Derivative Securities

Project description
3. The purpose of this question is to compute the credit value adjustment (CVA) for interest
rate swaps. Let 0 = t0 < t1 < t2 < · · · < tn be a sequence of times, and let Vi be the (possibly
uncertain) value of a financial asset at times ti
, where 1 ≤ i ≤ n. Then the CVA for the
financial asset may be approximated as follows:
CVA =
Xn
i=1
(1 − R)P(ti)D(ti−1, ti) max(Vi
, 0), (7)
where
• P(ti) is the discount factor to time ti
,
• R = 0.4 is the fraction of max(Vi
, 0) recovered when the counterparty defaults at time ti
and assumed not to depend on times ti
,
• D(ti−1, ti) is the probability of the counterparty defaulting between time ti−1 and ti
.
Suppose it is July 3, 2013. Assume that the discount factors are as given in Question 2,
you have 2 years remaining on a quarterly-quaterly vanilla swap receiving fixed at 3.50% with
notional $10,000,000. The 3-month forward rates and survival probabilities for a counterparty
are as follows:
Maturity 09/10/2013 08/01/2014 09/04/2014 09/07/2014
3-month Rate 2.6610% 2.6350% 2.6090% 2.6060%
Survial Prob 0.975310 0.951229 0.927743 0.904837
Maturity 08/10/2014 07/01/2015 08/04/2015 08/07/2015
3-month Rate 2.6090% 2.6350% 2.6750% 2.7330%
Survival Prob 0.882497 0.860708 0.839457 0.818731
(a) PV01 is the change in value of the fixed leg of the swap if the fixed rate is increased by
1 basis point (0.01%). Compute the PV01 of the (remaining) 2-year swap. [1 mark]
(b) Approximating Vi by the current forward swap values, compute the CVA for the 2-year
receiver swap as a multiple of PV01. Hand in a printout of the spreadsheet used to solve
this problem. [4 marks]
(c) Explain why it is more accurate to use swaption values in place of max(Vi
, 0) when
computing the CVA of a swap. [1 mark]
(d) Assuming the volatility of 25% for all swaptions, compute the CVA for the 2-year swap
using swaption values in place of max(Vi
, 0). Hand in a printout of the spreadsheet used
to solve this problem. [6 marks]
(e) Suppose you are a bank and, in absence of suitable arrangements, you charge upfront
CVA as calculated above. Devise and explain an arrangement by which the counterparty
can avoid paying the upfront CVA charge. [3 marks]
Hint: How do futures exchanges get around the problem of default risk when they take on
the settlement risk on all contracts?
4. For this question, you will need to download the data file msft.csv from Blackboard. This file
contains the daily closing prices for Microsoft shares obtained from Yahoo! Assume throughout
that Microsoft does not pay any dividends.5
(a) Use the data to estimate the historical per annum volatility of the Microsoft share price
rounded to 4 decimal places. In annualising the volatility, assume that there are 252
trading days per year. [2 marks]
(b) Use the estimated volatility to compute the up and down factors for the binomial tree
with time step 1 week. For simplicity, assume that 1 week corresponds to ∆t = 1/52 of a
year. [2 marks]
(c) Assuming that the continuously compounded risk free rate is 0.2% per annum, compute
the corresponding risk neutral probability. Again, assume that 1 week corresponds to
∆t = 1/52 of a year. [1 mark]
(d) A European up-and-out barrier option is one which becomes worthless when the underlying
asset price touches or crosses a fixed barrier B, and otherwise equivalent to a standard
European option. Use binomial trees to compute the price of a 12-week up-and-out barrier
put option on Microsoft with strike $25.13 and barrier B = $27.00 on 28 June, 2013. Hand
in a printout of the spreadsheet used to solve this problem. [5 marks]
(e) A European up-and-in barrier option is one which is worthless unless the underlying asset
price touches or crosses a fixed barrier B, after which it becomes a standard European
option. Determine and explain the relationship between the standard put option and the
two barrier put options, up-and-out and up-and in, described above. [3 marks]
(f) Use parts (d) and (e) to compute the price of the 12-week up-and-in barrier put option on
Microsoft with strike $25.13 and barrier B = $27.00 on 28 June, 2013. [2 marks


Are you looking for a writer who will provide you with A+ quality work? If yes, then click here to meet our highly talented and qualified writers
5. This question examines the SVI (stochastic volatility inspired) smile parameterisation due to
Gatheral. Suppose the forward price of an underlying asset is F, and consider the volatility
smile implied by options with expiry T. Under the SVI parameterisation the implied volatility,
σ(k), as a function of k = ln(F/K), where K is the strike, is given by the formula
2
σ
2
(k) = a + b
h
ρ(k − m) +
p
(k − m)
2 + σ
2
i
, (8)
where a, b, ρ, m, and σ are model parameters.
Suppose the 1 year forward price of an asset is F = 1.4045, and the following implied
volatilities are given
Strike K 1.1590 1.2929 1.4158 1.5315 1.6529
Implied volatility σ 16.08% 14.00% 12.56% 11.97% 12.18%
(a) Set a = 0.014, b = 0.0321, σ = 0.0515, ρ = 0.798, m = 0.0168, and compute the implied
volatilities according to the SVI model at values of k corresponding to the strikes given
in the table above. Hand in a printout of the spreadsheet used for this problem.[3 marks]
(b) For each given strike in the above table, compute the square of the error in the implied
volatility computed using the parameters in part (a). [1 mark]
(c) Calibrate the SVI model to the given volatility smile. For this, use Excel’s solver to
minimise the sum of the squared errors by varying the SVI parameters. [2 marks]
(d) Assume that the 1 year discount factor is 0.9904, and use the calibrated SVI parameters
to compute the price of a call option with strike 1.5. [4 marks]
Are you looking for a writer who will provide you with A+ quality work? If yes, then click here to meet our highly talented and qualified writers