Elements of financial risk management chapter 10

1 Option Pricing Elements of Financial Risk Management Chapter 10 Peter Christoffersen Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Overview • In this chapter we derive a no-arbitrage relationship between put and call prices on same underlying asset • Summarize binomial tree approach to option pricing • Establish an option pricing formula under simplistic assumption that daily returns on the underlying asset follow a normal distribution with constant variance • Extend the normal distribution model by allowing for skewness and kurtosis in returns • Extend the model by allowing for time-varying variance relying on the GARCH models • Introduce the ad hoc implied volatility function (IVF) approach to option pricing Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Basic Definitions • An European call option gives the owner the right but not the obligation to buy a unit of the underlying asset days from now at the price X • is the number of days to maturity • X is the strike price of the option • c is the price of the European option today • St is the price of the underlying asset today • is the price of the underlying at maturity Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Basic Definitions • A European put option gives the owner the option the right to sell a ~ now at the price X unit of the underlying asset days from T • p denotes the price of the European put option today • The European options restricts the owner from exercising the option before the maturity date • American options can be exercised any time before the maturity date • Note that the number of days to maturity is counted is calendar days and not in trading days ~ • A standard year has 365 calendar days but onlyTaround 252 trading days Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Basic Definitions • The payoffs (shown in Figure 10.1) are drawn as a function of the hypothetical price of the underlying asset at maturity of the option, • Mathematically, the payoff function for a call option is • and for a put option it is • Note the linear payoffs of stocks and bonds and the nonlinear payoffs of options from Figure 10.1 • We next consider the relationship between European call and put option prices Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Figure 10.1: Payoff as a Function of the Value of the Underlying Asset at Maturity Call Option, Put Option, Underlying Asset, Risk-Free Bond Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Basic Definitions • Put-call parity does not rely on any particular option pricing model It states • It can be derived from considering two portfolios: • One consists of underlying asset and put option and another consists of call option, and a cash position equal to the discounted value of the strike price • Whether underlying asset price at maturity, ends up below or above strike price X; both portfolios will have same value, namely , at maturity • Therefore they must have same value today, otherwise arbitrage opportunities would exist Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Basic Definitions • The portfolio values underlying this argument are shown in the following Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Basic Definitions • put-call parity suggests how options can be used in risk management • Suppose an investor who has an investment horizon of days owns a stock with current value St • Value of the stock at maturity of the option is which in the worst case could be zero • An investor who owns the stock along with a put option with a strike price of X is guaranteed the future portfolio value , which is at least X • The protection is not free however as buying the put option requires paying the current put option price Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen 10 Option Pricing Using Binomial Trees • We begin by assuming that the distribution of the future price of the underlying risky asset is binomial • This means that in a short interval of time, the stock price can only take on one of two values, up and down • Binomial tree approach is able to compute the fair market value of American options, which are complicated because early exercise is possible Elements of Financial Risk Management Second Edition â 2012 by Peter Christoffersen Model Implementation 72 For a given conditional variance σ 2t+1 , and parameters ω, α, β, λ *, we can use Monte Carlo simulation to create future hypothetical paths of the asset returns • We can illustrate the simulation of hypothetical daily returns from day t+1 to the maturity on day as • where the are obtained from a N(0,1) random number generator and where MC is the number of simulated return paths Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Model Implementation 73 • We need to calculate the expectation term E*t[*] in the option pricing formula using the risk-neutral process, thus, we calculate the simulated risk-neutral return in period t+j for simulation path i as • and the variance is updated by • the simulation paths in the first period all start out from the same σ2t+1; therefore, for all i, we have Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Model Implementation 74 • Once we have simulated, say, 5000 paths (MC=5000) each day until the maturity date , we can calculate hypothetical risk-neutral asset prices at maturity as • Option price is calculated by taking the average over future hypothetical payoffs and discounting them to the present as in • where GH denotes GARCH Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Model Implementation 75 • Thus, we use simulation to calculate the average future payoff, which is then used as an estimate of the expected value, E*t[*] • As number of Monte Carlo replications gets infinitely large, the average will converge to the expectation • Around 5000 replications provide a precise estimate • In theory, we can estimate all the parameters in the GARCH model using the maximum likelihood method on the underlying asset returns • But to obtain a better fit of the option prices, we can instead minimize the option pricing errors directly Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Model Implementation 76 • Treating the initial variance σ 2t+1 as a parameter to be estimated, we can estimate GARCH option pricing model on a daily sample of options by numerically minimizing the mean squared error • Note that for every new parameter vector the numerical optimizer tries, the GARCH options must all be repriced using the MC simulation technique • Thus the estimation can be quite time consuming Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Closed Form GARCH Option Pricing Model 77 • A drawback of the GARCH option pricing framework is that it does not provide us with a closed-form solution for the option price, which must instead be calculated through simulation • Although the simulation technique is straightforward, it does take computing time and introduces an additional source of error arising from approximation of the simulated average to the expected value • To overcome these issues, we introduce the closed-form GARCH or CFG model Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Closed Form GARCH Option Pricing Model 78 • Assume that returns are generated by the process • Note that the risk premium is now multiplied by the conditional variance not standard deviation • Also note that zt enters in the variance innovation term without being scaled by σt • Variance persistence in this model can be derived as αθ2 + β and the unconditional variance as (ω + α) / (1- αθ2 - β) Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Closed Form GARCH Option Pricing Model 79 • The risk-neutral version of this process is • To verify that the risky assets earn the risk-free rate under the risk-neutral measure, we check again that Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Closed Form GARCH Option Pricing Model 80 • The variance in the model can be verified as before • Under this special GARCH process for returns, the European option price can be calculated as • where formulas for P1 and P2 are given in appendix • Notice that the structure of the option pricing formula is identical to that of the BSM model • As in BSM model, P2 is the risk-neutral probability of exercise, and P1 is the delta of the option Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen 81 Implied Volatility Function (IVF) Models • This approach to European option pricing is completely static and ad hoc but it turns out to offer reasonably good fit to observed option prices • The idea behind the approach is that the implied volatility smile changes only slowly over time • If we can estimate a functional form on the smile today, then that functional form may work reasonably in pricing options in the near future as well • The implied volatility smiles and smirks mentioned earlier suggest that option prices may be well captured by the following four-step approach: Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen 82 Implied Volatility Function (IVF) Models – Calculate the implied BSM volatilities for all the observed option prices on a given day as – Regress the implied volatilities on a second-order polynomial in moneyness and maturity – That is, use ordinary least squares (OLS) to estimate the a parameters in the regression • where ei is an error term and where we have rescaled maturity to be in years rather than days Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Implied Volatility Function (IVF) Models • The rescaling is done to make the different a coefficients have roughly same order of magnitude • This will yield the implied volatility surface as a function of moneyness and maturity – Compute fitted values of implied volatility from regression Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen 83 84 Implied Volatility Function (IVF) Models – Calculate model option prices using fitted volatilities and BSM option pricing formula, as in • where the Max(*) function ensures that the volatility used in the option pricing formula is positive • Note that this option pricing approach requires only a sequence of simple calculations and it is thus easily implemented Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen 85 Implied Volatility Function (IVF) Models • To obtain much better model option prices, we can use the modified implied volatility function (MIVF) technique • We can use a numerical optimization technique to solve for a = {a 0, a1, a2, a3, a4, a5} by minimizing the mean squared error • The downside of this method is clearly that a numerical solution technique rather than simple OLS is needed to find the parameters Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Summary • • • • Binomial tree approach to option pricing Black-Scholes-Merton (BSM) model Gram-Charlier (GC) expansion Two types of GARCH option pricing models – Allowing for Dynamic Volatility – A Closed-Form GARCH option pricing model • Implied volatility function (IVF) approach Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen 86 ... Peter Christoffersen Figure 10.1: Payoff as a Function of the Value of the Underlying Asset at Maturity Call Option, Put Option, Underlying Asset, Risk- Free Bond Elements of Financial Risk Management. .. payoffs of stocks and bonds and the nonlinear payoffs of options from Figure 10.1 • We next consider the relationship between European call and put option prices Elements of Financial Risk Management. .. exist Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Basic Definitions • The portfolio values underlying this argument are shown in the following Elements of Financial