# Dynamic-Delta-Hedging

Implemented a dynamic delta hedging strategy for google stock.

Delta-hedging is a hedging strategy that aims to replicate the value of a financial derivative, such as a

Call option, written on a traded asset through dynamically managing a proper number of shares of

the underlying asset and risk free security. The complete project is written in C++ with keeping appropriate object oriented design considerations in mind.

The delta hedging process:

- Assuming a hedging window from t0 to tN .
- At t0, we sell a European call option contract with expiration date T, strike price K. Assuming the option is written on one share of stock and t0 < tN ≤ T.
- To hedge the short position in the European call and replicate a portfolio which will pay us the same payoff, we decide to buy δ shares of the underlying stock

at t0, where δ =

∂V

∂S is the rate of change of option value V with respect to changes in the underlying

price S. - As δ changes during the hedging period, we need to re-balance our portfolio everyday. Denoting δi as value of δ on ith day of the trading,

Every δi has to be calculated using implied volatility for each date. - Cumulative hedging error/ p/l out o strategy:

HEi = δi−1Si + Bi−1*exp(ri−1∆t) − Vi*exp(ri−1δt) − δiSi (i ≥ 1) and B0 = V0 − δ0S0.

where Bi = δi−1Si + Bi−1

Si, Vi, ri denoting the stock price, option price, risk-free rate at time ti

, i. ∆t represents 1 business day, which is 1/252 year.

The model used for pricing option is the Black)Scholes model, where the underlying stock moves in the following fashion:

Use the following model to simulate the price series {S0, S∆t, S2∆t, · · · , ST } at N equally-spaced

time points over time horizon [0, T] where ∆t = T/N:

St+∆t = St + µSt∆t + σSt*(√∆t)*Zt,

Complete following two tasks:

- Assuming some initial conditions, generate 100 sample paths (simulations), price delta, hedging error for each of them for every day, and calculate a final P/L for each path. (Result generated in the output csv files)
- Use the real market data from given files in the data folder, calculate Implied volatility, delta, hedging error everyday for the real time market data everyday for a given observation window.
- Calculate implied volatility smile for options with different Strike price