Analysis of the Models Used in Variance Swap Pricing Jason Vinar U of MN Workshop 2011
Workshop Goals Price variance swaps using a common rule of thumb used by traders, using Monte Carlo simulation with local volatility, and by replication from vanilla options Use two volatility parameterizations to describe the implied volatility smile Compare the pricing differences between approaches across the time series of data and term structure of maturities As time allows explore pricing adjustments for long dated maturities
Overview A variance swap is a popular way to add volatility exposure to a portfolio since It comes without the directional risk of the underlying security he variance swap quotes are based on the implied volatility while the payout is based on the realized volatility he variance is additive allowing the investor to Easily take a forward volatility position Mark an existing variance swap position in a transparent manner by coupling the to date realized variance with future variance Naturally, the attractive features come with drawbacks Pricing methods ultimately rely on prices for instruments that are not traded or non-existent Careful analysis of the models and understanding their shortcomings must be be done before taking a variance swap position
Definition of a Variance Swap Allows investors to trade future realized volatility against the current implied volatility Like other swaps the initial value is 0 with one party paying a fixed, pre-determined level and the other party paying a floating level he fixed side of the variance swap is the strike he floating side is determined by future movements in the assets and is measured in log returns he Pay-off P = N(σ 2 R K var ) where N is the notional amount, K var is the strike quoted in annualized variance, and σr 2 is the realized variance over the life of the contract defined as σ 2 R = 252 D D i=1 ( ln S i S i 1 with D being the number of trading days during the contract ) 2
Review P&L of a delta hedged vanilla option position he daily P&L of an option comes from changes in stock price (delta and gamma), time (theta), volatility (vega), rates (rho), dividends (mu), and other, e.g. second order changes Assuming constant rates, volatility, dividends and no other P&L this reduces to changes in stock price and time Further, assume that the position is delta hedged, i.e. = 0, then the daily P&L is PL D = 1 2 Γ( S)2 + Θ t Gamma and heta have opposite signs so you either make money from Gamma and lose from heta (long in options) or vice versa
Review P&L Continued heta can be approximated in terms of Gamma as Θ 1 2 ΓS 2 σ 2 imp Using the approximate heta in the daily P&L we get PL D 1 2 Γ(( S)2 S 2 σ 2 imp t) 1 2 ΓS 2 (( S S ) 2 (σ imp t) 2 he P&L is driven by two factors: dollar Gamma and the variance spread he total option P&L is PL = 1 2 D γ t [rt 2 (σ imp t) 2 ] t=1 where γ t = Γ t S 2 t, r t = ln St S t 1, and D is the number of trading days )
Pricing by Replication using Vanilla Options Compare the variance swap pay-off to the P&L for vanilla option Variance swap pay-off Vanilla option P&L P = 252 D PL = 1 2 D t=1 ( ln S t S t 1 ) 2 K 2 var D γ t [rt 2 (σ imp t) 2 ] t=1 he difference between the two results is the weighting: variance swaps are evenly weighted whereas the option is dollar-gamma weighted Can we find a combination of put and call options (same maturity but with different strikes) such that the aggregate dollar-gamma is constant, i.e. N put w p γ A = i=1 N call i γ p i + i=1 w c i γ c i = C
Pricing by Replication using Vanilla Options Using weights that are inversely proportional to the square of the strike produces a constant dollar gamma C. In a discrete setting the fair strike (variance swap price) is K vs = 2er [ N puts i=1 w i put(k i ) F N calls + i=1 ] call(k i ) w i F where w i = c K with c = 1 2 N 1, put(k) is the put price, and call(k) is the call price. Note, out of the money options (w.r.t. the forward) are not used. Commonly, this approach uses a range of percent strikes (of the forward) to find the dollar strike, e.g. 50% to 150% by 5% increments he put component uses strikes that are less than or equal to the forward and the call component uses strikes that are greater than or equal to the forward. his construction includes the same strike on both sides an therefore the respective weights (w N and w 1 ) are divided by 2.
Pricing by Replication using Vanilla Options An exact replication requires an infinite number of options Here the fair strike is [ Kvs 2e = r F 1 put(k) 0 K 2 dk + F F 1 call(k) K 2 F ] dk Either approach requires option prices (volatility) for strikes that are not actively traded or even available. A model used to fit the implied volatility surface is used to fill in the gaps and extrapolate beyond the range of traded strikes. As one would expect, the extrapolation leads to uncertainty in the price of the variance swap. Quants have adapted their volatility surface parameterizations and traders have developed rule-of-thumb pricing to circumvent this problem.
Volatility Parameterization he volatility parameterizations will be used to fill in option prices and volatilities for strikes that are not traded. In both parameterizations k = ln(k/f ), where K is the strike and F is the forward Each time slice (option expiration) has its own set of parameters Gatheral s SVI Model σ(f,, K; α) = α 1 + α 3 [α 4 (k α 2 ) + (k α 2 ) 2 + α 2 5 ]
Volatility Parameterization 7-parameter Skew Model A L e λlk + β 4 σ(f,, K; β) = β 1 + kβ2 + k2 β 3 2 A R e λ R k + β 6 : k L < k : k L k k R : k R > k where k L = β 1 β 5 σ kl = β 1 + k Lβ 2 dσ kl kr = β 1 β 7 + k2 L β3 2 σ kr = β 1 + k R β 2 + k2 R β3 2 = β2 + k Lβ 3 dσ kr = β2 + k R β 3 σ L = σ kl β 4 σ R = σ kr β 6 λ L = dσ k L σ L λ R = dσ k R σ R A L = σ L e λ L k L A R = σ R e λ R k R
Price by Rule of humb 90% put: take 90% of the spot value as the strike and plug that into the volatility function 25% delta put: search for the strike that has a -25% delta (on the put side) and plug that strike into the volatility function. Example
What Lies Ahead Prerequisite Skills, Methods, and Models used in this Workshop Programming Creating functions Non-linear least squares Numerical integration Data organization Black-Scholes Model Monte Carlo simulation Questions to be answered Do the rule of thumb methods come close to the market price? How about the Monte Carlo simulaiton? How about the theoretical price? Does one of the volatility parameterizations perform better than the other? Does it depend on the pricing method? How well do the pricing methods perform as the variance swap maturity increases?
References More han you Ever Wanted o Know About Variance Swaps; Demeterfi, Derman, Kamal, Zou; Goldman Sachs Quantitative Strategies (1999). Just What You Need o Know About Variance Swaps; Bossu, Guichard, Strasser; JPMorgan Equity Derivatives Report (2005).