Calibration exercise

A desk does not choose Heston's parameters; the market does. Here is a market: implied volatilities at five strikes and four maturities, produced by parameters you cannot see. Tune the five sliders until the model's smiles sit on the market's. Your score is the root-mean-square miss in volatility points. When you have had enough, let the optimiser try, then reveal the answer.

Your parameters

Today's volatility √v0: 20.0%

Long-run volatility √θ: 20.0%

Mean reversion κ: 2.0

Vol of variance σ: 0.30

Correlation ρ: -0.50

Hints: √v0 sets the short-dated level and √θ the long-dated one; κ decides how fast one becomes the other; ρ tilts the smile; σ bends it. Market: spot 100, rate 5.0%, yield 0.0%.

Try this

0 of 3 done

Level first

Set √v0 so the shortest smile sits at the right height, then √θ for the two-year smile.

What to look for: Most of the error disappears with two sliders. The term structure of at-the-money volatility is the first thing a calibration pins down.

Then the tilt

Move ρ until the left and right ends of the smiles miss by the same amount, then use σ for the curvature.

What to look for: ρ rotates the smile and σ bends it. When the miss table shows a symmetric pattern the correlation is right and only σ is left.

Beat the machine

Get under one vol point by hand, then let the optimiser run from your parameters and compare.

What to look for: Nelder-Mead usually shaves the last few tenths off, but it starts from where you left it: a good hand fit is what makes a fast automatic one.

Current score

9.06

vol points, root-mean-square over 20 quotes

Best so far

–

0 attempts

Verdict

Far off. Start with √v0 and √θ to get the level of the short and long maturities right.

Miss by quote, in vol points

T \ K	80	90	100	110	120
0.25y	+10.17	+10.23	+9.98	+7.63	+5.36
0.5y	+10.02	+10.08	+9.86	+8.14	+6.03
1y	+9.85	+9.88	+9.66	+8.70	+7.20
2y	+9.70	+9.67	+9.48	+9.03	+8.32

Positive means your model is too high at that quote. Look for patterns: a whole row off means the term structure; the left column off means the skew.

Beyond Black-Scholes, step 9 of 9

Calibrate a market

Give the class ten minutes with the sliders. Then run the optimiser and reveal. Discuss why kappa and sigma trade off against each other.