Questions and Answers of Market Practice In Financial Modelling

5. Scenario You are working on the equity derivatives desk of a financial institution and you have been asked to write a European style, at -the-money call option on a basket of shares in ABC Bank
4. If the guaranteed bond offers the opportunity to 'lock-in' to predefined higher values of the underlying as time passes describe how this feature could be built into the product's structure.
3. If the guaranteed bond offers the opportunity to 'lock-in' to the value of the underlying at certain future dates describe how this feature could be built into the product's structure.
2. Demonstrate how the guaranteed bond discussed in question 1 could be hedged using:(a) A zero coupon bond plus call options on the underlying index.(b) A dynamically adjusted delta hedge strategy.
1. Scenario A building society has just issued a special, limited issue bond. The bond offers the holder the opportunity of enjoying an equity index payoff over a 2-year period with limited downside
4. If the bond described in question 1 has a reset clause which is triggered should the share price fall below £6.50 explain how this feature could be incorporated into the CB price using a binomial
3. Examine how different values of lambda impact on the CB 's price. Discuss ways in which an appropriate value of lambda might be established.
2. If the convertible bond is callable after the first year when the prevailing price of the share exceeds the conversion price by more than 30% explain how a binomial or trinomial tree could be
1. Calculate the price of a convertible bond with four years to maturity when the risk-free rate of interest is 5.0%, the company issuing the bond faces a 200 basis point credit risk add-on, share
7. Explain how interest rates that vary over time might be incorporated into a binomial tree used to price equity options.
6. Use the same data as in question 5 to establish the credit spread on an 18-month to maturity corporate bond with an embedded call option at 12 months, and the time value associated with the call
5. Using the following par yields calculate the implied spot and implied forward rates, the benchmark and corporate bond spread adjusted rates where the corporate bond has a coupon of 10.00% and is
4. If a bullet corporate bond with 18 months to maturity is priced at 102.00 and offers a coupon of 7.50% what is the implied credit spread over the benchmark rate when the benchmark rates are those
3. Using the data in question 1 calibrate the benchmark bond forward rates up to a period of 24 months.
2. Without performing any calculations what should the value of an embedded call option be under this interest rate regime?
1. Using the following par yields calculate the implied spot and implied forward rates.Period 6months 12 months 18 months 24 months Comment on your results.Par yields 6.000%5.875%5.750%5.500%
6. Using Goal Seek fmd the value of lambda which will ensure a barrier of 480 is touched for a down-and-in put for the case where the current underlying security is quoted at 490, volatility is 30%
5. Explore the possibility of modifying the spreadsheet BINOM.XLS in order to price barrier options. Discuss fully the way in which the spreadsheet needs to be modified to capture breaches of any
4. Repeat question 3 using the TRINOM.XLS spreadsheet and create a table of call premia based on values of lambda ranging from 1 to 4. Comment on your results.
3. Price the call option described in question l(a) using 100 period in the spreadsheet BINOM.XLS. Comment on the premium and delta obtained using this larger dimension tree.
2. Use the same data as in question 1 to solve for the option's premium in a twoperiod trinomial tree.
1 .(a) Using a two-period binomial tree fmd the call premium for a 6-month option with a strike of 500, where the underlying security is currently quoted at 480 with annualised volatility of 25%,
3.(a) As a portfolio manager you are concerned that the FfSE 100 index has peaked after a recent display of strength and you want to hedge you portfolio's value which is £20,000,000. The FfSE 100
2. Analysts have advised the manager of an FfSE 100 index-tracking fund that the associated index is about to fall heavily. It is believed that as much as 8% of its current level could be lost before
1. Analysts have advised the manager of an FfSE 100 index-tracking fund that the associated index is about to fall heavily. It is believed that as much as 10% of its current level could be lost
List the key players in your organisation who influence the development and use of financial models.AppendixLO1
Prepare a model template for use in your organisation.AppendixLO1
Prepare a cost/benefit analysis (or risk assessment) to support your proposed model template.AppendixLO1
A firm has experienced the following sales, costs and gross profit figures over the last 3 years:Create a forecast of sales, costs and gross profits for the next 5 years.Assume growth will continue
In a management buyout situation, the financial model is used to assess the cash flows generated by the business to cover the loan interest and repayments, and the sums involved can be substantial.
Assuming an interest rate of 8%, calculate interest on the loan, and update the new cash balance to show the payment of the interest.If there are no funds available to pay the interest, the bank
Set up the interest on unpaid interest routine and amend the cash flow accordingly.AppendixLO1
Refine the calculator further by setting up a drop down list of loan amounts between 100,000 and 1,000,000.AppendixLO1
The bank offers different interest rates according to the amount borrowed, as shown:Up to 200,000 Base rate plus 3%200,000–499,999 Base rate plus 2%500,000–799,999 Base rate plus
Amend the loan repayment calculator so that the interest rate changes according to the amount selected from the drop down list.AppendixLO1
Finally, set up appropriate protection to prevent users from changing the inputs and calculations, but allowing them to use the calculator as intended.AppendixLO1
Think about the models that you have worked with recently and draft a standard audit sheet that you could use in your organisation. Identify the most relevant checks that could be usefully applied to
If you managed to prepare a model template for your organisation at the end of Chapter 1, how does your audit sheet fit in with it?AppendixLO1
Use the Excel PMT function to calculate the monthly repayment, over 5 years, on a loan of 1,000,000. The annual interest rate is 8%.AppendixLO1
Now set up a data table with the following interest rates: 5%, 6%, 7%, 8%, 9% and 10%.AppendixLO1
The respective probabilities are as follows: 0.05, 0.1, 0.15, 0.5, 0.15 and 0.05. Enter these values above the interest rates on the data table.AppendixLO1
An additional step is to multiply the interest rates against their respective probabilities to calculate the ‘most likely’ interest rate. What is the monthly repayment using this rate?AppendixLO1
Display the results as a chart.AppendixLO1
When dealing with probabilities it is important to have an audit check to confirm that the total probability is equal to 1. Write a conditional format that will flag up if the total is less than or
Write a macro to be stored in the circular model, so that on opening the workbook the message box informs the user of the iteration status. This should run automatically when the file is
Develop the macro further so that the user can respond to this prompt by switching the iteration off. This can be done using the MsgBox macro itself.AppendixLO1
Create a macro that will switch off iteration when the workbook is closed.AppendixLO1
Assume deterministic interest rates. Following our discussion in Section 12.1.4, derive the quanto effect of having a stochastic funding spread assuming that the asset follows a standard lognormal
Implement via Monte Carlo a long-dated FX model with stochastic interest rates and stochastic vol. The parameterisation in Question 2 of Chapter 10 is reasonable for interest rates and spot FX
Repeat the comparison in Question 3 of Chapter 10, except this time leave correlations constant (i.e. 0% between spot FX and interest rates). Instead, increase both interest rates vols to 1%. Next
Implement via Monte Carlo the CEV parameterisation of spot FX with Gaussian rates in Section 10.2.1. For spot FX , use 85, for CEV vol use 0.15 x 851−β Try it with β = 0.2, 0.5, 0.9. What values
Use the same parameters as in Question 2 except where specified otherwise. Compute the 10-year forward starting volatilities for expiries 1y, 2y,..., 10y. Now, increase the domestic-spot correlation
Consider the same parameterisation as in Question 1 but to be more specific, say the HJM processes correspond to a Hull-White model with constant vol of 0.7% and mean reversion speed of 2%. Take
Consider a lognormal Libor Market Model. For simplicity, assume that Libor rates apply over 1y periods. Take current Libor rates to be flat at 3%. Using the discussion on the volatility triangle in
For a Heston model, take mean reversion of 15%. Calibrate η, ρ, θ (t) to the prices of strangles and risk reversals over 5 years (using the characteristic function method for pricing European
Consider the Taylor expansion of the integrand of the cumulative normal function for x ≈ 0, i.e. Thereby, find an approximation for the price of a lognormal option close-to-the-money. Attempt
Compute the forward delta of a call option based on the normal formula. (The price is given in Section 1.2.4.) Contrast this with the forward delta for an at-the-money (i.e. forward = strike) call
Consider a very steep skew. Would SABR always be able to fit this?
Consider the Craig-Sneyd discretisation for three space variables. By performing the steps sequentially, show that the scheme is consistent (i.e. we do indeed recover a discretised form of the
Find a self-similarity solution, the Fokker-Planck equation can be transformed into a standard diffusion, so solving the above diffusion equation with the given initial and boundary conditions, is
Build a simple Monte Carlo framework based on stochastic vol. Convince yourself that the fair value strike of a vol swap is not very different from the at-the-money vol.
Consider the Hull-White model with SDE drt = κ,(t)(θ(t) – rt)dt + σ(t)dWt. Following the same logic as we did in the text, determine θ(t). Can you see why in practice calculating θ(t) directly
Risk reversals and strangles are described in Section 2.3.2 as packages of calls and puts with strikes on either side of the forward. Using the approximation to the SABR formula per Section 6.2.3,
Take the actual SABR formula per Section 6.2.1 with σ0 = 0.5%, β = 0, ρ = −25%, ν = 30%, and F = 4%. By considering an expiry of 20 years and strikes at 10 basis points increments from 1 % to
Show how the Black-Scholes PDE with final condition V(S,T) — VT(S) can be transformed into the initial condition. The exercise is more to convince you that the Black-Scholes PDE is a standard
Consider the diffusion equation that the scheme is stable. Show also that the implicit scheme is unconditionally stable. ди Ət = 82 u а ах² Let u ||
Suppose that each Libor rate is driven by a separate Wiener process dLi (t) = σi (Li(t),t) dWit (under the measure with numeraire being the discount bond maturing at the pay date of the Libor
Consider a two-factor short rate model described by dxt = where ƒt,T is the HJM instantaneous forward rate. -Kr(t) xidt + oz(t)dWF, dyt -Ky(t)yidt + oy(t)dW!', rt = y(t) + x + yt and dWƒdW? =
Consider again the Hull-White model. Using ƒtT = ETt [rT],obtain the volatility of the SDE of the discount bond D(t,T) = e–∫Tt ƒt,sds .
Consider the construction of a trinomial tree for the general short rate model of form dyt = κ(θ(t) − yt)dt + σdWt, rt = g (yt). The approach typically involves constructing the tree for dxt =
Consider again the Hull-White model with constant mean reversion κ = 1 %. Suppose also interest rates are flat at 4% so that discount bond prices are given by D(0,T) = e_0.04T First find the SDE for
Consider the correlation matrix Ω. We can perform an eigenvector decomposition so that Ω = SAST, where A is a diagonal matrix of eigenvalues and S is the eigenvector matrix whose determinant is 1.
Consider a two-factor lognormal Libor Market Model. Ignore the drift, so that we can write dLit = . . . dt + σiLit (ai1dWt1 + ai2dWt2), where a21i + a21i = 1 and dWt1 dWt2 = 0. Now, let us suppose
The Longstaff-Schwartz algorithm involves a choice of basis functions. Suppose we have n + 1 dependent variables {yi}ni=0 to fit corresponding to the explanatory variables {xi}ni=0 Show that a
Consider the following parameterisation based on Gaussian interest rates and lognormal spot FX : dSt = (rt – rƒt) Stdt + quanto Libor rate, i.e. where the Libor rate LT (set at time T, with
Assume lognormal dynamics for the underlying dSt = rStdt +σStdWt and constant interest rates r. Compute the price of a cliquet with payoff max (ST2/ST1 − 1,0) at time T2 > T1.
Consider the Hull-White model drt = κ(θ(t) − rt)dt + σdWt Find corr (ƒt,T,ƒu,T) where ƒt,T = ETt [rT] and t < u < T.
Consider the equation for the par swap rate under OIS discounting in Section 12.1.3. If the OIS-Libor spread is zero, show that this reduces to the classical equation D(0,T0) – D (0, TN) = Suppose
Due to the huge supply of yen, the funding bias for USD/JPY is such that it is cheaper to borrow yen versus dollars (i.e.sJPY BRL ≠ 0) Consider now a cross currency swap with one leg paying
Consider a flat Libor discount curve given by D(0, T) =e −0.03T. Suppose that the OIS discount curve is D (0,T) = e −0.028T. Compute the swap rate (annual fixed coupons, semi-annual floating
Notwithstanding the discussion in Section 12.1.4, since we forecast based on Libor, the martingale equation only holds for risky discounting at the credit-worthiness of Libor counterparts. As such,
The Ho-Lee model assumes that the short rate has stochastic differential equation drt = θ (t)dt + crdWt. Note that the money market account is defined via BT = e∫0T rudu and the discount bond
Consider a contract with payoff max at time T. From your answer to Question 2, what is the value of such a payoff?Question 2,Suppose that two assets SA and SB have stochastic differential equations
Consider the simple convexity adjustment (Question 3 of Chapter 3) and apply it to the 10-year and 2-year CMS rates setting in 30 years’ time. (Suppose for this example that both swap rates are
For a derivative V(XT) on an underlying XT as seen at time T, its value is given by V(Xt) = e –r (T– t) E[V(XT)] = e –r (T– t) ∫∞–∞ V(x)p(x)dx, where p(•) is the probability
Derive Dupire’s formula in Section 5.1.2 in terms of put prices rather than call prices.Section 5.1.2, More interesting is the question of how to recover (St, t) to fit all given market prices. Let
Derive the density for the Gumbel and Clayton copulae.
For simplicity, assume the Ho-Lee dynamics drt = θ(t)dt+σdWt. The forward rate is given via ƒ(t,T) = ETt [rT]. (All this will become clear when we discuss short rate models in Chapter 8.)
For a cash-settled swaption, the cash-annuity is defined by This is j ust some function of RT. Do a second order Taylor-expansion of Ac(RT) about Ro. Notice that under the T-forward measure, we get
Assume interest rates are lognormal. Compute the convexity adjustment for the fair value LIA rate. What would the convexity adjustment be if rates are normal? (In this question, assume semi-annual
Consider the Black-Scholes model. What is the gamma of a digital option with payoff 1ST>K at time T? Why would you not want to delta-hedge such an option?
Derive the formula for the Black-Scholes vega for a call option. What about a digital call option with payoff 1ST>K at time T?
Suppose we have a curve built from swaps with annual coupons and spanning maturities Suppose the swap rates are monotonically increasing (i.e. Si j for Ti j). Define annually compounded zero
Compute the Black-Scholes delta and gamma for a call-option with payoff max(ST — K,0) at time T.
Suppose you are given quotes on put options on interest rates of different strikes for a given maturity (i.e. payoffs are max( K – RT,0). For the 1% strike, the price is 0.002 and for the 2%
Find P(min0Wt ≤ m.
Find cov(Wt, Wt) for t < T.
Suppose that two assets SA and SB have stochastic differential equations dStA,B = rStA,B dt + σ A,BStA,B dWtA,B where dWtA dWtB = ρdt. Derive the formula for the option that pays max at time
Consider the SDE dXt = μXtdt + σXtdWt for the FX rate. By considering the domestic and foreign money market accounts with SDEs dBt = rtBtdt and dBƒt = rƒt Bƒt dt and using Girsanov’s
Show that the formula for the price of a put option with payoff of PT = max(K – ST, 0) at time T is Pt = e-r(T-t) KN (−d₁ + o√Tt) - StN(-d₁), where d₁= = log ()+(r+²) (T-t) o√T-t

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