Pricing a Forward-Start Cliquet with a Dupire Local Volatility Model

# Pricing forward start Cliquet option with implied volatility with Dupire


# 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.