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![Luc_Faucheux_2020
A swap rate is a weighted basket of forward rates
¨ At-the-money swap rate equation: ∑) 𝐷𝐶𝐹 𝑖 . 𝐷 𝑖 . 𝑁 𝑖 . 𝑓 𝑖 = ∑* 𝐷𝐶𝐹 𝑗 . 𝐷 𝑗 . 𝑁 𝑗 . 𝑅
¨ Above equation is valid at all times before the swap start, forwards and discount factors
being calculated on the then current discount curve the usual way, if the period I on the
float side starts at time ts(i) and ends at time te(i), and the forward is “aligned” with the
period (no swap in arrears or CMS like)
¨ 𝑅(𝑡) = ∑) 𝐷𝐶𝐹 𝑖 . 𝐷 𝑖 . 𝑁 𝑖 . 𝑓 𝑡, 𝑡𝑠 𝑖 , 𝑡𝑒(𝑖) /[∑* 𝐷𝐶𝐹 𝑗 . 𝐷 𝑗 . 𝑁 𝑗 ]
¨ “frozen numeraire” approximation, expand above equation in first order in forward rates but
keeping the discount factors constant
¨ 𝑑𝑅(𝑡) = ∑) 𝐷𝐶𝐹 𝑖 . 𝐷 𝑖 . 𝑁 𝑖 . 𝑑𝑓 𝑡, 𝑡𝑠 𝑖 , 𝑡𝑒(𝑖) /[∑* 𝐷𝐶𝐹 𝑗 . 𝐷 𝑗 . 𝑁 𝑗 ]
¨ Taking the square of the above yields the instantaneous volatility of the swap rate
¨ Λ+
" . 𝑑𝑡 =< 𝑑𝑅" >=
∑)! ∑)" 𝐷𝐶𝐹 𝑖1 . 𝐷 𝑖1 . 𝑁 𝑖1 . 𝐷𝐶𝐹 𝑖2 . 𝐷 𝑖2 . 𝑁 𝑖2 . < 𝑑𝑓 𝑖1 . 𝑑𝑓 𝑖2 > /
[∑*! ∑*" 𝐷𝐶𝐹 𝑗1 . 𝐷 𝑗1 . 𝑁 𝑗1 𝐷𝐶𝐹 𝑗2 . 𝐷 𝑗2 . 𝑁 𝑗2 ]
13](https://image.slidesharecdn.com/lf2020trees-200531142700/85/Lf-2020-trees-13-320.jpg)
![Luc_Faucheux_2020
A swap rate is a weighted basket of forward rates
¨ instantaneous volatility of the swap rate
¨ Λ+
"
. 𝑑𝑡 =< 𝑑𝑅" >=
∑)! ∑)" 𝐷𝐶𝐹 𝑖1 . 𝐷 𝑖1 . 𝑁 𝑖1 . 𝐷𝐶𝐹 𝑖2 . 𝐷 𝑖2 . 𝑁 𝑖2 . < 𝑑𝑓 𝑖1 . 𝑑𝑓 𝑖2 > /
[∑*! ∑*" 𝐷𝐶𝐹 𝑗1 . 𝐷 𝑗1 . 𝑁 𝑗1 𝐷𝐶𝐹 𝑗2 . 𝐷 𝑗2 . 𝑁 𝑗2 ]
¨ Where 𝑑𝑓 𝑖1 = 𝑑𝑓 𝑡, 𝑡𝑠 𝑖1 , 𝑡𝑒(𝑖1) and 𝑑𝑓 𝑖2 = 𝑑𝑓 𝑡, 𝑡𝑠 𝑖2 , 𝑡𝑒(𝑖2)
¨ In abbreviated notation
¨ < 𝑑𝑓 𝑖1 . 𝑑𝑓 𝑖2 >= 𝜎 𝑖1 . 𝜎 𝑖2 . 𝜌 𝑖1, 𝑖2 . 𝑑𝑡
¨ So to calculate the instantaneous volatility of the swap rate you need the instantaneous
volatility of each forward BUT ALSO the instantaneous correlation matrix between the
forward constituting the weighted basket.
14](https://image.slidesharecdn.com/lf2020trees-200531142700/85/Lf-2020-trees-14-320.jpg)
![Luc_Faucheux_2020
Parametrization of the volatility
¨ Usually assume a bell shape, with a maximum and a long term asymptote, with simple
analytical expression in order to integrate easily, and have numerically stable calibration
(sensitivity of the parameters to the market input are well behaved)
¨ We choose the following
– .𝜎 0 = .𝜎$%&
– .𝜎 ∞ = .𝜎'
– .𝜎 𝑇$() = .𝜎$()
– If 𝑡 < 𝑇$(), we use .𝜎*
𝑡 = .𝜎$%&
*
+ .𝜎$()
*
− .𝜎$%&
*
∗ (
+
,!"#
)
– If 𝑡 >= 𝑇$(), we use .𝜎*
𝑡 = .𝜎$()
*
+ .𝜎'
*
− .𝜎$()
*
∗ [1 − exp
- +-,!"#
,$
]
22](https://image.slidesharecdn.com/lf2020trees-200531142700/85/Lf-2020-trees-22-320.jpg)
![Luc_Faucheux_2020
Building a BDT tree IV
¨ We create a grid of forward yields 𝑋(𝑘, 𝑖)
¨ We chose for the “bottom” value 𝑋 𝑘, 0 =
3(#&,#&'!)
(!2
((&)
+
)^(1)
, where 𝑉(𝑘) is the volatility input
¨ The idea is to start with a distribution of forwards centered around the input one 𝑓(𝑡1, 𝑡1-!)
and with a standard deviation that will be close to the volatility input
¨ Each successive value for a given period going up the nodes in the tree is given by
¨ 𝑋 𝑘, 𝑖 + 1 = 𝑋 𝑘, 𝑖 . [1 + 𝑉 𝑘 ]1
42
Y_fwd 1 1.498501 1.994013 2.488784 2.97612 3.209678 3.427392
1.501499 2.005996 2.51125 3.024024 3.290726 3.573885
2.01805 2.533919 3.072699 3.373821 3.72664
2.556793 3.122158 3.459014 3.885924
3.172413 3.546359 4.052015
3.635909 4.225206
4.405799](https://image.slidesharecdn.com/lf2020trees-200531142700/85/Lf-2020-trees-42-320.jpg)
![Luc_Faucheux_2020
Standard Swap periods
¨ On the fixed side, coupon payment at the end of the period
– Period start date (psj)
– Adjusted period start date (psj_adj)
– Period end date (pej)
– Adjusted period end date (pej_adj)
– Payment date (pmj)
– PV of a period 𝑃𝑉 𝑗 = 𝐷𝐶𝐹 𝑗 . 𝐷 𝑗 . 𝑁 𝑗 . 𝑅 = 𝐷𝐶𝐹 𝑝𝑠𝑗./-, 𝑝𝑒𝑗_𝑎𝑑𝑗 . 𝐷 𝑝𝑚- . 𝑁 𝑗 . 𝑅
¨ On the float side, floating rate sets at the beginning of the period, and pays at the end (Libor in advance or
standard Libor swap, as opposed to Libor in arrears)
– PV of a period (swaplet) 𝑃𝑉 𝑖 = 𝐷𝐶𝐹 𝑖 . 𝐷 𝑖 . 𝑁 𝑖 . 𝑓 𝑖
– 𝐷𝐶𝐹 𝑖 = 𝐷𝐶𝐹 𝑝𝑠𝑖./-, 𝑝𝑒𝑖./- and 𝐷 𝑖 = 𝐷(𝑝𝑚𝑖)
– 𝐷 𝑝𝑒𝑖 = 𝐷 𝑝𝑠𝑖 ∗
0
01234 56,,58, .:(,)
or
– 𝐷𝐶𝐹 𝑝𝑒𝑖, 𝑝𝑠𝑖 . 𝑓 𝑖 = [1 −
2 58,
2 56,
]
59](https://image.slidesharecdn.com/lf2020trees-200531142700/85/Lf-2020-trees-59-320.jpg)
![Luc_Faucheux_2020
Floating leg of a swap
¨ A floating swaplet pays 𝐷𝐶𝐹 𝑖 . 𝑁 𝑖 . 𝑓 𝑖 and its PV is 𝑃𝑉 𝑖 = 𝐷𝐶𝐹 𝑖 . 𝐷 𝑖 . 𝑁 𝑖 . 𝑓 𝑖
¨ Where 𝐷𝐶𝐹 𝑝𝑒𝑖, 𝑝𝑠𝑖 . 𝑓 𝑖 = [1 −
2 58,
2 56,
]
¨ We know from the fixed rate leg that < 𝐷 𝑖 >= 𝐷 𝑖 , but what about < 𝐷 𝑖 . 𝑓 𝑖 > ?
¨ Note, to be exact < 𝐷 𝑖 >= 𝐷 𝑖 should really read ∏?#BC?$
𝐸𝑋𝑃{𝑡@, 𝑑(𝑡@, 𝑡@ + 1)}, where
𝐸𝑋𝑃 𝑡@, 𝑑 𝑡@, 𝑡@ + 1 is the expected value of the overnight discount between the time (tc) and (tc+1),
observed up until time tc (because it drops off the curve after tc, and before tc, no matter where you
observe it, its expected value is equal to today’s value)
¨ 𝐸𝑋𝑃 𝑡@, 𝑑 𝑡@, 𝑡@ + 1 = 𝐸𝑋𝑃 𝑡 < 𝑡@, 𝑑 𝑡@, 𝑡@ + 1 = 𝑑(𝑡@ + 1)
¨ Back to < 𝐷 𝑖 . 𝑓 𝑖 > , there is a little trick
63](https://image.slidesharecdn.com/lf2020trees-200531142700/85/Lf-2020-trees-63-320.jpg)
![Luc_Faucheux_2020
Legs of a swap in arrears
¨ A fixed leg of a swap is a series of fixed cash flows that pays
¨ 𝐷𝐶𝐹 𝑖 . 𝑁 𝑖 . 𝑆𝑅
¨ And its PV is:
¨ 𝐷𝐶𝐹 𝑖 . 𝑁 𝑖 . 𝑆𝑅. 𝐷(𝑖 − 1)
¨ A floating swaplet “in arrears” pays:
¨ 𝐷𝐶𝐹 𝑖 . 𝑁 𝑖 . 𝑓 𝑖
¨ and its PV is:
¨ 𝑃𝑉 𝑖 = 𝐷𝐶𝐹 𝑖 . 𝐷 𝑖 − 1 . 𝑁 𝑖 . 𝑓 𝑖
¨ Where 𝐷𝐶𝐹 𝑝𝑒𝑖, 𝑝𝑠𝑖 . 𝑓 𝑖 = [1 −
2 58,
2 56,
]
71](https://image.slidesharecdn.com/lf2020trees-200531142700/85/Lf-2020-trees-71-320.jpg)
![Luc_Faucheux_2020
Floating leg of an “in arrears” swap
¨ Because the forward f(i) sets at the beginning of the period, once we reach the period start, everything
is known about the payment, and it becomes a fixed cashflow.
¨ 𝑃𝑉 𝑖 = 𝐷𝐶𝐹 𝑖 . 𝐷 𝑖 − 1 . 𝑁 𝑖 . 𝑓 𝑖 = 𝐷𝐶𝐹 𝑖 . 𝑁 𝑖 . 𝐸𝑋𝑃 𝑝𝑠𝑖, 𝐷 𝑝𝑠𝑖 . 𝐸𝑋𝑃{𝑝𝑠𝑖, 1. 𝑓 𝑖 }
¨ 𝐸𝑋𝑃 𝑝𝑠𝑖, 𝐷 𝑝𝑠𝑖 = 𝐷(𝑝𝑠𝑖)
¨ 𝐷𝐶𝐹 𝑖 . 𝐸𝑋𝑃 𝑝𝑠𝑖, 1. 𝑓 𝑖 = 𝐸𝑋𝑃{𝑝𝑠𝑖, 𝐷𝐶𝐹 𝑖 . 𝑓 𝑖 }
¨ Where 𝐷𝐶𝐹 𝑝𝑒𝑖, 𝑝𝑠𝑖 . 𝑓 𝑖 = [1 −
2 58,
2 56,
]
¨ Now, magic trick,
D
01D
=
D10=0
01D
=
01D=0
01D
= 1 −
0
01D
, but that gets us nowhere
¨ So, 𝐸𝑋𝑃 𝑝𝑠𝑖, 𝐷𝐶𝐹 𝑖 . 𝑓 𝑖 = 𝐸𝑋𝑃 1 −
2 58,
2 56,
¨ This has not only a timing difference but a ratio, and we all know that 𝐸𝑋𝑃
0
D
<> 1/𝐸𝑋𝑃(𝑥)
¨ This can be done through the Taylor expansion (or also Jensen inequality)
¨ This is the famous Swap in arrears convexity trade of May 1995 (story time)
73](https://image.slidesharecdn.com/lf2020trees-200531142700/85/Lf-2020-trees-73-320.jpg)
![Luc_Faucheux_2020
Changing the drift by changing the probabilities IV
¨ So we match the process: 𝑑𝑋 = 𝑎. 𝑑𝑡 + 𝜎. 𝑑𝑊
¨ 𝛿𝑋 = 2𝜎. 𝛿𝑡!/"
¨ 𝜀 =
E
"=
𝛿𝑡!/"
¨ WAIT A MINUTE you should say, 𝜀 is only a number, not a complicated formula:
¨ 𝜀 =
E
"=
𝛿𝑡!/"
¨ 𝑎 is a drift and so scales as [
9
#
]
¨ 𝜎 is a volatility and so scales as [
9
#!/+], or easier 𝜎". [𝑡]scales as [𝑋"]
¨ SO.. 𝜀 scales as [
9
#
].
#
!
+
9
. [𝑡!/"] which is dimensionless and a number indeed !!
123](https://image.slidesharecdn.com/lf2020trees-200531142700/85/Lf-2020-trees-123-320.jpg)
![Luc_Faucheux_2020
Adding some more mean reversion to the BDT tree XII
¨ In the limit of small 𝜇:
¨ < ∆𝑋 > =< 𝑋12!|𝑋(𝑘, 𝑖) > − 𝑋(𝑘, 𝑖)
¨ < 𝑋 > =< 𝑋 𝜇 = 0 >. 1 − 𝜇 + 𝜇. ∑*$%
*$12!
𝑋 𝑘 + 1, 𝑗 . Π 𝑋(𝑘 + 1, 𝑗)
¨ < ∆𝑋 > = < ∆𝑋 𝜇 = 0 >. 1 − 𝜇 + 𝜇. 𝛿 < ∆𝑋 >
¨ With 𝛿 < ∆𝑋 > = ∑*$%
*$12!
{𝑋 𝑘 + 1, 𝑗 − 𝑋 𝑘, 𝑖 }. Π 𝑋(𝑘 + 1, 𝑗)
¨ Or again :
¨ 𝛿 < ∆𝑋 > = [∑*$%
*$12!
𝑋 𝑘 + 1, 𝑗 . Π 𝑋 𝑘 + 1, 𝑗 − 𝑋 𝑘, 𝑖 ]
¨ Since: ∑*$%
*$12!
𝑋 𝑘, 𝑖 . Π 𝑋 𝑘 + 1, 𝑗 = 𝑋 𝑘, 𝑖 . ∑*$%
*$12!
Π 𝑋 𝑘 + 1, 𝑗 = 𝑋 𝑘, 𝑖
143](https://image.slidesharecdn.com/lf2020trees-200531142700/85/Lf-2020-trees-143-320.jpg)
![Luc_Faucheux_2020
Adding some more mean reversion to the BDT tree XIII
¨ We now have:
¨ < ∆𝑋 > = < ∆𝑋 𝜇 = 0 >. 1 − 𝜇 + 𝜇. 𝛿 < ∆𝑋 >
¨ 𝛿 < ∆𝑋 > = [∑*$%
*$12!
𝑋 𝑘 + 1, 𝑗 . Π 𝑋 𝑘 + 1, 𝑗 − 𝑋 𝑘, 𝑖 ]
¨ 𝛿 < ∆𝑋 > = [𝑋?(𝑘 + 1) − 𝑋 𝑘, 𝑖 ]
¨ Remember that:
¨ 𝑋? 𝑘 + 1 = ∑*$%
*$12!
𝑋 𝑘 + 1, 𝑗 . Π 𝑋 𝑘 + 1, 𝑗
¨ In the limit of a dense tree 𝑋? 𝑘 = 𝑋(𝑘,
1
"
)
¨ And for each node 𝑋 𝑘, 𝑖 = 𝑋? 𝑘 + 𝑖 −
1
"
. 𝛿𝑋(𝑘)
144](https://image.slidesharecdn.com/lf2020trees-200531142700/85/Lf-2020-trees-144-320.jpg)
![Luc_Faucheux_2020
Adding some more mean reversion to the BDT tree XIV
¨ < ∆𝑋 > = < ∆𝑋 𝜇 = 0 >. 1 − 𝜇 + 𝜇. [𝑋?(𝑘 + 1) − 𝑋 𝑘, 𝑖 ]
¨ Again we take the liberty of setting 𝑋? 𝑘 + 1 = 𝑋? 𝑘 = 0
¨ (as we can always impose a constant drift later)
¨ And so : < ∆𝑋 > = 𝑋 𝑘, 𝑖 .
!
89 #
.
;89 #
;#
. 𝛿𝑡. 1 − 𝜇 − 𝜇. 𝑋 𝑘, 𝑖
¨ This looks almost like what we are after
¨ We need to check that the units are correct and consistent between the two terms, in
particular we look to be missing a 𝛿𝑡 factor in the second term
¨ Also we worked in the limit (𝜇 → 0) but we know that the BDT construction is algorithmic in
nature, so we are missing the cross-terms and higher order terms, we will need to be
conscious of that
¨ On the other hand we are trying to get the SDE, which is continuous limit, and not that
helpful, because we are using BDT anyways, and not looking for closed form solutions
145](https://image.slidesharecdn.com/lf2020trees-200531142700/85/Lf-2020-trees-145-320.jpg)
![Luc_Faucheux_2020
Trinomial trees – an example of stability constraint VI
¨ The solution is then:
¨ 𝑃A = 1 − {(𝛼. 𝛿𝑡)"+𝜎". 𝛿𝑡}.
89
"
-"
¨ 𝑃L =
!
"
. [
"M.8#
89
+ {(𝛼. 𝛿𝑡)"+𝜎". 𝛿𝑡}.
89
"
-"
]
¨ 𝑃K =
!
"
. [−
"M.8#
89
+ {(𝛼. 𝛿𝑡)"+𝜎". 𝛿𝑡}.
89
"
-"
]
¨ Note that it would seem sensible that all probabilities should be in [0,1]
¨ A negative probability in the tree is not necessarily a sign that something is wrong, but it is a
constraint that we would like to enforce (related to stability analysis)
155](https://image.slidesharecdn.com/lf2020trees-200531142700/85/Lf-2020-trees-155-320.jpg)
Uploaded byluc faucheux
Lf 2020 trees
The document discusses various numerical methods in financial modeling, specifically focusing on binomial trees, Monte Carlo simulations, and forward rates. It contrasts recombining and non-recombining binomial trees in terms of their computational efficiency and arbitrage compliance, while also elaborating on the complexities of modeling forward rates and swaps. Additionally, it addresses volatility calculations for options and swaps, emphasizing their respective dependencies on the underlying forward rates and market conditions.
Lf 2020 trees
- 1.
- 2. Luc_Faucheux_2020 Recombining Binomial trees 2 ¨Fast, memory efficient, numerically stable and well understood (number of nodes ~ N^2) ¨ Can be used to run Monte Carlo simulations on the trees ¨ Arbitrage is respected ONLY on average on a slice
- 3. Luc_Faucheux_2020 Recombining Binomial trees ¨Cropping ¨ Mean reversion ¨ Storing curve in memory versus recalculating on the fly (storing discounts versus calling exp()) ¨ Single volatility models for callable as an example 3
- 4. Luc_Faucheux_2020 Non-recombining binomial tree ¨Respect arbitrage free at EVERY node in the tree ¨ Simplest to implement ¨ Very close to Monte Carlo simulations ¨ Very expensive in CPU and memory (number of nodes ~ 2^N) 4
- 5. Luc_Faucheux_2020 Monte Carlo simulation ¨Bundling ¨ Regression ¨ Choice of regression factors ¨ Numerically noisy (accuracy ~ N^(-1/2)) ¨ Very CPU intensive ¨ But very intuitive, very flexible, and as we saw in the Skew module, you can run a Monte Carlo on a simpler model, or on an entire portfolio once you have created a portfolio map 5
- 6. Luc_Faucheux_2020 6 The life ofa forward rate
- 7. Luc_Faucheux_2020 The glorious lifeof a valiant forward ¨ f(t,t1,t2) is the forward rate between the time t1 and t2 on the curve observed at time t ¨ t, t1 and t2 are by convention in absolute ¨ f(t,t1,t2) evolves from (t) to (t+1) into f(t+1,t1,t2) with instantaneous volatility 𝜎(𝑡, 𝑡!, 𝑡") ¨ f(t,t1,t2) “dies” as the anchor overnight rate on the curve observed at time t2 ¨ “Rolling forward” convention as opposed to “constant forward” 7 f(0,0,1) f(0,1,2) f(0,2,3) f(0,3,4) f(0,4,5) f(0,5,6) f(0,6,7) f(0,7,8) f(0,8,9) f(0,9,10) f(0,10,11) f(0,11,12) f(1,1,2) f(1,6,7) f(2,2,3) f(2,6,7) f(3,3,4) f(3,6,7) f(4,4,5) f(4,6,7) f(5,5,6) f(5,6,7) f(6,6,7) f(7,7,8) f(8,8,9) f(9,9,10) f(10,10,11) f(11,11,12) f(12,12,13)
- 8. Luc_Faucheux_2020 The glorious lifeof a valiant forward ¨ Each line can be viewed as the new curve at time t ¨ Today (t=0) curve is defined by the successive forwards f(0,0,1), f(0,1,2)….. ¨ At time t the curve will then be defined by the successive forwards f(t,t,t+1), f(t,t+1,t+2),… ¨ Similar to our HJM spreadsheet but sliding down the curve back one every time 8 f(0,0,1) f(0,1,2) f(0,2,3) f(0,3,4) f(0,4,5) f(0,5,6) f(0,6,7) f(0,7,8) f(0,8,9) f(0,9,10) f(0,10,11) f(0,11,12) f(1,1,2) f(1,6,7) f(2,2,3) f(2,6,7) f(3,3,4) f(3,6,7) f(4,4,5) f(4,6,7) f(5,5,6) f(5,6,7) f(6,6,7) f(7,7,8) f(8,8,9) f(9,9,10) f(10,10,11) f(11,11,12) f(12,12,13)
- 9. Luc_Faucheux_2020 The glorious lifeof a valiant forward ¨ In practice, 𝜎 𝑡, 𝑡!, 𝑡" tends to 0 when (t=t1), and has a maximum in the “belly” of the curve ¨ In reality, 𝜎 𝑡, 𝑡!, 𝑡" is also dependent on the actual forward f(t,t1,t2) as well as previous instantaneous volatilities (GARCH for example) and previous forwards ¨ A common assumption is for the volatility 𝜎 𝑡, 𝑡!, 𝑡" to be stationary for the same class of forwards. A class of forward is defined as all forwards of equal maturity T: (t2-t1=T) ¨ 𝜎 𝑡, 𝑡!, 𝑡" = '𝜎 𝑡! − 𝑡 9 f(0,0,1) f(0,1,2) f(0,2,3) f(0,3,4) f(0,4,5) f(0,5,6) f(0,6,7) f(0,7,8) f(0,8,9) f(0,9,10) f(0,10,11) f(0,11,12) f(1,1,2) f(1,6,7) f(1,11,12) f(2,2,3) f(2,6,7) f(2,11,12) f(3,3,4) f(3,6,7) f(3,11,12) f(4,4,5) f(4,6,7) f(4,11,12) f(5,5,6) f(5,6,7) f(5,11,12) f(6,6,7) f(6,11,12) f(7,7,8) f(7,11,12) f(8,8,9) f(8,11,12) f(9,9,10) f(9,11,12) f(10,10,11) f(10,11,12) f(11,11,12) f(12,12,13)
- 10. Luc_Faucheux_2020 Regular Eurodollar optionsor caplet ¨ Average variance for the forward over the life, option expires at the same time that the forward ¨ 𝜎". 𝑡! = ∫#$% #$#! 𝜎" 𝑡, 𝑡!, 𝑡" . 𝑑𝑡 = ∫#$% #$#! '𝜎" 𝑡! − 𝑡 . 𝑑𝑡 ¨ Pricing different option for different strikes K, and expressing those option prices in a common model (say Lognormal or Normal) will return the skew and smile expressed within that model 10 f(0,0,1) f(0,1,2) f(0,2,3) f(0,3,4) f(0,4,5) f(0,5,6) f(0,6,7) f(0,7,8) f(0,8,9) f(0,9,10) f(0,10,11) f(0,11,12) f(1,1,2) f(1,11,12) f(2,2,3) f(2,11,12) f(3,3,4) f(3,11,12) f(4,4,5) f(4,11,12) f(5,5,6) f(5,11,12) f(6,6,7) f(6,11,12) f(7,7,8) f(7,11,12) f(8,8,9) f(8,11,12) f(9,9,10) f(9,11,12) f(10,10,11) f(10,11,12) f(11,11,12) f(12,12,13)
- 11. Luc_Faucheux_2020 Mid-curve Eurodollar optionsor forward caplets ¨ Average variance for the forward over the option, option expires BEFORE the forward at a time Texp ¨ 𝜎". 𝑡&'( = ∫#$% #$#"#$ 𝜎" 𝑡, 𝑡!, 𝑡" . 𝑑𝑡 = ∫#$% #$#"#$ '𝜎" 𝑡! − 𝑡 . 𝑑𝑡 ¨ Pricing different option for different strikes K, and expressing those option prices in a common model (say Lognormal or Normal) will return the skew and smile expressed within that model 11 f(0,0,1) f(0,1,2) f(0,2,3) f(0,3,4) f(0,4,5) f(0,5,6) f(0,6,7) f(0,7,8) f(0,8,9) f(0,9,10) f(0,10,11) f(0,11,12) f(1,1,2) f(1,11,12) f(2,2,3) f(2,11,12) f(3,3,4) f(3,11,12) f(4,4,5) f(4,11,12) f(5,5,6) f(5,11,12) f(6,6,7) f(6,11,12) f(7,7,8) Texpiry f(7,11,12) f(8,8,9) f(9,9,10) f(10,10,11) f(11,11,12) f(12,12,13)
- 12. Luc_Faucheux_2020 A swap isa weighted basket of forwards ¨ Consider a swap with swap rate R (at-the-money swap rate) – Nfloat periods on the Float side with forecasted forward f(i) – indexed by i, with – daycount fraction DCF(i), – discount D(i) – Notional N(i) – Nfixed periods on the Fixed side, – indexed by j, with – daycount fraction DCF(j), – discount D(j) – Notional N(j) ! ! 𝐷𝐶𝐹 𝑖 . 𝐷 𝑖 . 𝑁 𝑖 . 𝑓 𝑖 = ! " 𝐷𝐶𝐹 𝑗 . 𝐷 𝑗 . 𝑁 𝑗 . 𝑅 12
- 13. Luc_Faucheux_2020 A swap rateis a weighted basket of forward rates ¨ At-the-money swap rate equation: ∑) 𝐷𝐶𝐹 𝑖 . 𝐷 𝑖 . 𝑁 𝑖 . 𝑓 𝑖 = ∑* 𝐷𝐶𝐹 𝑗 . 𝐷 𝑗 . 𝑁 𝑗 . 𝑅 ¨ Above equation is valid at all times before the swap start, forwards and discount factors being calculated on the then current discount curve the usual way, if the period I on the float side starts at time ts(i) and ends at time te(i), and the forward is “aligned” with the period (no swap in arrears or CMS like) ¨ 𝑅(𝑡) = ∑) 𝐷𝐶𝐹 𝑖 . 𝐷 𝑖 . 𝑁 𝑖 . 𝑓 𝑡, 𝑡𝑠 𝑖 , 𝑡𝑒(𝑖) /[∑* 𝐷𝐶𝐹 𝑗 . 𝐷 𝑗 . 𝑁 𝑗 ] ¨ “frozen numeraire” approximation, expand above equation in first order in forward rates but keeping the discount factors constant ¨ 𝑑𝑅(𝑡) = ∑) 𝐷𝐶𝐹 𝑖 . 𝐷 𝑖 . 𝑁 𝑖 . 𝑑𝑓 𝑡, 𝑡𝑠 𝑖 , 𝑡𝑒(𝑖) /[∑* 𝐷𝐶𝐹 𝑗 . 𝐷 𝑗 . 𝑁 𝑗 ] ¨ Taking the square of the above yields the instantaneous volatility of the swap rate ¨ Λ+ " . 𝑑𝑡 =< 𝑑𝑅" >= ∑)! ∑)" 𝐷𝐶𝐹 𝑖1 . 𝐷 𝑖1 . 𝑁 𝑖1 . 𝐷𝐶𝐹 𝑖2 . 𝐷 𝑖2 . 𝑁 𝑖2 . < 𝑑𝑓 𝑖1 . 𝑑𝑓 𝑖2 > / [∑*! ∑*" 𝐷𝐶𝐹 𝑗1 . 𝐷 𝑗1 . 𝑁 𝑗1 𝐷𝐶𝐹 𝑗2 . 𝐷 𝑗2 . 𝑁 𝑗2 ] 13
- 14. Luc_Faucheux_2020 A swap rateis a weighted basket of forward rates ¨ instantaneous volatility of the swap rate ¨ Λ+ " . 𝑑𝑡 =< 𝑑𝑅" >= ∑)! ∑)" 𝐷𝐶𝐹 𝑖1 . 𝐷 𝑖1 . 𝑁 𝑖1 . 𝐷𝐶𝐹 𝑖2 . 𝐷 𝑖2 . 𝑁 𝑖2 . < 𝑑𝑓 𝑖1 . 𝑑𝑓 𝑖2 > / [∑*! ∑*" 𝐷𝐶𝐹 𝑗1 . 𝐷 𝑗1 . 𝑁 𝑗1 𝐷𝐶𝐹 𝑗2 . 𝐷 𝑗2 . 𝑁 𝑗2 ] ¨ Where 𝑑𝑓 𝑖1 = 𝑑𝑓 𝑡, 𝑡𝑠 𝑖1 , 𝑡𝑒(𝑖1) and 𝑑𝑓 𝑖2 = 𝑑𝑓 𝑡, 𝑡𝑠 𝑖2 , 𝑡𝑒(𝑖2) ¨ In abbreviated notation ¨ < 𝑑𝑓 𝑖1 . 𝑑𝑓 𝑖2 >= 𝜎 𝑖1 . 𝜎 𝑖2 . 𝜌 𝑖1, 𝑖2 . 𝑑𝑡 ¨ So to calculate the instantaneous volatility of the swap rate you need the instantaneous volatility of each forward BUT ALSO the instantaneous correlation matrix between the forward constituting the weighted basket. 14
- 15. Luc_Faucheux_2020 A swap evolvingto the first set ¨ Example above : a 5x12 swap evolving on the volatility surface up until the first set 15 f(0,0,1) f(0,1,2) f(0,2,3) f(0,3,4) f(0,4,5) f(0,5,6) f(0,6,7) f(0,7,8) f(0,8,9) f(0,9,10) f(0,10,11) f(0,11,12) f(1,1,2) f(1,5,6) f(1,6,7) f(1,7,8) f(1,8,9) f(1,9,10) f(1,10,11) f(1,11,12) f(2,2,3) f(2,5,6) f(2,6,7) f(2,7,8) f(2,8,9) f(2,9,10) f(2,10,11) f(2,11,12) f(3,3,4) f(3,5,6) f(3,6,7) f(3,7,8) f(3,8,9) f(3,9,10) f(3,10,11) f(3,11,12) f(4,4,5) f(4,5,6) f(4,6,7) f(4,7,8) f(4,8,9) f(4,9,10) f(4,10,11) f(4,11,12) f(5,5,6) f(5,6,7) f(5,7,8) f(5,8,9) f(5,9,10) f(5,10,11) f(5,11,12) f(6,6,7) f(7,7,8) f(8,8,9) f(9,9,10) f(10,10,11) f(11,11,12) f(12,12,13)
- 16. Luc_Faucheux_2020 A swap evolvingto maturity ¨ Example above : a 5x12 swap evolving on the volatility surface up until maturity 16 f(0,0,1) f(0,1,2) f(0,2,3) f(0,3,4) f(0,4,5) f(0,5,6) f(0,6,7) f(0,7,8) f(0,8,9) f(0,9,10) f(0,10,11) f(0,11,12) f(1,1,2) f(1,5,6) f(1,6,7) f(1,7,8) f(1,8,9) f(1,9,10) f(1,10,11) f(1,11,12) f(2,2,3) f(2,5,6) f(2,6,7) f(2,7,8) f(2,8,9) f(2,9,10) f(2,10,11) f(2,11,12) f(3,3,4) f(3,5,6) f(3,6,7) f(3,7,8) f(3,8,9) f(3,9,10) f(3,10,11) f(3,11,12) f(4,4,5) f(4,5,6) f(4,6,7) f(4,7,8) f(4,8,9) f(4,9,10) f(4,10,11) f(4,11,12) f(5,5,6) f(5,6,7) f(5,7,8) f(5,8,9) f(5,9,10) f(5,10,11) f(5,11,12) f(6,6,7) f(6,7,8) f(6,8,9) f(6,9,10) f(6,10,11) f(6,11,12) f(7,7,8) f(7,8,9) f(7,9,10) f(7,10,11) f(7,11,12) f(8,8,9) f(8,9,10) f(8,10,11) f(8,11,12) f(9,9,10) f(9,10,11) f(9,11,12) f(10,10,11) f(10,11,12) f(11,11,12) f(12,12,13)
- 17. Luc_Faucheux_2020 A swaption isa mid-curve on the basket of forwards ¨ Example above : a “5y7y” swaption, or a 5y option on a 7y swap, equating the year to the time units ¨ Option expires at time t5, underlying is a swap starting at time t5 and ending at time t12 ¨ Note that only the first forward gets to experience the “whole life” volatility, all the other forwards essentially will experience the “mid-curve” or truncated volatility up to the swaption expiry 17 f(0,0,1) f(0,1,2) f(0,2,3) f(0,3,4) f(0,4,5) f(0,5,6) f(0,6,7) f(0,7,8) f(0,8,9) f(0,9,10) f(0,10,11) f(0,11,12) f(1,1,2) f(1,5,6) f(1,6,7) f(1,7,8) f(1,8,9) f(1,9,10) f(1,10,11) f(1,11,12) f(2,2,3) f(2,5,6) f(2,6,7) f(2,7,8) f(2,8,9) f(2,9,10) f(2,10,11) f(2,11,12) f(3,3,4) f(3,5,6) f(3,6,7) f(3,7,8) f(3,8,9) f(3,9,10) f(3,10,11) f(3,11,12) f(4,4,5) f(4,5,6) f(4,6,7) f(4,7,8) f(4,8,9) f(4,9,10) f(4,10,11) f(4,11,12) f(5,5,6) f(5,6,7) f(5,7,8) f(5,8,9) f(5,9,10) f(5,10,11) f(5,11,12) f(6,6,7) f(7,7,8) f(8,8,9) f(9,9,10) f(10,10,11) f(11,11,12) f(12,12,13)
- 18. Luc_Faucheux_2020 Market practice assumption ¨Volatility is assumed to be stationary: 𝜎 𝑡, 𝑡!, 𝑡" = '𝜎 𝑡! − 𝑡 , 𝑡" ¨ Correlation is also stationary: 𝜌 𝑖1, 𝑖2 = 𝜌( 𝑡𝑠(𝑖1) − 𝑡𝑠(𝑖2) ) ¨ Parametrization of the volatility and correlation (Ribbonato, ..) ¨ 𝜌 𝑖1, 𝑖2 = 𝜌 𝑡𝑠 𝑖1 − 𝑡𝑠 𝑖2 = 𝐿, + 1 − 𝐿, . exp{ -|#/ )! -#/ )" | 0% } – 𝐿# is the long term (asymptotic) level of correlation – 𝑇# is the characteristic time scale (half-life) for the correlation to decrease from 100% perpendicular to the diagonal 18
- 19. Luc_Faucheux_2020 Correlation matrices –100% correlation 19 f(0,0,1) f(0,1,2) f(0,2,3) f(0,3,4) f(0,4,5) f(0,5,6) f(0,6,7) f(0,7,8) f(0,8,9) f(0,9,10) f(0,10,11) f(0,11,12) f(0,0,1) 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% f(0,1,2) 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% f(0,2,3) 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% f(0,3,4) 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% f(0,4,5) 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% f(0,5,6) 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% f(0,6,7) 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% f(0,7,8) 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% f(0,8,9) 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% f(0,9,10) 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% f(0,10,11) 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% f(0,11,12) 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% rho_inf 100% TC 4
- 20. Luc_Faucheux_2020 Correlation matrices –0% correlation 20 f(0,0,1) f(0,1,2) f(0,2,3) f(0,3,4) f(0,4,5) f(0,5,6) f(0,6,7) f(0,7,8) f(0,8,9) f(0,9,10) f(0,10,11) f(0,11,12) f(0,0,1) 100% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% f(0,1,2) 0% 100% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% f(0,2,3) 0% 0% 100% 0% 0% 0% 0% 0% 0% 0% 0% 0% f(0,3,4) 0% 0% 0% 100% 0% 0% 0% 0% 0% 0% 0% 0% f(0,4,5) 0% 0% 0% 0% 100% 0% 0% 0% 0% 0% 0% 0% f(0,5,6) 0% 0% 0% 0% 0% 100% 0% 0% 0% 0% 0% 0% f(0,6,7) 0% 0% 0% 0% 0% 0% 100% 0% 0% 0% 0% 0% f(0,7,8) 0% 0% 0% 0% 0% 0% 0% 100% 0% 0% 0% 0% f(0,8,9) 0% 0% 0% 0% 0% 0% 0% 0% 100% 0% 0% 0% f(0,9,10) 0% 0% 0% 0% 0% 0% 0% 0% 0% 100% 0% 0% f(0,10,11) 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 100% 0% f(0,11,12) 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 100% rho_inf 0% TC 0.1
- 21. Luc_Faucheux_2020 Correlation matrices –market correlation 21 f(0,0,1) f(0,1,2) f(0,2,3) f(0,3,4) f(0,4,5) f(0,5,6) f(0,6,7) f(0,7,8) f(0,8,9) f(0,9,10) f(0,10,11) f(0,11,12) f(0,0,1) 100% 88% 77% 69% 61% 55% 49% 45% 41% 38% 35% 33% f(0,1,2) 88% 100% 88% 77% 69% 61% 55% 49% 45% 41% 38% 35% f(0,2,3) 77% 88% 100% 88% 77% 69% 61% 55% 49% 45% 41% 38% f(0,3,4) 69% 77% 88% 100% 88% 77% 69% 61% 55% 49% 45% 41% f(0,4,5) 61% 69% 77% 88% 100% 88% 77% 69% 61% 55% 49% 45% f(0,5,6) 55% 61% 69% 77% 88% 100% 88% 77% 69% 61% 55% 49% f(0,6,7) 49% 55% 61% 69% 77% 88% 100% 88% 77% 69% 61% 55% f(0,7,8) 45% 49% 55% 61% 69% 77% 88% 100% 88% 77% 69% 61% f(0,8,9) 41% 45% 49% 55% 61% 69% 77% 88% 100% 88% 77% 69% f(0,9,10) 38% 41% 45% 49% 55% 61% 69% 77% 88% 100% 88% 77% f(0,10,11) 35% 38% 41% 45% 49% 55% 61% 69% 77% 88% 100% 88% f(0,11,12) 33% 35% 38% 41% 45% 49% 55% 61% 69% 77% 88% 100% rho_inf 20% TC 6
- 22. Luc_Faucheux_2020 Parametrization of thevolatility ¨ Usually assume a bell shape, with a maximum and a long term asymptote, with simple analytical expression in order to integrate easily, and have numerically stable calibration (sensitivity of the parameters to the market input are well behaved) ¨ We choose the following – .𝜎 0 = .𝜎$%& – .𝜎 ∞ = .𝜎' – .𝜎 𝑇$() = .𝜎$() – If 𝑡 < 𝑇$(), we use .𝜎* 𝑡 = .𝜎$%& * + .𝜎$() * − .𝜎$%& * ∗ ( + ,!"# ) – If 𝑡 >= 𝑇$(), we use .𝜎* 𝑡 = .𝜎$() * + .𝜎' * − .𝜎$() * ∗ [1 − exp - +-,!"# ,$ ] 22
- 23. Luc_Faucheux_2020 Volatility – case1 23 TMAX 2 SIG(MIN) 5% SIG(MAX) 20% SIG(INFINITY) 5% T(INFINITY) 2
- 24. Luc_Faucheux_2020 Volatility – case2 24 TMAX 2 SIG(MIN) 5% SIG(MAX) 20% SIG(INFINITY) 30% T(INFINITY) 2
- 25. Luc_Faucheux_2020 Volatility – case3 25 TMAX 2 SIG(MIN) 30% SIG(MAX) 20% SIG(INFINITY) 5% T(INFINITY) 2
- 26. Luc_Faucheux_2020 Volatility – case4 26 TMAX 1 SIG(MIN) 0% SIG(MAX) 20% SIG(INFINITY) 0% T(INFINITY) 0
- 27. Luc_Faucheux_2020 Calibration of theparameters to the market 27
- 28.
- 29. Luc_Faucheux_2020 Perfect decorrelation (0%except on diagonal) 29
- 30. Luc_Faucheux_2020 Flows and Marketparticipants on the Swaption Grid 30 Lots of overlapping information
- 31. Luc_Faucheux_2020 One swaption ¨ At5-t12 swaption evolving on the forward volatility surface 31 f(0,0,1) f(0,1,2) f(0,2,3) f(0,3,4) f(0,4,5) f(0,5,6) f(0,6,7) f(0,7,8) f(0,8,9) f(0,9,10) f(0,10,11) f(0,11,12) f(1,1,2) f(1,5,6) f(1,6,7) f(1,7,8) f(1,8,9) f(1,9,10) f(1,10,11) f(1,11,12) f(2,2,3) f(2,5,6) f(2,6,7) f(2,7,8) f(2,8,9) f(2,9,10) f(2,10,11) f(2,11,12) f(3,3,4) f(3,5,6) f(3,6,7) f(3,7,8) f(3,8,9) f(3,9,10) f(3,10,11) f(3,11,12) f(4,4,5) f(4,5,6) f(4,6,7) f(4,7,8) f(4,8,9) f(4,9,10) f(4,10,11) f(4,11,12) f(5,5,6) f(5,6,7) f(5,7,8) f(5,8,9) f(5,9,10) f(5,10,11) f(5,11,12) f(6,6,7) f(7,7,8) f(8,8,9) f(9,9,10) f(10,10,11) f(11,11,12) f(12,12,13)
- 32. Luc_Faucheux_2020 A caplet andtwo distinct swaptions on the grid ¨ A t5-t12 swaption (orange) ¨ A t7 caplet (mauve) ¨ A t9-t12 swaption (dark orange) ¨ On the typical swaption grid, the t5-t12 and the t9-t12 are different distinct points with no overlap ¨ The caplet is also one point. Even though those 3 structures pan overlapping areas of the forward volatility surface of different sizes, they all get condensed to one point on the swaption grid 32 f(0,0,1) f(0,1,2) f(0,2,3) f(0,3,4) f(0,4,5) f(0,5,6) f(0,6,7) f(0,7,8) f(0,8,9) f(0,9,10) f(0,10,11) f(0,11,12) f(1,1,2) f(1,5,6) f(1,6,7) f(1,7,8) f(1,8,9) f(1,9,10) f(1,10,11) f(1,11,12) f(2,2,3) f(2,5,6) f(2,6,7) f(2,7,8) f(2,8,9) f(2,9,10) f(2,10,11) f(2,11,12) f(3,3,4) f(3,5,6) f(3,6,7) f(3,7,8) f(3,8,9) f(3,9,10) f(3,10,11) f(3,11,12) f(4,4,5) f(4,5,6) f(4,6,7) f(4,7,8) f(4,8,9) f(4,9,10) f(4,10,11) f(4,11,12) f(5,5,6) f(5,6,7) f(5,7,8) f(5,8,9) f(5,9,10) f(5,10,11) f(5,11,12) f(6,6,7) f(6,7,8) f(6,9,10) f(6,10,11) f(6,11,12) f(7,7,8) f(7,9,10) f(7,10,11) f(7,11,12) f(8,8,9) f(8,9,10) f(8,10,11) f(8,11,12) f(9,9,10) f(9,10,11) f(9,11,12) f(10,10,11) f(11,11,12) f(12,12,13)
- 33. Luc_Faucheux_2020 You have pricedtoo much, and not enough ¨ The options market is incredibly hard to calibrate ¨ Some points have lots of overlapping market info ¨ Large gaps in between, requiring either – Interpolation on the market inputs (on the grid) with very little theoretical justification – Interpolation on the model parameters, running the risk of being “off market” 33
- 34. Luc_Faucheux_2020 34 An example oftree: BDT Black Derman Toy
- 35.
- 36. Luc_Faucheux_2020 An illustrated example: Black Derman Toy ¨ Work through a tree implementation of the BDT model (short rate model) ¨ Most widely used model when modeling fixed income securities ¨ Illustrate some of the relevant issues around numerical implementation ¨ Work through some of the math to illustrate some pitfalls when switching from: – The numerical tree implementation – To the dynamics (Stochastic Differential Equation or SDE) – To the PDE for the PDF (Partial Differential Equation for the Probability Density Function) – To the grid implementation 36
- 37. Luc_Faucheux_2020 The BDT modeltree implementation 37 t0 t1 t2 t3 t4 (0,0) (1,0) (1,1) (2,0) (2,1) (2,2) (3,0) (3,1) (3,2) (3,3) (4,0) (4,1) (4,2) (4,3) (4,4)
- 38. Luc_Faucheux_2020 The BDT modeltree implementation 38 tk tk+1 X(k,i) X(k+1,i+1) X(k+1,i) ¨ We index the stochastic variable 𝑋(𝑘, 𝑖), where 𝑘 is the period index and 𝑖 is the position index ¨ We choose a time step to be 𝛿𝑡 = 𝜏 ¨ At first we assume this time step to be constant ¨ The spacing for a given period is 𝑋 𝑘, 𝑖 + 1 − 𝑋 𝑘, 𝑖 = 𝛿𝑋 for all index 𝑖 ¨ For a given period, all nodes in the tree are equally spaced ¨ We start with the assumption that the spacing 𝛿𝑋 is a constant
- 39. Luc_Faucheux_2020 Building a BDTtree I ¨ For sake of simplicity, we go back to time spacing of unit 1, and neglect any holidays and daycount fraction details like we did in the curve module ¨ We have the forward yield curve as input 𝑓(𝑡1, 𝑡1-!) ¨ We can compute the spot discount factors and the forward discount factors ¨ Forward discount 𝑑1 = 𝑑(𝑡1) which here we chose to be 𝑑 𝑡1 = ! !23(#&,#&'!) ¨ The spot discount curve is such that 𝐷 0 = 1, 𝑑% = 𝑑 0 = 1 and 𝐷 𝑡12! = 𝐷 𝑡1 . 𝑑(𝑡12!) 39 0-1 1-2 2-3 3-4 4-5 5-6 6-7 Y0_fwd 1 1.5 2 2.5 3 3.25 3.5 D0_spot 0.990099 0.975467 0.95634 0.933015 0.90584 0.877327 0.847658 D0_fwd 0.990099 0.985222 0.980392 0.97561 0.970874 0.968523 0.966184 VOL_spot 0.2 0.3 0.3 0.4 0.5 0.7 STDEV 0.08 0.36 0.54 1.28 2.5 5.88
- 40. Luc_Faucheux_2020 Building a BDTtree II ¨ Forward discount 𝑑1 = 𝑑(𝑡1) which here we chose to be 𝑑 𝑡1 = ! !23(#&,#&'!) ¨ Note that this is more a choice on how we define the forward yield curve. ¨ As we kept repeating in the curve module, the only thing that matters are the discount factors. Yields are not unique and depend on conventions, holidays, daycount fraction and assumptions to compute them. Discount factors ARE unique ¨ We also have as input to the models the volatility curve for the variable we are using the build the BDT tree. In our case we will use the forward. ¨ Note that you could use another variable ¨ The goal is to calibrate the model inputs to the market on observable prices ¨ So if you build the tree on a given variable, you will get a different value for the volatility inputs after calibration but you will recover the market prices 40
- 41. Luc_Faucheux_2020 Building a BDTtree III 41 0-1 1-2 2-3 3-4 4-5 5-6 6-7 Y0_fwd 1 1.5 2 2.5 3 3.25 3.5 D0_spot 0.990099 0.975467 0.95634 0.933015 0.90584 0.877327 0.847658 D0_fwd 0.990099 0.985222 0.980392 0.97561 0.970874 0.968523 0.966184 VOL_spot 0.2 0.3 0.3 0.4 0.5 0.7 STDEV 0.08 0.36 0.54 1.28 2.5 5.88 ¨ In our case we chose the forward to be lognormally distributed, and so this volatility will be the annualized lognormal yield volatility in % per year ¨ Again, the point is that because there are going to be so many assumptions and numerical computations that will be different for each implementation of a tree, you cannot really compare the model inputs between two models, just like you cannot really compare a normal volatility to a lognormal one ¨ The only thing you can compare are the prices produced by the model
- 42. Luc_Faucheux_2020 Building a BDTtree IV ¨ We create a grid of forward yields 𝑋(𝑘, 𝑖) ¨ We chose for the “bottom” value 𝑋 𝑘, 0 = 3(#&,#&'!) (!2 ((&) + )^(1) , where 𝑉(𝑘) is the volatility input ¨ The idea is to start with a distribution of forwards centered around the input one 𝑓(𝑡1, 𝑡1-!) and with a standard deviation that will be close to the volatility input ¨ Each successive value for a given period going up the nodes in the tree is given by ¨ 𝑋 𝑘, 𝑖 + 1 = 𝑋 𝑘, 𝑖 . [1 + 𝑉 𝑘 ]1 42 Y_fwd 1 1.498501 1.994013 2.488784 2.97612 3.209678 3.427392 1.501499 2.005996 2.51125 3.024024 3.290726 3.573885 2.01805 2.533919 3.072699 3.373821 3.72664 2.556793 3.122158 3.459014 3.885924 3.172413 3.546359 4.052015 3.635909 4.225206 4.405799
- 43. Luc_Faucheux_2020 Building a BDTtree V ¨ Note that the goal is to start with a ”reasonable” state for the tree ¨ The calibration process will automatically adjust to the solution we need ¨ Note that the BDT model is somewhat different from the other models out there ¨ BDT does NOT start with an equation, BDT essentially starts from a numerical implementation and just “let the calibration takes care of things” ¨ This makes the underlying dynamics a somewhat complicated beast to express in the usual SDE and PDE ¨ We will show in a simple case how to do this, if only to point out the pitfalls and the gaps between the SDE world and the discrete numerical implementation world 43
- 44. Luc_Faucheux_2020 Building a BDTtree VI ¨ This pragmatic approach (again does not sound smart at first because you are just using Excel, you are not writing complicated equations), turns out to be the smart one in the end. ¨ This is also what makes BDT (and the tweaks like BK,..) one of the most commonly used model in Finance, it is intuitive, it is robust, it is fast ¨ Note that BDT also has drawbacks, the calibration process can be at time unstable, and because of the discrete nature of it, it is also numerically “noisy”. For computing the Greeks using a BDT, depending on the payoff of the option that you are pricing, you will get “jumps” in value when you are crossing a node in your valuation ¨ Note that the number of nodes in the BDT tree increases linearly with the number of time steps for a given period, and thus the total number of nodes scales as the square of the number of time steps, making reducing the noise by increasing the number of nodes and times steps a computationally expensive proposition 44
- 45. Luc_Faucheux_2020 Building a BDTtree VII ¨ From each forward 𝑋(𝑘, 𝑖) at each node we compute the Discount factor ¨ 𝑑 𝑘, 𝑖 = 1/(1 + 𝑋 𝑘, 𝑖 ) ¨ We now have built a tree where at each node we have the forward and the discount factor 45 D_fwd 0.990099 0.985236 0.98045 0.975717 0.971099 0.968901 0.966862 0.985207 0.980335 0.975503 0.970647 0.968141 0.965494 0.980219 0.975287 0.970189 0.967363 0.964072 0.975069 0.969724 0.966566 0.962594 0.969251 0.965751 0.961058 0.964917 0.959461 0.957801
- 46. Luc_Faucheux_2020 Building a BDTtree VIII ¨ We also attach to each node the usual binomial probability in order to recover the Gaussian distribution ¨ The probability at each node is the probability to end up at that given node ¨ We assume here for simplicity equal probability for “up” and “down” to be 0.5 ¨ Note that there are tons of numerical schemes out there to change the probabilities and thus affect the drift inside the tree, but this does not change the fundamentals of the model ¨ Note also that now that we have a tree, sometimes Monte-Carlo simulation are run “on the tree” in order ot price path-dependent structures 46 PROBA 1 0.5 0.25 0.125 0.0625 0.03125 0.015625 0.5 0.5 0.375 0.25 0.15625 0.09375 0.25 0.375 0.375 0.3125 0.234375 0.125 0.25 0.3125 0.3125 0.0625 0.15625 0.234375 0.03125 0.09375 0.015625
- 47.
- 48. Luc_Faucheux_2020 Calibrating a BDTtree I ¨ We now have built this tree, but we need to calibrate it to market ¨ This is usually a multi step process usually performed in a sequential manner (but not always) ¨ The first step is usually ensuring that we recover the discount curve, this will ensure that we will be pricing bonds and swaps and zero coupon bonds to market ¨ This is where the numerical implementation of a solver for example, is crucial, in particular for the stability of the solution 48
- 49. Luc_Faucheux_2020 Calibrating a BDTtree II ¨ We calculate the expectation for the discount (remember that it should be probability weighted) ¨ This is obviously not equal to our input discount curve ¨ We need to “shift” the forwards in order to solve for the discount 49
- 50. Luc_Faucheux_2020 Calibrating a BDTtree III 50 K 0.999899 1.000063 0.997004 0.99103 0.976221 0.950988 0.899574 NEW_Y 0.999899 1.498596 1.98804 2.466459 2.90535 3.052366 3.083193 1.501593 1.999986 2.488724 2.952115 3.129442 3.214975 2.012004 2.51119 2.999633 3.208465 3.352389 2.533859 3.047916 3.289482 3.495676 3.096976 3.372546 3.645088 3.457707 3.800886 3.963343 NEW_D 0.9901 0.985235 0.980507 0.975929 0.971767 0.97038 0.97009 0.985206 0.980392 0.975717 0.971325 0.969655 0.968852 0.980277 0.975503 0.970877 0.968913 0.967564 0.975288 0.970422 0.968153 0.966224 0.969961 0.967375 0.964831 0.966579 0.963383 0.961877 E(D) 0.9901 0.985221 0.980392 0.97561 0.970874 0.968524 0.966184 D_0_fwd 0.990099 0.985222 0.980392 0.97561 0.970874 0.968523 0.966184 distance 1.000001 0.999999 1 1 1 1.000001 1
- 51. Luc_Faucheux_2020 Calibrating a BDTtree IV ¨ This is where we get into the wonderful world of numerical solvers ¨ Note that we are “bootstrapping” the factors along the tree ¨ So a little error at one level needs to be corrected at the next level, potentially creating instabilities 51
- 52.
- 53. Luc_Faucheux_2020 Pricing a caplet ¨Payoff = MAX(Forward-Strike, 0) 53
- 54. Luc_Faucheux_2020 Pricing a caplet- II ¨ The cash flows are then discounted back inside the tree accumulating the cash flows 54
- 55. Luc_Faucheux_2020 Pricing a cap ¨From each forward 𝑋(𝑘, 𝑖) at each node we compute the discount factor ¨ 𝑑 𝑘, 𝑖 = 1/(1 + 𝑋 𝑘, 𝑖 ), (assuming the usual convention of unity for daycount fraction) ¨ At each node the payoff is 𝐶𝑎𝑝 𝑘, 𝑖 = 𝑀𝐴𝑋(𝑋 𝑘, 𝑖 − 𝐾, 0) ¨ The discounting back is given by: ¨ 𝐶𝑎𝑝 𝑘, 𝑖 = ! " . 𝑑 𝑘, 𝑖 . 𝐶𝑎𝑝 𝑘 + 1, 𝑖 + 1 + 𝐶𝑎𝑝 𝑘 + 1, 𝑖 + 𝑀𝐴𝑋 𝑋 𝑘, 𝑖 − 𝐾, 0 . 𝑑 ¨ If the cap is paid at the end of the period, 𝑑 = 𝑑(𝑘, 𝑖) otherwise 𝑑 = 1 55 tk tk+1 X(k,i) X(k+1,i+1) X(k+1,i)
- 56. Luc_Faucheux_2020 Pricing a swapin BDT ¨ In the ”swap” world you do not need to build a tree, you only need a discount curve and use simple formulas like SUMPRODUCT 56
- 57. Luc_Faucheux_2020 Pricing a swapin BDT - II ¨ We saw this when looking at the convexity ¨ This is because the right way (the right measure) to look at cashflows is the discount curve. ¨ A swap fixed leg is the sum of fixed payment times discount factors ¨ Same for the Float leg (a little more complicated) ¨ But in fact a swap PV is a linear sum of discount factors ¨ As such it exhibits ZERO convexity against the discount curve ¨ And so to price a swap you do not need a volatility curve, nor an option model, but if you are pricing a derivatives on a swap (like a swaption), you better make sure that your option model does recover the same price for the underlier ¨ This sounds obvious, but you would be surprised how many options models out there are not repricing the underlier correctly ¨ Let’s look back at the “yield curve” (zero vol) swap valuation 57
- 58. Luc_Faucheux_2020 A swap isa weighted basket of forwards: AT-THE-MONEY ¨ Consider a swap with swap rate R (at-the-money swap rate) – Nfloat periods on the Float side with forecasted forward f(i) – indexed by i, with – daycount fraction DCF(i), – discount D(i) – Notional N(i) – Nfixed periods on the Fixed side, – indexed by j, with – daycount fraction DCF(j), – discount D(j) – Notional N(j) 𝑃𝑉 𝐹𝐿𝑂𝐴𝑇 = ) , 𝐷𝐶𝐹 𝑖 . 𝐷 𝑖 . 𝑁 𝑖 . 𝑓 𝑖 = ) - 𝐷𝐶𝐹 𝑗 . 𝐷 𝑗 . 𝑁 𝑗 . 𝑅 = 𝑃𝑉(𝐹𝐼𝑋𝐸𝐷) 58
- 59. Luc_Faucheux_2020 Standard Swap periods ¨On the fixed side, coupon payment at the end of the period – Period start date (psj) – Adjusted period start date (psj_adj) – Period end date (pej) – Adjusted period end date (pej_adj) – Payment date (pmj) – PV of a period 𝑃𝑉 𝑗 = 𝐷𝐶𝐹 𝑗 . 𝐷 𝑗 . 𝑁 𝑗 . 𝑅 = 𝐷𝐶𝐹 𝑝𝑠𝑗./-, 𝑝𝑒𝑗_𝑎𝑑𝑗 . 𝐷 𝑝𝑚- . 𝑁 𝑗 . 𝑅 ¨ On the float side, floating rate sets at the beginning of the period, and pays at the end (Libor in advance or standard Libor swap, as opposed to Libor in arrears) – PV of a period (swaplet) 𝑃𝑉 𝑖 = 𝐷𝐶𝐹 𝑖 . 𝐷 𝑖 . 𝑁 𝑖 . 𝑓 𝑖 – 𝐷𝐶𝐹 𝑖 = 𝐷𝐶𝐹 𝑝𝑠𝑖./-, 𝑝𝑒𝑖./- and 𝐷 𝑖 = 𝐷(𝑝𝑚𝑖) – 𝐷 𝑝𝑒𝑖 = 𝐷 𝑝𝑠𝑖 ∗ 0 01234 56,,58, .:(,) or – 𝐷𝐶𝐹 𝑝𝑒𝑖, 𝑝𝑠𝑖 . 𝑓 𝑖 = [1 − 2 58, 2 56, ] 59
- 60. Luc_Faucheux_2020 Zero coupon bondson today’s curve with zero volatility ¨ Zero-coupon bonds 𝑃 𝑡0, 𝑡; = 𝐷 𝑡0, 𝑡; = ⁄𝐷(𝑡;) 𝐷(𝑡0) ¨ “𝑃 𝑡0, 𝑡; is the price at time 𝑡0of a zero-coupon bond maturing at time 𝑡;” ¨ “𝑃 𝑡0, 𝑡; is the price at time 𝑡0of a risk-free zero-coupon bond with principal $1 maturing at time 𝑡;” ¨ IT SHOULD REALLY SAY : “Using today’s discount curve at time 𝑡<, 𝑃 𝑡0, 𝑡; is the price of a risk-free zero- coupon bond with principal $1 maturing at time 𝑡;, and the value of that price has been forward discounted to time 𝑡0, again using today’s discount curve” ¨ People love the zero coupon bonds, in many cases they make those the stochastic drivers of the rates model (HJM for example) 60
- 61. Luc_Faucheux_2020 Expected Values ina non deterministic world ¨ Simply compounded spot interest rate: 𝐿(𝑡0, 𝑡;) ¨ 𝐿 𝑡0, 𝑡; = 0=>(?!,?") /@: ?!,?" .>(?!,?") or more simply 𝑃 𝑡0, 𝑡; = 0 01/@: ?!,?" .A(?!,?") ¨ Related to how to roll the curve forward at zero volatility, ¨ Method 2 – Compute the discount factors curve – Divide all discount factors by the overnight d(t0,t1)=d(t1) – Use new discount factor curve starting at t1 ¨ So at zero volatility, when t goes from t0 to t1, the price of a zero discount bonds 𝑃 𝑡0, 𝑡; is unchanged: 𝑃 𝑡@, 𝑡0, 𝑡; = 𝑃 0, 𝑡0, 𝑡; where 𝑡@ is the “curve” time in the future. ¨ NOW, if the volatility is non zero, 𝑃 1, 𝑡0, 𝑡; ≠ 𝑃 0, 𝑡0, 𝑡; ¨ It is only true ON AVERAGE < 𝑃 1, 𝑡0, 𝑡; >= 𝑃 0, 𝑡0, 𝑡; or EXP 𝑃 1, 𝑡0, 𝑡; = 𝑃 0, 𝑡0, 𝑡; where EXP is the Expected value (average). ¨ This is called the rolling numeraire or “bank account” numeraire: if you deposit 𝑃 0,0, 𝑡; today to get $1 at time t2, ON AVERAGE you should also be able to invest 𝑃 0,0, 𝑡; until time t1, then deposit it until time t2 and still get $1 61
- 62. Luc_Faucheux_2020 Fixed Leg ofa swap ¨ A fixed leg of a swap is a series of fixed cash flows. ¨ Now matter how the curve moves, ON AVERAGE the price of zero coupon bonds is conserved ¨ < 𝑃 𝑡@, 𝑡0, 𝑡; > = 𝑃 0, 𝑡0, 𝑡; and 𝑃 𝑡@, 𝑡0, 𝑡; = 2(?#,?") 2(?#,?!) ¨ In particular when t2=t1+1, 𝑃 𝑡@, 𝑡0, 𝑡0 + 1 = 2(?#,?!10) 2(?#,?!) = 𝑑(𝑡@, 𝑡0) ¨ So < 𝑃 𝑡@, 𝑡0, 𝑡0 + 1 > = < 2 ?#,?!10 2 ?#,?! > = < 𝑑(𝑡@, 𝑡0)> = 𝑃 0, 𝑡0, 𝑡0 + 1 = 𝑑(0, 𝑡0) ¨ Also by recurrence < 𝐷 1, 𝑡0 > ∗ 𝑑 0,1 = 𝐷(0, 𝑡0) at time t=0 ¨ At time t=1, 𝑑 1,2 is fixed and has zero volatility (will drop when t goes from 1 to 2) ¨ So < 𝐷 2, 𝑡0 > ∗ 𝑑 1,2 = 𝐷(1, 𝑡0) at time t=1 ¨ So at every point in the future, if you invest then a unit of currency and ”roll” it forward (bank numeraire), the expected gain is today’s gain if you had entered into the same contract. ¨ Still another way to say, if you invest one unit of currency for a given length of time t, it is equivalent to investing overnight and rolling the proceeds everyday (the arbitrage free framework does not take credit into consideration) 62
- 63. Luc_Faucheux_2020 Floating leg ofa swap ¨ A floating swaplet pays 𝐷𝐶𝐹 𝑖 . 𝑁 𝑖 . 𝑓 𝑖 and its PV is 𝑃𝑉 𝑖 = 𝐷𝐶𝐹 𝑖 . 𝐷 𝑖 . 𝑁 𝑖 . 𝑓 𝑖 ¨ Where 𝐷𝐶𝐹 𝑝𝑒𝑖, 𝑝𝑠𝑖 . 𝑓 𝑖 = [1 − 2 58, 2 56, ] ¨ We know from the fixed rate leg that < 𝐷 𝑖 >= 𝐷 𝑖 , but what about < 𝐷 𝑖 . 𝑓 𝑖 > ? ¨ Note, to be exact < 𝐷 𝑖 >= 𝐷 𝑖 should really read ∏?#BC?$ 𝐸𝑋𝑃{𝑡@, 𝑑(𝑡@, 𝑡@ + 1)}, where 𝐸𝑋𝑃 𝑡@, 𝑑 𝑡@, 𝑡@ + 1 is the expected value of the overnight discount between the time (tc) and (tc+1), observed up until time tc (because it drops off the curve after tc, and before tc, no matter where you observe it, its expected value is equal to today’s value) ¨ 𝐸𝑋𝑃 𝑡@, 𝑑 𝑡@, 𝑡@ + 1 = 𝐸𝑋𝑃 𝑡 < 𝑡@, 𝑑 𝑡@, 𝑡@ + 1 = 𝑑(𝑡@ + 1) ¨ Back to < 𝐷 𝑖 . 𝑓 𝑖 > , there is a little trick 63
- 64. Luc_Faucheux_2020 Floating leg ofa swap ¨ Because the forward f(i) sets at the beginning of the period, once we reach the period start, everything is known about the payment, and it becomes a fixed cashflow. ¨ 𝑃𝑉 𝑖 = 𝐷𝐶𝐹 𝑖 . 𝐷 𝑖 . 𝑁 𝑖 . 𝑓 𝑖 = 𝐷𝐶𝐹 𝑖 . 𝑁 𝑖 . 𝐸𝑋𝑃 𝑝𝑠𝑖, 𝐷 𝑝𝑠𝑖 . 𝐸𝑋𝑃{𝑝𝑠𝑖, 𝐷 𝑝𝑠𝑖, 𝑝𝑒𝑖 . 𝑓 𝑖 } ¨ 𝐸𝑋𝑃 𝑝𝑠𝑖, 𝐷 𝑝𝑠𝑖 = 𝐷(𝑝𝑠𝑖) ¨ 𝐷𝐶𝐹 𝑖 . 𝐸𝑋𝑃 𝑝𝑠𝑖, 𝐷 𝑝𝑠𝑖, 𝑝𝑒𝑖 . 𝑓 𝑖 = 𝐸𝑋𝑃{𝑝𝑠𝑖, 234(,) 01234 , .:(,) . 𝑓 𝑖 } ¨ Now, magic trick, D 01D = D10=0 01D = 01D=0 01D = 1 − 0 01D ¨ So, 𝐸𝑋𝑃 𝑝𝑠𝑖, 234 , 01234 , .: , . 𝑓 𝑖 = 𝐸𝑋𝑃 𝑝𝑠𝑖, 1 − 0 01234 , .: , = 𝐸𝑋𝑃 𝑝𝑠𝑖, 1 − 𝑃 𝑝𝑠𝑖, 𝑝𝑠𝑖, 𝑝𝑒𝑖 ¨ And because the price of zero coupon bond is respected: ¨ 𝐸𝑋𝑃 𝑝𝑠𝑖, 234 , 01234 , .: , . 𝑓 𝑖 = 1 − 𝑃 0, 𝑝𝑠𝑖, 𝑝𝑒𝑖 = 1 − 0 01234 , .: , = 234 , 01234 , .: , . 𝑓 𝑖 64
- 65. Luc_Faucheux_2020 Quick summary ¨ Inthe rolling numeraire measure, – PV of fixed cashflows are conserved (Expected value of a fixed cashflow as the curve evolves in a stochastic manner over time will converge to the fixed amount at the payment date) – Price of zero coupon bonds are conserved – Price of bonds are conserved (the price of a bond will change over time, but ON AVERAGE the price you should be willing to pay for this bond is the price you can compute today using today’s curve, because the price of a bond exhibits no convexity with respect to the discount curve changing, AS LONG as the discount curve changes in a manner that respect the Arbitrage free condition, that is that a contract where you invest X today to get Y at time T, is the same (equivalent, on average), as any contract where you invest X today, get the proceeds at some point in time in the future, then reinvest them until T.) – This is either painfully obvious or really deep, depending on how you look at it – The arbitrage free assumption does NOT know about credit – The arbitrage free assumption does NOT know about individual utility function (also called time-indifferent, it assumes that market participants are indifferent about receiving X today versus Y at time T, where the ratio Y/X is the price of today zero coupon bond maturing at time T, and that price will be conserved over time) – The Floating leg of a swap ALSO exhibit zero convexity against the discount factor curve, because it can be expressed as a linear function of discount factors, thanks to the amazing trick x=x+1-1 – It is surprising that of ALL the possible structures we could come up with a swap, there is almost zero probability that we would not need a volatility curve, and yet 99.9% of the swaps being traded are such that they are NOT convex in terms of the discount factors. Note that this breaks down once you have stochastic funding (forecasting and discounting on different curves ¨ Other markets (equity, commodities,..) do NOT have such a strong underlying constraint that needs to be respected.. In HJM for example, we will show that respecting the arbitrage enforces zero possible choice for the drift, once the volatility is known, everything else is. 65
- 66. Luc_Faucheux_2020 Pricing a swapin BDT - III 66
- 67. Luc_Faucheux_2020 Pricing a swapin BDT - IV ¨ NOTE : swaps payments are "in arrears" at the end of the period, and so you have to discount for that period ¨ index is set "in advance" and paid "in arrears" ¨ If you were dealing with a swap where the payment is set at the same time as the payment, value would be different ¨ in fact it would exhibit convexity ("in arrears" swap) ¨ you would need a BDT tree or an option model in order to value it ¨ But numerically quite easy to do, you just do not discount for that period 67
- 68. Luc_Faucheux_2020 Pricing a swapin BDT - V 68
- 69. Luc_Faucheux_2020 Pricing a swapin BDT - VI ¨ We recover the same swap rate ¨ Please note that we are really discounting cash flows, never a rate or a yield ¨ Note that in practice to ”take into account” all the numerical imprecisions and the many steps, the solver is sometimes done on the Swap Rate (Market observable) ¨ The Swap Rate is really only the value of the Fixed coupon so that the discounted value of all fixed payments is equal to the discounted value of all the floating payments ¨ Note that if we had priced the swap “in arrears” we would get a different swap rate than the one computed from the yield curve (even taking into account the timing difference) ¨ The difference between those two rates is the convexity adjustment ¨ In the SUMPRODUCT formula (deterministic world, or zero volatility), we just need to offset the discounts by one period. Float fixes at the beginning of the period and pays at the beginning, fixed also pays at the beginning of the period. This timing difference between the fixing of the float and the payment is what creates the convexity. ¨ Because we cannot do the little trick: ' !2' = '2!-! !2' = !2'-! !2' = 1 − ! !2' 69
- 70. Luc_Faucheux_2020 Pricing a swapin BDT - VI-b ¨ Another way is also to parametrize the shifts in order to allow for some tolerance, and globally minimize for the Market observables (Swap rates), and be careful over the overlap of swap durations ¨ This is sometimes done in practice in order to “correct” for a number of numerical imprecisions along the way (missing holidays, imperfect daycount fraction, small residuals in the minimization steps,..) 70
- 71. Luc_Faucheux_2020 Legs of aswap in arrears ¨ A fixed leg of a swap is a series of fixed cash flows that pays ¨ 𝐷𝐶𝐹 𝑖 . 𝑁 𝑖 . 𝑆𝑅 ¨ And its PV is: ¨ 𝐷𝐶𝐹 𝑖 . 𝑁 𝑖 . 𝑆𝑅. 𝐷(𝑖 − 1) ¨ A floating swaplet “in arrears” pays: ¨ 𝐷𝐶𝐹 𝑖 . 𝑁 𝑖 . 𝑓 𝑖 ¨ and its PV is: ¨ 𝑃𝑉 𝑖 = 𝐷𝐶𝐹 𝑖 . 𝐷 𝑖 − 1 . 𝑁 𝑖 . 𝑓 𝑖 ¨ Where 𝐷𝐶𝐹 𝑝𝑒𝑖, 𝑝𝑠𝑖 . 𝑓 𝑖 = [1 − 2 58, 2 56, ] 71
- 72. Luc_Faucheux_2020 Floating leg ofa regular swap ¨ Because the forward f(i) sets at the beginning of the period, once we reach the period start, everything is known about the payment, and it becomes a fixed cashflow. ¨ 𝑃𝑉 𝑖 = 𝐷𝐶𝐹 𝑖 . 𝐷 𝑖 . 𝑁 𝑖 . 𝑓 𝑖 = 𝐷𝐶𝐹 𝑖 . 𝑁 𝑖 . 𝐸𝑋𝑃 𝑝𝑠𝑖, 𝐷 𝑝𝑠𝑖 . 𝐸𝑋𝑃{𝑝𝑠𝑖, 𝐷 𝑝𝑠𝑖, 𝑝𝑒𝑖 . 𝑓 𝑖 } ¨ 𝐸𝑋𝑃 𝑝𝑠𝑖, 𝐷 𝑝𝑠𝑖 = 𝐷(𝑝𝑠𝑖) ¨ 𝐷𝐶𝐹 𝑖 . 𝐸𝑋𝑃 𝑝𝑠𝑖, 𝐷 𝑝𝑠𝑖, 𝑝𝑒𝑖 . 𝑓 𝑖 = 𝐸𝑋𝑃{𝑝𝑠𝑖, 234(,) 01234 , .:(,) . 𝑓 𝑖 } ¨ Now, magic trick, D 01D = D10=0 01D = 01D=0 01D = 1 − 0 01D ¨ So, 𝐸𝑋𝑃 𝑝𝑠𝑖, 234 , 01234 , .: , . 𝑓 𝑖 = 𝐸𝑋𝑃 𝑝𝑠𝑖, 1 − 0 01234 , .: , = 𝐸𝑋𝑃 𝑝𝑠𝑖, 1 − 𝑃 𝑝𝑠𝑖, 𝑝𝑠𝑖, 𝑝𝑒𝑖 ¨ And because the price of zero coupon bond is respected: ¨ 𝐸𝑋𝑃 𝑝𝑠𝑖, 234 , 01234 , .: , . 𝑓 𝑖 = 1 − 𝑃 0, 𝑝𝑠𝑖, 𝑝𝑒𝑖 = 1 − 0 01234 , .: , = 234 , 01234 , .: , . 𝑓 𝑖 72
- 73. Luc_Faucheux_2020 Floating leg ofan “in arrears” swap ¨ Because the forward f(i) sets at the beginning of the period, once we reach the period start, everything is known about the payment, and it becomes a fixed cashflow. ¨ 𝑃𝑉 𝑖 = 𝐷𝐶𝐹 𝑖 . 𝐷 𝑖 − 1 . 𝑁 𝑖 . 𝑓 𝑖 = 𝐷𝐶𝐹 𝑖 . 𝑁 𝑖 . 𝐸𝑋𝑃 𝑝𝑠𝑖, 𝐷 𝑝𝑠𝑖 . 𝐸𝑋𝑃{𝑝𝑠𝑖, 1. 𝑓 𝑖 } ¨ 𝐸𝑋𝑃 𝑝𝑠𝑖, 𝐷 𝑝𝑠𝑖 = 𝐷(𝑝𝑠𝑖) ¨ 𝐷𝐶𝐹 𝑖 . 𝐸𝑋𝑃 𝑝𝑠𝑖, 1. 𝑓 𝑖 = 𝐸𝑋𝑃{𝑝𝑠𝑖, 𝐷𝐶𝐹 𝑖 . 𝑓 𝑖 } ¨ Where 𝐷𝐶𝐹 𝑝𝑒𝑖, 𝑝𝑠𝑖 . 𝑓 𝑖 = [1 − 2 58, 2 56, ] ¨ Now, magic trick, D 01D = D10=0 01D = 01D=0 01D = 1 − 0 01D , but that gets us nowhere ¨ So, 𝐸𝑋𝑃 𝑝𝑠𝑖, 𝐷𝐶𝐹 𝑖 . 𝑓 𝑖 = 𝐸𝑋𝑃 1 − 2 58, 2 56, ¨ This has not only a timing difference but a ratio, and we all know that 𝐸𝑋𝑃 0 D <> 1/𝐸𝑋𝑃(𝑥) ¨ This can be done through the Taylor expansion (or also Jensen inequality) ¨ This is the famous Swap in arrears convexity trade of May 1995 (story time) 73
- 74.
- 75. Luc_Faucheux_2020 Pricing a swaption- II ¨ Careful that the only thing you should discount are cashflows, not rates ¨ You could look at the rates but then at each node you would need to weigh by the actual duration of that swap on that node to express it back into a cash flow ¨ Note that the Black’77 model assumes a single duration for all nodes, and so putting in the same input volatility in your BDT model and the Black’77 model will NOT recover the same option price ¨ The average of a product is not the product of the averages (or integrals) ¨ If you want to calibrate BDT to the implied volatilities from Black77 (or match the option prices that you see in the market), you would need to input a different volatilities. ¨ Just because it is called the same thing (volatility) and has the same units does not mean that you can equate the two 75
- 76. Luc_Faucheux_2020 Pricing a callable ¨We can now even price a callable swap ¨ This is the main reason to build a tree, as opposed to closed form models ¨ Note also that once you have the tree calibrated, it is very flexible in terms of any payoff of structure that you want to price ¨ A closed form solution would be hard to modify ¨ In practice, the most important step is the calibration step. Think about the yield curve moving in real time every seconds, the volatility curve moving maybe on a time scale of minutes, and so you need to build a calibration routine that can be done quickly and still be stable ¨ Sometimes tree models are calibrated carefully overnight and kept constant during the day, and trading desks adjust the price based on an estimate of the Greeks and the market moves 76
- 77.
- 78. Luc_Faucheux_2020 Pricing a callable- III ¨ Specific callable swap in the example is a 5y3ync1 ¨ Swap starts in 5year ¨ Swap ends in 8 years ¨ Call option at par (no fees either way) at year 6 and year 7 78
- 79. Luc_Faucheux_2020 Pricing a callableswap -IV ¨ From each forward 𝑋(𝑘, 𝑖) at each node we compute the discount factor ¨ 𝑑 𝑘, 𝑖 = 1/(1 + 𝑋 𝑘, 𝑖 ), (assuming the usual convention of unity for daycount fraction) ¨ At each node the payoff of the regular swap is SWAP 𝑘, 𝑖 = 𝑋 𝑘, 𝑖 − 𝐾 . 𝑑(𝑘, 𝑖) ¨ The discounting back is given by: ¨ SWAP 𝑘, 𝑖 = ! " . 𝑑 𝑘, 𝑖 . SWAP 𝑘 + 1, 𝑖 + 1 + SWAP 𝑘 + 1, 𝑖 + 𝑋 𝑘, 𝑖 − 𝐾 . 𝑑(𝑘, 𝑖) 79 tk tk+1 X(k,i) X(k+1,i+1) X(k+1,i)
- 80. Luc_Faucheux_2020 Pricing a callableswap -V ¨ At each node the payoff of the regular swap is SWAP 𝑘, 𝑖 = 𝑋 𝑘, 𝑖 − 𝐾 . 𝑑(𝑘, 𝑖) ¨ The discounting back is given by: ¨ SWAP 𝑘, 𝑖 = ! " . 𝑑 𝑘, 𝑖 . SWAP 𝑘 + 1, 𝑖 + 1 + SWAP 𝑘 + 1, 𝑖 + 𝑋 𝑘, 𝑖 − 𝐾 . 𝑑(𝑘, 𝑖) ¨ IF on an option node, the new payoff then becomes 𝑀𝐴𝑋(SWAP 𝑘, 𝑖 , 0) ¨ You can easily add a fee to cancel (>0 or <0), for example if there is a fee to cancel expressed as paid over the cancel period, the payoff becomes: 𝑀𝐴𝑋(SWAP 𝑘, 𝑖 , −fee) 80 tk tk+1 X(k,i) X(k+1,i+1) X(k+1,i)
- 81. Luc_Faucheux_2020 Pricing a callableswap -VI 81
- 82. Luc_Faucheux_2020 Pricing a callableswap -VII ¨ Because of the optionality, the payer of a fixed rate callable swap is willing to pay an above market rate (higher than the swap rate) to pay for the option ¨ We can use again a numerical solver (GoalSeek in Excel) ¨ Note: a great numerical solver is hard to find, and is used all the times in Finance ¨ Solve for 0 PV on the regular swap -> SR = 3.414 in our example ¨ Solve for 0 PV on the fixed callable ->SR = 3.448 in our example ¨ The callable swap offers a pick-up of 3.5 basis point running above the Swap rate ¨ The more optionality, the higher the pick-up ¨ The higher the volatility, the more valuable the option, the higher the pick-up ¨ If you calibrate your model to European swaptions, do you recover the price of callables observed in the market ? The answer is no (for many reasons, see the Structured ppt), but people try to adjust for that by introducing mean reversion in the BDT tree. Let’s see what that means and how that works. 82
- 83. Luc_Faucheux_2020 Pricing a callableswap -VIII ¨ Most of the times in practice, trading desks will keep an adjustment grids between the European and the Callable and adjust the volatility curve by that adjustment to price a callable. ¨ This is quite unsatisfactory for a lot of reason ¨ It is not consistent ¨ Where do you hedge the callable? On the shifted grid or the original one? ¨ The adjustment creates a PL difference, do you hold that as a reserve, do you release it? ¨ The PL difference will show some market directionality. Do you hedge that market directionality? ¨ If you hedge a Callable with European options, do you price and hedge those Europeans on the shifted grid or the original one? ¨ Different callables at different coupon will have different adjustments 83
- 84. Luc_Faucheux_2020 Pricing a callableswap -IX ¨ We have now all the building blocks of a full-fledged Fixed-Income front-to-back infrastructure ¨ To do the risk management, bump the yield curve and the vol curve and reprice all the trades AFTER recalibrating the BDT tree ¨ You can price pretty much any payoff that you want that is a function of the variables in the tree ¨ You can also do a “Monte-Carlo on the tree” for path-dependent options, you are just going forward in the tree, as opposed to discounting backwards 84
- 85. Luc_Faucheux_2020 Pricing a cappedfloater ¨ Break-even is we can pay MIN(Libor+100,2.562) against receiving Libor 85
- 86. Luc_Faucheux_2020 Pricing an inversefloater ¨ Break-even is we pay MAX(4.95-Libor,0) against receiving Libor 86
- 87. Luc_Faucheux_2020 87 A look atthe dynamics of the BDT
- 88. Luc_Faucheux_2020 The BDT dynamicsI ¨ The following question comes up: ¨ You have built something that reprices market observables ¨ Congratulations but what dynamics did you use ? ¨ When we use Black-Sholes, we can write the SDE for the underlier, and the PDE ¨ Can we do the same thing for BDT ? ¨ Answer is sort of yes unless you do numerical things to the BDT tree that are too drastic ¨ We are now giving some insights on this dynamics, if only to point out the difficulties in bridging the gap between the equations and the discrete numerical implementation of those 88
- 89. Luc_Faucheux_2020 The BDT dynamicsII 89 t0 t1 t2 t3 t4 (0,0) (1,0) (1,1) (2,0) (2,1) (2,2) (3,0) (3,1) (3,2) (3,3) (4,0) (4,1) (4,2) (4,3) (4,4)
- 90. Luc_Faucheux_2020 The BDT dynamicsIII ¨ We assume for now constant time step, constant spacing between nodes in the y axis 90
- 91. Luc_Faucheux_2020 The BDT dynamicsIV 91 tk tk+1 X(k,i) X(k+1,i+1) X(k+1,i) ¨ We index the stochastic variable 𝑋(𝑘, 𝑖), where 𝑘 is the period index and 𝑖 is the position index ¨ We choose a time step to be 𝛿𝑡 = 𝜏 ¨ At first we assume this time step to be constant ¨ The spacing for a given period is 𝑋 𝑘, 𝑖 + 1 − 𝑋 𝑘, 𝑖 = 𝛿𝑋 for all index 𝑖 ¨ For a given period, all nodes in the tree are equally spaced ¨ We start with the assumption that the spacing 𝛿𝑋 is a constant
- 92. Luc_Faucheux_2020 The BDT dynamicsV ¨ To the first order with no drift ¨ 𝑋 𝑘 + 1, 𝑖 = 𝑋 𝑘, 𝑖 − 𝛿𝑋/2 ¨ 𝑋 𝑘 + 1, 𝑖 + 1 = 𝑋 𝑘, 𝑖 + 𝛿𝑋/2 ¨ Probabilities Π 𝑘, 𝑖 to be at the node 𝑘, 𝑖 ¨ Using (1/2) probabilities up and down ¨ Π 𝑘 + 1, 𝑖 = ! " Π 𝑘, 𝑖 − 1 + Π 𝑘, 𝑖 ¨ This is also referred to as the master equation. ¨ Easier to work in the Π 𝑡, 𝑋 space to avoid the drift of the (𝑖) indices (zero drift means that the middle for 𝑋 would be the starting point but that is roughly the node (𝑘, 𝑖 = 1 " ) ¨ ALWAYS pay attention to the boundary conditions (borders), so for example ¨ Π 𝑘, 𝑖 = 0 IF 𝑖 > 𝑘 OR (𝑖 < 0) 92
- 93. Luc_Faucheux_2020 The BDT dynamicsVI ¨ < ∆𝑋 > = ! " . 𝑋 𝑘 + 1, 𝑖 + 1 + 𝑋 𝑘 + 1, 𝑖 = 𝑋(𝑘, 𝑖) ¨ So no drift in this simple example ¨ When writing 𝑑𝑋 = 𝜇. 𝑑𝑡 + 𝜎. 𝑑𝑊, we would have 𝜇 = 0 ¨ < ∆𝑋"> = ! " . (𝑋 𝑘 + 1, 𝑖 + 1 − 𝑋 𝑘, 𝑖 )^(2) + (𝑋 𝑘 + 1, 𝑖 − 𝑋 𝑘, 𝑖 )^(2) ¨ < ∆𝑋"> = ! " . (𝛿𝑋/2)" + (𝛿𝑋/2)" = 89+ : ¨ When writing 𝑑𝑋 = 𝜇. 𝑑𝑡 + 𝜎. 𝑑𝑊, we would have (𝜎". 𝛿𝑡) = 89+ : 93 X(k,i) X(k+1,i+1) X(k+1,i) 𝛿𝑋/2 𝛿𝑋/2
- 94. Luc_Faucheux_2020 The BDT dynamicsVII ¨ (𝜎". 𝛿𝑡) = 89+ : ¨ Again we see the scaling argument that if we want 𝜎to be well defined (does not go to 0, does not go to infinity), as we are reducing the size of the time step and the spacing in order to reach the continuous limit from the discrete implementation, we need ¨ 89+ :8# → 𝜎" = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 when 𝛿𝑋 → 0 and 𝛿𝑡 → 0 ¨ So the discrete tree implemented converges to the continuous dynamics described by the SDE (Stochastic Differential Equation): 𝑑𝑋 = 𝜇. 𝑑𝑡 + 𝜎. 𝑑𝑊 ¨ With 𝜇 = 0 ¨ And 𝜎 = lim( 89+ :8# ) 94
- 95. Luc_Faucheux_2020 The BDT dynamicsVIII ¨ From the SDE 𝑑𝑋 = 𝜇. 𝑑𝑡 + 𝜎. 𝑑𝑊 we should be able to recover the PDE for the PDF ¨ ;< (9,#) ;# = − ; ;9 𝜇. Π 𝑋, 𝑡 − ; ;9 =+ " . Π 𝑋, 𝑡 = =+ " . ;+< (9,#) ;9+ ¨ ;< (9,#) ;# = =+ " . ;+< (9,#) ;9+ ¨ Let’s rederive this directly from the tree and the master equation ¨ Π 𝑘 + 1, 𝑖 = ! " Π 𝑘, 𝑖 − 1 + Π 𝑘, 𝑖 ¨ Or more importantly ¨ Π 𝑋(𝑘 + 1, 𝑖) = ! " Π 𝑋(𝑘, 𝑖 − 1) + Π 𝑋(𝑘, 𝑖) 95
- 96. Luc_Faucheux_2020 The BDT dynamicsIX ¨ We have Π 𝑋(𝑘 + 1, 𝑖) = ! " Π 𝑋(𝑘, 𝑖 − 1) + Π 𝑋(𝑘, 𝑖) ¨ With: ¨ 𝑋 𝑘 + 1, 𝑖 = 𝑋 𝑘, 𝑖 − 𝛿𝑋/2 ¨ 𝑋 𝑘 + 1, 𝑖 + 1 = 𝑋 𝑘, 𝑖 + 𝛿𝑋/2 ¨ Taylor expanding in time and space the master equation around 𝑋(𝑘 + 1, 𝑖) which is the middle point (easier because we will perform symmetric calculations) 96 X(k,i) X(k+1,i) X(k,i-1) 𝛿𝑋/2 𝛿𝑋/2
- 97. Luc_Faucheux_2020 The BDT dynamicsX ¨ Π 𝑋(𝑘 + 1, 𝑖) = ! " Π 𝑋(𝑘, 𝑖 − 1) + Π 𝑋(𝑘, 𝑖) ¨ With ¨ 𝑋 𝑘 + 1, 𝑖 = 𝑋 𝑘, 𝑖 − 𝛿𝑋/2 ¨ 𝑋 𝑘 + 1, 𝑖 + 1 = 𝑋 𝑘, 𝑖 + 𝛿𝑋/2 ¨ Or ¨ 𝑋 𝑘, 𝑖 − 1 = 𝑋 𝑘 + 1, 𝑖 − 𝛿𝑋/2 ¨ 𝑋 𝑘, 𝑖 = 𝑋 𝑘 + 1, 𝑖 + 𝛿𝑋/2 ¨ Π 𝑋(𝑘 + 1, 𝑖) = Π 𝑡 + 𝛿𝑡, 𝑋(𝑘 + 1, 𝑖) = Π 𝑡, 𝑋(𝑘 + 1, 𝑖) + ;< #,9(12!,)) ;# . 𝛿𝑡 + 𝒪(𝛿𝑡") 97
- 98. Luc_Faucheux_2020 The BDT dynamicsXI ¨ 𝑋 𝑘, 𝑖 − 1 = 𝑋 𝑘 + 1, 𝑖 − 𝛿𝑋/2 ¨ 𝑋 𝑘, 𝑖 = 𝑋 𝑘 + 1, 𝑖 + 𝛿𝑋/2 ¨ Π 𝑋(𝑘, 𝑖 − 1) = Π 𝑡, 𝑋 𝑘 + 1, 𝑖 − 𝛿𝑋/2 ¨ Π 𝑋(𝑘, 𝑖 − 1) = Π 𝑡, 𝑋 𝑘 + 1, 𝑖 ) − (𝛿𝑋/2 . ;< #,9(12!,)) ;9 + ! " . 89+ : . ;+< #,9(12!,)) ;9+ + 𝒪(𝛿𝑋>) ¨ AND ¨ Π 𝑋(𝑘, 𝑖) = Π 𝑡, 𝑋 𝑘 + 1, 𝑖 + 𝛿𝑋/2 ¨ Π 𝑋(𝑘, 𝑖) = Π 𝑡, 𝑋 𝑘 + 1, 𝑖 ) + (𝛿𝑋/2 . ;< #,9(12!,)) ;9 + ! " . 89+ : . ;+< #,9(12!,)) ;9+ + 𝒪(𝛿𝑋>) ¨ Π 𝑋(𝑘 + 1, 𝑖) = ! " Π 𝑋(𝑘, 𝑖 − 1) + Π 𝑋(𝑘, 𝑖) 98
- 99. Luc_Faucheux_2020 The BDT dynamicsXII ¨ ! " Π 𝑋(𝑘, 𝑖 − 1) + Π 𝑋(𝑘, 𝑖) = Π 𝑡, 𝑋(𝑘 + 1, 𝑖) + ! " . 89+ : . ;+< #,9(12!,)) ;9+ + 𝒪(𝛿𝑋>) ¨ Π 𝑋(𝑘 + 1, 𝑖) = Π 𝑡, 𝑋(𝑘 + 1, 𝑖) + ;< #,9(12!,)) ;# . 𝛿𝑡 + 𝒪(𝛿𝑡") ¨ Π 𝑋(𝑘 + 1, 𝑖) = ! " Π 𝑋(𝑘, 𝑖 − 1) + Π 𝑋(𝑘, 𝑖) ¨ Yields: ¨ ;< #,9(12!,)) ;# . 𝛿𝑡 = ! " . 89+ : . ;+< #,9(12!,)) ;9+ ¨ Defining the volatility as (𝜎". 𝛿𝑡) = 89+ : ¨ ;< ;# = =+ " . ;+< ;9+ ¨ We recover the usual diffusion equation. 99
- 100. Luc_Faucheux_2020 100 The BDT meanreversion
- 101. Luc_Faucheux_2020 The BDT meanreversion I ¨ Everything we did so far assumed constant time step and spacing between nodes. ¨ We keep a constant time step ¨ Because of the algorithm construction of the BDT, there is a constant shift (up or down) in order to match the discount curve. ¨ This is just a drift on top of the diffusive process. ¨ More subtle is the mean reversion effect. Mean reversion is a crucial concept in finance. It indicates that things have a tendency when diffusing away from the equilibrium to ”revert back to the mean” or mean revert, hence the term “mean reversion” ¨ What is the intuition for it in the BDT model? ¨ Essentially at a given time indexed by k, there are k nodes. The spacing between the nodes is given by the volatility input. What happens when this spacing is not time-dependent? Meaning its value change from (k) to (k+1). We could end up in a situation like this one: 101
- 102. Luc_Faucheux_2020 The BDT meanreversion II ¨ Decreasing volatility input from (k) to (k+1). The tree gets compressed. Positive mean reversion (usual one) ¨ Note: in this case it is obvious that a BDT binomial model breaks the arbitrage locally, even though one average over the time slices the arbitrage is still respected ¨ The further away from the center the greater the pull-back : typical of mean reversion 102
- 103. Luc_Faucheux_2020 The BDT meanreversion III ¨ Increasing volatility input from (k) to (k+1). The tree explodes. Negative mean reversion (unusual one) ¨ Note: in this case it is obvious that a BDT binomial model breaks the arbitrage locally, even though one average over the time slices the arbitrage is still respected ¨ The further away from the center the greater the push-out : typical of negative mean reversion 103
- 104. Luc_Faucheux_2020 The BDT meanreversion IV ¨ It should not come as a surprise, because in some ways, in BDT for a given “splice” at a given point in time k, you just put k nodes equally spaced and then you just shift those up and down in order to recover the average discount factor, and then you connect those dots back to the previous time slice ¨ Local versus average arbitrage free ¨ BDT is said to be arbitrage free because when discounting back on the tree you recover the price of today’s discount curve. Note that it is only ON AVERAGE that the arbitrage is being respected. When there is mean reversion, locally at any point in the BDT tree the ”local” arbitrage is NOT being respected, and so the local dynamics of BDT (if we can figure it out” will NOT be arbitrage free (only in very limited case for example constant volatility). ¨ Another way to say it is that if you build a BDT tree out of any node in an existing BDT tree, this will NOT be a BDT tree, and will not respect arbitrage ¨ Let’s try to estimate how this impacts the dynamic equation. 104
- 105. Luc_Faucheux_2020 The BDT meanreversion V ¨ For k large enough the tree is centered around 𝑋 𝑘, 1 " = 𝑋?(𝑘) for middle ¨ For each node 𝑋 𝑘, 𝑖 = 𝑋? 𝑘 + 𝑖 − 1 " . 𝛿𝑋(𝑘) ¨ 𝑋 𝑘 + 1, 𝑖 = 𝑋? 𝑘 + 1 + 𝑖 − 12! " . 𝛿𝑋(𝑘 + 1) ¨ 𝑋 𝑘 + 1, 𝑖 + 1 = 𝑋? 𝑘 + 1 + 𝑖 + 1 − 12! " . 𝛿𝑋(𝑘 + 1) 105 X(k,i) X(k+1,i+1) X(k+1,i) 𝛿𝑋(𝑘 + 1)
- 106. Luc_Faucheux_2020 The BDT meanreversion VI ¨ Setting the drift of the overall tree to 0, 𝑋? 𝑘 + 1 = 𝑋? 𝑘 ¨ Note that we can always add a drift later constant for all nodes, we are trying to isolate the effect of the changing spacing between two time slices ¨ 𝑋 𝑘, 𝑖 = 𝑋? 𝑘 + 𝑖 − 1 " . 𝛿𝑋(𝑘) ¨ 𝑋 𝑘 + 1, 𝑖 = 𝑋? 𝑘 + 1 + 𝑖 − 12! " . 𝛿𝑋(𝑘 + 1) ¨ 𝑋 𝑘 + 1, 𝑖 + 1 = 𝑋? 𝑘 + 1 + 𝑖 + 1 − 12! " . 𝛿𝑋(𝑘 + 1) ¨ 𝛿𝑋 𝑘 + 1 = 𝛿𝑋 𝑘 + ;89 # ;# . 𝛿𝑡 + 𝒪(𝛿𝑡") ¨ < ∆𝑋 > = ! " . 𝑋 𝑘 + 1, 𝑖 + 1 + 𝑋 𝑘 + 1, 𝑖 106
- 107. Luc_Faucheux_2020 The BDT meanreversion VII ¨ < 𝑋 > = ! " . 𝑋 𝑘 + 1, 𝑖 + 1 + 𝑋 𝑘 + 1, 𝑖 ¨ < 𝑋 > = ! " . 𝑋? 𝑘 + 1 + 𝑖 + 1 − 12! " . 𝛿𝑋(𝑘 + 1) + ! " . 𝑋? 𝑘 + 1 + 𝑖 − 12! " . 𝛿𝑋(𝑘 + 1) ¨ < 𝑋 > = 𝑋? 𝑘 + 1 + ! " . { 𝑖 + 1 − 12! " + (𝑖 − 12! " )}. 𝛿𝑋(𝑘 + 1) ¨ < 𝑋 > = 𝑋? 𝑘 + 1 + 𝑖 − 1 " . 𝛿𝑋(𝑘 + 1) ¨ 𝑋 𝑘, 𝑖 = 𝑋? 𝑘 + 𝑖 − 1 " . 𝛿𝑋(𝑘) ¨ 𝑋? 𝑘 + 1 = 𝑋? 𝑘 ¨ < 𝑋 > = 𝑋? 𝑘 + 𝑖 − 1 " . 𝛿𝑋 𝑘 + 𝑖 − 1 " . ;89 # ;# . 𝛿𝑡 ¨ < 𝑋 > = 𝑋(𝑘, 𝑖) + 𝑖 − 1 " . ;89 # ;# . 𝛿𝑡 107
- 108. Luc_Faucheux_2020 The BDT meanreversion VIII ¨ Locally there is a drift term equal to 𝑖 − 1 " . ;89 # ;# . 𝛿𝑡 ¨ In the middle where 𝑖 = 1 " , this drift term is nul ¨ The farther we get away from the center, the greater this drift term (not equal for all nodes) ¨ The greater the derivative of the spacing with respect to time ;89 # ;# the greater the drift ¨ < 𝑋 > = 𝑋(𝑘, 𝑖) + 𝑖 − 1 " . ;89 # ;# . 𝛿𝑡 ¨ < ∆𝑋 > = 𝑖 − 1 " . ;89 # ;# . 𝛿𝑡 ¨ 𝑋 𝑘, 𝑖 = 𝑋? 𝑘 + 𝑖 − 1 " . 𝛿𝑋(𝑘) ¨ So we have < ∆𝑋 > = ( 9 1,) -9E(1) 89 # ). ;89 # ;# . 𝛿𝑡 108
- 109. Luc_Faucheux_2020 The BDT meanreversion IX ¨ < ∆𝑋 > = 9 1,) -9E 1 89 # . ;89 # ;# . 𝛿𝑡 = 𝑋 𝑘, 𝑖 − 𝑋? 𝑘 . ! 89 # . ;89 # ;# . 𝛿𝑡 ¨ For convenience sake, either we set 𝑋? 𝑘 = 0, or we define 𝑌 𝑘, 𝑖 = 𝑋 𝑘, 𝑖 − 𝑋? 𝑘 and work in the tree in 𝑌. ¨ If we set 𝑋? 𝑘 = 0 ¨ < ∆𝑋 > = 9 1,) 89 # . ;89 # ;# . 𝛿𝑡 = 𝑋 𝑘, 𝑖 . ! 89 # . ;89 # ;# . 𝛿𝑡 ¨ So when writing 𝑑𝑋 = 𝜇. 𝑑𝑡 + 𝜎. 𝑑𝑊 we would have ¨ 𝜇 = 𝑋. ! 89 # . ;89 # ;# ¨ We had previously (𝜎". 𝛿𝑡) = 89+ : , so looks like 𝜇 = 𝑋. ! = . ;= ;# , where 𝜎 = 𝜎(𝑡) ¨ So looks like 𝑑𝑋 = 𝑋. =@ = . 𝑑𝑡 + 𝜎. 𝑑𝑊 109
- 110. Luc_Faucheux_2020 The BDT meanreversion X ¨ We still need to check that the 𝜎 term in front of the Brownian driver is still the correct one that we obtain in the case of constant 𝜎 ¨ To do that we need to estimate: ¨ < ∆𝑋"> = ! " . (𝑋 𝑘 + 1, 𝑖 + 1 − < 𝑋 >)^(2) + (𝑋 𝑘 + 1, 𝑖 − < 𝑋 >)^(2) ¨ Where < 𝑋 > is the mid point between 𝑋 𝑘 + 1, 𝑖 + 1 and 𝑋 𝑘 + 1, 𝑖 ¨ < 𝑋 > = ! " . 𝑋 𝑘 + 1, 𝑖 + 1 + 𝑋 𝑘 + 1, 𝑖 ¨ And so < ∆𝑋"> = ! : . {(𝑋 𝑘 + 1, 𝑖 + 1 − 𝑋 𝑘 + 1, 𝑖 }"= ! : . {(𝛿𝑋 𝑘 + 1 }" ¨ This is the same as before (the intuition is that the drift does not pollute the volatility, because the drift is of order 1 in time, and so any drift when computing the second moment will appear as a term of order 2 in time, which will be neglected when compared to the stochastic term, which is only of order 1 in time 110
- 111. Luc_Faucheux_2020 The BDT meanreversion XI ¨ The numerical implementation of the BDT tree, by equally spacing k nodes at time k, and then forcing those nodes to connect in a recombining binomial manner, “artificially” creates a mean reversion drift ¨ The BDT dynamics is 𝑑𝑋 = 𝑋. =@ = . 𝑑𝑡 + 𝜎. 𝑑𝑊 ¨ At constant volatility, the drift term disappear ¨ I can bet some large amount of money that if we had started from the actual dynamics and being asked to build a tree that follows it we would have struggled quite a lot ¨ We went through this derivation to illustrate the potential pitfalls of going in between numerical tree implementation and continuous dynamics ¨ Note that usually BDT are built with a short term rate in lognormal space, 𝑋 = ln(𝑟) ¨ The dynamics then become 𝑑(ln 𝑟 ) = ln(𝑟). =@ = . 𝑑𝑡 + 𝜎. 𝑑𝑊 ¨ Note that the meaning of 𝜎 is now different, it is now the lognormal volatility 111
- 112. Luc_Faucheux_2020 The BDT meanreversion XII ¨ Note that as usual following the Ito convention (Ito lemma) in Ito calculus ¨ 𝑑 ln(𝑟) = ( ⁄𝑑𝑟 𝑟) − ( ⁄𝜎" 2). 𝑑𝑡 ¨ Let’s go back to 𝑑𝑋 = 𝑋. =@ = . 𝑑𝑡 + 𝜎. 𝑑𝑊 ¨ Also normal models are much more in favor now that rates have proven that they can go negative. ¨ The associated PDE for the PDF is given by ¨ ;< (9,#) ;# = − ; ;9 𝜇. Π 𝑋, 𝑡 − ; ;9 =+ " . Π 𝑋, 𝑡 ¨ ;< (9,#) ;# = − ; ;9 𝑋. =@ = . Π 𝑋, 𝑡 − ; ;9 =+ " . Π 𝑋, 𝑡 ¨ Note that 𝜎 is a function of the time t and not the stochastic variable, so we are safe from any Ito versus Stratonovitch controversy for now 112
- 113. Luc_Faucheux_2020 The BDT meanreversion XIII ¨ In the most general case with added drift (the multiplicative factors K that we used in the Excel implementation of the BDT) ¨ 𝑑𝑋 = {𝐾 𝑡 + 𝑋. =F # = # }. 𝑑𝑡 + 𝜎(𝑡). 𝑑𝑊 ¨ We can also split the drift term in order to make obvious the reversion to the mean 𝑋A(𝑡) ¨ 𝑑𝑋 = {𝐾 𝑡 + (𝑋 − 𝑋A 𝑡 ). =F # = # }. 𝑑𝑡 + 𝜎(𝑡). 𝑑𝑊 ¨ Or going into Lognormal space ¨ 𝑑(ln 𝑟 ) = {𝐾 𝑡 + (ln(𝑟) − ln(𝑟A 𝑡 )). =F # = # }. 𝑑𝑡 + 𝜎(𝑡). 𝑑𝑊 ¨ The usual textbook description of the BDT dynamics just goes with ¨ 𝑑 ln 𝑟 = {𝜃 𝑡 + ln(𝑟). =F = }. 𝑑𝑡 + 𝜎. 𝑑𝑊 ¨ With the drift 𝜃 𝑡 taking care of the arbitrage free constraint to recover the discounts 113
- 114. Luc_Faucheux_2020 The BDT meanreversion XIV ¨ Always be careful when dealing with numerical implementation of SDE ¨ In the example above, when having constant volatility, the dynamics is the one sometimes called the Ho-Lee: ¨ 𝑑𝑋 = 𝐾 𝑡 . 𝑑𝑡 + 𝜎. 𝑑𝑊 ¨ If 𝑋 is the short-rate, we have 𝐾 𝑡 = ;3(%,#) ;# + 𝜎". 𝑡 ¨ Where 𝑓(0, 𝑡) is the initial forward rate curve ¨ When 𝜎 becomes a function of time 𝜎(𝑡), we would be tempted to keep the tree that we built for Ho-Lee and just change the spacing at each time step in order to fit the volatility. ¨ What we showed is that doing this “blindly” introduces mean reversion and changes the dynamics (SDE) to: ¨ 𝑑𝑋 = {𝐾 𝑡 + 𝑋. =F # = # }. 𝑑𝑡 + 𝜎(𝑡). 𝑑𝑊 114
- 115. Luc_Faucheux_2020 115 Some notes onprobabilities and drift inside the tree
- 116. Luc_Faucheux_2020 A quick asideon probabilities ¨ Probability to go up 𝑃B, probability to go down 𝑃C, with 𝑃B + 𝑃C = 1 ¨ If both probability are equal to 1/2 ¨ < ∆𝑋 > = ! " . 𝑋 + 89 " + 𝑋 − 𝛿𝑋/2 − 𝑋 = 0 ¨ < ∆𝑋"> = ! " . ( 89 " )^(2) + ( 89 " )^(2) = ( 89+ : ) 116 X X + 𝛿𝑋/2 X - 𝛿𝑋/2 𝛿𝑋/2 𝛿𝑋/2 𝑃! 𝑃" 𝛿𝑡
- 117. Luc_Faucheux_2020 A quick asideon probabilities II ¨ < ∆𝑋 > = 0 ¨ < ∆𝑋"> = ( 89+ : ) ¨ We match the process 𝑑𝑋 = 0. 𝑑𝑡 + 𝜎. 𝑑𝑊 ¨ 𝜎". 𝛿𝑡 = ( 89+ : ) ¨ 𝛿𝑋 = 2𝜎. 𝛿𝑡 117
- 118. Luc_Faucheux_2020 Adding drift theeasy way (?) ¨ We keep the probabilities equal to (1/2) ¨ The only difference is the addition of the drift term 118 X 𝑃! 𝑃" 𝛿𝑡 𝑋 − #$ % + 𝑎. 𝛿𝑡 𝑋 + #$ % + 𝑎. 𝛿𝑡
- 119. Luc_Faucheux_2020 Adding drift theeasy way (?) II ¨ < ∆𝑋 > = ! " . 𝑋 + 89 " + 𝑎. 𝛿𝑡 + 𝑋 − 89 " + 𝑎. 𝛿𝑡 − 𝑋 = 𝑎. 𝛿𝑡 ¨ < ∆𝑋"> = ! " . ( 89 " + 𝑎. 𝛿𝑡)^(2) + (− 89 " + 𝑎. 𝛿𝑡)^(2) ¨ < ∆𝑋"> = (𝑎. 𝛿𝑡)"+( 89+ : ) ¨ We only keep the terms in order 1 in 𝛿𝑡 (small time steps limit) ¨ < ∆𝑋"> = ( 89+ : ) ¨ Note that locally, < ∆𝑋"> = ! " . ( 89 " )^(2) + ( 89 " )^(2) = ( 89+ : ) exactly ¨ We match the process 𝑑𝑋 = 𝑎. 𝑑𝑡 + 𝜎. 𝑑𝑊 119
- 120. Luc_Faucheux_2020 Changing the driftby changing the probabilities I ¨ We do not change the values in the space axis (the values for X have been picked already) ¨ We change the probabilities 𝑃B and 𝑃C ¨ This is quite common when building trees, as the node spacing has already been calibrated. ¨ This actually is a little more tricky to deal with (in particular when you change the probabilities, do you still recover a Gaussian for example? This is linked to the Girsanov theorem, just to tell you that this is not as simple as it seems) 120 X X + 𝛿𝑋/2 X - 𝛿𝑋/2 𝛿𝑋/2 𝛿𝑋/2 𝑃! 𝑃" 𝛿𝑡
- 121. Luc_Faucheux_2020 Changing the driftby changing the probabilities II ¨ We define the probabilities 𝑃B = ! " + 𝜀 and 𝑃C = ! " − 𝜀 ¨ < ∆𝑋 > = ! " + 𝜀 . 𝑋 + 89 " } + ! " − 𝜀 . {𝑋 − 89 " − 𝑋 = 𝜀. 𝛿𝑋 ¨ Now, 𝛿𝑋 scales as 𝛿𝑡!/", so we know that right off the bat, in order for the drift term to be linear in time (in order to model 𝑑𝑋 = 𝑎. 𝑑𝑡 + 𝜎. 𝑑𝑊), we are going to need to have that number 𝜀 to scale something like {𝑎. (𝛿𝑡)!/"} ¨ < ∆𝑋"> = ! " + 𝜀 . 89 " " + ! " − 𝜀 . 89 " " ¨ < ∆𝑋"> = ! " + 𝜀 . 89 " " + ! " − 𝜀 . 89 " " ¨ < ∆𝑋"> = 89+ : (1 + 2. 𝜀") 121
- 122. Luc_Faucheux_2020 Changing the driftby changing the probabilities III ¨ < ∆𝑋 > = 𝜀. 𝛿𝑋 = 𝑎. 𝛿𝑡 ¨ < ∆𝑋"> = 89+ : 1 + 2. 𝜀" = 𝜎". 𝛿𝑡 ¨ So 𝜀 = E.8# 89 and 𝛿𝑋 = 2𝜎. 𝛿𝑡!/" so we can also write 𝜀 = E "= 𝛿𝑡!/" ¨ So changing the probability does not affect the variance in the limit of small time steps (continuous limit). ¨ Not as straightforward as shifting the values in space and keeping the probabilities equal to (1/2) ¨ Changing the probabilities to affect the drift is NOT trivial (in particular we have to convince ourselves that we did not change the shape of the distribution, just the first moment) ¨ Girsanov theorem, Radon-Nikodym derivative if you want to use big words 122
- 123. Luc_Faucheux_2020 Changing the driftby changing the probabilities IV ¨ So we match the process: 𝑑𝑋 = 𝑎. 𝑑𝑡 + 𝜎. 𝑑𝑊 ¨ 𝛿𝑋 = 2𝜎. 𝛿𝑡!/" ¨ 𝜀 = E "= 𝛿𝑡!/" ¨ WAIT A MINUTE you should say, 𝜀 is only a number, not a complicated formula: ¨ 𝜀 = E "= 𝛿𝑡!/" ¨ 𝑎 is a drift and so scales as [ 9 # ] ¨ 𝜎 is a volatility and so scales as [ 9 #!/+], or easier 𝜎". [𝑡]scales as [𝑋"] ¨ SO.. 𝜀 scales as [ 9 # ]. # ! + 9 . [𝑡!/"] which is dimensionless and a number indeed !! 123
- 124. Luc_Faucheux_2020 124 Some tricks wecan play in BDT trees
- 125. Luc_Faucheux_2020 Adding some moremean reversion to the BDT tree ¨ Market practice is to do the following: – Build the tree (if needed to save memory, build also some “ghost nodes”) – Start with a given spacing in rates – Adjust the multiplication factor to match the discount curve – Adjust the spacing in order to match at-the-money caplet prices (this take into account the numerical noise aspect, and also the bias in the BDT discounting, which needs to be taken into account by adjusting the vol) – Recalibrate the multiplication factors to match the discount curve – Iterate a couple of times if needed – NOW comes the kicker: you will NOT recover the price of at-the-money European swaptions, NOR will you recover the price of callable options (Formosa), there are a lot of reasons why – The most used “trick” is to add another mean reversion in the BDT tree on top of the “natural” mean reversion : { .% . } 125
- 126. Luc_Faucheux_2020 Adding some moremean reversion to the BDT tree-b ¨ This is in practice a very important point: ¨ Models do get calibrated to a finite subset of instruments. ¨ For reasons that we went over in the Structured Powerpoint, a simple model will NEVER be able to capture the right price (and as a result risk and hedges) for caps and callables, or European swaptions ¨ A true model that should capture the callable prices should be able to incorporate the correlation between the curve steepness as stochastic driver, and the first factor in the volatility surface ¨ Such a model is quite a task to build (no one in the market currently has one) ¨ But as we also showed in the Skew Powerpoint, it is quite legit to use a simpler model, run it through a hedging scenario and capture the residual as the difference in pricing ¨ So you could say, why even bother with mean reversion? Take your model as is, and then run it through your scenario hedging 126
- 127. Luc_Faucheux_2020 Adding some moremean reversion to the BDT tree-c ¨ The answer to that is, yes you could do that, and that would be fine ¨ Introducing mean reversion to the current model brings it closer to the market price. ¨ So it might not be the actual dynamics, but at least you are starting closer to where you want to end, so any Taylor expansion / approximation will be smaller and more justified ¨ Also, even though we are a big proponent of the Scenario hedging, this is a class after all, so playing with the mean reversion is a nice way to introduce some of the concepts around tree pricing 127
- 128. Luc_Faucheux_2020 Adding some moremean reversion to the BDT tree II ¨ Let’s go through the derivation of that “trick” (not the only one but a rather popular one) ¨ More to illustrate the tree dynamics, and how a change in the algorithm affects the underlying dynamics, and how one needs to be careful about the small details ¨ In textbooks, sometimes you read that you need a trinomial tree in order to have an extra degree of freedom in order to accommodate mean reversion. ¨ I have never fully understood why that is the case, and what a trinomial tree has that a two- step binomial tree cannot exhibit ¨ We will also present a trinomial tree implementation as it is a neat example of something crucial when dealing with trees (or sometimes trees on a grid) : stability analysis and constraints on the discrete parameters 128
- 129. Luc_Faucheux_2020 Adding some moremean reversion to the BDT tree III ¨ 𝑋 𝑘, 𝑖 = 𝑋? 𝑘 + 𝑖 − 1 " . 𝛿𝑋(𝑘) ¨ 𝑋 𝑘, 1 " = 𝑋?(𝑘) ¨ Once the volatility structure 𝛿𝑋(𝑘) is given, there is no choice on the mean reversion parameter that is given by { =F = } ¨ Π 𝑋(𝑘 + 1, 𝑖) = ! " Π 𝑋(𝑘, 𝑖 − 1) + Π 𝑋(𝑘, 𝑖) 129 X(k,i) X(k+1,i+1) X(k+1,i) 𝛿𝑋/2 𝛿𝑋/2
- 130. Luc_Faucheux_2020 Adding some moremean reversion to the BDT tree IV ¨ Π 𝑋(𝑘, 𝑖) follows the binomial distribution, in the continuous limit we recover the Gaussian distribution and the diffusion equation. ¨ Π 𝑋(𝑘 + 1, 𝑖) = ! " Π 𝑋(𝑘, 𝑖 − 1) + Π 𝑋(𝑘, 𝑖) ¨ Π 𝑋(𝑘, 𝑖) = 2-1. ) 1 = ! "& . 1! )! 1-) ! ¨ Π 𝑋(𝑘, 𝑖 − 1) = 2-1. )-! 1 = ! "& . 1! ()-!)! 1-)2! ! ¨ We have : )-! 1 + ) 1 = ) 12! ¨ So by recurrence we also have Π 𝑋(𝑘 + 1, 𝑖) = 2-12!. ) 12! = ! "&H! . (12!)! )! 12!-) ! 130
- 131. Luc_Faucheux_2020 Adding some moremean reversion to the BDT tree V ¨ One introduces mean reversion in an algorithmic fashion in the following way: ¨ Usual tree: ¨ 𝑋 𝑘, 𝑖 → 𝑋 𝑘 + 1, 𝑖 with probability 𝑃C = 1/2 ¨ 𝑋 𝑘, 𝑖 → 𝑋 𝑘 + 1, 𝑖 + 1 with probability 𝑃B = 1/2 ¨ We introduce 𝜇 (note that a time dependent 𝜇(𝑡) is completely analogous, but for sake of simplicity we will drop the time dependence for now ¨ 𝑋 𝑘, 𝑖 → 𝑋 𝑘 + 1, 𝑖 with probability 𝑃C = ! " . (1 − 𝜇) ¨ 𝑋 𝑘, 𝑖 → 𝑋 𝑘 + 1, 𝑖 + 1 with probability 𝑃B = ! " . (1 − 𝜇) ¨ 𝑋 𝑘, 𝑖 → 𝑋 𝑘 + 1, 𝑗 with probability 𝑃 1,) →(12!,*) = 𝜇. Π 𝑋(𝑘 + 1, 𝑗) 131
- 132. Luc_Faucheux_2020 Adding some moremean reversion to the BDT tree VI ¨ The functional 𝑃 1,) →(12!,*) = 𝜇. Π 𝑋(𝑘 + 1, 𝑗) is chosen so that the probabilities on the tree are intact ¨ We do not change the variance, nor the shape of the distribution for a given time section in the BDT tree ¨ We only change the probability of jumping from one node back to another across time sections ¨ This will work either when jumping forward in time in the BDT tree (when computing things like Asian options that are path dependent, or for example the Arrow-Debreu prices of securities), but also when jumping backward in the BDT tree, when discounting back American options or pricing callable options. 132
- 133. Luc_Faucheux_2020 Adding some moremean reversion to the BDT tree VI-a ¨ Note that we do not lose any probability density ¨ 𝑋 𝑘, 𝑖 → 𝑋 𝑘 + 1, 𝑖 with probability 𝑃C = ! " . (1 − 𝜇) ¨ 𝑋 𝑘, 𝑖 → 𝑋 𝑘 + 1, 𝑖 + 1 with probability 𝑃B = ! " . (1 − 𝜇) ¨ 𝑋 𝑘, 𝑖 → 𝑋 𝑘 + 1, 𝑗 with probability 𝑃 1,) →(12!,*) = 𝜇. Π 𝑋(𝑘 + 1, 𝑗) ¨ So out of the node 𝑋 𝑘, 𝑖 we will transfer to the next time slice 𝑘 + 1 a total: ¨ 𝑃 = ! " . 1 − 𝜇 + ! " . 1 − 𝜇 + 𝜇. ∑*$% *$12! Π 𝑋 𝑘 + 1, 𝑗 ¨ And since ∑*$% *$12! Π 𝑋 𝑘 + 1, 𝑗 = 1 we have 𝑃 = 1 = 100% ¨ So 100% of the probability density in the node 𝑋 𝑘, 𝑖 gets transferred to the next time slice, we are not losing any probability density (note that we assumed that the sum of the next tree level probabilities add up to one, so a little self consistent) 133
- 134. Luc_Faucheux_2020 Adding some moremean reversion to the BDT tree VII ¨ The old master equation was: Π 𝑋(𝑘 + 1, 𝑖) = ! " Π 𝑋(𝑘, 𝑖 − 1) + Π 𝑋(𝑘, 𝑖) ¨ The new master equation is a little more complicated at first: ¨ So just to be rigorous, we will note Π′ 𝑋(𝑘 + 1, 𝑖) the new probability distribution ¨ Π 𝑋(𝑘 + 1, 𝑖) is the baseline or the 𝜇 = 0 solution ¨ Π 𝑋(𝑘 + 1, 𝑖) is the usual binomial distribution ¨ Π 𝑋(𝑘 + 1, 𝑖) = 2-12!. ) 12! = ! "&H! . (12!)! )! 12!-) ! 134
- 135. Luc_Faucheux_2020 Adding some moremean reversion to the BDT tree VII-b ¨ The new Master Equation now becomes: ¨ Π′ 𝑋(𝑘 + 1, 𝑖) = ! " 1 − 𝜇 . Π′ 𝑋 𝑘, 𝑖 − 1 + Π′ 𝑋(𝑘, 𝑖) + 𝜇. ∑* Π′ 𝑋 𝑘, 𝑗 . Π′( ) 𝑋(𝑘 + 1, 𝑖) ¨ We can use a recurrence argument here: ¨ Π@ 𝑋 0,0 = Π 𝑋 0,0 = 1 ¨ Π@ 𝑋 1,0 = ! " 1 − 𝜇 . Π 𝑋 0,0 + 𝜇. Π 𝑋(1,0) = ! " 1 − 𝜇 . 1 + 𝜇. ! " = ! " = Π 𝑋 1,0 ¨ Π@ 𝑋 1,1 = ! " 1 − 𝜇 . Π 𝑋 0,0 + 𝜇. Π 𝑋(1,1) = ! " 1 − 𝜇 . 1 + 𝜇. ! " = ! " = Π 𝑋 1,1 135
- 136. Luc_Faucheux_2020 Adding some moremean reversion to the BDT tree VII-c ¨ We now go from 𝑘 → (𝑘 + 1) ¨ We assume that Π@ 𝑋 𝑘, 𝑗 = Π 𝑋(𝑘, 𝑗) ¨ The master equation is ¨ Π′ 𝑋(𝑘 + 1, 𝑖) = !-I " Π′ 𝑋 𝑘, 𝑖 − 1 + Π′ 𝑋(𝑘, 𝑖) + 𝜇 ∑* Π′ 𝑋 𝑘, 𝑗 Π 𝑋(𝑘 + 1, 𝑖) ¨ Π′ 𝑋(𝑘 + 1, 𝑖) = !-I " Π 𝑋 𝑘, 𝑖 − 1 + Π 𝑋(𝑘, 𝑖) + 𝜇 ∑* Π 𝑋 𝑘, 𝑗 Π 𝑋(𝑘 + 1, 𝑖) ¨ Since ∑* Π 𝑋(𝑘, 𝑗) = 1 ¨ Π@ 𝑋 𝑘 + 1, 𝑖 = !-I " . Π 𝑋 𝑘, 𝑖 − 1 + Π 𝑋 𝑘, 𝑖 + 𝜇. Π 𝑋(𝑘 + 1, 𝑖) 136
- 137. Luc_Faucheux_2020 Adding some moremean reversion to the BDT tree VII-d ¨ Π@ 𝑋 𝑘 + 1, 𝑖 = !-I " . Π 𝑋 𝑘, 𝑖 − 1 + Π 𝑋 𝑘, 𝑖 + 𝜇. Π 𝑋(𝑘 + 1, 𝑖) ¨ Now note that if Π@ 𝑋 𝑘 + 1, 𝑖 = Π 𝑋(𝑘 + 1, 𝑖) ¨ The Master Equation now would become ¨ Or: Π′ 𝑋 𝑘 + 1, 𝑖 . (1 − 𝜇) = ! " 1 − 𝜇 . Π 𝑋(𝑘, 𝑖 − 1) + Π 𝑋(𝑘, 𝑖) ¨ Dividing both sides by the 1 − 𝜇 term we get: ¨ Π@ 𝑋 𝑘 + 1, 𝑖 = ! " Π 𝑋 𝑘, 𝑖 − 1 + Π 𝑋 𝑘, 𝑖 = Π 𝑋(𝑘 + 1, 𝑖) ¨ This is the same master equation, so probabilities are conserved, and the distribution is also conserved !! In particular, we also have : ∑*$% *$12! Π 𝑋 𝑘 + 1, 𝑗 = 1 137
- 138. Luc_Faucheux_2020 Adding some moremean reversion to the BDT tree VII-e ¨ Without guessing so much: ¨ Π@ 𝑋 𝑘 + 1, 𝑖 = !-I " . Π 𝑋 𝑘, 𝑖 − 1 + Π 𝑋 𝑘, 𝑖 + 𝜇. Π 𝑋(𝑘 + 1, 𝑖) ¨ And the binomial Master Equation ensures that ¨ !-I " . Π 𝑋 𝑘, 𝑖 − 1 + Π 𝑋 𝑘, 𝑖 = 1 − 𝜇 . Π 𝑋(𝑘 + 1, 𝑖) ¨ And so : ¨ Π@ 𝑋 𝑘 + 1, 𝑖 = 1 − 𝜇 . Π 𝑋(𝑘 + 1, 𝑖) + 𝜇. Π 𝑋(𝑘 + 1, 𝑖) = Π 𝑋(𝑘 + 1, 𝑖) ¨ Sounds pretty obvious, but our little trick did not change the probability distribution, and so it will not affect the price of caplets, and swaps, or any derivatives that can be discounted without any path dependency ¨ Swaptions and callables will change in pricing as it will have to be discounted back along the tree (say it another way, some of the State Prices will be affected) 138
- 139. Luc_Faucheux_2020 Adding some moremean reversion to the BDT tree VIII ¨ So numerically we can observe that this will change the price of callable and swaptions ¨ Will it affect the price of caplets? ¨ Can we write down the new dynamics for the BDT with this new parameter? ¨ Can we even write an equation for the continuous process (SDE) ? ¨ Let’s do the same exercise we did in order to get the new dynamics (if we can) 139
- 140. Luc_Faucheux_2020 Adding some moremean reversion to the BDT tree IX ¨ For k large enough the tree is centered around 𝑋 𝑘, 1 " = 𝑋?(𝑘) for middle ¨ For each node 𝑋 𝑘, 𝑖 = 𝑋? 𝑘 + 𝑖 − 1 " . 𝛿𝑋(𝑘) ¨ 𝑋 𝑘 + 1, 𝑖 = 𝑋? 𝑘 + 1 + 𝑖 − 12! " . 𝛿𝑋(𝑘 + 1) ¨ 𝑋 𝑘 + 1, 𝑖 + 1 = 𝑋? 𝑘 + 1 + 𝑖 + 1 − 12! " . 𝛿𝑋(𝑘 + 1) 140 X(k,i) X(k+1,i+1) X(k+1,i) 𝛿𝑋(𝑘 + 1)
- 141. Luc_Faucheux_2020 Adding some moremean reversion to the BDT tree X ¨ The big difference is that NOW the probabilities have changed to : ¨ 𝑋 𝑘, 𝑖 → 𝑋 𝑘 + 1, 𝑖 with probability 𝑃C = ! " . (1 − 𝜇) ¨ 𝑋 𝑘, 𝑖 → 𝑋 𝑘 + 1, 𝑖 + 1 with probability 𝑃B = ! " . (1 − 𝜇) ¨ 𝑋 𝑘, 𝑖 → 𝑋 𝑘 + 1, 𝑗 with probability 𝑃 1,) →(12!,*) = 𝜇. Π 𝑋(𝑘 + 1, 𝑗) ¨ Where before we had : ¨ < 𝑋 > =< 𝑋12!|𝑋(𝑘, 𝑖) > = ! " . 𝑋 𝑘 + 1, 𝑖 + 1 + 𝑋 𝑘 + 1, 𝑖 ¨ We now have to sum over all the possible nodes in the tree at time (𝑘 + 1) ¨ < 𝑋12!|𝑋(𝑘, 𝑖) > = ∑*$% *$12! 𝑋 𝑘 + 1, 𝑗 . 𝑃 1,) →(12!,*) 141
- 142. Luc_Faucheux_2020 Adding some moremean reversion to the BDT tree XI ¨ We can make things easier for us and work in the limit of (𝜇 → 0) ¨ We know that when (𝜇 = 0) we have: ¨ < ∆𝑋(𝜇 = 0) > = 9 1,) -9E 1 89 # . ;89 # ;# . 𝛿𝑡 = 𝑋 𝑘, 𝑖 − 𝑋? 𝑘 . ! 89 # . ;89 # ;# . 𝛿𝑡 ¨ For convenience sake, either we set 𝑋? 𝑘 = 0, or we define 𝑌 𝑘, 𝑖 = 𝑋 𝑘, 𝑖 − 𝑋? 𝑘 and work in the tree in 𝑌. ¨ If we set 𝑋? 𝑘 = 0 (we can always add a constant tree drift after) ¨ < ∆𝑋(𝜇 = 0) > = 9 1,) 89 # . ;89 # ;# . 𝛿𝑡 = 𝑋 𝑘, 𝑖 . ! 89 # . ;89 # ;# . 𝛿𝑡 142
- 143. Luc_Faucheux_2020 Adding some moremean reversion to the BDT tree XII ¨ In the limit of small 𝜇: ¨ < ∆𝑋 > =< 𝑋12!|𝑋(𝑘, 𝑖) > − 𝑋(𝑘, 𝑖) ¨ < 𝑋 > =< 𝑋 𝜇 = 0 >. 1 − 𝜇 + 𝜇. ∑*$% *$12! 𝑋 𝑘 + 1, 𝑗 . Π 𝑋(𝑘 + 1, 𝑗) ¨ < ∆𝑋 > = < ∆𝑋 𝜇 = 0 >. 1 − 𝜇 + 𝜇. 𝛿 < ∆𝑋 > ¨ With 𝛿 < ∆𝑋 > = ∑*$% *$12! {𝑋 𝑘 + 1, 𝑗 − 𝑋 𝑘, 𝑖 }. Π 𝑋(𝑘 + 1, 𝑗) ¨ Or again : ¨ 𝛿 < ∆𝑋 > = [∑*$% *$12! 𝑋 𝑘 + 1, 𝑗 . Π 𝑋 𝑘 + 1, 𝑗 − 𝑋 𝑘, 𝑖 ] ¨ Since: ∑*$% *$12! 𝑋 𝑘, 𝑖 . Π 𝑋 𝑘 + 1, 𝑗 = 𝑋 𝑘, 𝑖 . ∑*$% *$12! Π 𝑋 𝑘 + 1, 𝑗 = 𝑋 𝑘, 𝑖 143
- 144. Luc_Faucheux_2020 Adding some moremean reversion to the BDT tree XIII ¨ We now have: ¨ < ∆𝑋 > = < ∆𝑋 𝜇 = 0 >. 1 − 𝜇 + 𝜇. 𝛿 < ∆𝑋 > ¨ 𝛿 < ∆𝑋 > = [∑*$% *$12! 𝑋 𝑘 + 1, 𝑗 . Π 𝑋 𝑘 + 1, 𝑗 − 𝑋 𝑘, 𝑖 ] ¨ 𝛿 < ∆𝑋 > = [𝑋?(𝑘 + 1) − 𝑋 𝑘, 𝑖 ] ¨ Remember that: ¨ 𝑋? 𝑘 + 1 = ∑*$% *$12! 𝑋 𝑘 + 1, 𝑗 . Π 𝑋 𝑘 + 1, 𝑗 ¨ In the limit of a dense tree 𝑋? 𝑘 = 𝑋(𝑘, 1 " ) ¨ And for each node 𝑋 𝑘, 𝑖 = 𝑋? 𝑘 + 𝑖 − 1 " . 𝛿𝑋(𝑘) 144
- 145. Luc_Faucheux_2020 Adding some moremean reversion to the BDT tree XIV ¨ < ∆𝑋 > = < ∆𝑋 𝜇 = 0 >. 1 − 𝜇 + 𝜇. [𝑋?(𝑘 + 1) − 𝑋 𝑘, 𝑖 ] ¨ Again we take the liberty of setting 𝑋? 𝑘 + 1 = 𝑋? 𝑘 = 0 ¨ (as we can always impose a constant drift later) ¨ And so : < ∆𝑋 > = 𝑋 𝑘, 𝑖 . ! 89 # . ;89 # ;# . 𝛿𝑡. 1 − 𝜇 − 𝜇. 𝑋 𝑘, 𝑖 ¨ This looks almost like what we are after ¨ We need to check that the units are correct and consistent between the two terms, in particular we look to be missing a 𝛿𝑡 factor in the second term ¨ Also we worked in the limit (𝜇 → 0) but we know that the BDT construction is algorithmic in nature, so we are missing the cross-terms and higher order terms, we will need to be conscious of that ¨ On the other hand we are trying to get the SDE, which is continuous limit, and not that helpful, because we are using BDT anyways, and not looking for closed form solutions 145
- 146. Luc_Faucheux_2020 Adding some moremean reversion to the BDT tree XV ¨ < ∆𝑋 > = 𝑋 𝑘, 𝑖 . ! 89 # . ;89 # ;# . 𝛿𝑡. 1 − 𝜇 − 𝜇. 𝑋 𝑘, 𝑖 ¨ We have (𝜎". 𝛿𝑡) = 89+ : , so ! 89 # . ;89 # ;# = ! = . ;= ;# = =@ = ¨ For 𝜇 = 0 , the BDT dynamics is 𝑑𝑋 = 𝑋. =@ = . 𝑑𝑡 + 𝜎. 𝑑𝑊 ¨ In the continuous limit the binomial distribution Π 𝑋(𝑘, 𝑖) = 2-1. ) 1 = ! "& . 1! )! 1-) ! converges to the Gaussian ℎ 𝑋, 𝑡 = ! "J=+# . exp( -9+ "=+# ) 146
- 147. Luc_Faucheux_2020 Adding some moremean reversion to the BDT tree XVI ¨ We still need to calculate the standard deviation ¨ We could say the following: our trick did not change the probability distribution and it did not change the spacing. ¨ And so the variance of the distribution for a given “slice” (all nodes at the same time) has not changed ¨ This is not completely correct ¨ Because for the dynamics what we care about is the LOCAL variance (for the process) ¨ This is quite an important and sometimes concept in trees implementation , where people sometimes get confused between local and average concept ¨ For example a BDT tree respects the arbitrage free constraints ON AVERAGE (averaging over a slice) ¨ But the BDT tree will NOT respect the arbitrage free constraints locally (as a matter of fact the very existence of mean reversion in the SDE proves it) 147
- 148. Luc_Faucheux_2020 Adding some moremean reversion to the BDT tree XVII ¨ Why is the binomial distribution converging to a Gaussian ¨ Taylor expansion of the Master equation (Bachelier thesis) ¨ Stirling approximation (Bachelier thesis once again) 148
- 149. Luc_Faucheux_2020 149 An example ofstability in the tree
- 150. Luc_Faucheux_2020 Trinomial trees –an example of stability constraint I ¨ Again, not quite sure why that is so different from a two-step binomial, by cutting the time step by half (remember that 𝛿𝑋 = 2𝜎. 𝛿𝑡) 150 X(k,i) X(k+1,i+1) X(k+1,i-1) 𝛿𝑋/2 𝛿𝑋/2 X(k+1,i) X(k,i) X(k+2,i+1) X(k+2,i-1) 𝛿𝑋/2 X(k+2,i) 𝛿𝑋/2
- 151. Luc_Faucheux_2020 Trinomial trees –an example of stability constraint II ¨ We saw in the binomial that in order to match the process 𝑑𝑋 = 0. 𝑑𝑡 + 𝜎. 𝑑𝑊 ¨ 𝜎". 𝛿𝑡 = ( 89+ : ) ¨ 𝛿𝑋 = 2𝜎. 𝛿𝑡 ¨ So we just have to be a little careful, if we divide the time steps in half, we should only adjust the space step (vertical distance or spacing between nodes) by the square root of it. ¨ And now adding mean reversion in that binomial tree is also a little tricky ¨ But again, barring the fact that trinomial trees are maybe easier to use (and also easier to put on a grid from the usual PDE numerical methods like Explicit / Implicit / Crank- Nicholson), I am still not convinced that trinomial trees present a true qualitative difference from binomial trees. ¨ But again I have been known to be wrong before, and would love to be challenged on that issue 151
- 152. Luc_Faucheux_2020 Trinomial trees –an example of stability constraint III ¨ We are going to model the process 𝑑𝑋 = 𝛼. 𝑑𝑡 + 𝜎. 𝑑𝑊 ¨ We will then extend to something like 𝛼 = 𝛼 𝑋, 𝑡 = 𝜃 𝑡 − 𝑘 𝑡 . 𝑋 with mean reversion ¨ The trinomial trees usually are “on a grid”, meaning that the spacing is kept constant and the probabilities are adjusted to reflect the drift, the mean reversion if any, and the volatility ¨ This is useful as the same grid can be used for the PDE solving (implicit and explicit, Crank- Nicholson,..) for the diffusion equation 152 X(k,i) X(k+1,i+1) X(k+1,i-1) 𝛿𝑋/2 𝛿𝑋/2 X(k+1,i) 𝑃& 𝑃' 𝑃( 𝛿𝑡
- 153. Luc_Faucheux_2020 Trinomial trees –an example of stability constraint IV ¨ We have the 3 unknowns 𝑃K, 𝑃A, 𝑃L ¨ We have the 3 equations : ¨ 𝑃K + 𝑃A + 𝑃L = 1 ¨ < ∆𝑋 > = 𝑃K. − 89 " + 𝑃A. 0 + 𝑃L. ( 89 " ) = 𝛼. 𝛿𝑡 ¨ < ∆𝑋"> = 𝑃K. 89 " " + 𝑃A. 0 + 𝑃L. 89 " " = (𝛼. 𝛿𝑡)"+𝜎". 𝛿𝑡 ¨ We have 3 equations, we have 3 unknowns, we have a fighting chance 153
- 154. Luc_Faucheux_2020 Trinomial trees –an example of stability constraint V ¨ 𝑃K + 𝑃A + 𝑃L = 1 ¨ < ∆𝑋 > = (𝑃L − 𝑃K). 89 " = 𝛼. 𝛿𝑡 ¨ < ∆𝑋"> = (𝑃L + 𝑃K). 89 " " = (𝛼. 𝛿𝑡)"+𝜎". 𝛿𝑡 ¨ Or again : ¨ 𝑃K + 𝑃A + 𝑃L = 1 ¨ (𝑃L − 𝑃K) = "M.8# 89 ¨ (𝑃L + 𝑃K) = {(𝛼. 𝛿𝑡)"+𝜎". 𝛿𝑡}. 89 " -" 154
- 155. Luc_Faucheux_2020 Trinomial trees –an example of stability constraint VI ¨ The solution is then: ¨ 𝑃A = 1 − {(𝛼. 𝛿𝑡)"+𝜎". 𝛿𝑡}. 89 " -" ¨ 𝑃L = ! " . [ "M.8# 89 + {(𝛼. 𝛿𝑡)"+𝜎". 𝛿𝑡}. 89 " -" ] ¨ 𝑃K = ! " . [− "M.8# 89 + {(𝛼. 𝛿𝑡)"+𝜎". 𝛿𝑡}. 89 " -" ] ¨ Note that it would seem sensible that all probabilities should be in [0,1] ¨ A negative probability in the tree is not necessarily a sign that something is wrong, but it is a constraint that we would like to enforce (related to stability analysis) 155
- 156. Luc_Faucheux_2020 Trinomial trees –an example of stability constraint VII ¨ In the simple case where 𝛼 = 0 ¨ 𝑃A = 1 − 𝜎". 𝛿𝑡. 89 " -" ¨ 𝑃L = ! " . 𝜎". 𝛿𝑡. 89 " -" = 𝑃K ¨ 𝑃K = ! " . 𝜎". 𝛿𝑡. 89 " -" = 𝑃L ¨ And we have: 𝑃A > 0 implies 1 − 𝜎". 𝛿𝑡. 89 " -" > 0 or 𝛿𝑋 > 2𝜎 𝛿𝑡 ¨ So if we chose a spacing in 𝛿𝑋 smaller than (2𝜎 𝛿𝑡) we will create a negative probability for the middle jump 𝑃A (meaning the phase spacing is too small, and so the probabilities will push for more on the wings and will deplete the center in order to still verify the variance or second moment 156
- 157. Luc_Faucheux_2020 Trinomial trees –an example of stability constraint VIII ¨ Note that this does not mean that the probability density at the node 𝑋(𝑘 + 1, 𝑖) will be negative, but it will be depleted relative to the wings 𝑋(𝑘 + 1, 𝑖 + 1) and 𝑋(𝑘 + 1, 𝑖 − 1) ¨ If the spacing is really too small we can run into ¨ 𝑃L = ! " . 𝜎". 𝛿𝑡. 89 " -" = 𝑃K > 1 when 𝛿𝑋 < 𝜎 2𝛿𝑡 ¨ Again, mathematically there is nothing wrong there yet, it is just a little weird to have negative jump probabilities. Also a little hard to see what happens with more than one level ¨ But one can see how that effect will propagate along the tree and create a situation where only the 2 nodes on the wings have infinite positive probability and every other nodes in between will have negative infinite probability. This will also violate the added constraint that the probability distribution has to follow the heat equation and converges to a Gaussian. ¨ This is essentially what stability analysis is all about : the propagation of small errors, does that grow exponentially and diverges or is it “under control” ? 157