1. Simulating Delta Hedging
Matt Davison
Universita degli studi di Verona May

2. 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

3. 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

4. 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.

5. 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.

6. 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.

7. 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.

8. 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

9. Matlab code I (note: don’t trust 100%)
%% Test Options Hedging Simulator
% dS = r*S dt + sigma*S dW, S(0) = Szero (Risk neutral dynamics)
% r = 5%, sigma = 20%, solve on interval [0,10]
%randn('state',100)
r = 0.05; sigma = 0.2; Szero = 10; %problem parameters
T =1; N = 10000; dt = T/N; K = 10*exp(r*T);
M = 100;
HedgePL = zeros(1,M);
for j = 1:M
dW = sqrt(dt)*randn(N); %Brownian increments
W = cumsum(dW); % discretized Brownian path
R = Szero*exp((r - 0.5*sigma^2)*([dt:dt:T])+sigma*W);
ST = R(N);
S = [Szero,R(1:N-1)];

10. Matlab code 2 %Vectorized calculation of Delta
% gives Delta(0)to Delta(T-dt))
top = log(S/K) + (r + 0.5*sigma^2)*(T:-dt:dt);
bottom = sigma*sqrt(T:-dt:dt);
d1 = top./bottom;
d2 = d1 - sigma*sqrt(T:-dt:dt);
Delta = 1/2 + 1/2*erf(d1/sqrt(2));
Nd2 = 1/2 + 1/2*erf(d2/sqrt(2));
% normsdist(x) = 1/2(1 + erf(x/sqrt(2)

11. Matlab Code 3
CallPrice = S(1)*Delta(1) - K*exp(-r*T)*Nd2(1); balance = S(1)*Delta(1)-CallPrice; % Initial cash balance in trade for k = 1:N-1 cash(k) = balance*exp(r*dt) + (Delta(k+1)-Delta(k))*S(k+1); balance = cash(k); end % final amount will be cash - K if stock expires in the money then option pays S-K but -S needs to be paid to repay % Delta(N)=1 short or cash if stock expires out of the money, then call is worthless and % short position is D(N) = 0. if ST > K cash(N) = cash(N-1)*exp(r*dt) -K; else cash(N) = cash(N-1)*exp(r*dt); HedgePL(j) = cash(N);

12. Possible project
Simulate the hedging done by the Black Scholes formula but with prices coming from a different random variable.
More advanced: consider simulating two correlated price time series and hedging an Exchange option priced on them. Compute all this using the standard Black Scholes framework. Then test hedge against stochastic process governed by another price process.