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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.

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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.

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  • $\begingroup$ thanks for your answer. I will do a masterthesis. So I thought about describing the stock with the CEV model and to price options. What do you mean "not to historical data". Should I not look back like 12 months for an option to calibrate gamma and delta and then use this to plug into the volatility formula I provided? Furthermore, I will use stochastic interest rate to desire "r". $\endgroup$
    – Gaussen
    Nov 29 '21 at 8:50
  • $\begingroup$ Models are usually calibrated to options market prices. Delta and gamma are not calibrated directly, they can be calculated once you have calibrate d the model. I suggest you discuss with your thesis advisor the best way to proceed. $\endgroup$ Nov 29 '21 at 9:01
  • $\begingroup$ of course I will, I just don't understand what you mean by "can be calculated once you have calibrate d the model"? The main calibration is, for the CEV, to determine gamma and delta from older option prices, right? $\endgroup$
    – Gaussen
    Nov 29 '21 at 9:25
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    $\begingroup$ @Gaussen he means once you have your calibration, you have the pricing function $P(S_{t},r,K,t)$; then delta, for example, can be calculated as: $(P(S_{t}*1.001,r,K,t)-P(S_{t},r,K,t)/0.001*S_{t})$ and so forth. You seem to be confused in very basic things and as Frido suggested, please speak to your advisor. $\endgroup$
    – user121416
    Nov 30 '21 at 7:50
  • $\begingroup$ No, I am not confused. I just wondered some basic stuff regarding the calibration. I asked how I can get $\delta$ and $\gamma$. I wondered what kind of historical data will I use for this purpose. Example call options on the pfizer stock or what? Then I can just use least square estimates or something similar to minimise as: min(Call_market - Call_Cev)^2 to obtain $\delta$ and $\gamma$. In some literature they say I get $\delta$ and $\gamma$ directly. Therefore I am confused. But thanks for your answer :) $\endgroup$
    – Gaussen
    Nov 30 '21 at 9:04

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