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<h1>DiscreteHedging.cpp</h1>This is an example of using the QuantLib Monte Carlo framework.<p>
<div class="fragment"><pre class="fragment"><span class="comment">/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */</span>
<span class="comment">/* This example computes profit and loss of a discrete interval hedging</span>
<span class="comment"> strategy and compares with the results of Derman & Kamal's (Goldman Sachs</span>
<span class="comment"> Equity Derivatives Research) Research Note: "When You Cannot Hedge</span>
<span class="comment"> Continuously: The Corrections to Black-Scholes"</span>
<span class="comment"> http://www.ederman.com/emanuelderman/GSQSpapers/when_you_cannot_hedge.pdf</span>
<span class="comment"></span>
<span class="comment"> Suppose an option hedger sells an European option and receives the</span>
<span class="comment"> Black-Scholes value as the options premium.</span>
<span class="comment"> Then he follows a Black-Scholes hedging strategy, rehedging at discrete,</span>
<span class="comment"> evenly spaced time intervals as the underlying stock changes. At</span>
<span class="comment"> expiration, the hedger delivers the option payoff to the option holder,</span>
<span class="comment"> and unwinds the hedge. We are interested in understanding the final</span>
<span class="comment"> profit or loss of this strategy.</span>
<span class="comment"></span>
<span class="comment"> If the hedger had followed the exact Black-Scholes replication strategy,</span>
<span class="comment"> re-hedging continuously as the underlying stock evolved towards its final</span>
<span class="comment"> value at expiration, then, no matter what path the stock took, the final</span>
<span class="comment"> P&L would be exactly zero. When the replication strategy deviates from</span>
<span class="comment"> the exact Black-Scholes method, the final P&L may deviate from zero. This</span>
<span class="comment"> deviation is called the replication error. When the hedger rebalances at</span>
<span class="comment"> discrete rather than continuous intervals, the hedge is imperfect and the</span>
<span class="comment"> replication is inexact. The more often hedging occurs, the smaller the</span>
<span class="comment"> replication error.</span>
<span class="comment"></span>
<span class="comment"> We examine the range of possibilities, computing the replication error.</span>
<span class="comment">*/</span>
<span class="comment">// the only header you need to use QuantLib</span>
<span class="preprocessor">#include <ql/quantlib.hpp></span>
<span class="preprocessor">#ifdef BOOST_MSVC</span>
<span class="preprocessor"></span><span class="comment">/* Uncomment the following lines to unmask floating-point</span>
<span class="comment"> exceptions. Warning: unpredictable results can arise...</span>
<span class="comment"></span>
<span class="comment"> See http://www.wilmott.com/messageview.cfm?catid=10&threadid=9481</span>
<span class="comment"> Is there anyone with a definitive word about this?</span>
<span class="comment">*/</span>
<span class="comment">// #include <float.h></span>
<span class="comment">// namespace { unsigned int u = _controlfp(_EM_INEXACT, _MCW_EM); }</span>
<span class="preprocessor">#endif</span>
<span class="preprocessor"></span>
<span class="preprocessor">#include <boost/timer.hpp></span>
<span class="preprocessor">#include <iostream></span>
<span class="preprocessor">#include <iomanip></span>
<span class="keyword">using namespace </span>QuantLib;
<span class="preprocessor">#if defined(QL_ENABLE_SESSIONS)</span>
<span class="preprocessor"></span><span class="keyword">namespace </span>QuantLib {
<a name="a0"></a><a class="code" href="group__types.html#gb9c87440c314438e51a899a03d2442d0" title="integer number">Integer</a> sessionId() { <span class="keywordflow">return</span> 0; }
}
<span class="preprocessor">#endif</span>
<span class="preprocessor"></span>
<span class="comment">/* The ReplicationError class carries out Monte Carlo simulations to evaluate</span>
<span class="comment"> the outcome (the replication error) of the discrete hedging strategy over</span>
<span class="comment"> different, randomly generated scenarios of future stock price evolution.</span>
<span class="comment">*/</span>
<span class="keyword">class </span>ReplicationError
{
<span class="keyword">public</span>:
ReplicationError(Option::Type type,
<a name="a1"></a><a class="code" href="group__types.html#g14fb8fca43a68f4168654e1f9f7e22f7" title="continuous quantity with 1-year units">Time</a> maturity,
<a name="a2"></a><a class="code" href="group__types.html#g4bdf4bfe76b9ffa6fa64c47d8bfa0c78" title="real number">Real</a> strike,
<a class="code" href="group__types.html#g4bdf4bfe76b9ffa6fa64c47d8bfa0c78" title="real number">Real</a> s0,
<a name="a3"></a><a class="code" href="group__types.html#gaa95ab7fe66935e3f7535413fad2a7d3" title="volatility">Volatility</a> sigma,
<a name="a4"></a><a class="code" href="group__types.html#gede435af51236692b1107d7639581d39" title="interest rates">Rate</a> r)
: maturity_(maturity), payoff_(type, strike), s0_(s0),
sigma_(sigma), r_(r) {
<span class="comment">// value of the option</span>
<a name="a5"></a><a class="code" href="group__types.html#g642a971a0bcbbd2fb26c35e1a06e5761" title="discount factor between dates">DiscountFactor</a> rDiscount = std::exp(-r_*maturity_);
<a class="code" href="group__types.html#g642a971a0bcbbd2fb26c35e1a06e5761" title="discount factor between dates">DiscountFactor</a> qDiscount = 1.0;
<a class="code" href="group__types.html#g4bdf4bfe76b9ffa6fa64c47d8bfa0c78" title="real number">Real</a> forward = s0_*qDiscount/rDiscount;
<a class="code" href="group__types.html#g4bdf4bfe76b9ffa6fa64c47d8bfa0c78" title="real number">Real</a> stdDev = std::sqrt(sigma_*sigma_*maturity_);
boost::shared_ptr<StrikedTypePayoff> payoff(
<span class="keyword">new</span> <a name="_a6"></a><a class="code" href="class_quant_lib_1_1_plain_vanilla_payoff.html" title="Plain-vanilla payoff.">PlainVanillaPayoff</a>(payoff_));
<a name="_a7"></a><a class="code" href="class_quant_lib_1_1_black_calculator.html" title="Black 1976 calculator class.">BlackCalculator</a> black(payoff,forward,stdDev,rDiscount);
std::cout << <span class="stringliteral">"Option value: "</span> << black.value() << std::endl;
<span class="comment">// store option's vega, since Derman and Kamal's formula needs it</span>
vega_ = black.vega(maturity_);
std::cout << std::endl;
std::cout << std::setw(8) << <span class="stringliteral">" "</span> << <span class="stringliteral">" | "</span>
<< std::setw(8) << <span class="stringliteral">" "</span> << <span class="stringliteral">" | "</span>
<< std::setw(8) << <span class="stringliteral">"P&L"</span> << <span class="stringliteral">" | "</span>
<< std::setw(8) << <span class="stringliteral">"P&L"</span> << <span class="stringliteral">" | "</span>
<< std::setw(12) << <span class="stringliteral">"Derman&Kamal"</span> << <span class="stringliteral">" | "</span>
<< std::setw(8) << <span class="stringliteral">"P&L"</span> << <span class="stringliteral">" | "</span>
<< std::setw(8) << <span class="stringliteral">"P&L"</span> << std::endl;
std::cout << std::setw(8) << <span class="stringliteral">"samples"</span> << <span class="stringliteral">" | "</span>
<< std::setw(8) << <span class="stringliteral">"trades"</span> << <span class="stringliteral">" | "</span>
<< std::setw(8) << <span class="stringliteral">"mean"</span> << <span class="stringliteral">" | "</span>
<< std::setw(8) << <span class="stringliteral">"std.dev."</span> << <span class="stringliteral">" | "</span>
<< std::setw(12) << <span class="stringliteral">"formula"</span> << <span class="stringliteral">" | "</span>
<< std::setw(8) << <span class="stringliteral">"skewness"</span> << <span class="stringliteral">" | "</span>
<< std::setw(8) << <span class="stringliteral">"kurtosis"</span> << std::endl;
std::cout << std::string(78, <span class="charliteral">'-'</span>) << std::endl;
}
<span class="comment">// the actual replication error computation</span>
<span class="keywordtype">void</span> compute(<a name="a8"></a><a class="code" href="group__types.html#gf38bdb4c54463b1f456655efa95b5c77" title="size of a container">Size</a> nTimeSteps, <a class="code" href="group__types.html#gf38bdb4c54463b1f456655efa95b5c77" title="size of a container">Size</a> nSamples);
<span class="keyword">private</span>:
<a class="code" href="group__types.html#g14fb8fca43a68f4168654e1f9f7e22f7" title="continuous quantity with 1-year units">Time</a> maturity_;
<a class="code" href="class_quant_lib_1_1_plain_vanilla_payoff.html" title="Plain-vanilla payoff.">PlainVanillaPayoff</a> payoff_;
<a class="code" href="group__types.html#g4bdf4bfe76b9ffa6fa64c47d8bfa0c78" title="real number">Real</a> s0_;
<a class="code" href="group__types.html#gaa95ab7fe66935e3f7535413fad2a7d3" title="volatility">Volatility</a> sigma_;
<a class="code" href="group__types.html#gede435af51236692b1107d7639581d39" title="interest rates">Rate</a> r_;
<a class="code" href="group__types.html#g4bdf4bfe76b9ffa6fa64c47d8bfa0c78" title="real number">Real</a> vega_;
};
<span class="comment">// The key for the MonteCarlo simulation is to have a PathPricer that</span>
<span class="comment">// implements a value(const Path& path) method.</span>
<span class="comment">// This method prices the portfolio for each Path of the random variable</span>
<span class="keyword">class </span>ReplicationPathPricer : <span class="keyword">public</span> <a name="_a9"></a><a class="code" href="class_quant_lib_1_1_path_pricer.html" title="base class for path pricers">PathPricer</a><Path> {
<span class="keyword">public</span>:
<span class="comment">// real constructor</span>
ReplicationPathPricer(Option::Type type,
<a class="code" href="group__types.html#g4bdf4bfe76b9ffa6fa64c47d8bfa0c78" title="real number">Real</a> strike,
<a class="code" href="group__types.html#gede435af51236692b1107d7639581d39" title="interest rates">Rate</a> r,
<a class="code" href="group__types.html#g14fb8fca43a68f4168654e1f9f7e22f7" title="continuous quantity with 1-year units">Time</a> maturity,
<a class="code" href="group__types.html#gaa95ab7fe66935e3f7535413fad2a7d3" title="volatility">Volatility</a> sigma)
: type_(type), strike_(strike),
r_(r), maturity_(maturity), sigma_(sigma) {
<a name="a10"></a><a class="code" href="errors_8hpp.html#7a9bcab8006882bc7d5302a0861ab1a6" title="throw an error if the given pre-condition is not verified">QL_REQUIRE</a>(strike_ > 0.0, <span class="stringliteral">"strike must be positive"</span>);
<a class="code" href="errors_8hpp.html#7a9bcab8006882bc7d5302a0861ab1a6" title="throw an error if the given pre-condition is not verified">QL_REQUIRE</a>(r_ >= 0.0,
<span class="stringliteral">"risk free rate (r) must be positive or zero"</span>);
<a class="code" href="errors_8hpp.html#7a9bcab8006882bc7d5302a0861ab1a6" title="throw an error if the given pre-condition is not verified">QL_REQUIRE</a>(maturity_ > 0.0, <span class="stringliteral">"maturity must be positive"</span>);
<a class="code" href="errors_8hpp.html#7a9bcab8006882bc7d5302a0861ab1a6" title="throw an error if the given pre-condition is not verified">QL_REQUIRE</a>(sigma_ >= 0.0,
<span class="stringliteral">"volatility (sigma) must be positive or zero"</span>);
}
<span class="comment">// The value() method encapsulates the pricing code</span>
<a class="code" href="group__types.html#g4bdf4bfe76b9ffa6fa64c47d8bfa0c78" title="real number">Real</a> operator()(<span class="keyword">const</span> <a name="_a11"></a><a class="code" href="class_quant_lib_1_1_path.html" title="single-factor random walk">Path</a>& path) <span class="keyword">const</span>;
<span class="keyword">private</span>:
Option::Type type_;
<a class="code" href="group__types.html#g4bdf4bfe76b9ffa6fa64c47d8bfa0c78" title="real number">Real</a> strike_;
<a class="code" href="group__types.html#gede435af51236692b1107d7639581d39" title="interest rates">Rate</a> r_;
<a class="code" href="group__types.html#g14fb8fca43a68f4168654e1f9f7e22f7" title="continuous quantity with 1-year units">Time</a> maturity_;
<a class="code" href="group__types.html#gaa95ab7fe66935e3f7535413fad2a7d3" title="volatility">Volatility</a> sigma_;
};
<span class="comment">// Compute Replication Error as in the Derman and Kamal's research note</span>
<span class="keywordtype">int</span> main(<span class="keywordtype">int</span>, <span class="keywordtype">char</span>* []) {
<span class="keywordflow">try</span> {
boost::timer timer;
std::cout << std::endl;
<a class="code" href="group__types.html#g14fb8fca43a68f4168654e1f9f7e22f7" title="continuous quantity with 1-year units">Time</a> maturity = 1.0/12.0; <span class="comment">// 1 month</span>
<a class="code" href="group__types.html#g4bdf4bfe76b9ffa6fa64c47d8bfa0c78" title="real number">Real</a> strike = 100;
<a class="code" href="group__types.html#g4bdf4bfe76b9ffa6fa64c47d8bfa0c78" title="real number">Real</a> underlying = 100;
<a class="code" href="group__types.html#gaa95ab7fe66935e3f7535413fad2a7d3" title="volatility">Volatility</a> <a name="a12"></a><a class="code" href="group__manips.html#gc402ef7c87f63f7c603ee87210b5750c" title="output volatilities as percentages">volatility</a> = 0.20; <span class="comment">// 20%</span>
<a class="code" href="group__types.html#gede435af51236692b1107d7639581d39" title="interest rates">Rate</a> riskFreeRate = 0.05; <span class="comment">// 5%</span>
ReplicationError rp(Option::Call, maturity, strike, underlying,
volatility, riskFreeRate);
<a class="code" href="group__types.html#gf38bdb4c54463b1f456655efa95b5c77" title="size of a container">Size</a> scenarios = 50000;
<a class="code" href="group__types.html#gf38bdb4c54463b1f456655efa95b5c77" title="size of a container">Size</a> hedgesNum;
hedgesNum = 21;
rp.compute(hedgesNum, scenarios);
hedgesNum = 84;
rp.compute(hedgesNum, scenarios);
<a class="code" href="group__types.html#g4bdf4bfe76b9ffa6fa64c47d8bfa0c78" title="real number">Real</a> seconds = timer.elapsed();
<a class="code" href="group__types.html#gb9c87440c314438e51a899a03d2442d0" title="integer number">Integer</a> hours = int(seconds/3600);
seconds -= hours * 3600;
<a class="code" href="group__types.html#gb9c87440c314438e51a899a03d2442d0" title="integer number">Integer</a> minutes = int(seconds/60);
seconds -= minutes * 60;
std::cout << <span class="stringliteral">" \nRun completed in "</span>;
<span class="keywordflow">if</span> (hours > 0)
std::cout << hours << <span class="stringliteral">" h "</span>;
<span class="keywordflow">if</span> (hours > 0 || minutes > 0)
std::cout << minutes << <span class="stringliteral">" m "</span>;
std::cout << std::fixed << std::setprecision(0)
<< seconds << <span class="stringliteral">" s\n"</span> << std::endl;
<span class="keywordflow">return</span> 0;
} <span class="keywordflow">catch</span> (std::exception& e) {
std::cout << e.what() << std::endl;
<span class="keywordflow">return</span> 1;
} <span class="keywordflow">catch</span> (...) {
std::cout << <span class="stringliteral">"unknown error"</span> << std::endl;
<span class="keywordflow">return</span> 1;
}
}
<span class="comment">/* The actual computation of the Profit&Loss for each single path.</span>
<span class="comment"></span>
<span class="comment"> In each scenario N rehedging trades spaced evenly in time over</span>
<span class="comment"> the life of the option are carried out, using the Black-Scholes</span>
<span class="comment"> hedge ratio.</span>
<span class="comment">*/</span>
<a class="code" href="group__types.html#g4bdf4bfe76b9ffa6fa64c47d8bfa0c78" title="real number">Real</a> ReplicationPathPricer::operator()(<span class="keyword">const</span> <a class="code" href="class_quant_lib_1_1_path.html" title="single-factor random walk">Path</a>& path)<span class="keyword"> const </span>{
<a class="code" href="group__types.html#gf38bdb4c54463b1f456655efa95b5c77" title="size of a container">Size</a> n = path.<a name="a13"></a><a class="code" href="class_quant_lib_1_1_path.html#0004e5f7457adee64d3a05ee24509aaa">length</a>()-1;
<a class="code" href="errors_8hpp.html#7a9bcab8006882bc7d5302a0861ab1a6" title="throw an error if the given pre-condition is not verified">QL_REQUIRE</a>(n>0, <span class="stringliteral">"the path cannot be empty"</span>);
<span class="comment">// discrete hedging interval</span>
<a class="code" href="group__types.html#g14fb8fca43a68f4168654e1f9f7e22f7" title="continuous quantity with 1-year units">Time</a> dt = maturity_/n;
<span class="comment">// For simplicity, we assume the stock pays no dividends.</span>
<a class="code" href="group__types.html#gede435af51236692b1107d7639581d39" title="interest rates">Rate</a> stockDividendYield = 0.0;
<span class="comment">// let's start</span>
<a class="code" href="group__types.html#g14fb8fca43a68f4168654e1f9f7e22f7" title="continuous quantity with 1-year units">Time</a> t = 0;
<span class="comment">// stock value at t=0</span>
<a class="code" href="group__types.html#g4bdf4bfe76b9ffa6fa64c47d8bfa0c78" title="real number">Real</a> stock = path.<a name="a14"></a><a class="code" href="class_quant_lib_1_1_path.html#df80e67a5954b0635539898d1de0bac3" title="initial asset value">front</a>();
<span class="comment">// money account at t=0</span>
<a class="code" href="group__types.html#g4bdf4bfe76b9ffa6fa64c47d8bfa0c78" title="real number">Real</a> money_account = 0.0;
<span class="comment">/************************/</span>
<span class="comment">/*** the initial deal ***/</span>
<span class="comment">/************************/</span>
<span class="comment">// option fair price (Black-Scholes) at t=0</span>
<a class="code" href="group__types.html#g642a971a0bcbbd2fb26c35e1a06e5761" title="discount factor between dates">DiscountFactor</a> rDiscount = std::exp(-r_*maturity_);
<a class="code" href="group__types.html#g642a971a0bcbbd2fb26c35e1a06e5761" title="discount factor between dates">DiscountFactor</a> qDiscount = std::exp(-stockDividendYield*maturity_);
<a class="code" href="group__types.html#g4bdf4bfe76b9ffa6fa64c47d8bfa0c78" title="real number">Real</a> forward = stock*qDiscount/rDiscount;
<a class="code" href="group__types.html#g4bdf4bfe76b9ffa6fa64c47d8bfa0c78" title="real number">Real</a> stdDev = std::sqrt(sigma_*sigma_*maturity_);
boost::shared_ptr<StrikedTypePayoff> payoff(
<span class="keyword">new</span> <a class="code" href="class_quant_lib_1_1_plain_vanilla_payoff.html" title="Plain-vanilla payoff.">PlainVanillaPayoff</a>(type_,strike_));
<a class="code" href="class_quant_lib_1_1_black_calculator.html" title="Black 1976 calculator class.">BlackCalculator</a> black(payoff,forward,stdDev,rDiscount);
<span class="comment">// sell the option, cash in its premium</span>
money_account += black.value();
<span class="comment">// compute delta</span>
<a class="code" href="group__types.html#g4bdf4bfe76b9ffa6fa64c47d8bfa0c78" title="real number">Real</a> delta = black.delta(stock);
<span class="comment">// delta-hedge the option buying stock</span>
<a class="code" href="group__types.html#g4bdf4bfe76b9ffa6fa64c47d8bfa0c78" title="real number">Real</a> stockAmount = delta;
money_account -= stockAmount*stock;
<span class="comment">/**********************************/</span>
<span class="comment">/*** hedging during option life ***/</span>
<span class="comment">/**********************************/</span>
<span class="keywordflow">for</span> (<a class="code" href="group__types.html#gf38bdb4c54463b1f456655efa95b5c77" title="size of a container">Size</a> step = 0; step < n-1; step++){
<span class="comment">// time flows</span>
t += dt;
<span class="comment">// accruing on the money account</span>
money_account *= std::exp( r_*dt );
<span class="comment">// stock growth:</span>
stock = path[step+1];
<span class="comment">// recalculate option value at the current stock value,</span>
<span class="comment">// and the current time to maturity</span>
rDiscount = std::exp(-r_*(maturity_-t));
qDiscount = std::exp(-stockDividendYield*(maturity_-t));
forward = stock*qDiscount/rDiscount;
stdDev = std::sqrt(sigma_*sigma_*(maturity_-t));
<a class="code" href="class_quant_lib_1_1_black_calculator.html" title="Black 1976 calculator class.">BlackCalculator</a> black(payoff,forward,stdDev,rDiscount);
<span class="comment">// recalculate delta</span>
delta = black.delta(stock);
<span class="comment">// re-hedging</span>
money_account -= (delta - stockAmount)*stock;
stockAmount = delta;
}
<span class="comment">/*************************/</span>
<span class="comment">/*** option expiration ***/</span>
<span class="comment">/*************************/</span>
<span class="comment">// last accrual on my money account</span>
money_account *= std::exp( r_*dt );
<span class="comment">// last stock growth</span>
stock = path[n];
<span class="comment">// the hedger delivers the option payoff to the option holder</span>
<a class="code" href="group__types.html#g4bdf4bfe76b9ffa6fa64c47d8bfa0c78" title="real number">Real</a> optionPayoff = <a class="code" href="class_quant_lib_1_1_plain_vanilla_payoff.html" title="Plain-vanilla payoff.">PlainVanillaPayoff</a>(type_, strike_)(stock);
money_account -= optionPayoff;
<span class="comment">// and unwinds the hedge selling his stock position</span>
money_account += stockAmount*stock;
<span class="comment">// final Profit&Loss</span>
<span class="keywordflow">return</span> money_account;
}
<span class="comment">// The computation over nSamples paths of the P&L distribution</span>
<span class="keywordtype">void</span> ReplicationError::compute(<a class="code" href="group__types.html#gf38bdb4c54463b1f456655efa95b5c77" title="size of a container">Size</a> nTimeSteps, <a class="code" href="group__types.html#gf38bdb4c54463b1f456655efa95b5c77" title="size of a container">Size</a> nSamples)
{
<a class="code" href="errors_8hpp.html#7a9bcab8006882bc7d5302a0861ab1a6" title="throw an error if the given pre-condition is not verified">QL_REQUIRE</a>(nTimeSteps>0, <span class="stringliteral">"the number of steps must be > 0"</span>);
<span class="comment">// hedging interval</span>
<span class="comment">// Time tau = maturity_ / nTimeSteps;</span>
<span class="comment">/* Black-Scholes framework: the underlying stock price evolves</span>
<span class="comment"> lognormally with a fixed known volatility that stays constant</span>
<span class="comment"> throughout time.</span>
<span class="comment"> */</span>
<a name="_a15"></a><a class="code" href="class_quant_lib_1_1_calendar.html" title="calendar class">Calendar</a> calendar = <a name="_a16"></a><a class="code" href="class_quant_lib_1_1_t_a_r_g_e_t.html" title="TARGET calendar">TARGET</a>();
<a name="_a17"></a><a class="code" href="class_quant_lib_1_1_date.html" title="Concrete date class.">Date</a> today = Date::todaysDate();
<a name="_a18"></a><a class="code" href="class_quant_lib_1_1_day_counter.html" title="day counter class">DayCounter</a> dayCount = <a name="_a19"></a><a class="code" href="class_quant_lib_1_1_actual365_fixed.html" title="Actual/365 (Fixed) day count convention.">Actual365Fixed</a>();
<a name="_a20"></a><a class="code" href="class_quant_lib_1_1_handle.html" title="Shared handle to an observable.">Handle<Quote></a> stateVariable(
boost::shared_ptr<Quote>(<span class="keyword">new</span> <a name="_a21"></a><a class="code" href="class_quant_lib_1_1_simple_quote.html" title="market element returning a stored value">SimpleQuote</a>(s0_)));
<a class="code" href="class_quant_lib_1_1_handle.html" title="Shared handle to an observable.">Handle<YieldTermStructure></a> riskFreeRate(
boost::shared_ptr<YieldTermStructure>(
<span class="keyword">new</span> <a name="_a22"></a><a class="code" href="class_quant_lib_1_1_flat_forward.html" title="Flat interest-rate curve.">FlatForward</a>(today, r_, dayCount)));
<a class="code" href="class_quant_lib_1_1_handle.html" title="Shared handle to an observable.">Handle<YieldTermStructure></a> dividendYield(
boost::shared_ptr<YieldTermStructure>(
<span class="keyword">new</span> <a class="code" href="class_quant_lib_1_1_flat_forward.html" title="Flat interest-rate curve.">FlatForward</a>(today, 0.0, dayCount)));
<a class="code" href="class_quant_lib_1_1_handle.html" title="Shared handle to an observable.">Handle<BlackVolTermStructure></a> <a class="code" href="group__manips.html#gc402ef7c87f63f7c603ee87210b5750c" title="output volatilities as percentages">volatility</a>(
boost::shared_ptr<BlackVolTermStructure>(
<span class="keyword">new</span> <a name="_a23"></a><a class="code" href="class_quant_lib_1_1_black_constant_vol.html" title="Constant Black volatility, no time-strike dependence.">BlackConstantVol</a>(today, calendar, sigma_, dayCount)));
boost::shared_ptr<StochasticProcess1D> diffusion(
<span class="keyword">new</span> <a name="_a24"></a><a class="code" href="class_quant_lib_1_1_black_scholes_merton_process.html" title="Merton (1973) extension to the Black-Scholes stochastic process.">BlackScholesMertonProcess</a>(stateVariable, dividendYield,
riskFreeRate, volatility));
<span class="comment">// Black Scholes equation rules the path generator:</span>
<span class="comment">// at each step the log of the stock</span>
<span class="comment">// will have drift and sigma^2 variance</span>
<a name="_a25"></a><a class="code" href="class_quant_lib_1_1_inverse_cumulative_rsg.html" title="Inverse cumulative random sequence generator.">PseudoRandom::rsg_type</a> rsg =
PseudoRandom::make_sequence_generator(nTimeSteps, 0);
<span class="keywordtype">bool</span> brownianBridge = <span class="keyword">false</span>;
<span class="keyword">typedef</span> <a name="_a26"></a><a class="code" href="struct_quant_lib_1_1_single_variate.html" title="default Monte Carlo traits for single-variate models">SingleVariate<PseudoRandom>::path_generator_type</a> generator_type;
boost::shared_ptr<generator_type> myPathGenerator(<span class="keyword">new</span>
generator_type(diffusion, maturity_, nTimeSteps,
rsg, brownianBridge));
<span class="comment">// The replication strategy's Profit&Loss is computed for each path</span>
<span class="comment">// of the stock. The path pricer knows how to price a path using its</span>
<span class="comment">// value() method</span>
boost::shared_ptr<PathPricer<Path> > myPathPricer(<span class="keyword">new</span>
ReplicationPathPricer(payoff_.optionType(), payoff_.strike(),
r_, maturity_, sigma_));
<span class="comment">// a statistics accumulator for the path-dependant Profit&Loss values</span>
<a name="_a27"></a><a class="code" href="class_quant_lib_1_1_generic_risk_statistics.html" title="empirical-distribution risk measures">Statistics</a> statisticsAccumulator;
<span class="comment">// The Monte Carlo model generates paths using myPathGenerator</span>
<span class="comment">// each path is priced using myPathPricer</span>
<span class="comment">// prices will be accumulated into statisticsAccumulator</span>
<a name="_a28"></a><a class="code" href="class_quant_lib_1_1_monte_carlo_model.html" title="General-purpose Monte Carlo model for path samples.">MonteCarloModel<SingleVariate,PseudoRandom></a>
MCSimulation(myPathGenerator,
myPathPricer,
statisticsAccumulator,
<span class="keyword">false</span>);
<span class="comment">// the model simulates nSamples paths</span>
MCSimulation.addSamples(nSamples);
<span class="comment">// the sampleAccumulator method</span>
<span class="comment">// gives access to all the methods of statisticsAccumulator</span>
<a class="code" href="group__types.html#g4bdf4bfe76b9ffa6fa64c47d8bfa0c78" title="real number">Real</a> PLMean = MCSimulation.sampleAccumulator().mean();
<a class="code" href="group__types.html#g4bdf4bfe76b9ffa6fa64c47d8bfa0c78" title="real number">Real</a> PLStDev = MCSimulation.sampleAccumulator().standardDeviation();
<a class="code" href="group__types.html#g4bdf4bfe76b9ffa6fa64c47d8bfa0c78" title="real number">Real</a> PLSkew = MCSimulation.sampleAccumulator().skewness();
<a class="code" href="group__types.html#g4bdf4bfe76b9ffa6fa64c47d8bfa0c78" title="real number">Real</a> PLKurt = MCSimulation.sampleAccumulator().kurtosis();
<span class="comment">// Derman and Kamal's formula</span>
<a class="code" href="group__types.html#g4bdf4bfe76b9ffa6fa64c47d8bfa0c78" title="real number">Real</a> theorStD = std::sqrt(M_PI/4/nTimeSteps)*vega_*sigma_;
std::cout << std::fixed
<< std::setw(8) << nSamples << <span class="stringliteral">" | "</span>
<< std::setw(8) << nTimeSteps << <span class="stringliteral">" | "</span>
<< std::setw(8) << std::setprecision(3) << PLMean << <span class="stringliteral">" | "</span>
<< std::setw(8) << std::setprecision(2) << PLStDev << <span class="stringliteral">" | "</span>
<< std::setw(12) << std::setprecision(2) << theorStD << <span class="stringliteral">" | "</span>
<< std::setw(8) << std::setprecision(2) << PLSkew << <span class="stringliteral">" | "</span>
<< std::setw(8) << std::setprecision(2) << PLKurt << std::endl;
}
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