Interest rate models


Ito Process Basics

Ito Process for short term interest rate

$dr(t) = a(r) dt  + \sigma (r) dZ(t)$   [True Probability measure]

$dr(t) = [a(r) +\underbrace{ \sigma (r) \phi (r,t)}]dt  + \sigma (r) d\widetilde{Z}(t)$   [Risk-Neutral]

| interest rate risk premium | + * $\text{Sharp ratio} \phi = \frac{\text{risk premium}}{\sigma}$

$d\widetilde{Z}(t) = dZ(t) – \phi (r,t) dt$  + *Note sharp ratio is NEGATIVE

$\widetilde{Z}(T) = Z(t) – \int_{0}^{t} \phi (r, s) ds$

Ito process for a Zero Coupon Bond (ZCB) Price

$\frac{dP(r,t,T)}{P(r,t,T)} = \alpha (r,t,T)dt – q(r,t,T)dZ(t)$ [True Probability measure]

$\frac{dP(r,t,T)}{P(r,t,T)} = rdt – q(r,t,T)d\widetilde{Z}(t)$ [Risk Neutral]

$\alpha (r,t,T) = \frac{1}{P}[a(r)P_r  + 0.5[\sigma(r)]^2P_{rr}  + P_t]$

+  [icon name=”bomb” class=””] Drift of Derivative = Delta*(Drift of risk-neutral process) + 0.5*Gamma*(Volatility of risk-neutral process)$^2$ + theta

$q(r,t,T) = – \frac{P_r}{P}\sigma(r)$

 Sharp Ratio for a ZCB that expires at T

$\phi(r,t) = \frac{\alpha(r,t,T) – r}{q(r,t,T)}$ + Note that though volatility is negative in the ito process, while calculating the sharp ratio here we are not using the negative sign for q(r,t,T). This is the only exception. The general rule – If the Ito process has negative volatility, only in the sharp ratio calculation the negative sign is used; everywhere else the negative sign is NOT used. 

Partial Differential equation for Bonds

$rP = [a(r) + \sigma(r)\phi(r,t)]P_r + 0.5[\sigma(r)]^2 P_{rr} + P_t$

+ $\alpha (r,t,T)P = [a(r)]P_r  + 0.5[\sigma(r)]^2P_{rr}  + P_t$

$ \Delta\Gamma\Theta $ approximation for bonds

$ P(t+h, T) – P(t,T) = \Delta\epsilon + 0.5\Gamma\epsilon^2 + \theta h$

$\epsilon = r(t+h) – r(t) $

$\Delta =\frac{\partial P}{\partial r}$  |   $\Gamma =\frac{\partial^2 P}{\partial r^2}$  | $\Theta =\frac{\partial P}{\partial t}$

+ *This is similar to the $\Delta\Gamma\Theta$ approximation for options:
$ V(t+h, T) – V(t,T) = \Delta\epsilon + 0.5\Gamma\epsilon^2 + \theta h$
$\epsilon = S(t+h) – S(t) $
$\Delta =\frac{\partial V}{\partial S}$  |   $\Gamma =\frac{\partial^2 V}{\partial S^2}$  | $\Theta =\frac{\partial S}{\partial t}$

Rendlemen-Bartter Model

Ito Process for short term interest rate

$dr(t) = a.r(t) dt  + \sigma. r(t) dZ(t)$

Ito process is in the form of GBM

$ln \frac{r(t)}{r(0)} \sim N[m = (a-0.5\sigma ^2)t, v^2 = \sigma ^2 t]$

$r(t) = r(0) e^{m+ \sigma Z(t)}$

Characteristics of short rate in RB Model

[icon name=”thumbs-o-down” class=””]   Mean Reverting? * Since drift = ar(t) is not in the form a(b-x)

[icon name=”thumbs-o-down” class=””]  r can go negative? *Since volatility contains ‘r’ term

[icon name=”thumbs-o-up” class=””]  volatility varies with r? * Since volatility contains ‘r’ term

 

Vasicek Model

Ito Process for short term interest rate

$dr(t) = a.[b-r(t)] dt  + \sigma dZ(t)$

Characteristics of short rate

[icon name=”thumbs-o-up” class=””]  Mean Reverting? * Since drift = ar(t) is in the form a(b-x)

 [icon name=”thumbs-o-up” class=””] r can go negative? *Since volatility does NOT contain ‘r’ term

[icon name=”thumbs-o-down” class=””] volatility varies with r? *Since volatility does NOT contain ‘r’ term

Price of a ZCB

$P(r,t,T) = A(t,T). e^{-B(t,T).r}$

$A(t,T) $ – Do not memorize

$B(t,T) = \frac{1-e^{-a(T-t)}}{a}$

  • $A(t \pm c, T \pm c) = A(t,T)$ | $B(t \pm c, T \pm c) = B(t,T)$
  • $q(r,t,T) = B(t,T).\sigma $

Yield on infinite bond: $\overline r = b + \phi [\frac{\sigma}{a}] – 0.5 [\frac{\sigma}{a}]^2$

 

 

Ornstein-Uhlenbeck Process

Differential Form

$dX(t) = \lambda[\alpha – X(t)]dt + \sigma dZ(t)$

$\Leftrightarrow  X(t) = \alpha (1- e ^{-\lambda t})  +  X(0) e^{-\lambda t} + \sigma \int_{0}^{t}e^{-\lambda(t-s)}dZ(s) $

[icon name=”thumbs-o-up” class=””]  Mean Reverting?

As $\lambda$ increases, speed of mean reversion increases

Cox-Ingersoll-Ross Model

Ito Process for short term interest rate

$dr(t) = a.[b-r(t)] dt  + \sigma \sqrt{r} dZ(t)$

Characteristics of short rate

[icon name=”thumbs-o-up” class=””]  Mean Reverting? * Since drift = ar(t) is in the form a(b-x)

[icon name=”thumbs-o-down” class=””]  r can go negative? *Since volatility  contains ‘r’ term

[icon name=”thumbs-o-up” class=””] volatility varies with r? *Since volatility contains ‘r’ term

Price of a ZCB

$P(r,t,T) = A(t,T). e^{-B(t,T).r}$

$A(t,T) $ – Do not memorize

$B(t,T) $ –  Do not memorize

  • $A(t \pm c, T \pm c) = A(t,T)$ | $B(t \pm c, T \pm c) = B(t,T)$
  • $q(r,t,T) = B(t,T).\sigma \sqrt{r}$
  • $\frac{\alpha(r_1,t_1,T_1)}{r_1} = \frac{\alpha(r_2,t_2,T_2)}{r_2} = 1-\overline {\phi} B(t_1,T_1) = 1-\overline {\phi} B(t_2,T_2)$

Useful facts

  1. $\sigma (r) \propto \sqrt{r}$
  2. $\phi(r,t) = \frac{\overline{\phi}}{\sigma} \sqrt{r}$
  3. $\gamma ^2 = (a-\overline{\phi})^2 + 2\sigma ^2$
  4. As the maturity of a ZCB approaches $\infty$, its yield approaches: $\overline{r} =\underset{T \to \infty}{Lim}[\frac{-ln[P(r,t,T)]}{T-t}] = \frac{2ab}{a – \overline{\phi} + \gamma} $

 

Black Derman Toy Model

Basics:

  • All interest rates are effective
  • $p^* = 0.5$
  • $\sigma_i = \text{short rate volatility at time i}$2

 

Yield Volatility, Volatility in 1 yr, Volatility after 1 yr:

Volatility in bond yields is called yield volatility

$Y(1,T,r) = \text{Yield of a T yr bond at time 1 with short rate r}$

$(\text{Yield Volatility})_T = \frac{1}{2\sqrt{h}}ln[\frac{y(1,T,r_u)}{y(1,T,r_d)}]$

Note

  • $(\text{Yield Volatility})_2 = \sigma_1$
  • But $(\text{Yield Volatility})_3 \neq \sigma_2$

Steps to determine Yield Volatility:

  1. Find Bond prices – $P(1,T,r_u)$ and $P(1,T,r_d)$ – Use the BDT tree to find 2 possible T-year bond prices at Time 1
  2. Find corresponding yields – $y(1,T,r_u)$ and $y(1,T,r_d)$
    • $y(1,T,r_u) = [\frac{1}{P(1,T,r_u)}]^{\frac{1}{T-1}} – 1$
    • $y(1,T,r_d) = [\frac{1}{P(1,T,r_d)}]^{\frac{1}{T-1}} – 1$
  3. Now find $(\text{Yield Volatility})_T = \frac{1}{2\sqrt{h}}ln[\frac{y(1,T,r_u)}{y(1,T,r_d)}]$

Caplets and Caps:

Caplet: Call option on an interest rate. A caplet pays if interest rate $(R_T) \geq \text{Strike rate} (K_R)$

Value of (T+1) year caplet at time T+1  = Notional * max(0, $R_T – K_R$)

Value of (T+1) year caplet at time T= $\frac{Notional * max(0, R_T – K_R)}{(1+R_T)}$

Cap: Series of Caplets

$Caps = \sum_{i=1}^{t} PV(Caplet)_i$

Example: 2-Year Cap

——-$R_0$———–$R_1$——– + Note that these interest rates are forward looking – $R_T$ is the interest rate for time t=T to t= T+1 !

0——————1——————2

Caplet1        Caplet 2

$Cap = Caplet_1  + \frac{Caplet_2}{(1+R_0)}$

Duration and Delta Hedging for bonds

Duration Hedging:

Hedge Ratio = N = $ – \frac{(T_o – t)P(t,T_o)}{(T_h – t)P(t,T_h)}$ + o – refers to original bond, h  – refers to the bond that is being used to hedge

Delta Hedging:

Hedge Ratio = N = $ – \frac{\Delta_{P(t,T_o)}}{\Delta_{P(t,T_h)}}$ + o – refers to original bond, h  – refers to the bond that is being used to hedge

Black Formula

Pricing options on Bonds

$F = F_{0,T}[T, T+s] = \frac{P_0(0,T+s)}{P_0(0,T)}$ + Forward price agreed upon at time 0, to be paid at T for a ZCB bond that expires at time T+s

$\sigma = \sqrt {\frac{Var(ln F)}{t}}$ + Volatility of Forward price

$C = P(0,T)[+F.N(+d_1) – K.N(+d_2)]$

$P = P(0,T)[-F.N(-d_1) + K.N(-d_2)]$

$d_1 = \frac{ln(\frac{F}{K})+0.5V^2}{v}$ | $d_2 = d_1 – v$

Application of Black Formula

Caplet Price = $(1+K) [Max(0,\frac{1}{1+K_R} – P( T, T+s)] = (1+K_R). Put(K = \frac{1}{1+K_R})$

 

 

 


 

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