Simulating Delta Hedging Matt Davison Universita degli studi di Verona May 24 2018

Delta Hedging If you sell a call you hedge it (in the Black-Scholes world) by Buying Δ = N(d1) shares of the underlying security Dynamically (continuously) rebalancing that Delta ratio

In practice You don’t rebalance continuously, but instead daily or even less frequently The assumptions for Black Scholes don’t hold, in particular Trading is not cost free Stocks don’t follow GBM

How much does this matter? To get better insights about Delta hedging And to investigate the impact of violating some of its assumptions We build a simulation model.

Spreadsheet logic We basically just implement Delta hedging. We simulate stock prices, using the exact solution of GBM, on a daily basis. (or whatever time period we want). At each time step we buy or sell stocks according to the trading rule given by the Delta hedging algorithm. So, if we are short the call, we buy (positive)/sell(negative) Δ(Sk+1,tk+1) - Δ(Sk,tk) – shares of stock at Sk+1 For proceeds [Δ(Sk+1,tk+1) - Δ(Sk,tk) ] Sk+1 When transaction costs are included, we need to incorporate the sign of this trade to allow for the fact that we get less than this for a sale/pay more than this for a buy.

At the final time We cover the option position. Then we can reckon the profit/loss of the trading sequence (P/L) incorporating the interest we get on our bank account. We can see how this works on various stock price simulations.

Stylized results With no transaction costs, The more frequently you rebalance, the closer the present value of the option trading strategy gets to the Black Scholes prices This makes sense because in the infinite trading limit it converges to the BS price Transaction costs really kill you as you trade more and more, even if they are small.

Changing the random process In the exact solution of GBM there is a place to put Zt, an N(0,1) draw. We can replace this with a Uniform draw (calibrated to have mean 0 and standard deviation 1 still) to model thin tailed stock price moves. This is 2*√3(U(0,1)-0.5) Or a double exponential, again with mean 0 and s.d. 1, to model fat tailed price moves. Here the exponential has θ = 1/√2