Estimating an Illiquid Gilt Futures Price from Sparse Rates and FX Data

# Estimating the price of an illiquid 5y bond futures contract


# Estimating the price of an illiquid 5y bond futures contract












Say I know the price of 10y Gilt futures, 10y Treasury futures, 5y Treasury futures, and GBPUSD futures.

I am asked to produce a quote on 5y Gilt futures using only this data. What is a sensible approach to this problem?

WLOG we may assume the risk of default is zero.

## Answer by Attack68 (score 5)

https://quant.stackexchange.com/a/79588

That is not a lot of information and is certainly not enough to do this with sufficient accuracy for trading, but none-the-less we can do the exercise.

#### Information

I will assume that you also know additional information so your full suite is something like:



- The current SONIA rate: 5.25%



- Some forward GBPUSD rates (from your futures), e.g. 1y fwd: 1.276 2y fwd: 1.279 3y fwd: 1.281

- 10y and 5y US treasury rates: 4.52% and 4.51%

- 10y UK gilt rate: 4.32%

#### US Curve

Proceed by first building a US treasury curve, from the three US rates you have. Since this is particularly sparse I will use log-cubic spline interpolation and derive something like the following:

```
# PYTHON 
from rateslib import *  # requires >= 1.3.0

usd = Curve(
    # Discount factor nodes at 6M, 5Y and 10Y
    nodes={dt(2000, 1, 1): 1.0, dt(2000, 7, 1): 1.0, dt(2005, 1, 1): 1.0, dt(2010, 1, 5): 1.0},
    # Specify knots for a cubic spline
    t=[dt(2000, 1, 1), dt(2000, 1, 1), dt(2000, 1, 1), dt(2000, 1, 1), 
       dt(2000, 7, 1), dt(2005, 1, 1), 
       dt(2010, 1, 5), dt(2010, 1, 5), dt(2010, 1, 5), dt(2010, 1, 5)],
)
solver = Solver(
    curves=[usd],
    instruments=[
        IRS(dt(2000, 1, 4), "1b", spec="usd_irs", curves=usd),
        (FixedRateBond(dt(2000, 1, 1), "5y", spec="ust", curves=usd, fixed_rate=2.0), (), {"metric": "ytm"}),
        (FixedRateBond(dt(2000, 1, 1), "10y", spec="ust", curves=usd, fixed_rate=2.0), (), {"metric": "ytm"}),
    ],
    s=[5.25, 4.52, 4.51]  # <- SOFR rate + US treasury yields
)
SUCCESS: `func_tol` reached after 5 iterations (levenberg_marquardt), `f_val`: 4.9611913959284605e-12, `time`: 0.0374s
```

This will produce an overnight rates curve that looks as follows:

```
usd.plot("1b")
```

#### GBP Curve

Now is the tricky part. You are making 2 key assumptions that are very poor in practice:

- Treasury rates are affecting FX parity calculations. (RFR rates are usually used for FX parity calculations not Treasury rates, since theTreasury and RFR rates diverge over tenors)

- The cross currency basis is zero. FX parity equations are usually impacted by currency basis but you have none of that information so we assume it is zero. It can be quite significant, however.

So we will build a GBP curve that tries to assert FX parity under these assumptions.

```
gbp = Curve(
    # Same discount factor nodes at 6M, 5Y and 10Y
    nodes={dt(2000, 1, 1): 1.0, dt(2000, 7, 1): 1.0, dt(2005, 1, 1): 1.0, dt(2010, 1, 5): 1.0},
    # Same spline knot points
    t=[dt(2000, 1, 1), dt(2000, 1, 1), dt(2000, 1, 1), dt(2000, 1, 1), 
       dt(2000, 7, 1), dt(2005, 1, 1), 
       dt(2010, 1, 5), dt(2010, 1, 5), dt(2010, 1, 5), dt(2010, 1, 5)],
)
# Create an FX forward market from spot and the rates curves.
fxf = FXForwards(
    fx_rates=FXRates({"gbpusd": 1.275}, settlement=dt(2000, 1, 1)),
    fx_curves={"gbpgbp": gbp, "usdusd": usd, "gbpusd": gbp}
)
# Solve everything matching all the known instruments.
solver2 = Solver(
    pre_solvers=[solver],
    curves=[gbp],
    instruments=[
        IRS(dt(2000, 1, 4), "5b", spec="gbp_irs", curves=gbp),
        (FixedRateBond(dt(2000, 1, 1), "10y", spec="ukt", curves=gbp, fixed_rate=2.0), (), {"metric": "ytm"}),
        FXExchange(pair="gbpusd", settlement=dt(2001, 1, 1), curves=[None, gbp, None, usd]),
        FXExchange(pair="gbpusd", settlement=dt(2002, 1, 1), curves=[None, gbp, None, usd]),
        FXExchange(pair="gbpusd", settlement=dt(2003, 1, 1), curves=[None, gbp, None, usd]),
    ],
    s=[5.25, 4.32, 1.276, 1.279, 1.281],
    fx=fxf,
)
SUCCESS: `conv_tol` reached after 10 iterations (levenberg_marquardt), `f_val`: 1.3783402897407275e-06, `time`: 0.0493s
```

The process of solving tries to find a suitable GBP curve that matches all of the known information and will reprice the known GBP rates and also solve the FX parity equations (albeit not exactly in this case since the curve is overspecified)

Your curves now look like this:

```
usd.plot("1b", comparators=[gbp], labels=["usd", "gbp"])
```

With this curve you can estimate the yield of the 5y Gilt (and convert that to a potential futures price)

```
FixedRateBond(dt(2000, 1, 1), "5y", spec="ukt", fixed_rate=2.0).rate(gbp, metric="ytm")
# 4.296175
```

So the best guess for all this information is 4.296%.