Phase Transition in a Log-Normal Interest Rate Model

Motivation Black, Derman, Toy model Model with log-normal rates in the terminal measure Phase Transition

Phase Transition in a Log-normal Interest Rate Model

Dan Pirjol1

1J. P. Morgan, New York

17 Oct. 2011

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Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model Motivation Black, Derman, Toy model Model with log-normal rates in the terminal measure Phase Transition Outline

◮ Introduction to interest rate modeling ◮ Black-Derman-Toy model ◮ Generalization with continuous state variable ◮ Binomial tree ◮ BDT model with log-normal short rate in the terminal measure ◮ Analytical solution ◮ Surprising large volatility behaviour ◮ Phase transition ◮ Summary and conclusions

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Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model Motivation Black, Derman, Toy model Model with log-normal rates in the terminal measure Phase Transition [1] F. Black, E. Derman, W. Toy A One-Factor Model of Interest Rates Fin. Analysts Journal, 24-32 (1990). [2] P. Hunt, J. Kennedy and A. Pelsser, Markov-functional interest rate models Finance and Stochastics, 4, 391-408 (2000) [3] A. Pelsser Eﬃcient methods for valuing interest rate derivatives Springer Verlag, 2000. [4] D. Pirjol, Phase transition in a log-normal Markov functional model J. Math. Phys 52, 013301 (2011). [5] D. Pirjol, Nonanalytic behaviour in a log-normal Markov functional model, arXiv:1104.0322, 2011. fsu-logo

Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model Motivation Black, Derman, Toy model Model with log-normal rates in the terminal measure Phase Transition Interest rates

Interest rates are a measure of the time value of money: what is the value today of $1 paid in the future? 0.04 0.035 USD Zero Curve 0.03 27-Sep-2011 0.025 0.02 0.015 0.01 0.005

0 5 10 15 20 25

Example fsu-logo Zero curve R(t) for USD as of 27-Sep-2011. Gives the discount curve as D(t) = exp( R(t)t). − Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model Motivation Black, Derman, Toy model Model with log-normal rates in the terminal measure Phase Transition

Interest rates evolve in time 0.05 USD Zero Curve 0.04 22-Jun to 27-Sep

0.03

0.02

0.01

0 5 10 15 20 25

Example

The range of movement for the USD zero curve R(t) between fsu-logo 22-Jun-2011 and 27-Sep-2011.

Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model Motivation Black, Derman, Toy model Model with log-normal rates in the terminal measure Phase Transition

Yield curve inversion 6

5.75 USD Zero Rate

5.5 1-Nov-07

5.25

4.75

4.5

4.25

0 5 10 15 20 25 30

Example

Between 2006 and 2008 the USD yield curve was ‘inverted’ - rates for 2 fsu-logo years were lower than the rates for shorter maturities

Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model Motivation Black, Derman, Toy model Model with log-normal rates in the terminal measure Phase Transition Interest rate modeling

◮ The need to hedge against movements of the interest rates contributed to the creation of interest rate derivatives. ◮ Corporations, banks, hedge funds can now enter into many types of contracts aiming to mitigate or exploit/leverage the eﬀects of the interest rate movements ◮ Well-developed market. Daily turnover for1: ◮ interest rate swaps $295bn ◮ forward rate agreements $250bn ◮ interest rate options (caps/ﬂoors, swaptions) $70bn ◮ This requires a very good understanding of the dynamics of the interest rates markets: interest rate models

fsu-logo 1The FX and IR Derivatives Markets: Turnover in the US, 2010, Federal Reserve Bank of New York Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model Motivation Black, Derman, Toy model Model with log-normal rates in the terminal measure Phase Transition Setup and deﬁnitions

Consider a model for interest rates deﬁned on a set of discrete dates (tenor)

0= t0 < t1 < t2 tn 1 < tn · · · − At each time point we have a yield curve. Zero coupon bonds Pi,j : price of bond paying $1 at time tj , as of time ti . Pi,j is known only at time ti

Li = Libor rate set at time ti for the period (ti , ti+1) 1 1 Li = 1 τ Pi,i+1 −

L i

fsu-logo 0 1... i i+1... n

Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model Motivation Black, Derman, Toy model Model with log-normal rates in the terminal measure Phase Transition

Simplest interest rate derivatives: caplets and ﬂoorlets

Caplet on the Libor Li with strike K pays at time ti+1 the amount

Pay = max(Li (ti ) K, 0) −

Similar to a call option on the Libor Li

Caplet prices are parameterized in terms of caplet volatilities σi via the Black caplet formula

fwd Caplet(K)= P0,i+1CBS(Li , K, σi , ti )

Analogous to the Black-Scholes formula.

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Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model Motivation Black, Derman, Toy model Model with log-normal rates in the terminal measure Phase Transition Analogy with equities

Deﬁne the forward Libor Li (t) at time t, not necessarily equal to ti

1 Pt,i Li (t)= 1 τ Pt,i+1 − This is a stochastic variable similar to a stock price, and its evolution can be modeled in analogy with an equity

Assuming that Li (ti ) is log-normally distributed one recovers the Black caplet pricing formula

Generally the caplet price is the convolution of the payoﬀ with Φ(L), the Libor probability distribution function

∞ Caplet K dL L L K ( )= Φ( )( )+ fsu-logo Z0 −

Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model Motivation Black, Derman, Toy model Model with log-normal rates in the terminal measure Phase Transition Caplet volatility - term structure

σ ATM

Volatility hump at the short end

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Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model Motivation Black, Derman, Toy model Model with log-normal rates in the terminal measure Phase Transition ATM caplet volatility - term structure

120 3m caplet vols 100 27-Sep-11

0 2 4 6 8 10

fsu-logo Actual 3m USD Libor caplet yield (log-normal) volatilities as of 27-Sep-2011

σ (K)

σ ATM

Fwd Strike

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Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model Motivation Black, Derman, Toy model Model with log-normal rates in the terminal measure Phase Transition Modeling problem

Construct an interest rate model compatible with a given yield curve P0,i and caplet volatilities σi (K)

1. Short rate models. Model the distribution of the short rates Li (ti ) at the setting time ti . ◮ Black, Derman, Toy model ◮ Hull, White model - equivalent with the Linear Gaussian Model (LGM) ◮ Markov functional model 2. Market models. Describe the evolution of individual forward Libors Li (t) ◮ Libor Market Model, or the BGM model. fsu-logo

Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model Motivation Black, Derman, Toy model Model with log-normal rates in the terminal measure Phase Transition The natural (forward) measure

Each Libor Li has a diﬀerent “natural” measure Pi+1

Numeraire = Pt,i+1, the zero coupon bond maturing at time ti+1 The forward Libor Li (t)

1 Pt,i Li (t)= 1 τ Pt,i+1 −

is a martingale in the Pi+1 measure fwd E Li (0) = Li = [Li (ti )]

This is the analog for interest rates of the risk-neutral measure for equities

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Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model Motivation Black, Derman, Toy model Model with log-normal rates in the terminal measure Phase Transition Simple Libor market model

Simplest model for the forward Libor Li (t) which is compatible with a given yield curve P0,i and log-normal caplet volatilities σi Log-normal diﬀusion for the forward Libor Li (t): each Li (t) driven by its own separate Brownian motion Wi (t)

dLi (t)= Li (t)σi dWi (t)

fwd with initial condition Li (0) = Li , and Wi (t) is a Brownian motion in the measure Pi+1

Problem: each Libor Li (t) is described in a diﬀerent measure.

We would like to describe the joint dynamics of all rates in a common measure.

This line of argument leads to the Libor Market Model. Simpler fsu-logo approach: short rate models

Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model Motivation Black, Derman, Toy model Model with log-normal rates in the terminal measure Phase Transition The Black, Derman, Toy model

Describe the joint distribution of the Libors Li (ti ) at their setting times ti

L L L 0 LL1 i n−2 n−1

0 1... i i+1 ... n

Libors (short rate) Li (ti ) are log-normally distributed

1 2 σi x(ti ) σ ti Li (ti )= L˜i e − 2 i

where L˜i are constants to be determined such that the initial yield curve is correctly reproduced (calibration)

x(t) is a Brownian motion. A given path for x(t) describes a particular fsu-logo realization of the Libors Li (ti )

Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model Motivation Black, Derman, Toy model Model with log-normal rates in the terminal measure Phase Transition Black, Derman, Toy model

The model is formulated in the “spot” measure, where the numeraire B(t) is the discrete version of the money market account.

B(t0)=1

B(t1)=1+ L0τ

B(t2)=(1+ L0τ)(1 + L1τ)

· · · Model parameters: ◮ Volatility of Libor Li is σi ◮ Coeﬃcients L˜i

The volatilities σi are calibrated to the caplet volatilities (e.g. ATM vols), fsu-logo and the L˜i are calibrated such that the yield curve is correctly reproduced.

Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model Motivation Black, Derman, Toy model Model with log-normal rates in the terminal measure Phase Transition Black, Derman, Toy model - calibration

Price of a zero coupon bond paying $1 at time ti is given by an expectation value in the spot measure 1 1 P0,i = E = E h B(ti )i h(1 + L0τ)(1 + L1(x1)τ) (1 + Li 1(xi 1)τ) i · · · − −

The coeﬃcients L˜j can be determined by a forward induction:

1 P0,1 = L˜0 1+ L˜0τ → + 1 ∞ dx 1 x2 1 P = P E = P e− 2t1 0,2 0,1 0,1 ψ x 1 ψ2t h 1+ L1(x1)τ i Z √2πt1 1+ L˜1e 1 − 2 1 1 τ −∞ L˜ → 1 and so on... Requires solving a nonlinear equation at each time step fsu-logo

Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model Motivation Black, Derman, Toy model Model with log-normal rates in the terminal measure Phase Transition BDT tree

The model was originally formulated on a tree. Discretize the Markovian driver x(t), such that from each x(t) it can jump only to two values at t = t + τ

Mean and variance xup(t + τ) 1 1 E[x(t + τ) x(t)] = √τ √τ =0 − 2 − 2 x(t) @ 1 1 E[(x(t + τ) x(t))2]= τ + τ = τ @ − 2 2 @R xdown(t + τ)

Choose xup = x + √τ, xdown = x √τ with equal probabilities such that − the mean and variance of a Brownian motion are correctly reproduced fsu-logo

Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model Motivation Black, Derman, Toy model Model with log-normal rates in the terminal measure Phase Transition BDT tree - Markov driver x(t)

+2 Inputs:

+1 1. Zero coupon bonds P0,i Q Equivalent with zero rates R Q i Q 0 0 1 Q P = Q 0,i i Q (1 + Ri ) 1 − 2. Caplet volatilities σi Q Q Q 2 −

t - fsu-logo 0 1 2

Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model Motivation Black, Derman, Toy model Model with log-normal rates in the terminal measure Phase Transition BDT tree - the short rate r(t)

2 2σ2 2σ r2e − 2 Calibration: Determine r0, r1, r2, such that the zero coupon· · · prices are 1 2 σ1 σ r e 2 1 correctly reproduced 1 − @ @ @ Zero coupon bonds P0,i 2σ2 prices r r e 2 0 @ 2 − @ 1 P0,i = E @ Bi 1 2 h i σ1 σ r1e− − 2 1 @ Money market account B(t) @ - node and path dependent @ 2 2σ2 2σ r2e− − 2 B =1 t 0 - B1 =1+ r(1) fsu-logo 0 1 2 B2 = (1+ r(1))(1 + r(2)) . Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model Motivation Black, Derman, Toy model Model with log-normal rates in the terminal measure Phase Transition Calibration in detail

t = 1: no calibration needed 1 P0,1 = 1+ r0

t = 2: solve a non-linear equation for r1

1 1 1 1 1 P = + 0,2 σ 1 σ2 σ 1 σ2 1+ r0 2 1+ r1e 1− 2 1 2 1+ r1e− 1− 2 1 and so on for r , . 2 · · ·

Once ri are known, the tree can be populated with values for the short rate r(t) at each time t

Products (bond options, swaptions, caps/ﬂoors) can be priced by working fsu-logo backwards through the tree from the payoﬀ time to the present

Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model Motivation Black, Derman, Toy model Model with log-normal rates in the terminal measure Phase Transition Pricing a zero coupon bond maturing at t =2

up 1 P1,2 = r upτ 1+ 1 @ @ @ P 1 0,2 @ @ @ down 1 P1,2 = r downτ 1+ 1 @ @ @ 1 up σ 1 σ2 down σ 1 σ2 We know r = r e 1− 2 1 and r = r e− 1− 2 1 can ﬁnd the bond fsu-logo 1 1 1 1 → prices Pt,2 for all t

Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model Motivation Black, Derman, Toy model Model with log-normal rates in the terminal measure Phase Transition

BDT model in the terminal measure

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Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model Motivation Black, Derman, Toy model Model with log-normal rates in the terminal measure Phase Transition BDT model in the terminal measure

Keep the same log-normal distribution of the short rate Li as in the BDT model, but work in the terminal measure

1 2 ψi x(ti ) ψ ti Li (ti )= L˜i e − 2 i

Li (ti ) = Libor rate set at time ti for the period (ti , ti+1)

Numeraire in the terminal measure: Pt,n, the zero coupon bond maturing at the last time tn

L i

0 1... i i+1... n

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Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model Motivation Black, Derman, Toy model Model with log-normal rates in the terminal measure Phase Transition Why the terminal measure?

Why formulate the Libor distribution in the terminal measure?

◮ Numerical convenience. The calibration of the model is simpler than in the spot measure: no need to solve a nonlinear equation at each time step

◮ The model is a particular parametric realization of the so-called Market functional model (MFM), which is a short rate model aiming to reproduce exactly the caplet smile. MFM usually formulated in the terminal measure.

◮ More general functional distributions can be considered in the Markov functional model Li (ti )= L˜i f (xi ), parameterized by an arbitrary function f (x). This allows more general Libor distributions. fsu-logo

Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model Motivation Black, Derman, Toy model Model with log-normal rates in the terminal measure Phase Transition Preview of results

1. The BDT model in the terminal measure can be solved analytically for the case of uniform Libor volatilities ψi = ψ. Solution possible (in principle) also for arbitrary ψi , but messy results.

2. The analytical solution has a surprising behaviour at large volatility: ◮ The convexity adjustment explodes at a critical volatility, such that the average Libors in the terminal measure (convexity-adjusted Libors) L˜i become tiny (below machine precision) ◮ This is very unusual, as the convexity adjustments are “supposed” to be well-behaved (increasing) functions of volatility ◮ The Libor distribution function in the natural measure collapses to very small values (plus a long tail) above the critical volatility ◮ Caplet volatility has a jump at the critical volatility, after which it decreases slightly fsu-logo

Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model Motivation Black, Derman, Toy model Model with log-normal rates in the terminal measure Phase Transition What do we expect to ﬁnd?2

Convexity adjusted Libor L˜i = related to the price of an instrument paying Li (ti , ti+1) set at time ti and paid at time tn

Price = P0,nEn[Li ]= P0,nL˜i

Floating payment with delay: the convexity adjustment depends on the correlation between Li and the delay payment rate L(ti+1, tn)

L i

0 1... i i+1... n

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2Argument due to Radu Constantinescu. Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model Motivation Black, Derman, Toy model Model with log-normal rates in the terminal measure Phase Transition Convexity adjustment

Consider an instrument paying the rate Lab at time c. The price is proportional to the average of Lab in the c-forward measure

Price = P0,c Ec [Lab]

L ab L ab L bc

0 a b c

Can be computed approximatively by assuming log-normally distributed Lab and Lbc in the b-forward measure, with correlation ρ

E fwd fwd ρσab σbc √ab fwd 2 c [Lab] Lab 1 Lbc (c b)(e 1)+ O((Lbc (c b)) ) ≃ − − − − fsu-logo

Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model Motivation Black, Derman, Toy model Model with log-normal rates in the terminal measure Phase Transition Convexity adjustment

The convexity adjustment is negative if the correlation ρ between Lab and Lbc is positive

E fwd fwd ρσab σbc √ab fwd 2 c [Lab] Lab 1 Lbc (c b)(e 1)+ O((Lbc (c b)) ) ≃ − − − −

1.4 Hc-bLHexp-1L Convexity-adjusted Libors L˜i 1.2 for 3m Libors (b = a +0.25 ) 1

0.8 0.6 Parameters: 0.4 σab = σbc = 40% 0.2 a ρ = 20%, c = 10 years 0 2 4 6 8 Naive expectation: The convexity adjustment is largest in the middle of the time simulation interval, and vanishes near the beginning and the end. fsu-logo

Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model Motivation Black, Derman, Toy model Model with log-normal rates in the terminal measure Phase Transition Convexity adjusted Libors - analytical solution

The solution for the convexity adjusted Libors L˜i for several values of the volatility ψ fwd Simulation parameters: Li = 5%, n = 40, τ =0.25

5 ψ=0.2 4

2 ψ=0.3

ψ=0.4 1

fsu-logo 5 10 15 20 25 30 35 40

Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model Motivation Black, Derman, Toy model Model with log-normal rates in the terminal measure Phase Transition Surprising results

◮ For suﬃciently small volatility ψ, the convexity-adjusted Libors L˜i agree with expectations from the general convexity adjustment formula

◮ For volatility larger than a critical value ψcr, the convexity adjustment grows much faster. ◮ The model has two regimes, of low and large volatility, separated by a sharp transition ◮ Practical implication: the convexity-adjusted Libors L˜i become very small, below machine precision, and the simulation truncates them to zero

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Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model Motivation Black, Derman, Toy model Model with log-normal rates in the terminal measure Phase Transition Explanation

The size of the convexity adjustment is given by the expectation value

1 2 ψx ψ ti Ni = E[Pˆi,i+1e − 2 ]

˜ fwd Recall that the convexity-adjusted Libors are Li = Li /Ni

10 Plot of log Ni vs the volatility ψ 8

6 Simulation with n = 40 quarterly time steps 4

2 i = 30, t =7.5, r0 = 5%

0 0.1 0.2 0.3 0.4 0.5 ψ fsu-logo Note the sharp increase after a critical volatility ψcr 0.33 ∼

Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model Motivation Black, Derman, Toy model Model with log-normal rates in the terminal measure Phase Transition Explanation

The expectation value as integral

+ 1 2 ∞ dx 1 2 1 2 ψx ψ ti 2t x ψx ψ ti Ni = E[Pˆi,i+1e − 2 ]= e− i Pˆi,i+1(x)e − 2 Z √2πti −∞ The integrand 2 Simulation with n = 20 quarterly 1.5 time steps i = 10, ti =2.5

0.5 0.4 (solid) ψ =  0.5 (dashed)  0 0.52 (dotted) 5 10 15 17.5 x  Note the secondary maximum which appears for super-critical volatility fsu-logo at x 10√ti . This will be missed in usual simulations of the model. ∼ Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model Motivation Black, Derman, Toy model Model with log-normal rates in the terminal measure Phase Transition More surprises: Libor probability distribution

Above the critical volatility ψ > ψcr, the Libor distribution in the natural measure collapses to very small values, and develops a long tail

60 ψ = 0.35 ψ = 0.05

20 ψ = 0.1

10 ψ = 0.2

0 0.02 0.04 0.06 0.08 0.1 L

Simulation with n = 40 quarterly time steps τ =0.25. The plot refers to the Libor L30 set at time ti =7.5. fsu-logo

Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model Motivation Black, Derman, Toy model Model with log-normal rates in the terminal measure Phase Transition Caplet Black volatility

1.5 Black curve: ATM caplet volatility

1.25 Red curve: equivalent log-normal 1 Libor volatility 0.75 2 0.5 E[L ] σ2 t = log 1 LN i E 2 0.25 [L1]

0 0 0.1 0.2 0.3 0.4 Simulation with n = 40 quarterly time steps τ =0.25. The plot refers to the Libor L30 set at time ti =7.5

For small volatilities, the ATM caplet vol is equal with the Libor vol ψi . Above the critical volatility, the ATM caplet vol increases suddenly fsu-logo

Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model Motivation Black, Derman, Toy model Model with log-normal rates in the terminal measure Phase Transition Caplet smile

Above the critical volatility the caplet implied volatility develops a smile

1.4 ψ= 0.35 1.2

1 ψ= 0.34

0.8

0.6

0.4 ψ= 0.3 0.2 ψ= 0.2 ψ= 0.1 0.01 0.02 0.03 0.04 0.05 0.06 Κ

Simulation with n = 40 quarterly time steps τ =0.25. The plot refers to the Libor L30 set at time ti =7.5. fsu-logo

Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model Motivation Black, Derman, Toy model Model with log-normal rates in the terminal measure Phase Transition Conclusions

◮ For suﬃciently small volatility, the model with log-normally distributed Libors in the terminal measure produces a log-normal caplet smile ◮ The probability distribution for the Libors in the natural (forward) measure is log-normal

◮ Above a critical volatility ψcr the Libor probability distribution collapses at very small values, and develops a fat tail ◮ The caplet Black volatility increases suddenly above the critical volatility, and develops a smile ◮ These eﬀects are due to a coherent superposition of convexity adjustments In practice we would like to use the model only in the sub-critical regime.

Under what conditions does this transition appear, and how can we ﬁnd fsu-logo the critical volatility?

Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model Motivation Black, Derman, Toy model Model with log-normal rates in the terminal measure Phase Transition

Phase Transition

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Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model Motivation Black, Derman, Toy model Model with log-normal rates in the terminal measure Phase Transition The generating function

We would like to investigate the nature of the discontinuous behaviour observed at the critical volatility, and to calculate its value

(i) Deﬁne a generating function for the coeﬃcients cj giving the one-step zero coupon bond

n i 1 (i) − − (i) j f (x)= cj x Xj=0

This was motivated by a simpler solution for the recursion relation

The expectation value which displays the discontinuity is simply

n i 1 − − 2 2 (i) jψ ti (i) ψ ti Ni = c e = f (e ) j fsu-logo Xj=0

Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model Motivation Black, Derman, Toy model Model with log-normal rates in the terminal measure Phase Transition Properties of the generating function

f (i)(x) is a polynomial in x of degree n i 1 − − (i) (i) (i) 2 (i) n i 1 f (x)=1+ c1 x + c2 x + + cn i 1x − − · · · − − where the coeﬃcients are all positive and decrease with j

Can be found in closed form in the zero and inﬁnite volatility limits ψ 0, → ∞ It has no real positive zeros, but has n i 1 complex zeros. They are located on a curve surrounding the origin.− −

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Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model Motivation Black, Derman, Toy model Model with log-normal rates in the terminal measure Phase Transition Complex zeros of the generating function

Example: simulation with n = 40 quarterly time steps τ =0.25, ﬂat forward short rate r0 = 5%

i=30, psi=0.3 4 The zeros of the generating function at i = 30, corresponding to the 2 Libor set at ti =7.5 years

0 Number of zeros = n i 1=9 -2 − − Volatility ψ = 30% -4

-4 -2 0 2 4 Key mathematical result: The generating function f (i)(x) is continuous but its derivative has a jump at the point where the zeros pinch the real positive axis fsu-logo

i=30, psi=0.3 i=30, psi=0.31 4 4

10 2 2

0 0 8

-2 -2 6

-4 -4 4 -4 -2 0 2 4 -4 -2 0 2 4

i=30, psi=0.32 i=30, psi=0.33 4 4 2

2 2 0 0.1 0.2 0.3 0.4 0.5 0 0 ψ

-2 -2 2 Solid curve: plot of f (i)(eψ ti ) -4 -4

-4 -2 0 2 4 -4 -2 0 2 4

Criterion for determining the critical volatility: The critical point at which

the convexity adjustment increases coincides with the volatility where the fsu-logo 2 ψ ti complex zeros cross the circle of radius R1 = e

i=30, psi=0.3 i=30, psi=0.31 4 4

2 2 1.5

0 0 1.25

1 -2 -2

0.75 -4 -4

-4 -2 0 2 4 -4 -2 0 2 4 0.5

i=30, psi=0.32 i=30, psi=0.33 4 4 0.25

0 2 2 0 0.1 0.2 0.3 0.4

0 0

-2 -2 Black curve: ATM caplet volatility vs ψ -4 -4

-4 -2 0 2 4 -4 -2 0 2 4

Simulation with n = 40 quarterly time steps, at i = 30

The turning point in σATM coincides with the volatility where the fsu-logo 2 ψ ti complex zeros cross the circle of radius R1 = e

Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model Motivation Black, Derman, Toy model Model with log-normal rates in the terminal measure Phase Transition Phase transition

The model has discontinuous behaviour at a critical volatility ψcr ◮ The critical volatility at time ti is given by that value of the model volatility ψ for which the complex zeros of the generating function 2 f (i)(z) cross the circle of radius eψ ti ◮ The position of the zeros and thus the critical volatility depend on the shape of the initial yield curve P0,i ◮ r0ti r0τ For a ﬂat forward short rate P0,i = e− the zeros are zk = e− xk , where xk are the complex zeros of the simple polynomial

1 2 n i 1 Pn(x)= + x + x + + x − − 1 e r0τ · · · − − ◮ Approximative solution for the critical volatility at time ti

2 1 1 ψcr log fsu-logo ≃ i(n i 1)τ r0τ − −

Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model Motivation Black, Derman, Toy model Model with log-normal rates in the terminal measure Phase Transition Phase transition - qualitative features

◮ The critical volatility decreases as the size of the time step τ is reduced, approaching a very small value in the continuum limit ◮ The critical volatility increases as the forward short rate r0 is reduced, approaching a very large value as the rate r0 becomes very small the applicability range of the model is wider in the small rates regime→ The phenomenon is very similar with a phase transition in condensed matter physics, e.g. steam-liquid water condensation/evaporation, or water freezing

The Lee-Yang theory of phase transitions relates such phenomena to the properties of the complex zeros of the grand canonical partition function. fsu-logo

Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model Motivation Black, Derman, Toy model Model with log-normal rates in the terminal measure Phase Transition Numerical results

tn = 10 tn = 20 tn = 30 r0 τ =0.25 τ =0.5 τ =0.25 τ =0.5 τ =0.25 τ =0.5 1% 24.48% 32.55% 12.24% 16.28% 8.16% 10.85% 2% 23.02% 30.35% 11.51% 15.17% 7.67% 10.12% 3% 22.12% 28.98% 11.06% 14.49% 7.37% 9.66% 4% 21.46% 27.97% 10.73% 13.99% 7.15% 9.32% 5% 20.93% 27.16% 10.47% 13.58% 6.98% 9.05%

Table: The maximal Libor volatility ψ for which the model is everywhere below the critical volatility ψcr, for several choices of the total tenor tn, time step τ and the level of the interest rates r0 .

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Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model Motivation Black, Derman, Toy model Model with log-normal rates in the terminal measure Phase Transition Conclusions

◮ The Black, Derman, Toy model with log-normally distributed Libor in the terminal measure can be solved exactly in the limit of constant and uniform rate volatility ◮ The analytical solution shows that the model has two regimes at low- and high-volatility, with very diﬀerent qualitative properties ◮ The solution displays discontinuous behaviour at a certain critical volatility ψcr ◮ Low volatility regime ◮ Log-normal caplet smile ◮ Well-behaved Libor distributions ◮ High volatility regime ◮ The convexity adjustment explodes ◮ Libor pdf is concentrated at very small values, and has a long tail fsu-logo ◮ A non-trivial caplet smile is generated

Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model Motivation Black, Derman, Toy model Model with log-normal rates in the terminal measure Phase Transition Comments and questions

◮ A similar behaviour is expected also in a model with non-uniform Libor volatilities (time dependent), but the form of the analytical solution is more complicated

◮ Is this phenomenon generic for models with log-normally distributed short rates, e.g. the BDT model, or is it rather a consequence of working in the terminal measure?

◮ Are there other interest rate models displaying similar behaviour?

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Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model Motivation Black, Derman, Toy model Model with log-normal rates in the terminal measure Phase Transition

BDT model in the terminal measure - calibration and exact solution

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Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model Motivation Black, Derman, Toy model Model with log-normal rates in the terminal measure Phase Transition Calibrating the model by backward recursion

The non-arbitrage condition in the terminal measure tells us that the zero coupon bonds divided by the numeraire should be martingales

Pi,j 1 = E[ i ] , for all pairs (i, j) Pi,n Pj,n |F Denote the numeraire-rebased zero coupon bonds as

Pi,j Pˆi,j = Pi,n Choose the two cases ◮ (i, j) = (i, i + 1)

Pî,i (xi )= E[Pî ,i (1 + Li τ) i ] +1 +1 +2 +1 |F ◮ (i, j)=(0, i) fsu-logo Pˆ0,i = E[Pî,i+1(1 + Li τ)]

Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model Motivation Black, Derman, Toy model Model with log-normal rates in the terminal measure Phase Transition

Recursion for Pˆi,i+1(xi ), L˜i

The two non-arbitrage relations can be used to construct recursively Pˆi,i+1(xi ) and L˜i working backwards from the initial conditions

ˆ ˜ fwd Pn 1,n(x)=1 , Ln 1 = Ln 1 − − − No root ﬁnding is required at any step.

The calculation of the expectation values requires an integration over xi+1 at each step. Usual implementation methods: 1. Tree. Construct a discretization for the Brownian motion x(t)

2. SALI tree. Interpolate the function Pˆi,i+1(xi ) between nodes and perform the integrations numerically 3. Monte Carlo implementation fsu-logo

Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model Motivation Black, Derman, Toy model Model with log-normal rates in the terminal measure Phase Transition Analytical solution for uniform ψ

Consider the limit of uniform Libor volatilities ψi = ψ The model can be solved in closed form starting with the ansatz

n i 1 − − (i) 1 2 2 ˆ jψx 2 j ψ ti Pi,i+1(x)= cj e − Xj=0

(i) Matrix of coeﬃcients cj is triangular (e.g. for n = 5)

(n 1) c0 − 0 0 00  (n 2) (n 2)  c0 − c1 − 0 0 0 (n 3) (n 3) (n 3)  c − c − c − 0 0  cˆ =  0 1 2     ···(1) ···(1) ···(1) ···(1) ···   c0 c1 c2 c3 0   (0) (0) (0) (0) (0)    fsu-logo  c0 c1 c2 c3 c4 

Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model Motivation Black, Derman, Toy model Model with log-normal rates in the terminal measure Phase Transition c(i) Recursion relation for the coeﬃcients j

(i) ˜ The coeﬃcients cj and convexity adjusted Libors Li can be determined by a recursion

i i i 2 ( ) ( +1) ˜ ( +1) (j 1)ψ ti+1 cj = cj + Li+1τcj 1 e − − ˆ ˆ ˆ P0,i P0,i+1 fwd P0,i+1 L˜i = − = L (i) 2 i (i) 2 n i 1 jψ ti n i 1 jψ ti τΣj=0− − cj e Σj=0− − cj e

starting with the initial condition

L˜n 1τ = Pˆ0,n 1 1 , Pˆn 1,n(x)=1 − − − −

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(i) Linear recursion for the coeﬃcients cj . They can be determined (n 1) backwards from the last time point starting with c0 − =1

(i) (i+1) (i+1) 2 ˜ (j 1)ψ ti+1 cj = cj + Li+1τcj 1 e − −

(n 1) i = n 1 : c − − 0 @I 6 @ @ (n 2) (n 2) i = n 2 : c − c − − 0 1 @I @I 6 @ 6 @ @ @ (n 3) (n 3) (n 3) i = n 3 : c − c − c − fsu-logo − 0 1 2

Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model Motivation Black, Derman, Toy model Model with log-normal rates in the terminal measure Phase Transition Solution of the model

Knowing Pˆi,i+1(x) and L˜i one can ﬁnd all the zero coupon bond prices

Pˆi,j (x) Pi,j (x)= 1 2 ψx ψ ti Pˆi,i+1(x)(1 + L˜i τi e − 2 )

where

1 2 1 ψxj ψ tj ) Pˆi,j (x) = E i = E[Pˆj,j+1(1 + L˜j τj e − 2 ) i ] h Pj,n |F i |F n j 1 − − (j) 1 2 kψx 2 (kψ) ti = ck e − Xk=0 n j 1 − − (j) 1 2 2 2 ˜ (k+1)ψx 2 (k +1)ψ ti +kψ (tj ti ) +Lj τj ck e − − Xk=0 fsu-logo All Libor and swap rates can be computed along any path x(t)

Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model