Calibrating the CEV Model and Accounting for Jump Risk
Summary
The document asks how to estimate the CEV model’s parameters, which determine the scale and price dependence of stock volatility, for simulations of CPPI and OBPI portfolio strategies. The answer recommends calibrating the model to vanilla equity or index option prices using its closed-form pricing solution, rather than estimating parameters from historical stock data alone. It notes that implied volatility skew can be reflected by a CEV elasticity parameter below one.
The response also highlights a limitation for portfolio insurance strategies: jumps can cause losses that continuous diffusion models may not capture, including a CPPI cash lock after an overnight move. It suggests considering a Merton jump-diffusion model, or a Bates model when stochastic volatility is also needed, and notes that a Merton model with deterministic equity volatility may be more convenient alongside stochastic interest rates. These are recommendations, not a fitted model or empirical comparison; the document does not provide data requirements, calibration results, or detailed estimation steps.
Key ideas
- The CEV model links stock volatility to the stock price through its parameters.
- The answer recommends fitting CEV parameters to vanilla equity or index option prices.
- Option skew may be represented by a CEV elasticity parameter below one.
- Jump risk can materially affect CPPI and OBPI simulations, especially through abrupt losses.
- Jump-diffusion alternatives may be more suitable when jumps or stochastic volatility matter.
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Full text
# Calibration/estimation of the CEV model
# Calibration/estimation of the CEV model
The CEV model for a stock price $S(t)$, interest rate $r$ and variance $\delta$
$dS(t)=rS(t)dt+\delta S(t)^{\gamma}dW(t)$
where the volatility for the stock is given by $\sigma(t)=\delta S(t)^{\gamma -1}$
Is there any method for calibration/parameter estimation of: $\gamma$ and $\delta$? And what historical data will I need for this purpose?
Note: I will use a stochastic $r$ instead, hence $r(t)$. But that is another problem.
The whole purpose is to simulate the two portfolio strategies: CPPI and OBPI.
CPPI: consists of risky asset (stock) and risk free (zero coupon bond)
OBPI: consists of risky asset (stock) and a put option of it.
If something is unclear let me clarify.
Hope you can help me out.
Edit: More info. I will price the call option via the CEV model. Then I will use the put-call parity for obtaining the put-price.
Moreover, I will price the ZCB (zero coupon bond) via the SDE describing the interest rate. This is as mentioned not decided yet. But for instance via the CIR process or Vasicek.
Maybe this additional info can make it easier to answer me.
## Answer by user34971 (score 3)
https://quant.stackexchange.com/a/68938
The CEV model has closed form solutions. See for example Schorder's paper.
Models are typically calibrated to vanilla equity or equity index options, and not to historical data. So you can use the closed-form solution of the CEV model to fit it to vannilla options data. As these exhibit skew, the $\gamma$ will probably be less than 1.
In my experience, for OBPI and CPPI the jump component is not to be ignored. Afterall, an overnight jump may lead to a potential cash-lock of your CPPI. Instead of CEV, personally I would probably choose the Merton jump-diffusion model instead (or the Bates model if you also want stochastic volatility). I think MJD with deterministic equity volatility may also be easier to use with stochastic interest rates than the CEV model.
Hope this helps.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.