QuantCandy's Blog

This is great, free textbook Applied Analysis by John Hunter, Bruno Nachtergaele
here.   The book covers the topics:

Metric and Normed Spaces
Continuous Functions
The Contraction Mapping Theorem
Topological Spaces
Banach Spaces
Hilbert Spaces
Fourier Series
Bounded Linear Operators on a Hilbert Space
The Spectrum of Bounded Linear Operators
Linear Differential Operators and Green’s Functions
Distributions and the Fourier Transform
Measure Theory and Function Spaces
Differential Calculus and Variational Methods

Good grief!  Here I am on Memorial Day stuck with two massive assignments in credit derivatives which require simulation in matlab.  A friend emailed me about something and I was going through some old notes from a class I took at the University of Chicago (Financial Math program) taught by the illustrious Jack Cowan

As I was going through the notes, I saw some old matlab homework I’d done.  I didn’t even remember this, it was some kind of neural network code.  And at the top of the homework I had written, “Memorial Day 1999”.  Good grief!  Here it is 9 years later and I am still doing matlab on Memorial Day.  Guess it’s my Memorial Day tradition.  Hey, who needs parades and picnics when you can be coding up some matlab simulations.

“I’ve never seen a martingale, I never hope to see one.  But I can tell you anyhow, I’d rather see than be one.”

Once I finish this massive set of homework for credit derivatives (two days into it and I am only on about item #5 of 15 for one homework, have not even started the second assignment!) I hope I never have to see a martingale again in my life.   (That’s affine martingale you’ve got there, missy.) 

Commencement was excellent!  Thank you, Columbia and Columbia Video Network!  Now on to the PhD…

http://www.columbia.edu/cu/ceremonies/commencement/docs/events/commencement/index.html

Upcoming Online Information Sessions:
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This simple matlab code was for one of the homework problems. I wanted to generate n monte carlo paths, but have each one colored at random. I could modify this to rank order the final values and color accordingly, but was in a hurry to submit it. This code calculates the P&L from hedging at implied volatility but the underlying converges to realized volatility.

 
clf
num_paths = 100; % number of paths to average
hold on
kfrac = 0.95 ; % K = 95% of stock
rf = 0.10; % risk free rate, per annum
sig_i = 0.10; % implied volatility per annum
sig_r = 0.30; % realized volatility per annum
d = 0; % dividend yield, percent
T = 0.5; % time to expiry, years
num_days = 250; %number of days in year
mu = 0; % drift, percent per annum
S0 = 100; % initial stock price
%% assume stock price follows GBM dS = mu S dt + sigma*S*dZ where dZ is a
%% draw from standard normal distribution
hedging_frequency = 1; % hedging frequency in days, so 1 corresponds to daily
dt = hedging_frequency/(T*num_days); %% eg 1/250 for daily over a one year period
t = [0:hedging_frequency:T*num_days];
%% evolve stock price according to GBM
S=zeros(j,T*num_days);
gamma = S; %% initialization
profit = S;
totalprofit = zeros(1,num_paths);
for j=1:num_paths
profit(1,j) = 0;
S(1,j) =S0;
d1 = (log(1/kfrac)+(rf-d+0.5*sig_i^2*T))/(sig_i*sqrt(T));
gamma(1) = exp(-d1^2/2)/(S(1)*sig_i*sqrt(2*pi*T));
for i = 1:T*num_days-1
S(i+1,j) =S(i,j)*exp((rf-1/2*sig_i^2)*dt + sig_i*randn*sqrt(dt));
tau = T*num_days – i*dt;
d1 = (log(1/kfrac)+(rf-d+0.5*sig_i^2*tau))/(sig_i*sqrt(tau));
gamma(i+1,j) = exp(-d1^2/2)/(S(i,j)*sig_i*sqrt(2*pi*tau));
profit(i+1,j) =profit(i,j)+ gamma(i,j)*S(i,j)^2*(sig_r^2-sig_i^2)*1/T*exp(-rf*tau);
end

plot(S(:,j),’Color’,[0.01+.99*abs(rand),0.01+.99*abs(rand),0.01+.99*abs(rand)])
%%title(‘Simulated Stock Price paths by GBM’);
totalprofit(j) = 1/2*profit(i,j);
end
mean(totalprofit)
%%printf(‘Average expected profit = $%n’,mean(profit))

A downside of listening to lectures on the go is that you can get so caught up in them, you miss your stop.  That happened to me the other night when I was so fascinated by a point Derman was making on smile-consistent models that I missed my stop and ended up having to walk home almost 5 miles because I was too stubborn to wait for the train going back to where I could make the right connection. 

It’s not that I like to multitask but I have to multitask.  I don’t want to give up my morning swim, so I ordered a waterproof video iPod case from h20audio so I can study while I swim. I haven’t tested it yet but plan on it soon.

I just came across this album while – what else – procrastinating instead of studying. It makes a great accompaniment to Derman homework, though. Check it out: www.songsforicecreamtrucks.com.
I guarantee this will put a smile on your face, even in the dead of winter when you are really behind in homework. (What else is new?)