Advanced Derivatives
FI830
• Team Size: 2. You can use all resources and discuss the problems with other teams, but the problems should be solved and written up by each team. There will be penalties for not complying with this regulation, i.e. I will subtract points in case the same work is submitted by several teams (note that not only the students copying, but also the students providing their work to be copied will be penalized)!
• Submit your results in report (pdf!!). Do not send me emails with spreadsheets or computer code! However, I reserve the right to request code or further information from every student on the days following the submission deadline, and it is your responsibility to deliver the requested information within 24 hours of the request.
It is the beginning of January 2022 and you just started your first job at OTHERWHERECORP, an investment company specialized in providing investment solutions for wealthy individuals. Your supervisor tells you that several of their customers complain about not being internationally diversified, and the company intends to address this by providing an international investment opportunity. Specifically, they want to provide an investment opportunity in the “iShares Core FTSE 100 UCITS ETF GBP (Acc) (CUKX LN).” This is a tracker ETF for the FTSE 100 total return index traded in British pounds, which takes reinvestments of dividends into account.1 However, since their investors typically are worried about downside risk, they want to add on a guarantee feature that pays back the invested principal compounded at a 2% annual interest rate. Specifically, she has in mind to make available a product that, at maturity T, pays the maximum of the fund value – which is determined by their initial investment L (in US Dollars), converted to British pounds, multiplied by the return on the CUKX index over the time period – and their initial investment L multiplied by (1 + 2%)T . In addition to the deposit L, the investor will be required to pay an initial option fee G for the guarantee. Your supervisor would like to offer the product for initial deposits of L = $200, 000, over a period of T = 5 years.
It is your first assignment to derive suitable prices for the guarantee amount G and to illustrate the riskiness of this undertaking from the company’s point of view. Clearly, you rely on dynamic risk-management in the form of hedging the guarantee, and since your company has large trade accounts, you will not have to worry about transaction costs.
1. You realize that you can represent the payoff in British Pounds,
??L?ST ?L? T?
Payoff=max x S , x ×1.02 , (1)
000
1You can download the fund series at https://www.ishares.com/uk/professional/en/products/253716/ ishares-ftse-100-ucits-etf-acc-fund.
Final Project
• Please submit your project (electronically via Canvas) by 11:59pm on Friday, May 13, 2022.
• Total attainable points: 80 plus a potential bonus for particularly nice projects.
FI830
Final Project 2
2.
where x0 is the exchange rate from British Pounds to US Dollars at time zero (so that 1/x0 is the exchange rate from Dollars to British Pounds), via simple option prices.2
As a first approach, you decide to keep it simple and rely on the basic Black-Scholes model with constant coefficients for modeling the index, which of course is denominated in British Pounds:
dSt =St (μdt+σdWt), S0 >0, anddBt =BtrUK dt, B0 =1.
You pull daily price data for the CUKX index for the last 10 years (1/1/2012-1/1/2022) and for the
interest rate you rely on the prevailing 5-year rate on (UK!) government bonds.
(a) Based on these data, determine an estimate for the (realized) volatility σ. You can use the standard estimators for the Normal-distributed log-returns, i.e. it is not necessary to rely on highly sophisticated methods for volatility estimation. Since the long term Sharpe ratio is about λ = 25% and you think the last decade was a rather turbulent one, rather than estimating your μ fromthedatayouassumeμ=rUK +λ×σ.
(b) Determine a formula for the guarantee premium G in the Black-Scholes model and calculate the premium based on your estimates and your specification (1). Simply determine the guarantee price in British Pounds and convert it to US Dollars.
(c) Using the data from 2012 through 2016, and from 2017 through 2021, set up a backtest that demonstrates how your hedge would have performed in these time periods assuming that you rebalance your hedging portfolio daily.
When you report your first results to your supervisor, she rolls her eyes: “I would have expected more from a UW MSFE graduate. Your specification in Equation (1) is wrong, since at time T the minimum to pay is L dollars rather than L/x0 Pounds. That is different, of course. Your specification ignores exchange rate risk.” She points out that the payoff at time T is, rather:
??L? T?
Payoff=max xT x S ST,L×1.02 . (2)
00
A little depressed, you go back to your office – but gladly you remember that we can use the same ideas as in the basic Black-Scholes model for various risk factors, in particular foreign exchange rate risk. You discover that in the well-known Garman & Kohlhagen (1983) model, the exchange rate is modeled via a geometric Brownian motion with constant coefficients,
dxt =xt(αdt+ψdZt), x0 >0,
where α ∈ R, ψ > 0, and Z is a second Brownian motion with dZt dWt = ρdt.
(a) Show that the time-zero price of the guarantee associated with Payoff (2) can be represented as:
T −rUST rUS−21?σ2+ψ2+2ρσψ?√ ! G = L×1.02 ×e ×Φ − pσ2 +ψ2 +2ρσψ T
rUS+21?σ2+ψ2+2ρσψ?√ ! −L×Φ − pσ2 +ψ2 +2ρσψ T
,
where Φ(·) denotes the cumulative distribution function of the standard Normal distribution and rU S is the risk-free rate in the US (remember, set up a portfolio consisting of different securities and derive a PDE).
2Historical exchange rates can be found on FRED, http://research.stlouisfed.org/fred2/.
FI830
Final Project 3
3.
When you approach your supervisor with your new calculations she seems happier, although she complains that you did not provide ideas for a better risk-management strategy than simple Delta- Hedging. Back at your desk, you decide to produce a back-test that takes into account (daily) Delta- and Exchange-rate hedging.
To delta-hedge, use the underlying; for the exchange rate risk, assume that the company has two savings/bank accounts, one in London denominated in pounds compounded at rUK and one in New York denominated in US Dollars compounded at rUS. Derive the appropriate hedging ratios, i.e. the amounts invested in each asset to perfectly offset the risk.
Again using data from 2012 through 2016, and from 2017 through 2021, set up a backtest that demon- strates how your hedge would have performed in these time periods assuming that you rebalance your hedging portfolio daily. Interpret your overall results.
Your supervisor is happy with your performance. However, she has one last task for you. She would like to obtain an estimate of the guarantee premium that also accounts for interest rate risk and stochas- tic volatility.
Set up a more advanced model with stochastic stock, interest, volatility, and exchange rate. Use sensible calibrations for the different model components. You can use Monte Carlo simulations to determine an estimate for G.
4.
(b)
(c)
A helpful reference here might be Benninga et al. (2002), but note that your supervisor only has prior knowledge about this material from our FI830 class, so you should show the formula to her in terms of material we have discussed in class.
Pull suitable daily exchange rate data, also for the period 1/1/2012-1/1/2022, and determine parameters for the model including exchange rates (i.e. determine parameter values for α, ψ, and ρ). Assume the prevailing 5-year rate on US government bonds for the US interest rate. Determine the option premium based on your parameters.
Using this option premium, set up an analysis of how your Delta-Hedge from part 1 would have performed when taking exchange rate risk into account, i.e. simulate stock and exchange rate paths based on the model here to simulate the correct payoff (under P) and then Delta-Hedge using the Black Scholes model from 1. Report your profit and loss distribution.
