1 Option Risk Management Elements of Financial Risk Management Chapter 11 Peter Christoffersen Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Overview • In this chapter, we try to incorporate derivative securities into the portfolio risk model • The chapter is structured as follows: • We define the delta of an option, which provides a linear approximation to the nonlinear option price We then present delta formulas from the various models introduced in the previous chapter • We establish the delta-based approach to portfolio risk management The idea behind this approach is to linearize the option return and thereby make it fit into the risk models The downside of this approach is that it ignores the key asymmetry in option payoffs Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Overview • We define the gamma of an option, which gives a secondorder approximation of the option price as a function of the underlying asset price • We use gamma of an option to construct a quadratic model of the portfolio return distribution We discuss two implementations of the quadratic model: one relies on the Cornish-Fisher approximation and the other relies on the Monte Carlo simulation technique Elements of Financial Risk Management Second Edition â 2012 by Peter Christoffersen Overview We will measure the risk of options using the full valuation method, which relies on an accurate but computationally intensive version of the Monte Carlo simulation technique • We illustrate all the suggested methods in a simple example We then discuss a major pitfall in the use of the linear and quadratic approximations in another numerical example This pitfall, in turn, motivates the use of the full valuation model Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Option Delta • The delta of an option is defined as the partial derivative of the option price with respect to the underlying asset price, St • For puts and calls, we define • The option price for a generic underlying asset price, S, is approximated by • where St is the current price of the underlying asset Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Option Delta • The delta of an option provides a linear approximation to the nonlinear option price, where the approximation is reasonably good for asset prices close to the current price but gets gradually worse for prices that deviate significantly from the current price • To a risk manager, the poor approximation of delta to the true option price for large underlying price changes is clearly unsettling • Risk management is all about large price changes, and we will therefore consider more accurate approximations here Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Figure 11.1: Call Option Price and Delta Approximation Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Black-Scholes-Merton Model • The Black-Scholes-Merton (BSM) formula for a European call option price • where (*) is the cumulative density of a standard normal variable, and • Using basic calculus, we can take the partial derivative of the option price with respect to the underlying asset price, St, as follows: Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Black-Scholes-Merton Model • We refer to this as the delta of the option, and it has the interpretation that for small changes in St the call option price will change by (d) • Notice that as (*) is the normal cumulative density function, which is between zero and one, we have • so that the call option price in the BSM model will change in the same direction as the underlying asset price, but the change will be less than one-for-one Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Black-Scholes-Merton Model 10 • For a European put option, we have the put-call parity stating that • so that we can easily derive • Notice that we have • so that BSM put option price moves in the opposite direction of underlying asset, and again option price will change by less than the underlying asset price Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen A Simple Example • Using the and calculated earlier, we find • Notice again that due to the relatively high of this option, the quadratic VaR is more than 50% higher than the linear VaR Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen 67 A Simple Example 68 • Finally, we can use the full valuation approach to find the most accurate VaR • Using the simulated asset returns to calculate hypothetical future stock prices, , we calculate the simulated option portfolio value changes as • where 14 is the number of calendar days in the 10-tradingday risk horizon • We then calculate the full valuation VaR as Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen A Simple Example • In this example, the full valuation VaR is slightly higher than the quadratic VaR • The quadratic VaR thus provides a pretty good approximation in this simple portfolio of one option Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen 69 Figure 11.6: Histogram of Portfolio Value Changes Using the Delta-Based Model Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen 70 Figure 11.7: Histogram of Portfolio Value Changes Using the Gamma-Based Model Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen 71 Figure 11.8: Histogram of Portfolio Value Changes Using Full Valuation Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen 72 73 Pitfall in Delta and Gamma Approaches • Here we use an example to illustrate that the gamma approximation can sometimes be highly misleading • Consider an option portfolio that consists of types of options all on the same asset, and that has a price of St = 100, all with calendar days to maturity • The risk-free rate is 0.02/365 and the volatility is 0.015 per calendar day • We take a short position in put with a strike of 95, a short position in 1.5 calls with a strike of 95, and a long position in 2.5 calls with a strike of 105 Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen 74 Pitfall in Delta and Gamma Approaches • Using the BSM model to calculate the delta and gamma of the individual options, we get Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen 75 Pitfall in Delta and Gamma Approaches • Now we try to assess the accuracy of the delta and gamma approximation for portfolio over a five trading day or equivalently seven calendar day horizon • Rather than computing VaRs, we will take a closer look at the complete payoff profile of the portfolio for different future values of underlying asset price, St+5 • We refer to the value of the portfolio today as VPFt and to the hypothetical future value as VPFt+5(St+5) Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen 76 Pitfall in Delta and Gamma Approaches • We first calculate the value of the portfolio today as • The delta of the portfolio is similarly • Now, the delta approximation to the portfolio value in five trading days is easily calculated as Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen 77 Pitfall in Delta and Gamma Approaches • The gamma of the portfolio is • and the gamma approximation to the portfolio value in five trading days is Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen 78 Pitfall in Delta and Gamma Approaches • Finally, relying on full valuation, we must calculate the future hypothetical portfolio values as • where we subtract seven calendar days from the time to maturity corresponding to the risk management horizon of five trading days Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen 79 Figure 11.9 Future portfolio values for option portfolio Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Pitfall in Delta and Gamma Approaches 80 • The important lesson of this three-option example is as follows: • The different strike prices and the different exposures to the underlying asset price around the different strikes create higher order nonlinearities, which are not well captured by the simple linear and quadratic approximations • In realistic option portfolios consisting of thousands of contracts, there may be no alternative to using the full valuation method Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Summary 81 • This chapter has presented three methods for incorporating options into the risk management model – Delta-based (linear) risk models – Gamma-based (quadratic) risk models – Full Valuation • A simple example • Pitfalls in Delta and Gamma-based risk models • The main lesson from the chapter is that for nontrivial options portfolios and for risk management horizons beyond just a few days, the full valuation approach may be the only reliable choice Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen ... approximations here Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Figure 11.1: Call Option Price and Delta Approximation Elements of Financial Risk Management Second... (bottom) Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen 12 Figure 11.3: The Delta of Three Call Options In-the-money At-the-money Out -of- the-money Elements of Financial. .. option payoffs Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Overview • We define the gamma of an option, which gives a secondorder approximation of the option