Derivatives 1 Course Work

Derivatives 1 Coursework Orange County Case Management Report Shenchao He Yuikit Yiu DeQiang Wu Ao Luo 11/26/2009 MSc in Mathematical Trading and Finance

Question 1 The relationship between unlevered excess return Ru-r and unlevered excess return RL-r is derived below: Let B be the “borrowed funds”, then we have: V0=OF+B RL=OF+B×Ru-BrOF →RL-r=OF+B×Ru-rOF+BOF Because we haveV0=OF+B: →RL-r=V0OFRu-r=2.7Ru-r

For volatility, we have: VarRL-r=Var2.7Ru-r VarRL-r=VarRL-Varr+CovRL,r Var2.7Ru-r=Var2.7Ru-Var2.7r+CovRu,r Since r is the known “risk-free” borrowing rate and contains no variability, which means: Varr=Var2.7r=0, CovRu,r=CovRL,r=0 ∴ VarRL=Var2.7Ru → σ2RL=2.72σ2Ru → σRL=2.7σRu

For the relationship between the Sharpe ratio for the levered and the unlevered portfolio: SRL=RL-rσL=2.7Ru-r2.7σu=Ru-rσu=SRu This makes perfect sense that leveraged return will also bring leveraged risk into the portfolio.

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Question 2 a) VCV/Delta-Normal method Since dy(t)=yt-yt-1, we could calculate the sample standard deviation of monthly change in the yield and use it as the forecast of volatility, σ (dy)=0.4037% p.m. Using VCV method, we assume yield changes are normally distributed and accept duration approximation, Then, monthly $ VaR of bond portfolio (5% lower tail) is: VaRp = Vp*1.65*Dp*σ (dy) = 7.7*1.65*2.74*0.4037% = $ 0.3794 bn Where Vp = market value of bond portfolio, Dp = (effective) duration of bond portfolio

b) Historic Simulation method We use historic simulation method when we do not assume a particular distribution for monthly change in yield. Firstly, calculate the actual monthly gain(loss) at time-t in excel, $ ∆Vpt= Vp*Dp*dy(t) Figure 1.

Then order ∆Vp in ascending order (of 504 numbers) or make a histogram (see figure 1).

So VaR forecasting using HS, for next month at 1% tail ( 5th most negative) = -$0.2287 bn. As can be obviously seen, In absolute term, VaR(VCV, 5% tail) > VaR (HS, 1% tail), where VaR is always reported as a positive number. 3

Through root T-rule, σT = √Tσ, then, VaRVCV = Vp*1.65*Dp*√T *σ (dy) = 7.7*1.65*2.74*12*0.4037% = $ 1.3143 bn VaRHS = 0.2287*12 = $ 0.7922 bn. The assumption behind root T-rule is that yields monthly changes are identically and independently distributed and have a constant variance. However, it’s unrealistic since our forecast is based on assumption of a time varying variance. Besides, the rule can be only used for relatively short horizons as the accuracy drop down as T increases. To conclude compared with VaR by using historical simulation, the figure for annual VaR based on VCV is much more consistent with $ 1.6 billion actual loss.

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Question 3 a) Using EWMA model, we computed that the time-varying volatility of the monthly change in yield, σt2=0.116418013 Hence we obtain the VaR p.a. = $1.11094 bn, which is approximately equals to the VaR obtained by VCV method. b) By Backtesting the EWMA Model, we observed that actual Profit and Loss in each tail is 4.83% which is less than 5%. Hence our EWMA forecast is acceptable. Next, we used EWMA model to produce the forecast from Dec 1994 to present on 50year US T0Bond yields. We observed that the actual Profit and Loss in each tail is 3.17%. c)

5-day VaR

Bootstrapping

Monte Carlo simulation

$3521724.9

$41049.66

There are two key differences between using the bootstrap method and Monte Carlo method: 1. Monte Carlo method uses real statistical sample with no sample limitation, whereas bootstrap method is re-sampling based on the relatively limited population, which cannot represent the real situation. 2. However, Monte Carlo method has an assumption that the underlying time series follow an identical and independent distribution and thus bring model risk into the result. While on the other hand, bootstrap method has no assumption at all and eliminates this kind of risk.

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Question 4 (1063 words) a) I am a US investor who holds a portfolio of $10m US BBBCorporate bonds with 3 years to maturity. I am concerned that interest rates are expected to be highly volatile over the next year. I decided to use interest rate futures contract to hedge my position over the next year. We assume that the change in the yield is the same for all maturities, only parallel shifts can occur in the yield curve. If interest rate movements differ, the price movement risk is not hedged. Secondly, we assume that future contracts are held to maturity. Estimation must be made on which of the available bonds is likely to be cheapest to deliver at the time the hedge is implemented. Finally, we assume no default risk and no arbitrage opportunities. As prices in both bonds (the underlying assets) and interest rate futures are almost perfectly correlated, we want to create a negative correlation to make the hedge possible. In this case, we require a short position in futures to hedge the bond portfolio. If interest rates go up, a gain will be made on the short futures position, which offsets the loss in the bond portfolio. If interest rates fall, a loss will be made on the short position, but there will be a gain on the bond portfolio. We want to choose the future contract so that duration of the underlying asset is as close as possible to the duration of the asset being hedged. Bond futures are very liquid future contracts and among the most traded futures contracts. Two types of futures can be considered here: Eurodollar and T-bond futures. Eurodollar futures tend to be used for short-term interest rates exposures; while T-bond futures are used for long-term. As we want to hedge the position over next year, we are going to use T-bond futures in our scenario. In order to hedge my position using interest rate futures contract, we are going to use the method called duration-based hedge ratio to calculate the number of contracts required to hedge against the uncertainty of change in yield. The duration based hedge ratio is Nf=DPPPDFPF DP and DF = durations of the bond portfolio and futures contract 6

PP and PF = the prices of the bond portfolio and futures contract

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Portfolio

Instrument

Price

Duration

$10m bonds

Short futures

120’25

9.10 years

Bond

$10m

7.4 years

Number of bond futures: 79 contracts

Bond futures often bear an additional risk often referred to as the basis risk. Basis risk arises when imperfect correlation between two investments creates the potential for excess gains or loss in a hedging strategy. In our case, there is a risk that our bond portfolio and T-bond will not fluctuate identically. Additionally, margin calls may be required. b) To hedge/insure my $10m portfolio of 5-year US BBB-Corporate bonds, we can buy put options on the bond portfolio. This approach is referred to as a protective put strategy. By using the protective put strategy, we are locked in a minimum price, at which sell the bonds if interest rates rises and buy back at lower spot price. If interest rate falls, we then do not exercise the put options. This will effectively immunize the portfolio against interest rates rises without losing the benefits of rate reductions. The hedging with bond options is not very popular; the hedging with futures options is more commonly applied. The main reason appears to be that a futures contract is more liquid and easier to trade than the underlying asset. Furthermore, a futures price is known immediately from trading on the futures exchange CBOT, whereas the spot price of a bond can only be obtained by contacting brokers/dealers. Exercising a futures option does not usually lead to delivery of the underlying asset as the underlying future contract is closed out prior to delivery. Futures options are therefore normally settled in cash. Transaction costs are lower than spot options in most circumstances. The most important options on interest rate futures are: the options on T-Bond futures, T-Note futures and Eurodollar futures etc. In our case, we are concerned that interest rate over next year will rise, which causes the value in bond portfolio will fall. To hedge our position, we will need to long 100 put future options with strike price at $100,000 plus long futures contracts to buy $10m bond in one year time. We illustrate the payoff at the table below.

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Current Value

Payoff at expiration

Payoff at expiration

FT≤K

FT≥K

Portfoli o

Instrument

$10m bonds

Long put future option

p

K-FT

0

Long futures

0

FT-F0

FT-F0

Bond

F0e-rT

F0

F0

Total

p+F0e-rT

K

FT

There are some fundamental difference between the two strategies in a) and b). Hedging bond portfolio with futures contracts attempt to equalize the volatilities of the bond and futures positions so that the net changes in portfolio values are as close to zero as possible. With strategy applied in b), investors lock in the minimum price with opportunity to benefit from rise in value of the bond portfolio. However, investors can enter into futures contract with no upfront cost whereas buying options require the premium. c) Based on this situation, one could possibly sell 10m AA-rated corporate bonds and buy 10m in FTSE “All Share Index”. However, the transaction fee might be quite high due to its large notional amount. A better solution would be to use Equity Swap. An equity swap is a financial derivative contract where a set of future cash flows are exchanged between two parties that allows each party to diversify its income, while still holding its original assets. The two sets of equal cash flows which are exchanged involve an equity-based cash flow and that is traded for another fixed-income benchmark cash flow. In our scenario, one can buy an equity swap with payoff tracked to return on FTSE “All Share Index” and swap for either a fixed or floating rate depending upon whichever coupon type of the AA-rated corporate bond and with one year tenor. Ideally, the coupon from bonds could match the cashflow in the swap deal in return for the index return. The advantage here is that one can gain exposure to UK companies through the index return from equity swap but without selling the bonds and holding 10m worth of equity portfolio which in turn introduce a substantial amount of transaction, stamp duty and 9

custody fees. The key risk of this approach is that you are not completely decoupled from exposure of bond price over the life time of the swap as you are still holding them. An example of the marked-to-market value of the equity swap over its life time is illustrated in the spreadsheet.

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Reference John C. Hull. “Duration-Based Hedging Strategies,” Options, Futures, and Other Derivatives”, Sixth Edition, 142-144.

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