constant  θ(t)  defines the average direction that r moves at time t

 Market price of risk proves to be irrelevant when pricing IR derivatives  Average direction of the short rate will be moving in the future is almost equal to the slope of instantaneous forward curve

 No-arbitrage yield curve model  Parameters are consistent with bond prices implied in the zero coupon yield curve  In absence of default risk, bond price must pull towards par as it approaches maturity.  Assumes SR is normally distributed & subject to mean reversion  MR  ensures consistency with empirical observation: long rates are less volatile than short rates.  HWM generalized by Vasicek  θ(t)  deterministic function of time which calibrated against the theoretical bond price  V(t)  Brownian motion under the risk-neutral measure  a  speed of mean-reversion

 Input parameters for HWM  a : relative volatility of LR and SR  σ : volatility of the short rate  Not directly provided by the market (inferred from data of IR derivatives)

 Call option, two step, Δt=1, strike price =0.40. Our account amount \$100. Probabilities: 0.25,0.5 & 0.25  Payoff at the end of second time step:  Rollback precedure as: (pro1*valu1+...+pro3*valu3)e -rΔt 0.00% E 4.40%(4) 0.00%B 3.81%(0.963)F 3.88% A 3.23%(0.233)C 3.29%G 3.36% 0.00%D 2.76%H 2.83% 0.00% I 2.31% 0.00%

 Alternative branching possibilities  The pattern upward is useful for incorporating mean reversion when interest rates are very low and Downward is for interest rates are very high.

 HWM for instantaneous rate r:  First Step assumptions:  all time steps are equal in size Δt  rate of Δt,(R) follows the same procedure:  New variable called R* (initial value 0)

 : spacing between interest rates on the tree  for error minimization.  Define branching techniques  Upwards a << 0  Downwards a >> 0  Normal a = 0  Define probabilities(depends on branching)  probabilities are positive as long as: Straight / Normal Branching

 With initial parameters: σ = 0.01, a = 0.1,Δt = 1 ΔR=0.0173,j max =2,we get:

 Second step  convert R* into R tree by displacing the nodes on the R*-tree  Define α i as α(iΔt),Qi,j as the present value of a security that payoff \$1 if node (I,j) is reached and 0.Otherwise,forward induction  With continuously compounded zero rates in the first stage  Q 0,0 is 1  α 0 =3.824%  α 0 right price for a zero-coupon bond maturing at time Δt  Q 1,1 =probability *e -rΔt =0.1604  Q 1,0 =0.6417 and Q 1,-1 =0.1604. MaturityRate(%) 0.53.43 1.03.824 1.54.183 2.04.512 2.54.812 3.05.086

 Bond price(initial structure) = e -0.04512x2 =0.913  Solving for alfa1= 0.05205  It means that the central node at time Δt in the tree for R corresponds to an interest rate of 5.205%  Using the same method, we get:  Q2,2=0.0182,Q2,0=0.4736,Q2,-1=0.2033 and Q2,-2=0.0189.  Calculate α 2,Q 3,j ’s will be found as well. We can then find α 3 and so on…

 Finally we get:

 Underlying interest rate  Payoff date  American options

4,0 % 3,5 % 3,2 % 4,0 % 5,0 % 5,4 % 4,0 % 4,6 % 3,0 % 3,7 %

Thank You!!!

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