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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"><a name="l00001"></a>00001 <span class="comment">/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */</span>
<a name="l00002"></a>00002
<a name="l00021"></a>00021 <span class="comment">/* This example computes profit and loss of a discrete interval hedging</span>
<a name="l00022"></a>00022 <span class="comment"> strategy and compares with the results of Derman & Kamal's (Goldman Sachs</span>
<a name="l00023"></a>00023 <span class="comment"> Equity Derivatives Research) Research Note: "When You Cannot Hedge</span>
<a name="l00024"></a>00024 <span class="comment"> Continuously: The Corrections to Black-Scholes"</span>
<a name="l00025"></a>00025 <span class="comment"> http://www.ederman.com/emanuelderman/GSQSpapers/when_you_cannot_hedge.pdf</span>
<a name="l00026"></a>00026 <span class="comment"></span>
<a name="l00027"></a>00027 <span class="comment"> Suppose an option hedger sells an European option and receives the</span>
<a name="l00028"></a>00028 <span class="comment"> Black-Scholes value as the options premium.</span>
<a name="l00029"></a>00029 <span class="comment"> Then he follows a Black-Scholes hedging strategy, rehedging at discrete,</span>
<a name="l00030"></a>00030 <span class="comment"> evenly spaced time intervals as the underlying stock changes. At</span>
<a name="l00031"></a>00031 <span class="comment"> expiration, the hedger delivers the option payoff to the option holder,</span>
<a name="l00032"></a>00032 <span class="comment"> and unwinds the hedge. We are interested in understanding the final</span>
<a name="l00033"></a>00033 <span class="comment"> profit or loss of this strategy.</span>
<a name="l00034"></a>00034 <span class="comment"></span>
<a name="l00035"></a>00035 <span class="comment"> If the hedger had followed the exact Black-Scholes replication strategy,</span>
<a name="l00036"></a>00036 <span class="comment"> re-hedging continuously as the underlying stock evolved towards its final</span>
<a name="l00037"></a>00037 <span class="comment"> value at expiration, then, no matter what path the stock took, the final</span>
<a name="l00038"></a>00038 <span class="comment"> P&L would be exactly zero. When the replication strategy deviates from</span>
<a name="l00039"></a>00039 <span class="comment"> the exact Black-Scholes method, the final P&L may deviate from zero. This</span>
<a name="l00040"></a>00040 <span class="comment"> deviation is called the replication error. When the hedger rebalances at</span>
<a name="l00041"></a>00041 <span class="comment"> discrete rather than continuous intervals, the hedge is imperfect and the</span>
<a name="l00042"></a>00042 <span class="comment"> replication is inexact. The more often hedging occurs, the smaller the</span>
<a name="l00043"></a>00043 <span class="comment"> replication error.</span>
<a name="l00044"></a>00044 <span class="comment"></span>
<a name="l00045"></a>00045 <span class="comment"> We examine the range of possibilities, computing the replication error.</span>
<a name="l00046"></a>00046 <span class="comment">*/</span>
<a name="l00047"></a>00047
<a name="l00048"></a>00048 <span class="comment">// the only header you need to use QuantLib</span>
<a name="l00049"></a>00049 <span class="preprocessor">#define BOOST_LIB_DIAGNOSTIC</span>
<a name="l00050"></a>00050 <span class="preprocessor"></span><span class="preprocessor"># include <ql/quantlib.hpp></span>
<a name="l00051"></a>00051 <span class="preprocessor">#undef BOOST_LIB_DIAGNOSTIC</span>
<a name="l00052"></a>00052 <span class="preprocessor"></span>
<a name="l00053"></a>00053 <span class="preprocessor">#ifdef BOOST_MSVC</span>
<a name="l00054"></a>00054 <span class="preprocessor"></span><span class="comment">/* Uncomment the following lines to unmask floating-point</span>
<a name="l00055"></a>00055 <span class="comment"> exceptions. Warning: unpredictable results can arise...</span>
<a name="l00056"></a>00056 <span class="comment"></span>
<a name="l00057"></a>00057 <span class="comment"> See http://www.wilmott.com/messageview.cfm?catid=10&threadid=9481</span>
<a name="l00058"></a>00058 <span class="comment"> Is there anyone with a definitive word about this?</span>
<a name="l00059"></a>00059 <span class="comment">*/</span>
<a name="l00060"></a>00060 <span class="comment">// #include <float.h></span>
<a name="l00061"></a>00061 <span class="comment">// namespace { unsigned int u = _controlfp(_EM_INEXACT, _MCW_EM); }</span>
<a name="l00062"></a>00062 <span class="preprocessor">#endif</span>
<a name="l00063"></a>00063 <span class="preprocessor"></span>
<a name="l00064"></a>00064 <span class="preprocessor">#include <boost/timer.hpp></span>
<a name="l00065"></a>00065 <span class="preprocessor">#include <iostream></span>
<a name="l00066"></a>00066 <span class="preprocessor">#include <iomanip></span>
<a name="l00067"></a>00067
<a name="l00068"></a>00068 <span class="keyword">using namespace </span>QuantLib;
<a name="l00069"></a>00069
<a name="l00070"></a>00070 <span class="preprocessor">#if defined(QL_ENABLE_SESSIONS)</span>
<a name="l00071"></a>00071 <span class="preprocessor"></span><span class="keyword">namespace </span>QuantLib {
<a name="l00072"></a>00072
<a name="l00073"></a>00073 <a name="a0"></a><a class="code" href="group__types.html#gb9c87440c314438e51a899a03d2442d0">Integer</a> sessionId() { <span class="keywordflow">return</span> 0; }
<a name="l00074"></a>00074
<a name="l00075"></a>00075 }
<a name="l00076"></a>00076 <span class="preprocessor">#endif</span>
<a name="l00077"></a>00077 <span class="preprocessor"></span>
<a name="l00078"></a>00078
<a name="l00079"></a>00079 <span class="comment">/* The ReplicationError class carries out Monte Carlo simulations to evaluate</span>
<a name="l00080"></a>00080 <span class="comment"> the outcome (the replication error) of the discrete hedging strategy over</span>
<a name="l00081"></a>00081 <span class="comment"> different, randomly generated scenarios of future stock price evolution.</span>
<a name="l00082"></a>00082 <span class="comment">*/</span>
<a name="l00083"></a>00083 <span class="keyword">class </span>ReplicationError
<a name="l00084"></a>00084 {
<a name="l00085"></a>00085 <span class="keyword">public</span>:
<a name="l00086"></a>00086 ReplicationError(Option::Type type,
<a name="l00087"></a>00087 <a name="a1"></a><a class="code" href="group__types.html#g14fb8fca43a68f4168654e1f9f7e22f7">Time</a> maturity,
<a name="l00088"></a>00088 <a name="a2"></a><a class="code" href="group__types.html#g4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> strike,
<a name="l00089"></a>00089 <a class="code" href="group__types.html#g4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> s0,
<a name="l00090"></a>00090 <a name="a3"></a><a class="code" href="group__types.html#gaa95ab7fe66935e3f7535413fad2a7d3">Volatility</a> sigma,
<a name="l00091"></a>00091 <a name="a4"></a><a class="code" href="group__types.html#gede435af51236692b1107d7639581d39">Rate</a> r)
<a name="l00092"></a>00092 : maturity_(maturity), payoff_(type, strike), s0_(s0),
<a name="l00093"></a>00093 sigma_(sigma), r_(r) {
<a name="l00094"></a>00094
<a name="l00095"></a>00095 <span class="comment">// value of the option</span>
<a name="l00096"></a>00096 <a name="a5"></a><a class="code" href="group__types.html#g642a971a0bcbbd2fb26c35e1a06e5761">DiscountFactor</a> rDiscount = std::exp(-r_*maturity_);
<a name="l00097"></a>00097 <a class="code" href="group__types.html#g642a971a0bcbbd2fb26c35e1a06e5761">DiscountFactor</a> qDiscount = 1.0;
<a name="l00098"></a>00098 <a class="code" href="group__types.html#g4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> forward = s0_*qDiscount/rDiscount;
<a name="l00099"></a>00099 <a class="code" href="group__types.html#g4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> variance = sigma_*sigma_*maturity_;
<a name="l00100"></a>00100 boost::shared_ptr<StrikedTypePayoff> payoff(
<a name="l00101"></a>00101 <span class="keyword">new</span> <a name="_a6"></a><a class="code" href="class_quant_lib_1_1_plain_vanilla_payoff.html">PlainVanillaPayoff</a>(payoff_));
<a name="l00102"></a>00102 <a name="_a7"></a><a class="code" href="class_quant_lib_1_1_black_formula.html">BlackFormula</a> black(forward,rDiscount,variance,payoff);
<a name="l00103"></a>00103 std::cout << <span class="stringliteral">"Option value: "</span> << black.value() << std::endl;
<a name="l00104"></a>00104
<a name="l00105"></a>00105 <span class="comment">// store option's vega, since Derman and Kamal's formula needs it</span>
<a name="l00106"></a>00106 vega_ = black.vega(maturity_);
<a name="l00107"></a>00107
<a name="l00108"></a>00108 std::cout << std::endl;
<a name="l00109"></a>00109 std::cout <<
<a name="l00110"></a>00110 <span class="stringliteral">" | | P&L \t| P&L | Derman&Kamal | P&L"</span>
<a name="l00111"></a>00111 <span class="stringliteral">" \t| P&L"</span> << std::endl;
<a name="l00112"></a>00112
<a name="l00113"></a>00113 std::cout <<
<a name="l00114"></a>00114 <span class="stringliteral">"samples | trades | Mean \t| Std Dev | Formula |"</span>
<a name="l00115"></a>00115 <span class="stringliteral">" skewness \t| kurt."</span> << std::endl;
<a name="l00116"></a>00116
<a name="l00117"></a>00117 std::cout << <span class="stringliteral">"---------------------------------"</span>
<a name="l00118"></a>00118 <span class="stringliteral">"----------------------------------------------"</span> << std::endl;
<a name="l00119"></a>00119 }
<a name="l00120"></a>00120
<a name="l00121"></a>00121 <span class="comment">// the actual replication error computation</span>
<a name="l00122"></a>00122 <span class="keywordtype">void</span> compute(<a name="a8"></a><a class="code" href="group__types.html#gf38bdb4c54463b1f456655efa95b5c77">Size</a> nTimeSteps, <a class="code" href="group__types.html#gf38bdb4c54463b1f456655efa95b5c77">Size</a> nSamples);
<a name="l00123"></a>00123 <span class="keyword">private</span>:
<a name="l00124"></a>00124 <a class="code" href="group__types.html#g14fb8fca43a68f4168654e1f9f7e22f7">Time</a> maturity_;
<a name="l00125"></a>00125 <a class="code" href="class_quant_lib_1_1_plain_vanilla_payoff.html">PlainVanillaPayoff</a> payoff_;
<a name="l00126"></a>00126 <a class="code" href="group__types.html#g4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> s0_;
<a name="l00127"></a>00127 <a class="code" href="group__types.html#gaa95ab7fe66935e3f7535413fad2a7d3">Volatility</a> sigma_;
<a name="l00128"></a>00128 <a class="code" href="group__types.html#gede435af51236692b1107d7639581d39">Rate</a> r_;
<a name="l00129"></a>00129 <a class="code" href="group__types.html#g4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> vega_;
<a name="l00130"></a>00130 };
<a name="l00131"></a>00131
<a name="l00132"></a>00132 <span class="comment">// The key for the MonteCarlo simulation is to have a PathPricer that</span>
<a name="l00133"></a>00133 <span class="comment">// implements a value(const Path& path) method.</span>
<a name="l00134"></a>00134 <span class="comment">// This method prices the portfolio for each Path of the random variable</span>
<a name="l00135"></a>00135 <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">PathPricer</a><Path> {
<a name="l00136"></a>00136 <span class="keyword">public</span>:
<a name="l00137"></a>00137 <span class="comment">// real constructor</span>
<a name="l00138"></a>00138 ReplicationPathPricer(Option::Type type,
<a name="l00139"></a>00139 <a class="code" href="group__types.html#g4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> strike,
<a name="l00140"></a>00140 <a class="code" href="group__types.html#gede435af51236692b1107d7639581d39">Rate</a> r,
<a name="l00141"></a>00141 <a class="code" href="group__types.html#g14fb8fca43a68f4168654e1f9f7e22f7">Time</a> maturity,
<a name="l00142"></a>00142 <a class="code" href="group__types.html#gaa95ab7fe66935e3f7535413fad2a7d3">Volatility</a> sigma)
<a name="l00143"></a>00143 : type_(type), strike_(strike),
<a name="l00144"></a>00144 r_(r), maturity_(maturity), sigma_(sigma) {
<a name="l00145"></a>00145 <a name="a10"></a><a class="code" href="errors_8hpp.html#7a9bcab8006882bc7d5302a0861ab1a6">QL_REQUIRE</a>(strike_ > 0.0, <span class="stringliteral">"strike must be positive"</span>);
<a name="l00146"></a>00146 <a class="code" href="errors_8hpp.html#7a9bcab8006882bc7d5302a0861ab1a6">QL_REQUIRE</a>(r_ >= 0.0,
<a name="l00147"></a>00147 <span class="stringliteral">"risk free rate (r) must be positive or zero"</span>);
<a name="l00148"></a>00148 <a class="code" href="errors_8hpp.html#7a9bcab8006882bc7d5302a0861ab1a6">QL_REQUIRE</a>(maturity_ > 0.0, <span class="stringliteral">"maturity must be positive"</span>);
<a name="l00149"></a>00149 <a class="code" href="errors_8hpp.html#7a9bcab8006882bc7d5302a0861ab1a6">QL_REQUIRE</a>(sigma_ >= 0.0,
<a name="l00150"></a>00150 <span class="stringliteral">"volatility (sigma) must be positive or zero"</span>);
<a name="l00151"></a>00151
<a name="l00152"></a>00152 }
<a name="l00153"></a>00153 <span class="comment">// The value() method encapsulates the pricing code</span>
<a name="l00154"></a>00154 <a class="code" href="group__types.html#g4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> operator()(<span class="keyword">const</span> <a name="_a11"></a><a class="code" href="class_quant_lib_1_1_path.html">Path</a>& path) <span class="keyword">const</span>;
<a name="l00155"></a>00155
<a name="l00156"></a>00156 <span class="keyword">private</span>:
<a name="l00157"></a>00157 Option::Type type_;
<a name="l00158"></a>00158 <a class="code" href="group__types.html#g4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> strike_;
<a name="l00159"></a>00159 <a class="code" href="group__types.html#gede435af51236692b1107d7639581d39">Rate</a> r_;
<a name="l00160"></a>00160 <a class="code" href="group__types.html#g14fb8fca43a68f4168654e1f9f7e22f7">Time</a> maturity_;
<a name="l00161"></a>00161 <a class="code" href="group__types.html#gaa95ab7fe66935e3f7535413fad2a7d3">Volatility</a> sigma_;
<a name="l00162"></a>00162 };
<a name="l00163"></a>00163
<a name="l00164"></a>00164
<a name="l00165"></a>00165 <span class="comment">// Compute Replication Error as in the Derman and Kamal's research note</span>
<a name="l00166"></a>00166 <span class="keywordtype">int</span> main(<span class="keywordtype">int</span>, <span class="keywordtype">char</span>* [])
<a name="l00167"></a>00167 {
<a name="l00168"></a>00168 <span class="keywordflow">try</span> {
<a name="l00169"></a>00169 <a name="a12"></a><a class="code" href="group__misc_macros.html#g11d8e34f54505b0bf6f550508278673d">QL_IO_INIT</a>
<a name="l00170"></a>00170
<a name="l00171"></a>00171 boost::timer timer;
<a name="l00172"></a>00172 std::cout << std::endl;
<a name="l00173"></a>00173
<a name="l00174"></a>00174 <a class="code" href="group__types.html#g14fb8fca43a68f4168654e1f9f7e22f7">Time</a> maturity = 1.0/12.0; <span class="comment">// 1 month</span>
<a name="l00175"></a>00175 <a class="code" href="group__types.html#g4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> strike = 100;
<a name="l00176"></a>00176 <a class="code" href="group__types.html#g4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> underlying = 100;
<a name="l00177"></a>00177 <a class="code" href="group__types.html#gaa95ab7fe66935e3f7535413fad2a7d3">Volatility</a> <a name="a13"></a><a class="code" href="group__manips.html#gc402ef7c87f63f7c603ee87210b5750c">volatility</a> = 0.20; <span class="comment">// 20%</span>
<a name="l00178"></a>00178 <a class="code" href="group__types.html#gede435af51236692b1107d7639581d39">Rate</a> riskFreeRate = 0.05; <span class="comment">// 5%</span>
<a name="l00179"></a>00179 ReplicationError rp(Option::Call, maturity, strike, underlying,
<a name="l00180"></a>00180 volatility, riskFreeRate);
<a name="l00181"></a>00181
<a name="l00182"></a>00182 <a class="code" href="group__types.html#gf38bdb4c54463b1f456655efa95b5c77">Size</a> scenarios = 50000;
<a name="l00183"></a>00183 <a class="code" href="group__types.html#gf38bdb4c54463b1f456655efa95b5c77">Size</a> hedgesNum;
<a name="l00184"></a>00184
<a name="l00185"></a>00185 hedgesNum = 21;
<a name="l00186"></a>00186 rp.compute(hedgesNum, scenarios);
<a name="l00187"></a>00187
<a name="l00188"></a>00188 hedgesNum = 84;
<a name="l00189"></a>00189 rp.compute(hedgesNum, scenarios);
<a name="l00190"></a>00190
<a name="l00191"></a>00191 <a class="code" href="group__types.html#g4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> seconds = timer.elapsed();
<a name="l00192"></a>00192 <a class="code" href="group__types.html#gb9c87440c314438e51a899a03d2442d0">Integer</a> hours = int(seconds/3600);
<a name="l00193"></a>00193 seconds -= hours * 3600;
<a name="l00194"></a>00194 <a class="code" href="group__types.html#gb9c87440c314438e51a899a03d2442d0">Integer</a> minutes = int(seconds/60);
<a name="l00195"></a>00195 seconds -= minutes * 60;
<a name="l00196"></a>00196 std::cout << <span class="stringliteral">" \nRun completed in "</span>;
<a name="l00197"></a>00197 <span class="keywordflow">if</span> (hours > 0)
<a name="l00198"></a>00198 std::cout << hours << <span class="stringliteral">" h "</span>;
<a name="l00199"></a>00199 <span class="keywordflow">if</span> (hours > 0 || minutes > 0)
<a name="l00200"></a>00200 std::cout << minutes << <span class="stringliteral">" m "</span>;
<a name="l00201"></a>00201 std::cout << std::fixed << std::setprecision(0)
<a name="l00202"></a>00202 << seconds << <span class="stringliteral">" s\n"</span> << std::endl;
<a name="l00203"></a>00203
<a name="l00204"></a>00204 <span class="keywordflow">return</span> 0;
<a name="l00205"></a>00205 } <span class="keywordflow">catch</span> (std::exception& e) {
<a name="l00206"></a>00206 std::cout << e.what() << std::endl;
<a name="l00207"></a>00207 <span class="keywordflow">return</span> 1;
<a name="l00208"></a>00208 } <span class="keywordflow">catch</span> (...) {
<a name="l00209"></a>00209 std::cout << <span class="stringliteral">"unknown error"</span> << std::endl;
<a name="l00210"></a>00210 <span class="keywordflow">return</span> 1;
<a name="l00211"></a>00211 }
<a name="l00212"></a>00212 }
<a name="l00213"></a>00213
<a name="l00214"></a>00214
<a name="l00215"></a>00215 <span class="comment">/* The actual computation of the Profit&Loss for each single path.</span>
<a name="l00216"></a>00216 <span class="comment"></span>
<a name="l00217"></a>00217 <span class="comment"> In each scenario N rehedging trades spaced evenly in time over</span>
<a name="l00218"></a>00218 <span class="comment"> the life of the option are carried out, using the Black-Scholes</span>
<a name="l00219"></a>00219 <span class="comment"> hedge ratio.</span>
<a name="l00220"></a>00220 <span class="comment">*/</span>
<a name="l00221"></a>00221 <a class="code" href="group__types.html#g4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> ReplicationPathPricer::operator()(<span class="keyword">const</span> <a class="code" href="class_quant_lib_1_1_path.html">Path</a>& path)<span class="keyword"> const </span>{
<a name="l00222"></a>00222
<a name="l00223"></a>00223 <a class="code" href="group__types.html#gf38bdb4c54463b1f456655efa95b5c77">Size</a> n = path.<a name="a14"></a><a class="code" href="class_quant_lib_1_1_path.html#022887254632280a41166a9e97b87e0c">length</a>()-1;
<a name="l00224"></a>00224 <a class="code" href="errors_8hpp.html#7a9bcab8006882bc7d5302a0861ab1a6">QL_REQUIRE</a>(n>0, <span class="stringliteral">"the path cannot be empty"</span>);
<a name="l00225"></a>00225
<a name="l00226"></a>00226 <span class="comment">// discrete hedging interval</span>
<a name="l00227"></a>00227 <a class="code" href="group__types.html#g14fb8fca43a68f4168654e1f9f7e22f7">Time</a> dt = maturity_/n;
<a name="l00228"></a>00228
<a name="l00229"></a>00229 <span class="comment">// For simplicity, we assume the stock pays no dividends.</span>
<a name="l00230"></a>00230 <a class="code" href="group__types.html#gede435af51236692b1107d7639581d39">Rate</a> stockDividendYield = 0.0;
<a name="l00231"></a>00231
<a name="l00232"></a>00232 <span class="comment">// let's start</span>
<a name="l00233"></a>00233 <a class="code" href="group__types.html#g14fb8fca43a68f4168654e1f9f7e22f7">Time</a> t = 0;
<a name="l00234"></a>00234
<a name="l00235"></a>00235 <span class="comment">// stock value at t=0</span>
<a name="l00236"></a>00236 <a class="code" href="group__types.html#g4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> stock = path.<a name="a15"></a><a class="code" href="class_quant_lib_1_1_path.html#f78ff4861aaf9bf0524b958dbb4d1b18">front</a>();
<a name="l00237"></a>00237
<a name="l00238"></a>00238 <span class="comment">// money account at t=0</span>
<a name="l00239"></a>00239 <a class="code" href="group__types.html#g4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> money_account = 0.0;
<a name="l00240"></a>00240
<a name="l00241"></a>00241 <span class="comment">/************************/</span>
<a name="l00242"></a>00242 <span class="comment">/*** the initial deal ***/</span>
<a name="l00243"></a>00243 <span class="comment">/************************/</span>
<a name="l00244"></a>00244 <span class="comment">// option fair price (Black-Scholes) at t=0</span>
<a name="l00245"></a>00245 <a class="code" href="group__types.html#g642a971a0bcbbd2fb26c35e1a06e5761">DiscountFactor</a> rDiscount = std::exp(-r_*maturity_);
<a name="l00246"></a>00246 <a class="code" href="group__types.html#g642a971a0bcbbd2fb26c35e1a06e5761">DiscountFactor</a> qDiscount = std::exp(-stockDividendYield*maturity_);
<a name="l00247"></a>00247 <a class="code" href="group__types.html#g4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> forward = stock*qDiscount/rDiscount;
<a name="l00248"></a>00248 <a class="code" href="group__types.html#g4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> variance = sigma_*sigma_*maturity_;
<a name="l00249"></a>00249 boost::shared_ptr<StrikedTypePayoff> payoff(
<a name="l00250"></a>00250 <span class="keyword">new</span> <a class="code" href="class_quant_lib_1_1_plain_vanilla_payoff.html">PlainVanillaPayoff</a>(type_,strike_));
<a name="l00251"></a>00251 <a class="code" href="class_quant_lib_1_1_black_formula.html">BlackFormula</a> black(forward,rDiscount,variance,payoff);
<a name="l00252"></a>00252 <span class="comment">// sell the option, cash in its premium</span>
<a name="l00253"></a>00253 money_account += black.value();
<a name="l00254"></a>00254 <span class="comment">// compute delta</span>
<a name="l00255"></a>00255 <a class="code" href="group__types.html#g4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> delta = black.delta(stock);
<a name="l00256"></a>00256 <span class="comment">// delta-hedge the option buying stock</span>
<a name="l00257"></a>00257 <a class="code" href="group__types.html#g4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> stockAmount = delta;
<a name="l00258"></a>00258 money_account -= stockAmount*stock;
<a name="l00259"></a>00259
<a name="l00260"></a>00260 <span class="comment">/**********************************/</span>
<a name="l00261"></a>00261 <span class="comment">/*** hedging during option life ***/</span>
<a name="l00262"></a>00262 <span class="comment">/**********************************/</span>
<a name="l00263"></a>00263 <span class="keywordflow">for</span> (<a class="code" href="group__types.html#gf38bdb4c54463b1f456655efa95b5c77">Size</a> step = 0; step < n-1; step++){
<a name="l00264"></a>00264
<a name="l00265"></a>00265 <span class="comment">// time flows</span>
<a name="l00266"></a>00266 t += dt;
<a name="l00267"></a>00267
<a name="l00268"></a>00268 <span class="comment">// accruing on the money account</span>
<a name="l00269"></a>00269 money_account *= std::exp( r_*dt );
<a name="l00270"></a>00270
<a name="l00271"></a>00271 <span class="comment">// stock growth:</span>
<a name="l00272"></a>00272 stock = path[step+1];
<a name="l00273"></a>00273
<a name="l00274"></a>00274 <span class="comment">// recalculate option value at the current stock value,</span>
<a name="l00275"></a>00275 <span class="comment">// and the current time to maturity</span>
<a name="l00276"></a>00276 rDiscount = std::exp(-r_*(maturity_-t));
<a name="l00277"></a>00277 qDiscount = std::exp(-stockDividendYield*(maturity_-t));
<a name="l00278"></a>00278 forward = stock*qDiscount/rDiscount;
<a name="l00279"></a>00279 variance = sigma_*sigma_*(maturity_-t);
<a name="l00280"></a>00280 <a class="code" href="class_quant_lib_1_1_black_formula.html">BlackFormula</a> black(forward,rDiscount,variance,payoff);
<a name="l00281"></a>00281
<a name="l00282"></a>00282 <span class="comment">// recalculate delta</span>
<a name="l00283"></a>00283 delta = black.delta(stock);
<a name="l00284"></a>00284
<a name="l00285"></a>00285 <span class="comment">// re-hedging</span>
<a name="l00286"></a>00286 money_account -= (delta - stockAmount)*stock;
<a name="l00287"></a>00287 stockAmount = delta;
<a name="l00288"></a>00288 }
<a name="l00289"></a>00289
<a name="l00290"></a>00290 <span class="comment">/*************************/</span>
<a name="l00291"></a>00291 <span class="comment">/*** option expiration ***/</span>
<a name="l00292"></a>00292 <span class="comment">/*************************/</span>
<a name="l00293"></a>00293 <span class="comment">// last accrual on my money account</span>
<a name="l00294"></a>00294 money_account *= std::exp( r_*dt );
<a name="l00295"></a>00295 <span class="comment">// last stock growth</span>
<a name="l00296"></a>00296 stock = path[n];
<a name="l00297"></a>00297
<a name="l00298"></a>00298 <span class="comment">// the hedger delivers the option payoff to the option holder</span>
<a name="l00299"></a>00299 <a class="code" href="group__types.html#g4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> optionPayoff = <a class="code" href="class_quant_lib_1_1_plain_vanilla_payoff.html">PlainVanillaPayoff</a>(type_, strike_)(stock);
<a name="l00300"></a>00300 money_account -= optionPayoff;
<a name="l00301"></a>00301
<a name="l00302"></a>00302 <span class="comment">// and unwinds the hedge selling his stock position</span>
<a name="l00303"></a>00303 money_account += stockAmount*stock;
<a name="l00304"></a>00304
<a name="l00305"></a>00305 <span class="comment">// final Profit&Loss</span>
<a name="l00306"></a>00306 <span class="keywordflow">return</span> money_account;
<a name="l00307"></a>00307 }
<a name="l00308"></a>00308
<a name="l00309"></a>00309
<a name="l00310"></a>00310 <span class="comment">// The computation over nSamples paths of the P&L distribution</span>
<a name="l00311"></a>00311 <span class="keywordtype">void</span> ReplicationError::compute(<a class="code" href="group__types.html#gf38bdb4c54463b1f456655efa95b5c77">Size</a> nTimeSteps, <a class="code" href="group__types.html#gf38bdb4c54463b1f456655efa95b5c77">Size</a> nSamples)
<a name="l00312"></a>00312 {
<a name="l00313"></a>00313 <a class="code" href="errors_8hpp.html#7a9bcab8006882bc7d5302a0861ab1a6">QL_REQUIRE</a>(nTimeSteps>0, <span class="stringliteral">"the number of steps must be > 0"</span>);
<a name="l00314"></a>00314
<a name="l00315"></a>00315 <span class="comment">// hedging interval</span>
<a name="l00316"></a>00316 <span class="comment">// Time tau = maturity_ / nTimeSteps;</span>
<a name="l00317"></a>00317
<a name="l00318"></a>00318 <span class="comment">/* Black-Scholes framework: the underlying stock price evolves</span>
<a name="l00319"></a>00319 <span class="comment"> lognormally with a fixed known volatility that stays constant</span>
<a name="l00320"></a>00320 <span class="comment"> throughout time.</span>
<a name="l00321"></a>00321 <span class="comment"> */</span>
<a name="l00322"></a>00322 <a name="_a16"></a><a class="code" href="class_quant_lib_1_1_date.html">Date</a> today = Date::todaysDate();
<a name="l00323"></a>00323 <a name="_a17"></a><a class="code" href="class_quant_lib_1_1_day_counter.html">DayCounter</a> dayCount = <a name="_a18"></a><a class="code" href="class_quant_lib_1_1_actual365_fixed.html">Actual365Fixed</a>();
<a name="l00324"></a>00324 <a name="_a19"></a><a class="code" href="class_quant_lib_1_1_handle.html">Handle<Quote></a> stateVariable(
<a name="l00325"></a>00325 boost::shared_ptr<Quote>(<span class="keyword">new</span> <a name="_a20"></a><a class="code" href="class_quant_lib_1_1_simple_quote.html">SimpleQuote</a>(s0_)));
<a name="l00326"></a>00326 <a class="code" href="class_quant_lib_1_1_handle.html">Handle<YieldTermStructure></a> riskFreeRate(
<a name="l00327"></a>00327 boost::shared_ptr<YieldTermStructure>(
<a name="l00328"></a>00328 <span class="keyword">new</span> <a name="_a21"></a><a class="code" href="class_quant_lib_1_1_flat_forward.html">FlatForward</a>(today, r_, dayCount)));
<a name="l00329"></a>00329 <a class="code" href="class_quant_lib_1_1_handle.html">Handle<YieldTermStructure></a> dividendYield(
<a name="l00330"></a>00330 boost::shared_ptr<YieldTermStructure>(
<a name="l00331"></a>00331 <span class="keyword">new</span> <a class="code" href="class_quant_lib_1_1_flat_forward.html">FlatForward</a>(today, 0.0, dayCount)));
<a name="l00332"></a>00332 <a class="code" href="class_quant_lib_1_1_handle.html">Handle<BlackVolTermStructure></a> <a class="code" href="group__manips.html#gc402ef7c87f63f7c603ee87210b5750c">volatility</a>(
<a name="l00333"></a>00333 boost::shared_ptr<BlackVolTermStructure>(
<a name="l00334"></a>00334 <span class="keyword">new</span> <a name="_a22"></a><a class="code" href="class_quant_lib_1_1_black_constant_vol.html">BlackConstantVol</a>(today, sigma_, dayCount)));
<a name="l00335"></a>00335 boost::shared_ptr<StochasticProcess1D> diffusion(
<a name="l00336"></a>00336 <span class="keyword">new</span> <a name="_a23"></a><a class="code" href="class_quant_lib_1_1_black_scholes_merton_process.html">BlackScholesMertonProcess</a>(stateVariable, dividendYield,
<a name="l00337"></a>00337 riskFreeRate, volatility));
<a name="l00338"></a>00338
<a name="l00339"></a>00339 <span class="comment">// Black Scholes equation rules the path generator:</span>
<a name="l00340"></a>00340 <span class="comment">// at each step the log of the stock</span>
<a name="l00341"></a>00341 <span class="comment">// will have drift and sigma^2 variance</span>
<a name="l00342"></a>00342 <a name="_a24"></a><a class="code" href="class_quant_lib_1_1_inverse_cumulative_rsg.html">PseudoRandom::rsg_type</a> rsg =
<a name="l00343"></a>00343 PseudoRandom::make_sequence_generator(nTimeSteps, 0);
<a name="l00344"></a>00344
<a name="l00345"></a>00345 <span class="keywordtype">bool</span> brownianBridge = <span class="keyword">false</span>;
<a name="l00346"></a>00346
<a name="l00347"></a>00347 <span class="keyword">typedef</span> <a name="_a25"></a><a class="code" href="struct_quant_lib_1_1_single_variate.html">SingleVariate<PseudoRandom>::path_generator_type</a> generator_type;
<a name="l00348"></a>00348 boost::shared_ptr<generator_type> myPathGenerator(<span class="keyword">new</span>
<a name="l00349"></a>00349 generator_type(diffusion, maturity_, nTimeSteps,
<a name="l00350"></a>00350 rsg, brownianBridge));
<a name="l00351"></a>00351
<a name="l00352"></a>00352 <span class="comment">// The replication strategy's Profit&Loss is computed for each path</span>
<a name="l00353"></a>00353 <span class="comment">// of the stock. The path pricer knows how to price a path using its</span>
<a name="l00354"></a>00354 <span class="comment">// value() method</span>
<a name="l00355"></a>00355 boost::shared_ptr<PathPricer<Path> > myPathPricer(<span class="keyword">new</span>
<a name="l00356"></a>00356 ReplicationPathPricer(payoff_.optionType(), payoff_.strike(),
<a name="l00357"></a>00357 r_, maturity_, sigma_));
<a name="l00358"></a>00358
<a name="l00359"></a>00359 <span class="comment">// a statistics accumulator for the path-dependant Profit&Loss values</span>
<a name="l00360"></a>00360 <a name="_a26"></a><a class="code" href="class_quant_lib_1_1_generic_risk_statistics.html">Statistics</a> statisticsAccumulator;
<a name="l00361"></a>00361
<a name="l00362"></a>00362 <span class="comment">// The OneFactorMontecarloModel generates paths using myPathGenerator</span>
<a name="l00363"></a>00363 <span class="comment">// each path is priced using myPathPricer</span>
<a name="l00364"></a>00364 <span class="comment">// prices will be accumulated into statisticsAccumulator</span>
<a name="l00365"></a>00365 <a name="_a27"></a><a class="code" href="class_quant_lib_1_1_monte_carlo_model.html">OneFactorMonteCarloOption</a> MCSimulation(myPathGenerator,
<a name="l00366"></a>00366 myPathPricer,
<a name="l00367"></a>00367 statisticsAccumulator,
<a name="l00368"></a>00368 <span class="keyword">false</span>);
<a name="l00369"></a>00369
<a name="l00370"></a>00370 <span class="comment">// the model simulates nSamples paths</span>
<a name="l00371"></a>00371 MCSimulation.addSamples(nSamples);
<a name="l00372"></a>00372
<a name="l00373"></a>00373 <span class="comment">// the sampleAccumulator method of OneFactorMonteCarloOption_old</span>
<a name="l00374"></a>00374 <span class="comment">// gives access to all the methods of statisticsAccumulator</span>
<a name="l00375"></a>00375 <a class="code" href="group__types.html#g4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> PLMean = MCSimulation.sampleAccumulator().mean();
<a name="l00376"></a>00376 <a class="code" href="group__types.html#g4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> PLStDev = MCSimulation.sampleAccumulator().standardDeviation();
<a name="l00377"></a>00377 <a class="code" href="group__types.html#g4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> PLSkew = MCSimulation.sampleAccumulator().skewness();
<a name="l00378"></a>00378 <a class="code" href="group__types.html#g4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> PLKurt = MCSimulation.sampleAccumulator().kurtosis();
<a name="l00379"></a>00379
<a name="l00380"></a>00380 <span class="comment">// Derman and Kamal's formula</span>
<a name="l00381"></a>00381 <a class="code" href="group__types.html#g4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> theorStD = std::sqrt(M_PI/4/nTimeSteps)*vega_*sigma_;
<a name="l00382"></a>00382
<a name="l00383"></a>00383
<a name="l00384"></a>00384 std::cout << std::fixed
<a name="l00385"></a>00385 << nSamples << <span class="stringliteral">"\t| "</span>
<a name="l00386"></a>00386 << nTimeSteps << <span class="stringliteral">"\t | "</span>
<a name="l00387"></a>00387 << std::setprecision(3) << PLMean << <span class="stringliteral">" \t| "</span>
<a name="l00388"></a>00388 << std::setprecision(2) << PLStDev << <span class="stringliteral">" \t | "</span>
<a name="l00389"></a>00389 << std::setprecision(2) << theorStD << <span class="stringliteral">" \t | "</span>
<a name="l00390"></a>00390 << std::setprecision(2) << PLSkew << <span class="stringliteral">" \t| "</span>
<a name="l00391"></a>00391 << std::setprecision(2) << PLKurt << std::endl;
<a name="l00392"></a>00392 }
</pre></div>
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