The Skorokhod Embedding Problem and Model Independent Bounds for Options Prices. David Hobson University of Warwick

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1 The Skorokhod Embedding Problem and Model Independent Bounds for Options Prices David Hobson University of Warwick Summer School in Financial Mathematics, Ljubljana, September 211

2 Overview Lecture 1: A philosophy for model-independent pricing Lecture 2: The Skorokhod Embedding Problem Lecture 3: An application to Variance Swaps

3 Model-independent option pricing The classical approach The classical approach to derivative pricing The standard approach is to postulate a model (or a parametric family of models) (Ω,F,P,F,S). Price a contingent claim via expectation under the equivalent martingale measure Q: We have replication: E Q [e rt F T ] T e rt F T = E Q [e rt F T ]+ θ t d(s t e rt ) P/Qa.s.

4 Model-independent option pricing The classical approach Example For the family of exponential Brownian motion models ds t /S t = µdt +σdw t, for a call with payoff (S T K) +, the price is C BS where C BS = C BS (T,K;,S ;r,µ,σ) For convex payoffs (eg puts and calls) C BS (σ) is increasing in σ. Hence, in practice the model is calibrated using a liquid option. C BS (T,K) is an extrapolation device.

5 Model-independent option pricing Mis-specified models In a complete market the price and hedge are uniquely specified and replication is perfect provided the model is a perfect description of the real world. With an imperfect model (for notational simplicity, suppose r = ) Theorem Suppose the goal is to price a European claim on S with convex payoff F(S T ). Suppose the claim is priced and hedged under a diffusion model ds t = S t σ(s t )db t and that S is a martingale. Suppose the real world dynamics are such that ds t = S t σ t db t Suppose σ t σ(s t ). Then we have super-replication T E Q [F(S T )]+ θ(t,s t )ds t F(S T ) Q a.s.

6 Model-independent option pricing Pricing criteria in incomplete markets Pricing criteria in incomplete markets Price under a martingale measure Price under a risk measure Super-replication pricing Utility-indifference pricing Marginal utility-indifference pricing All these criteria are model based.

7 Model-independent option pricing A reverse approach: recovering models from prices Breeden and Litzenberger Lemma Suppose, for fixed T, call prices are known for every strike K (, ). Then, assuming we have C(T,K) = E Q [e rt (S T K) + ] Q(S T > K) = e rt K C(T,K) Q(S T dk) = e rt 2 2 K C(T,K)

8 Model-independent option pricing A reverse approach: recovering models from prices Corollary Any co-maturing European claim can be priced and hedged perfectly. F(S T ) = F(x)+(S T x)f (x) + x F (k)(s T k) + dk + x F (k)(k S T ) + dk E[e rt F(S T )] = e rt F(x)+(E[e rt S T ] e rt x)f (x) + x F (k)c(t,k) + dk + x F (k)p(t,k)dk

9 Model-independent option pricing A reverse approach: recovering models from prices Dupire Lemma Suppose call prices are known for every strike K (, ) and every T (,T]. Assuming C(T,K) is sufficiently differentaible, there exists a unique diffusion of the form ds t = S t σ(t,s t )db t +rs t dt such that C(T,K) = E[e rt (S T K) + ] In particular σ(t, s) solves = 1 2 σ(t,k)2 K K C(T,K) rk K C(T,K) T C(T,K)

10 Model-independent option pricing A reverse approach: recovering models from prices More generally, given a set of option prices is there a model consistent with those option prices? is there a model with continuous paths? is there a model with paths of finite quadratic variation? if there is, is it unique? if there is not, is there an arbitrage? is there a model-independent (strong) arbitrage? is there a model-dependent (weak) arbitrage?? A consistent model is a quintuple (Ω,F,Q,F,S) such that E Q [e rt F T ] = f for each traded claim with associated traded price f. Prices of underlyings and vanillas { Models/ Classes of models } Prices and hedges for exotics

11 Model-independent option pricing The pricing problem The pricing problem Suppose we are given a family {F α ;α A} of traded payoffs. Let V be the linear space with basis {F α ;α A}. Suppose we are also given a map (pricing functional) L : V R such that L(λ 1 F 1 +λ 2 F 2 ) = λ 1 L(F 1 )+λ 2 L(F 2 ) F 1 F 2 L(F 1 ) L(F 2 ) Suppose we now add a family of payoffs {F α ; α Ã}. When can L be extended to a linear function L : Ṽ R where Ṽ is the linear space with basis {F α ;α A} {F α ; α Ã}? The family {F α } is the set of traded securities, with traded prices {f α }. The family {F α } is the set of exotics.

12 Model-independent option pricing The pricing problem We split the class of traded securities into underlyings and vanillas and A into A U A V Underlyings may be traded with a simple strategy. If S is an underlying (e r(t τ) S T S τ ) {F α ;α A U } Such a security is costless. Moreover we allow positions (ie multiples) δ τ = δ(s t ; t τ) which depend on the price history up to τ Vanillas (typically calls and puts) may only be traded at time zero.

13 Model-independent option pricing The pricing problem Definition A semi-static superhedging strategy Π for ˆF is a portfolio of traded instruments such that Π = m U i=1 δ α i F αi + m V j=1 π α j F αj ˆF. Lemma Suppose L(F α ) = f α α A. Suppose Π = m U i=1 δ α i F αi + m V j=1 π α j F αj ˆF. Then, for any extension L, L(ˆF) m V j=1 π α j f αj.

14 Model-independent option pricing The pricing problem Henceforth we assume a single underlying, and that the set of vanilla options is the set of calls and puts with maturity T and with the full continuum of strikes K. No other vanilla securities trade. The aim is to give (the range of) no-arbitrage prices for a co-maturing exotic.

15 Model-independent option pricing The canonical example One-touch digitals Let S be a forward price. We assume it is possible to trade forwards on S. Given the prices of calls with maturity T, what is the price of a one-touch digital option paying ˆF = I {sup t T S t B}? Note C(K) is a decreasing convex function with C() = S. Suppose S < B else the problem is trivial. Let H B = inf{t : S t B}. Then ˆF = I {HB T} and we have, for each K < B, Then ˆF (S T K) + 1 B K +(S 1 H B S T ) B K I {H B T} C(K) L(ˆF) inf K<BB K

16 Model-independent option pricing The canonical example Is there a model for which the bound is attained? If so then the bound is a best bound. Write K = K (B) = arginf{c(k)/(b K)}. Write µ for the law of S T. Suppose µ has a density with respect to Lebesgue measure. Let b = E µ [S T S T > K ] = y>k yµ(dy)/ y>k µ(dy). Let a = E µ [S T S T < K ] = y<k yµ(dy)/ y<k µ(dy).

17 Model-independent option pricing The canonical example Consider a process (S t ; t T) such that S = yµ(dy) S is constant on (,T/2) S T/2 = { b with probability (S a)/(b a) a with probability (b S )/(b a) S is constant on (T/2,T) conditional on S T/2 = b, S T has law µ restricted to (K, ); conditional on S T/2 = a, S T has law µ restricted to (,K ). Lemma S is a martingale with S T µ. (S T K ) = (H b T) = (S T > K ) b = B P(H B T) = E[ˆF] = C(K )/(B K )

18 Model-independent option pricing The canonical example Do other models exist? Are there continuous models? Consider S over [,T/2). Let W = S = yµ(dy). Let Ha,b W = inf{u : W u / (a,b)}. Then { b with probability (S a)/(b a) W H W = a,b a with probability (b S )/(b a) Let Λ be an increasing continuous function such that Λ() = and lim t T/2 Λ(t) =. Now set S t = W min{h W a,b,λ(t)}. Then (S t) t T/2 is a martingale with the correct distribution at time T/2. Is there a similar construction over [T/2,T]? The same time-change argument will work, so we can convert this to a question about Brownian motion.

19 Model-independent option pricing The canonical example Suppose (S t ) t T is a martingale such that S T µ. Then there exists a Brownian motion W and a time-change A such that S t = W At. If S is continuous then the Dambis-Dubins-Schwarz Theorem gives that S t = W S,S t Then W AT µ. Conversely, suppose W τ µ. Then S t = W t/(t t) τ is such that S T = W τ µ. Hence we can identify price processes (S t ) t T for which S T µ with stopped Brownian motions with W τ µ. The problem of finding stopping times for Brownian motion for which W τ µ is called the Skorokhod embedding problem.

20 Skorokhod embeddings Skorokhod embeddings

21 Model-independent bounds for variance swaps Model-independent bounds for variance swaps Overview Continuous monitoring, continuous price process Semimartingale case Non-probabilistic case Continuous monitoring, no continuity assumption on price process Discrete monitoring

22 Continuous monitoring, continuous price process Variance swaps: Neuberger/Dupire Suppose we have a model (Ω,F,F,P) supporting an asset price F = (F t ) t T. Think of F as a forward price. Suppose F is a continuous semimartingale. Denote the quadratic variation by [F] t. Suppose we know the prices of vanilla options with maturity T. We wish to price a variance swap, a path-dependent security paying T d[f](t)/f2 t [logf] T. Then, by Itô s formula for semimartingales T d[f] F 2 t = 2lnF T +2lnF + T 2 F t df t There is a model-independent price for a variance swap, and a perfect hedge consisting of a static portfolio of calls and a dynamic

23 Continuous monitoring, continuous price process Variance swaps: Neuberger/Dupire Suppose we have (Ω,F,F,P) supporting F = F t. Suppose F is a continuous semimartingale. Suppose that the asset price process is continuous (f(t)) t T. Suppose price paths have a quadratic variation [f](t). Suppose we know the prices of vanilla options with maturity T. We wish to price a variance swap, a path-dependent security paying T d[f](t)/f(t)2 [logf] T. Then, by Föllmer s pathwise Itô s formula for semimartingales T d[f] f(t) 2 = 2lnf(T)+2lnf()+ T 2 f(t) df(t) There is a model-independent price for a variance swap, and a perfect hedge consisting of a static portfolio of calls and a dynamic strategy in the underlying.

24 Continuous monitoring, continuous price process In Brownian terms Suppose we now work on a stochastic basis (Ω,F,P,F) on which the price realisation is a stochastic process X t (ω). WLOG we may assume X t is a martingale. If we write X t = B At for some continuous time-change, (and B = X = 1 [WLOG]) then and T (dx t ) 2 X 2 t = AT T ds B 2 s (db At ) 2 B 2 A t = AT AT = 2lnB AT + (db s ) 2 B 2 s = 2 B s db s AT Writing τ for A T, and if X T B τ µ, then [ τ ] ds E = 2E[lnB τ ] = E[ 2lnZ Z µ]. B 2 s ds B 2 s

25 Continuously monitored processes with jumps Continuously monitored processes with jumps What if we no longer assume continuity of our price process? The problem Assume given the continuum of call prices with maturity T; moreover assume calls are available as hedging instruments. Place upper/lower bounds on the price of the variance swap contract. Write µ for the marginal law at time T implied by vanilla prices.

26 Continuously monitored processes with jumps Time changed BM and the variance swap We have X t = B At but the time-change A may have jumps. X T B AT µ. Define J t = inf s t X s and R t = sup s t X s. Define I t = inf s t B s and S t = sup s t B s. Then I At J t X t R t S At and T Taking the lower bound T (dx t ) 2 T (X t ) 2 (dx t ) 2 T (R t ) 2 (dx t ) 2 T (R t ) 2 (dx t ) 2 T (X t ) 2 (dx t ) 2 (J t ) 2 (db At ) 2 T (M) (S At ) 2 = da AT t (S At ) 2 du (S u ) 2

27 Skorokhod embedding problems A Skorokhod embedding problem for Brownian motion Let B be Brownian motion (with B = 1). Suppose µ is a distribution with mean 1. The problem Over Skorokhod embeddings τ (of µ in B) find the embedding which minimises τ du S 2 u (in expectation).

28 Skorokhod embedding problems Optimality for the SEP Let B be a Brownian motion. Let κ be a decreasing function κ : [, ) [,1], and let G 1 be a random variable Define τ G κ = min(inf{u > : B u G},inf{u > : B u κ(s u )}) Theorem (Perkins, 1985) There exists κ,g such that τκ G is an embedding of µ. (Hobson-Klimmek, 21) τκ G minimises E[ τ S 2 t dt] over solutions of the SEP for µ.

29 Skorokhod embedding problems S t B t 1 τ t κ(s t )

30 Skorokhod embedding problems A lower bound on the price of a variance swap Thus we have a model independent bound for the price of the variance swap in the presence of jumps. Question Can this bound be attained? For the bound to be attained we must have (at jumps times) X t = R t = S At = S At (if µ has a density, else we must care about atoms) (R T x) (X T x) (X T < γ(x)) for a monotonic decreasing function γ [Recall we have reduced the problem to a functional of B τ and S τ alone]

31 Skorokhod embedding problems Suppose B = 1. Define H x = inf{u > : B u x}. Theorem Define M t = B(H 1+t/(T t) τ G κ ) Then M t is a martingale for which M T has law µ and for which the price of the variance swap attains the model-independent lower bound.

32 Skorokhod embedding problems G Figure: A price realisation

33 Discrete monitoring Discrete monitoring Definition H : R + R + R is a Variance Swap Kernel if for all x,y R + H(x,y), H(x,x) =, lim y x x 2 H(x,y) (y x) 2 = 1, Examples: H D (x,y) = (y x) 2 /x 2, H L (x,y) = (lny lnx) 2, H B = 2 ( log(y/x) ( y x)) x. Roughly speaking H(x,x(1+ǫ)) = ǫ 2 +o(ǫ 2 ).

34 Discrete monitoring Definition A partition P on [,T] is a set of times = t < t 1 <... < t N = T. Definition The payoff of a variance swap with kernel H for a price realisation f and partition P is V H (f,p) = N 1 k= H(f(t k ),f(t k+1 )). We concentrate on H = H D where H D (x,y) = (y x) 2 /x 2.

35 Hedging Hedging Suppose we can find ψ,δ : R + R such that Then H(x,y) ψ(y) ψ(x)+(y x)δ(x) (*) H(f(t k ),f(t k+1 )) ψ(f(t k+1 )) ψ(f(t k ))+δ(f(t k ))(f(t k+1 ) f(t k )) N 1 V H (f,p) ψ(f(t)) ψ(f())+ δ(f(t k ))(f(t k+1 ) f(t k )) k= If ψ is differentiable, then by considering y x in (*) we see that we may take δ(x) = ψ (x). So we want to find ψ for which H(x,y) ψ(y) ψ(x) (y x)ψ (x) By scaling we assume initial price is unitary. Then we may take ψ(1) = ψ (1) =.

36 Hedging ψ κ(x) 1 Figure: Upper and lower bounds for the uniform case - together with the price under a continuity assumption x

37 Hedging For the best ψ we expect equality at y = x and at y = κ(x) where κ = κ µ was determined earlier. Let κ : (1, ) (,1) be decreasing. Suppose ψ = ψ K solves H(x,y) ψ(y) ψ(x) (y x)ψ (x) with equality at y = κ(x). Then, for x > 1, H(x,κ(x)) = ψ(κ(x)) ψ(x) (κ(x) x)ψ (x) (1) H y (x,κ(x)) = ψ (κ(x)) ψ (x) Then differentiating (1) we get H x (x,κ(x)) = ψ (x)(x κ(x)) i.e. ψ (x) = H x (x,κ(x))/(x κ(x)).

38 Hedging Let k = κ 1. Define ψ = ψ κ by { x ψ(x) = 1 (x u)hx(u,κ(u)) u κ(u) du x > 1 H(k(x),x) +ψ(k(x))+(k(x) x)ψ (k(x)) x < 1

39 Hedging Definition H is an increasing (decreasing) variance swap kernel if H 1. x(u,y) u y is monotone increasing (decreasing) in y, 2. H(a,b)+H y (a,b)(c b) ( )H(a,c) H(b,c) for all a > b. Thus H D is an increasing and H L a decreasing variance swap kernel. The kernel H B is both an increasing and a decreasing kernel. Define L(x,y) = H(x,y) ψ(y)+ψ(x)+ψ (x)(y x). Lemma If H is an increasing (decreasing) variance swap kernel then L(x,y), for all x,y >.

40 Hedging Corollary V HD (f,p) ψ κ (f(t))+σ N 1 i= ψ κ(f(t i ))(f(t i+1 f(t i )) This result holds for any decreasing κ We can optimise over κ. When we do so we find the cheapest subreplicating hedge is obtained by κ = κ µ. Converse results exist for upper bounds and for H L. For H N the upper bound equals the lower bound. This part of the story involves no probability. To show attainment of the bounds, and therefore that the bounds are best possible we need to introduce a stochastic model.

41 Numerical results Figure: Upper and lower bounds for the uniform case - relative to the price under the continuity assumption

42 Summary and Open issues Figure: Upper and lower bounds for the lognormal case - relative to the price under the continuity assumption. Horizontal axis denotes volatility.

43 Summary and Open issues Summary We have found bounds on the price of volatility swaps which apply accross all models consistent with an observed set of co-maturing puts and calls. The bounds can be enforced with a simple hedging strategy. Depending on the form of the kernal, prices may be lower/higher than the continuous case if the jumps are downward. (For H D, jumps down mean that the standard Dupire/Neuberger argument overstates the fair price.)

44 Summary and Open issues Extensions: the forward start problem Suppose we know call prices at T 1 > T > and we want to consider the forward starting variance swap with payoff V H (f,p) = N 1 1 k=n H(f(t k ),f(t k+1 )) where P = {t N,...t N1 } and T = t N < t N +1 < < t N1 = T 1.

45 Summary and Open issues Extensions: the problem with intermediate data Suppose that in addition to call prices at T we also know call prices at intermediate times = T < T 1 < < T K = T, where T k P. Let P k = {t n P;T k 1 t n T k }. Then V H (f,p) = K V H (f,p k ) k=1 and we can get a best bound for the problem with intermediate data from adding together the best bounds for the forward starting problem.

46 Summary and Open issues ?? continuous case 1