# 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:

1. Assuming a hedging window from t0 to tN .
2. 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.
3. 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.
4. 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.
5. Cumulative hedging error/ p/l out o strategy:
HEi = δi−1Si + Bi−1 exp(ri−1∆t) − Vi
where Bi = δi−1Si + Bi−1
exp(ri−1δt) − δiSi (i ≥ 1) and B0 = V0 − δ0S0.
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,