## Transcribed Text

1. Caplet Q2. HJM model evolves the forward curve using the system of same-form SDEs. To ob- tain an expectation of LIBOR rate in the future select the rate from corresponding tenor column = Ti+1 � Ti of the HJM output. Convert to the simple annualised rate using L = m �ef =m � 1 where m is compounding frequency per year.
Forward LIBOR L(t; Ti; Ti+1) expires at Ti when the rate becomes a LIBOR Öx and matures at Ti+1 when the casháow is paid.
Design and code the Monte Carlo pricing of a áoorlet option written on 6M LIBOR L(0; 0:5; 1), DFOIS(0;1) max(K � L;0)
with parameters K = 6%, as follows from the task (Notional is absent). To obtain the discounting curve, assume no traded OIS instruments available but the LIBOR-OIS spread is 80bps. Subtract that constant spread from each simulated forward curve to obtain discounting curve and DFOIS.
Use Black formulae to obtain option volatility. The solution is only possible numerically, ether by setting up the Solver problem on a spreadsheet or coding/using a root-Önding procedure.
Floorlet = DFOIS [KN(�d2)�LN(�d1)] ln(L=K) 0:52T
Refer to HJM Model MC.xlsm spreadsheet on how to evolve the forward curve. Recode the SDE simulation in the suitable environment using the same volatility functions ñthat is, without recalibration.
Convergence diagrams must be provided together with a brief error analysis. Consider using low-discrepancy sequences of random numbers.
Vary the strike K and graphically present the model-dependent volatility skew (if any).
Example of an option payo§ calculation is given on Caplet tab in HJM Model MC.xlsm. Use Python, VBA, C++ or other suitable environment. It is part of the exam task to interpret these instructions and choose correct rate and DF from the simulated curves output.
d1;2 = pT
2. We wish to Önd the approximate value of a casháow for a áoorlet on the one month LIBOR, when using the Vasicek model. Show that this is given by
max�rf �r� 1 (� r);0; 24
where rf is the áoor rate and r the spot rate.
3. Consider the following interest rate data
for which we wish to obtain a model of the form
dr=u(r)+r dX:
Outline a method for doing this. Your account should be no longer than two and a half sides of A4 paper and include details of: capturing the volatility structure w (r); functional form of the drift u (r) ; slope of the yield curve to calculate the market price of risk (r) :
4. Consider a spot rate model given by
dr=(� r)dt+( r+ )dW;
where all parameters are constant. By looking for a solution of the form Z (r; t) = e(A(t;T )�rB(t;T )) of the Bond Pricing Equation; show that the resulting pair of Örst order ordinary di§erential equations are
with
By solving
show that
where
dt
A (T ; T ) = B (T ; T ) = 0:
B(t;T)=
2�e 1(T�t)�1
( + 1)(e1(T�t)�1)+2 1
p 2
1= +2:
dA
dt dB
= B � 12 B 2
= 12 B 2 + B � 1 ;
dB=12 B2+ B�1; dt
5. A two factor interest rate model depends on the spot rate r and another quantity l: The state variables r and l follow the SDEs, in turn,
dr = u(r;t)dt+w(r;t)dW1 (t) dl = p(r;t)dt+q(r;t)dW2 (t);
where the Brownian motions are correlated with E[dW1dW2] = dt: A bond with maturity T has value V (r; l; t; T ):
a. For the following question, you are required to present detailed working.
Consider a portfolio where one bond of maturity T is hedged with two others of maturities T1 and T2 which is given by
= V (r; l; t; T ) � 1V1(r; l; t; T1) � 2V2(r; l; t; T2):
Use ItÙís lemma to show that in one time step the change in the portfolio value is given by
l respectively. c. Given that
d =
L(V)dt+ @V dr+ @V dl�1 L(V1)dt+ @V1dr+ @V1dl
@r @l @r @l L(V2)dt+ @V2dr+ @V2dl ;
�2
b. Using the No Arbitrage Principle show that the problem reduces to the inconsistent system
where the operator
@r @l
@12@2 12@2 @2
where
Hence obtain the two factor interest rate model
L@t+2w @r2 +2q @l2 +wq@r@l:
@V�@V1�@V2 = 0; @r 1 @r 2 @r
@V�@V1�@V2 = 0; @l 1 @l 2 @l
L0(V ) � 1L0(V1) � 2L0(V2) = 0
L0(V ) = L(V ) � rV:
@V + 1w2@2V +wq@2V + 1q2@2V +
@t 2 @r2 @r@l 2 @l2 (u�rw)@V +(p�lq)@V �rV =0;
(5.1) where the arbitrary functions r and l represent the market price of risk associated with r and
@r @l
u�rw = 0=p�lq;
w = q= a+br+cl;
where a; b and c are constants, derive a set of Örst order equations and boundary conditions forA; BandCsuchthatabondV isoftheform
V =exp(A(t;T)� rB(t;T)� lC(t;T));
is a solution of the Bond Pricing Equation (5:1) ; with redemption value
V (r;l;T;T) = 1:
This concerns questions 3 and 4. They should read as follows: 3.
dr=u(r)dt+r dW:
4. Consider a spot rate model given by
dr=(� r)dt+p r+ dW; where all parameters are constant.

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