# Pricing forward start Cliquet option with implied volatility with Dupire

I have the following implied volatility matrix of a stock index downloaded the 15th February 2019, the value of the stock was 3188.44 at the time:

           0.800 0.900 0.950 0.975 1.000 1.025 1.050 1.100 1.200
2019-02-22 34.74 26.15 21.39 17.00 13.52 11.83 12.21 14.22 17.66
2019-03-01 31.90 26.66 19.65 16.28 13.57 11.62 11.39 12.07 15.20
2019-03-08 34.31 24.62 18.50 15.82 13.61 11.65 11.39 13.23 15.41
2019-03-15 35.03 22.84 17.83 15.43 13.46 11.59 11.00 11.46 12.28
2019-04-18 27.60 20.07 16.66 15.16 13.76 12.33 11.56 11.70 11.97
2019-05-17 24.54 18.46 15.69 14.45 13.23 12.23 11.75 11.77 12.31
2019-06-21 23.05 17.75 15.45 14.41 13.40 12.60 12.14 12.04 12.67
2019-07-19 22.47 17.53 15.49 14.64 14.70 12.92 12.36 12.21 12.99
2019-09-20 21.46 17.26 15.45 14.72 14.07 13.49 12.93 11.85 11.81
2019-12-20 20.53 17.06 15.62 14.96 14.31 13.71 13.18 12.40 12.21
2020-06-19 19.55 16.78 15.76 15.24 14.85 14.52 14.20 13.60 12.97
2020-12-18 19.01 16.86 16.08 15.68 15.33 15.06 14.81 14.43 13.87
2021-06-18 18.56 16.77 16.07 15.78 15.52 15.27 15.05 14.71 14.95
2021-12-17 18.39 17.94 16.43 16.18 15.95 15.74 15.55 15.23 14.75
2022-12-16 17.90 16.82 16.39 16.19 16.02 15.86 15.72 15.49 15.16
2023-12-15 17.66 16.79 16.41 16.26 16.12 16.00 15.89 15.71 15.46


the rows correspond to the maturity of the call option and the columns the moneyness of the strike. E.g. : the column 0.8 corresponds to a strike $$K$$ of $$0.8*S_0$$

In a world without interest rate and dividend, I am trying to price an infinite sum of yearly coupons given by:

$$C_0 \sum_{i=1}^{\infty} (\frac{S_T}{S_{T-1}}- 5\% * i)^+$$

with $$C_0 > 0$$

I was going to compute it numerically, hoping that the sum would rapidly converges.

Based on this answer: https://quant.stackexchange.com/a/21919/31546, I would like to simulate paths of the underlying, and then for each path compute the yearly returns for $$T = 1, 2, 3...$$.

I could then compute a numerical mean corresponding to:

$$C_0 * \mathbb{E}\left[ (\frac{S_T}{S_{T-1}}- 5\%)^+ + (\frac{S_T}{S_{T-1}}- 10\%)^+ +(\frac{S_T}{S_{T-1}}- 15\%)^+ + ... \right]$$

My underlying will have this dynamic in the risk-neutral world, since interest rate $$r=0$$

$$dS_t = \sigma(S_t, t) S_tt dW_t$$

My goal is to determine $$\sigma(S_t, t)$$ : to do so I will use Dupire volatility function given by : $$\sigma(S_t, t)^2 = 2*\frac{\frac{\delta C}{\delta T}} {K^2 \frac{\delta^2C}{\delta K^2}}$$

To get the derivatives I will interpolate my matrix of implied volatilities, in order to have a volatility for each step of one day of maturity, and steps of 0.005*S_0 for the strike.

Then for any points of my surface $$\sigma(K,t)$$ I can get a call price $$C^{BS}(K,T)$$ using the standard B-S formula. Applying the first and second order finite differences methods all along my surface I will get the derivatives.

My question is: Can I price the product correctly if I run 1000's paths of the underlying $$S$$ using my Dupire volatility function?

This is part of an homework. So I would be grateful if someone could point what is wrong in my reasonning.

• Local volatility is not a good model for forward start options. Comment back at me and I'll write up why tomorrow. – will May 11 '19 at 23:04
• Hello @will thanks for the comment. indeed multiple answers on this site point that volatility should be stochastic as well. But this is an homework question and it should be treated using deterministic Dupire volatility function. I am sure that there are many issues in my answer (e.g. I do not plan to treat butterfly arbitrage yet), I was wondering if I was on the right track. – RandowMalk May 12 '19 at 4:35
• fair enough. So, to begin with, your pricing function should be orthogonal to the process which generate the paths, so i would begin by making a pure BS diffuser and trying to price on that. Then move onto local vol. For the time differentials in duoire you'll need to interpolate your surface in the time space, there's a technique in gatheral's paper on arbitrage free vol surfaces that is reccomend for this. After you have your surface, check that you reprice all the vanillas used to calibrate it. – will May 12 '19 at 9:32