Option Pricing
May 25, 2018
Link to complete code on Github
Introduction
This program was written in June 2016 when I took the course Financial Computation in National Taiwan University. Unlike normal optins, lookback options are path dependent and its strike price is determined at maturity. Rather than being calculated according to a fixed strike price, the strike price of a lookback option equals to the maximum or minimum price of underlying assets over its lifetime. Therefore, the pay off of an European lookback option can be written as:
1. Call:
ct = max ( St - mt , 0 ) , mt = min { Su | 0 <= u <= t }
2. Put:
pt = max ( mt - St , 0 ) , mt = max { Su | 0 < = u < = t }
1. Call:
ct = max ( St - mt , 0 ) , mt = min { Su | 0 <= u <= t }
2. Put:
pt = max ( mt - St , 0 ) , mt = max { Su | 0 < = u < = t }
Pricing Algorithm– Monte Carlo Simulation
The pricing algorithm can be sepreated into process as follows:
1. Generate a uniformly distributed random variable between 0 and 1, and compute the value under normal distribution using the inverse cummulative distribution function. The inverse cummulative distribution function can be written in different ways which is often discussed in the course Numerical Analysis.
1. Generate a uniformly distributed random variable between 0 and 1, and compute the value under normal distribution using the inverse cummulative distribution function. The inverse cummulative distribution function can be written in different ways which is often discussed in the course Numerical Analysis.
u=rand()/(double)RAND_MAX; // Generate a uniformly distributed random variable
normInv = NormalCDFInverse(u);
normInv=normInv*standardDev-mean;
stockPrice[i]=stockPrice[i]*exp(normInv);
smax=max(smax,stockPrice[i]);
2. Recall that if Z ~ N(0,1), a lognormal stock price can be written
ST = S0 * e (α - δ - 0.5 * σ * σ ) * T + σ * sqrt(T) * Z
Therefore, we can derive the price as follow, and later find averages of payoffs with numerous times of simulations.
ST = S0 * e (α - δ - 0.5 * σ * σ ) * T + σ * sqrt(T) * Z
Therefore, we can derive the price as follow, and later find averages of payoffs with numerous times of simulations.
//European lookback put
stockPrice[i]=max(smax-stockPrice[i],0)*exp(-iRate*tOption);
average1= average1+stockPrice[i];
Pricing Program Applications
In addition to setting spot price equals 50 at t0, risk free rate 0.1, yield rate 0, volatility of the underlying stock price 0.4, time to maturity of option 0.25, and time interval of 300, you can enter the number of times of simulation. After 9999 times of simulation, for example, the program would demonstrate the price of the European lookback put.
