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Derivatives Analytics with Python & Numpy

Dr. Yves J. Hilpisch

24 June 2011

EuroPython 2011

# CV  Yves Hilpisch

1  1993–1996 Dipl.-Kfm. (“MBA”) at Saarland University (Banks and Financial Markets)

2  1996–2000 . (“Ph.D.”) at Saarland University (Mathematical Finance)

3  1997–2004 Management Consultant Financial Services & Insurance Industry

4  2005–present Founder and MD of Visixion GmbH

I     management and technical consulting work

I DEXISION—Derivatives Analytics on Demand ()

5  2010–present Lecturer Saarland University

I     Course: “Numerical Methods for the Market-Based Valuation of Options”

I Book Project: “Market-Based Valuation of Equity Derivatives—From Theory to

Implementation in Python”

Derivatives Analytics and Python

Data Analysis

Time Series

Cross-Sectional Data

Monte Carlo Simulation

Model Economy

European Options American Options

Speed-up of 480+ Times

DEXISION—Our Analytics Suite

Capabilities

Technology

Derivatives Analytics is concerned with the valuation, hedging and risk management of derivative financial instruments

In contrast to ordinary financial instruments which may have an intrinsic value (like the stock of a company), derivative instruments derive their values from other instruments

Tyical tasks in this context are

I simulation

I data analysis (historical, current, simulated data)

I discounting

I arithmetic operations (summing, averaging, etc.)

I linear algebra (vector and matrix operations, regression)

I solving optimization problems I visualization I

Python can do all this quite well—but C, C++, C#, Matlab, VBA, JAVA and other languages still dominate the financial services industry

Why Python for Derivatives Analytics?

1  Open Source: Python and the majority of available libraries are completely open source

2  Syntax: Python programming is easy to learn, the code is quite compact and in general highly readable (= fast development + easy maintenance)

3  Multi-Paradigm: Python is as good at functional programming as well as at object oriented programming

4  Interpreted: Python is an interpreted language which makes rapid prototyping and development in general a bit more convenient

5  Libraries: nowadays, there is a wealth of powerful libraries available and the supply grows steadily; there is hardly a problem which cannot be easily attacked with an existing library

6  Speed: a common prejudice with regard to interpreted languages—compared to compiled ones like C++ or C—is the slow speed of code execution; however, financial applications are more or less all about matrix/array manipulations and other operations which can be done at the speed of C code with the essential library Numpy

What does the financial market say about Python?

in the London area (mainly financial services) the number of Python developer contract offerings evolved as follows (respectively for the three months period ending on 22 April)

I 142 in year 2009

I 245 in year 2010

I 644 in year 2011 these figures imply a more than fourfold demand for the Python skill in 2011 as compared to 2009

over the same period, the average daily rate for contract work increased from 400 GBP to 475 GBP

obviously, Python is catching up at a rapid pace in the financial services industry

# In Derivatives Analytics you have to analyze different types of data

Fundamental types of data to be analyzed

I time series

I cross sections

Python libraries suited to analyze and visualize such data

I xlrd ): reading data from Excel files

I Numpy (): array manipulations of any kind

I Pandas (): time series analysis, cross-sectional data analysis

I matplotlib (): 2d and 3d plotting

# DAX time series—index level and daily log returns3

3

Source: , 29 Apr 2011

from xlrd import open_workbook from pandas import * from datetime import * from matplotlib.pyplot import * from numpy import *

xls = open_workbook('') for s in xls.sheets():

datesDAX = []; quoteDAX = [] for row in range(s.nrows-1,0,-1): year = int(s.cell(row,0).value) month = int(s.cell(row,1).value) day          = int(s.cell(row,2).value) datesDAX.append(date(year,month,day)) quoteDAX.append(float(s.cell(row,8).value)) print

# and plotting it with Pandas and matplotlib

DAXq = Series(quoteDAX,index=datesDAX)

DAXr = Series(log(DAXq/DAXq.shift(1)),index=datesDAX)

DAXr = where(isnull(DAXr),0.0,DAXr)

# Data Frames for Quotes and Returns data = {'QUO':DAXq,'RET':DAXr,'RVO':rv} DAX = DataFrame(data,index=DAXq.index)

# Graphical Output

figure() subplot(211) plot(DAX.index,DAX['QUO']) ylabel('DAX Daily Quotes') grid(True);axis('tight') subplot(212) plot(DAX.index,DAX['RET']) ylabel('DAX Daily Log Returns') grid(True);axis('tight')

# DAX time series—252 moving mean return, volatility and correlation between both4

Source: , 29 Apr 2011

# Pandas provides a number of really convenient functions

# mean return, volatility and correlation (252 days moving = 1 year) figure() subplot(311)

mr252 = Series(rolling_mean(DAX['RET'],252)*252,index=DAX.index) ();grid(True);ylabel('Return (252d Mov)')

x,y=REG(mr252,0);plot(x,y)

subplot(312)

vo252 = Series(rolling_std(DAX['RET'],252)*sqrt(252),index=DAX.index) ();grid(True);ylabel('Vola (252d Mov)')

x,y=REG(vo252,0);plot(x,y);vx=axis()

subplot(313)

co252 = Series(rolling_corr(mr252,vo252,252),index=DAX.index) ();grid(True);ylabel('Corr (252d Mov)')

x,y=REG(co252,0);plot(x,y);cx=axis()

axis([vx,vx,cx,cx])

Cross-Sectional Data

# DAX cross-sectional data—implied volatility surface

maturities: 21 (red dots), 49 (green crosses), 140 (blue triangles), 231 (yellow stones) and 322 days (purple hectagons)

Model Economy

# Model economy—Black-Scholes-Merton continuous time

economy with final date T,0 < T <

uncertainty is represented by a filtered probability space {Ω,F,F,P} for 0 ≤ tT the risk-neutral index dynamics are given by the SDE

(1)

St index level at date t, r constant risk-less short rate, σ constant volatility of the index and Zt standard Brownian motion

the process S generates the filtration F, i.e. Ft ≡ F(S0≤st) a risk-less zero-coupon bond satisfies the DE

(2)

the time t value of a zero-coupon bond paying one unit of currency at T with

0 ≤ t < T is Bt(T) = er(Tt)

Model Economy

# Model economy—Black-Scholes-Merton discrete time

to simulate the financial model, i.e. to generate numerical values for St, the SDE

(1) has to be discretized

to this end, divide the given time interval [0,T] in equidistant sub-intervals ∆t such that now t ∈ {0,t,2∆t, ,T}, i.e. there are M +1 points in time with MT/t a discrete version of the continuous time market model (1)–(2) is

(3)

(4)

for t ∈ {∆t, ,T} and standard normally distributed zt

this scheme is an Euler discretization which is known to be exact for the geometric Brownian motion (1)

Option valuation by simulation—European options  a European put option on the index S pays at maturity T

h(ST) ≡ max[KST,0]

for a fixed strike price K

to value such an option, simulate I paths of St such that you get I values ST,i,i ∈ {1, ,I}

the Monte Carlo estimator for the put option value then is

# Simulating the index level for European option valuation

20 simulated index level paths; thick blue line = average drift

# Numpy offers all you need for an efficient implementation (I)

#

# Valuation of European Put Option

# by Monte Carlo Simulation

#

from numpy import *

from numpy.random import standard_normal,seed from time import time t0=time()

## Parameters -- American Put Option

S0 = 36.                                                  # initial stock level

K = 40. # strike price T = 1.0 # time-to-maturity vol= 0.2 # volatility r = 0.06 # short rate

## Simulation Parameters

seed(150000)                                   # seed for Python RNG

M = 50                                            # time steps

I = 50000                                           # simulation paths

dt = T/M                                                 # length of time interval

df = exp(-r*dt) # discount factor per time interval

# Numpy offers all you need for an efficient implementation (II)

## Index Level Path Generation

S=zeros((M+1,I),'d')                                                             # index value matrix

S[0,:]=S0                                                                                # initial values

for t in range(1,M+1,1): # stock price paths ran=standard_normal(I) # pseudo-random numbers

S[t,:]=S[t-1,:]*exp((r-vol**2/2)*dt+vol*ran*sqrt(dt))

## Valuation

h=maximum(K-S[-1],0)     # inner values at maturity V0=exp(-r*T)*sum(h)/I  # MCS estimator

## Output

print"Time elapsed in Seconds              %8.3f" %(time()-t0) print"----------------------------------------" print"European Put Option Value %8.3f" %V0 print"----------------------------------------"

# American options—solving optimal stopping problems (I)

to value American options by Monte Carlo simulation, a discrete optimal stopping problem has to be solved:

V0 =             sup                           E                                               (5)

τ∈{0,t,2∆t, ,T}

it is well-known that the value of the American option at date t is then given by

Vt(s) = max[ht(s),Ct(s)]                                                 (6)

i.e. the maximum of the payoff ht(s) of immediate exercise and the expected payoff Ct(s) of not exercising; this quantity is given as

Ct(s) = EQt (ertVt+∆t(St+∆t)|St = s)                                         (7)

American options—solving optimal stopping problems (II) problem: given a date t and a path i, you do not know the expected value in

(7)—you only know the single simulated continuation value Yt,i

solution of Longstaff and Schwartz (2001): estimate the continuation values Ct,i by ordinary least-squares regression—given the I simulated index levels St,i and continuation values Yt,i (use cross section of simulated data at date t)  their algorithm is called Least Squares Monte Carlo (LSM)

# implementation is straightforward (I)

#

# Valuation of American Put Option

# with Least-Squares Monte Carlo

#

from numpy import *

from numpy.random import standard_normal,seed from matplotlib.pyplot import * from time import time t0=time()

## Simulation Parameters

seed(150000)                                  # seed for Python RNG

M = 50                                            # time steps

I = 4*4096   # paths for valuation reg= 9                    # no of basis functions AP = True          # antithetic paths

MM = True                                      # moment matching of RN

## Parameters -- American Put Option

r = 0.06         # short rate vol= 0.2          # volatility S0 = 36.            # initial stock level

T = 1.0                                               # time-to-maturity

V0_right=4.48637 # American Put Option (500 steps bin. model) dt = T/M    # length of time interval

df = exp(-r*dt) # discount factor per time interval

# implementation is straightforward (II)

## Function Definitions def RNG(I):

if AP == True: ran=standard_normal(I/2) ran=concatenate((ran,-ran)) else:

ran=standard_normal(I) if MM == True:

ran=ran-mean(ran) ran=ran/std(ran)

return ran

def GenS(I):

S=zeros((M+1,I),'d')                                                  # index level matrix

S[0,:]=S0                                                                          # initial values

for t in range(1,M+1,1):                # index level paths ran=RNG(I)

S[t,:]=S[t-1,:]*exp((r-vol**2/2)*dt+vol*ran*sqrt(dt)) return S

def IV(S):

return maximum(40.-S,0)

# implementation is straightforward (III)

## Valuation by LSM

S=GenS(I) # generate stock price paths h=IV(S) # inner value matrix V=IV(S) # value matrix for t in range(M-1,-1,-1):

rg=polyfit(S[t,:],V[t+1,:]*df,reg)  # regression at time t C=polyval(rg,S[t,:])                     # continuation values

V[t,:]=where(h[t,:]>C,h[t,:],V[t+1,:]*df) # exercise decision V0=sum(V[0,:])/I # LSM estimator

## Output

print"Time elapsed in Seconds              %8.3f" %(time()-t0) print"----------------------------------------"

print"Right Value                                                                      %8.3f" %V0_right

print"----------------------------------------" print"LSM Value for Am. Option %8.3f" %V0 print"Absolute Error %8.3f" %(V0-V0_right)

print"Relative Error in Percent %8.3f" %((V0-V0_right)/V0_right*100) print"----------------------------------------"

# The Challenge—“dozens of minutes” in Matlab

realistic market models generally include multiple sources of randomness which are possibly correlated

the simulation of such complex models in combination with Least Squares Monte Carlo is computationally demanding and time consuming

in their research paper, Medvedev and Scaillet (2009) analyze the valuation of American put options in the presence of stochastic volatility and stochastic short rates

Medvedev and Scaillet (2009) write on page 16:

“To give an idea of the computational advantage of our method, aMatlab code implementing the algorithm of Longstaff and Schwartz (2001) takes dozens of minutes to compute a single option pricewhile our approximation takes roughly a tenth of a second.”

# The Results—“only seconds” in Python

Python is well-suited to implement efficent, i.e. fast and accurate, numerical valuation algorithms

I MCS/LSM with 25 steps/35,000 paths:

180 megabytes of data crunched in 1.5 seconds I MCS/LSM with 50 steps/100,000 paths:

980 megabytes of data crunched in 8.5 seconds

reported times are from my 3 years old notebook

the speed-up compared to the times reported in Medvedev and Scaillet (2009) is

480+ times (1.5 seconds vs. 720+ seconds) to reach this speed-up, our algorithm mainly uses variance reductions techniques (like moment matching and control variates) which allows to reduce the number of time steps and paths significantly

# Results from 3 simulation runs for the 36 American put options of Medvedev and Scaillet (2009)

---------------------------------------------------Start Calculations        2011-06-22 13:43:02.163000 ---------------------------------------------------Name of Simulation                       Base_3_25_35_TTF_2.5_1.5

Seed Value for RNG                                                                             150000

Number of Runs            3 Time Steps                        25 Paths        35000

Control Variates             True Moment Matching   True Antithetic Paths        False Option Prices            108

Absolute Tolerance                                                                               0.0250

Relative Tolerance                                                                                0.0150

Errors        0 Error Ratio                         0.0000 Aver Val Error        -0.0059 Aver Abs Val Error                       0.0154 Time in Seconds   135.7890 Time in Minutes                      2.2631

Time per Option            1.2573 ---------------------------------------------------End Calculations  2011-06-22 13:45:17.952000 ----------------------------------------------------

Capabilities

# DEXISION can handle a number of financial derivatives ranging from plain vanilla to complex and exotic

Example products:

I plain vanilla options

I American options

I Asian options

I certificates (bonus, express, etc.)

I swaps, swaptions

I real options

I portfolios of options

I life insurance contracts  Example underlyings:

I indices

I stocks

I bonds

I interest rates

I currencies

I commodities

Capabilities

# DEXISION can be beneficially applied in a number of areas

financial research: researchers, lecturers and students in (mathematical) finance find in DEXISION an easy-to-learn tool to model, value and analyze financial derivatives financial engineering: financial engineers and risk managers in investment banks, hedge funds, etc. can use DEXISION to quickly model and value diverse financial products, to cross-check valuations and to assess risks of complex derivatives portfolios  actuarial calculations: those responsible for the design, valuation and risk management of market-oriented insurance products can engineer, value and test new and existing products easily  financial reporting: IFRS and other reporting standards require the use of formal (option) pricing models when there are no market prices; DEXISION considerably simplifies the modelling, valuation and risk assessment for illiquid, complex, non-traded financial instruments and embedded options  real options valuation: DEXISION offers unique capabilities to account for the specifics of real options (as compared to financial options)

# DEXISION is based on a Python-LAMP environment and makes heavy use of Numpy

Suse Linux 11.1 as 64 bit operating system

Apache 2 as Web server

MySQL 5.0.67 as relational database

Python 2.6 as core language (integrated via mod_python in Apache)

Numpy 1.3.0 as fast linear algebra library

Dojo 1.0 as JavaScript framework for the GUI

SVG for all custom graphics

MoinMoin Wiki (Python powered) for Web documentation

# Our aim is to make DEXISION the Google of Derivatives Analytics

recently, Visixion added Web services to DEXISION’s functionalities which allow to integrate it into any environment once a structure is modeled in DEXISION, updates of valuations can be received in real-time via these Web services (with data delivered e.g. in XML format) during the Web service call, data/variables can also be provided a call to value an American put option on the DAX index could look like:

?company=X&user=Y&pwd= Z&paths=50000&steps=150&portfolio=DAX/DAX_Am_Put_Dec_2011&DAX_ current=7200&DAX_vola=0.175&rate=0.03&strike=6800

# Contact

Dr. Yves J. Hilpisch

Visixion GmbH

Rathausstrasse 75-79

66333 Voelklingen Germany

— Derivatives Analytics and Python Programming — Derivatives Analytics On Demand

E

T/F +49 3212 1129194

Source: all figures from on 24 April 2011.

 Notably, this library was developed by a hedge fund.

 Source: , 29 Apr 2011

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