Arbitrage Bounds for Volatility Derivatives as Free Boundary Problem Bruno Dupire Bloomberg L.P. NY bdupire@bloomberg.net PDE and Mathematical Finance, KTH, Stockholm August 16, 25
Variance Swaps Vanilla options are complex bets on S/σ Variance Swaps capture volatility independently of S Payoff: (ln S t i+1 S ti ) 2 V S Realized Variance Replicable from Vanilla option (if no jump): lns T = lns + T ds S + 1 2 T (d lns) 2 2
Options on Realized Variance Over the past couple of years, massive growth of Call on Realized Variance: ( (ln S t i+1 S ti ) 2 L) + Put on Realized Variance: (L (ln S t i+1 S ti ) 2 ) + Cannot be replicated by Vanilla options 3
Classical Models Classical approach: To price an option on X: Model the dynamics of X, in particular its vol perform dynamic hedging For options on realized variance: Hypothesis on the vol of VS Dynamic hedge with VS But Skew contains important information and we will examine how to exploit it to obtain bounds for the option prices. 4
Link with Skorokhod Problem Option prices of maturity T Risk Neutral density of S T : ϕ(k) = 2 K 2E[(S T K) + ] Skorokhod problem: For a given probability density function such that xϕ(x)dx =, x 2 ϕ(x)dx = v <, find a stopping time τ of finite expectation such that the density of a Brownian motion W stopped at τ is ϕ A continuous martingale S is a time changed Brownian Motion: W t S inf{u:<s>u >t} is a BM, and S t = W <S>t 5
Solution of Skorokhod Calibrated Martingale τ solution of Skorokhod: W τ ϕ Then S T W τ t/(t t) satisfies S T = W τ ϕ If S T ϕ, then τ < S > T is a solution of Skorokhod as W τ = W <S>T = S T ϕ 6
ROOT Solution Possibly simplest solution τ : exit time W t 7
Barrier Density B = {K,T (K,T) Barrier} W ϕ = Density of W τ PDE: φ T (K, T) = 2 φ K2(K, T) on BC φ(k, t) = ϕ(k) on B t BUT: How about Density Barrier? 8
PDE construction of ROOT (1) Given ϕ, define f(k) x K ϕ(x)dx If dx s = σ(x s, s)dw s, Cf(K, t) E[ X t K ] satisfies f t = σ2 (K, t) 2 f 2 K 2 Apply the previous equation with σ(k, t) = 1 until f(k,t K ) = f(k). Then for t > t K, σ(k,t) ( f(k, t) = f(k)) 9
PDE computation of ROOT (2) Define τ as the hitting time of B {(K,t) : σ(k, t) = }, X t = W τ t Then f(k, t) E[ X t K ] = E[ W τ t K ] f(k, ) = lim t f(k, t) = E[ W τ K ] Thus W τ ϕ, and B is the ROOT barrier 1
PDE computation of ROOT (3) Interpretation within Potential Theory 11
ROOT Exemples K K K time t time t time t K K time t time t 12
min(< S > T L) + max E[W 2 δ L ] realized Variance RV =< S > T = τ Call on RV : (τ L) + Ito: W 2 τ = W 2 τ L + τ τ L W tdw t + τ τ L taking expectation, v = E[W 2 τ L ] + E[(τ L)+ ] Minimize one expectation amounts to maximize the other one 13
Link τ / LVM Suppose dy t = σ t dw t, then define f Y (K, T) E[ Y T K ] f Y satisfies fy T = 1 2 E[σ2 T Y T = K] 2 f Y K 2 (Dupire 96) Let τ be a stopping time. For X t = W τ t, one has σ t = I τ>t and E[σ 2 T X T = K] = P[τ > T X T = K] dy t = a(y t, t)dw t where a(y, t) P[τ > t X t = y] generates the same prices as X: f Y (K, T) = f X (K, T) for all (K, T) For our purpose, τ identified by a(x, t)=p[τ > t X t = x] 14
Optimality of ROOT As E[ W 2 τ L ] = (E[ W τ L K ] W K )dk, to maximize E[ W τ L K ] = to maximize E[ W 2 τ L ] to minimize E[(τ L) + ] E[ W τ L K ] = f(k, L) and a(x, t) = P[τ > t W t = x], f satisfies: f 2 = 1 2 a2 (x, t)f 11 for a > f 2 = for a = f is maximum for ROOT time, where a = 1 in B C and a = in B 15
ROST Filling Scheme 16
PDE interpretation/construction Given, ϕ, define C(K, ) x K ϕ(x)dx Now define µ (K) = C(K, ) K µ t+δt = min(µ t P δt, µ t ), where µ t P δt (x) = µ t (x + y) e y2 /(2δt) 2πδt dy 17
PDE construction of ROST Define B(t) = {K : µ t P δt (K) = µ t (K)} B is the frontier of the reverse barrier of ROST 18
ROST Examples - Normal density: Gaussian Mixture ROOT Barrier ROST Barrier - Asymmetric density: 19
CONCLUSION Skorokhod problem is the right framework to analyze range of exotic prices constrained by Vanilla prices Barrier solutions provide canonical mapping of densities into barriers They give the range of prices for option on realized variance The Root solution diffuses as much as possile until it is constrained The Rost solution stops as soon as possible We provide explicit construction of these barriers 2