Recombining tree approximations for Game Options in Local Volatility models
RRECOMBINING TREE APPROXIMATIONS FOR GAMEOPTIONS IN LOCAL VOLATILITY MODELS
BENJAMIN GOTTESMAN BERDAHHEBREW UNIVERSITY
Abstract.
In this paper we introduce a numerical method for optimal stop-ping in the framework of one dimensional diffusion. We use the Skorokhod em-bedding in order to construct recombining tree approximations for diffusionswith general coefficients. This technique allows us to determine convergencerates and construct nearly optimal stopping times which are optimal at thesame rate. Finally, we demonstrate the efficiency of our scheme with severalexamples of game options. Introduction
Game contingent claim (GCC) or game option, which have been introduced byKifer [8], is a derivative contract between the seller and the buyer of the option,where both have the right to exercise it at any time before a maturity date T . Thebuyer pays an initial amount, which correspond to the price of the option. If thebuyer exercises the contract at time t , before the seller cancels, then he receives thepayment Y t , but if the seller cancels the contract before the buyer then the latterreceives X t . The difference ∆ t = X t − Y t is called the penalty which the sellerneeds to pay to the buyer for the contract cancellation. Concretely, if the seller willexercise at a stopping time σ ≤ T and the buyer at a stopping time τ ≤ T then theformer pays to the latter the amount H ( σ, τ ) where(1.1) H ( γ, τ ) := X γ I γ<τ + Y τ I τ ≤ γ with I Q = 1 if an event Q occurs and I Q = 0 if not. Without loss of generality weassume that the payoff H ( γ, τ ) is discounted.Consider a local volatility model with time horizon T < ∞ , which consists ofa riskless savings account with constant interest rate and of a risky asset whosediscounted value at time t satisfies the stochastic differential equation (SDE)(1.2) dS t S t = σ ( S t ) dW t with a given initial value S > Date : July 14, 2020.
Key words and phrases.
Dynkin games, Game options, Local Volatility, Skorokhod embedding. a r X i v : . [ q -f i n . M F ] J u l B. Gottesman Berdah [11, 14] dealt with finite maturity game options. These two papers considered the(constant volatility) Black–Scholes model.It is well known that pricing game options (see [9] and the references there) leadsto Dynkin games. For finite maturity Dynkin games there are no explicit solutionseven in the relatively simple framework where the diffusion process is a standardBrownian motion, and so it is important to obtain efficient numerical schemes.In this article we extend the results from [2] which studied numerical schemesfor American options in local volatility models. We construct recombing tree ap-proximations for Dynkin games in local volatility models and we obtain the sameerror estimates as in [2]. Namely, our method allows to compute the correspondingvalue and the optimal control with complexity O ( n ) and error estimates of order O ( n − / ), where n is the number of time steps. Finally, we apply our techniqueand provide several numerical results.2. Preliminaries and the Main Result
Consider a complete probability space (Ω , F , P ) together with a standard one–dimensional Brownian motion { W t } ∞ t =0 , and the filtration F t = σ { W s | s ≤ t } com-pleted by the null sets.We consider the model given by (1.2). Set Z t := ln S t . From the Itˆo’s formula(2.1) dZ t = ψ ( Z t ) dW t − ψ ( Z t )2 dt, Z = ln S . where ψ ( z ) := σ ( e z ), z ∈ R . Assumption 2.1.
We assume that ψ : R → R + is a Lipschitz continuous functionsuch that ψ, ψ are bounded. In particular, assumption (2.1) implies that the SDE (2.1) has a unique strongsolution . Moreover, since σ is uniformly bounded then the market model which isgiven by (1.2) is complete. Thus, from [8, 9] we obtain that the price of the gamecontingent claim given by (1.1) equals to V := inf γ ∈T T sup τ ∈T T E [ H ( γ, τ )] = sup τ ∈T T inf γ ∈T T E [ H ( γ, τ )] . (2.2)Assume that the payoffs are given by X t := g ( t, S t ), Y t := f ( t, S t ) where g, f :[0 , T ] × R + → R satisfy g ≥ f and, the following Lipschitz condition | f ( t , x ) − f ( t , x ) | + | g ( t , x ) − g ( t , x ) | + | h ( t , x ) − h ( t , x ) |≤ L ((1+ | x | ) | t − t | + | x − x | ) , t , t ∈ [0 , T ] , x , x ∈ R + for some constant L .We aim to approximate efficiently the value V . As in Section 2.3 our main toolwill be the Skorokhod embedding technique.2.1. Skorokhod embedding.
Fix n ∈ N and denote h := Tn . Set σ := sup z ∈ R ψ ( z ).We want to construct a sequence of stopping times (on the Brownian probabilityspace) 0 < θ ( n )1 < ... < θ ( n ) n such that for any k (2.3) Z θ ( n ) k +1 − Z θ ( n ) k ∈ {− σ √ h, , σ √ h } and(2.4) E ( θ ( n ) k +1 − θ ( n ) k | F θ ( n ) k ) = h + O ( n − / ) . ecombining tree approximations for Game Options in Local Volatility models 3 To that end, we apply the results from Section 2.3 in [2]. For any A ∈ [0 , σ √ h ]consider the stopping times ρ Z A = inf { t : | Z t − Z | = A } and κ Z A = (cid:88) i =1 I Z ρY A = Z +( − i A inf { t ≥ ρ Z A : Z t = Z or Z t = Z + ( − i σ √ h } . Lemma 2.2.
Define the stopping times θ ( n )1 , ..., θ ( n ) n by the following recursive re-lations θ ( n )0 := 0 θ ( n ) k := κ Z θ ( n ) k − σ (cid:32) Z θ ( n ) k − (cid:33) √ h/σ , for k = 1 , ..., n. Then the stopping times θ ( n )0 , ..., θ ( n ) n satisfy (2.3)–(2.4).Proof. The proof follows from Section 2.3 in [2] (see Remark 2.2 there). (cid:3)
Next, introduce the functions p (1) ( z ) := (cid:16) − e − σ z √ h/σ (cid:17) (cid:16) e σ z √ h/σ − (cid:17)(cid:16) e σ z √ h/σ − e − σ z √ h/σ (cid:17) (cid:16) e σ √ h − (cid:17) , z ∈ R ,p ( − ( z ) = e σ √ h p (1) ( z ) , z ∈ R and p (0) ( z ) = 1 − p (1) ( z ) − p ( − ( z ) , z ∈ R . Observe that the support of the random variable Z ρ Z A and Z κ Z A − Z ρ Z A | Z ρ Z A consist of only two points. Thus, from the strong Markov property of Z and thefact that e Z is a martingale we obtain P (cid:16) Z θ ( n )1 = Z + ( − i σ √ h (cid:17) = p ( i ) ( Z ) , i = − , , . By applying again the strong Markov property we conclude that for any k (2.5) P (cid:16) Z θ ( n ) k +1 = Z θ ( n ) k + ( − i σ √ h |F θ ( n ) k (cid:17) = p ( i ) ( Z k ) , i = − , , . Dynkin Games for Trinomial Models.
For a given n , denote by S n theset of all stopping time with respect to the filtration {F θ ( n ) k } nk =0 , with values in theset { , , ..., n } . Introduce the Dynkin game value(2.6) V n : = inf ζ ∈S n sup η ∈S n E (cid:104) g (cid:16) ζh, S θ ( n ) ζ (cid:17) I ζ<η + f (cid:16) ηh, S θ ( n ) η (cid:17) I η ≤ ζ (cid:105) = sup η ∈S n inf ζ ∈S n E (cid:104) g (cid:16) ζh, S θ ( n ) ζ (cid:17) I ζ<η + f (cid:16) ηh, S θ ( n ) η (cid:17) I η ≤ ζ (cid:105) . Recall that S t := e Z t . Hence, the process { S θ ( n ) k } nk =0 lies on the grid S exp (cid:16) σ √ hi (cid:17) , i = − n, − n, ..., , , ..., n .By combining standard dynamical programming for Dynkin games (see [13]),thestrong Markov property of S and the transition probabilities given by (2.5) wecompute V n by the following backward recursion. B. Gottesman Berdah
Define the functions J ( n ) k : { Z + σ √ h {− k, − k, ..., , , ..., k }} → R , k = 0 , ..., nJ ( n ) n ( z ) = f ( T, e z )and for k = 0 , , ..., n − J ( n ) k ( z ) = max f ( kh, e z ) , min g ( kh, e z ) , (cid:88) i = − , , p ( i ) J ( n ) k +1 ( z + iσ √ h ) . We get that V n = J ( n )0 ( Z ) . Moreover, the stopping time given by η ∗ n := n ∧ min (cid:26) k : J ( n ) k ( Z θ ( n ) k ) = f (cid:18) kh, e Z θ ( n ) k (cid:19)(cid:27) is an optimal stopping time for the buyer and ζ ∗ n := n ∧ min (cid:26) k : J ( n ) k ( Z θ ( n ) k ) = g (cid:18) kh, e Z θ ( n ) k (cid:19)(cid:27) is an optimal stopping time for the seller.Namely,(2.7) V n : = sup η ∈S n E (cid:20) g (cid:18) ζ ∗ n h, S θ ( n ) ζ ∗ n (cid:19) I ζ ∗ n <η + f (cid:16) ηh, S θ ( n ) η (cid:17) I η ≤ ζ ∗ n (cid:21) = inf ζ ∈S n E (cid:20) g (cid:16) ζh, S θ ( n ) ζ (cid:17) I ζ<η ∗ n + f (cid:18) η ∗ n h, S θ ( n ) η ∗ n (cid:19) I η ∗ n ≤ ζ (cid:21) . As in Section 2.2 the grid structure allows to compute V n with complexity O ( n ).We arrive to the approximation result. Theorem 2.3.
The values V and V n defined respectively by (2.2) and (2.6) satisfy | V − V n | = O ( n − / ) . Moreover, for the stopping times τ ∗ n := T ∧ θ ( n ) η ∗ n and γ ∗ n := T I ζ ∗ n = n + ( T ∧ θ ( n ) ζ ∗ n ) I ζ ∗ n Fix n ∈ N and denote h := Tn . Let τ ∈ T T . Define the map ϕ n : T T → S n by ϕ n ( τ ) = n ∧ min { k : θ ( n ) k ≥ τ } , τ ∈ T T . Observe that for any τ ∈ T T we have | τ − θ ( n ) ϕ n ( τ ) | ≤ | T − θ ( n ) n | + max ≤ k ≤ n θ k − θ k − ≤ h + 3 max ≤ k ≤ n | θ ( n ) k − kh | . Thus, by applying (2.3)–(2.4) and using the exactly the same arguments as inSection 2.4 we obtain(2.8) sup τ ∈T T E P (cid:20)(cid:12)(cid:12)(cid:12)(cid:12) f (cid:18) ϕ n ( τ ) h, S θ ( n ) ϕn ( τ ) (cid:19) − f ( τ, S τ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) = O ( n − / ) . ecombining tree approximations for Game Options in Local Volatility models 5 Next, we notice that | ζ ∗ n h − γ ∗ n | ≤ h + max ≤ k ≤ n | θ ( n ) k − kh | . Thus, (again we use the same arguments as in Section 2.4 )(2.9) E P (cid:20)(cid:12)(cid:12)(cid:12)(cid:12) g (cid:18) ζ ∗ n h, S θ ( n ) ζ ∗ n (cid:19) − g (cid:0) γ ∗ n , S γ ∗ n (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:21) = O ( n − / ) . From the definitions it is clear that for a given stopping time τ ∈ T T the inequality γ ∗ n < τ implies ζ ∗ n < ϕ n ( τ ). Namely, { γ ∗ n < τ } ⊂ { ζ ∗ n < ϕ n ( τ ) } . Thus, from (2.7)–(2.9) sup τ ∈T E P [ H ( γ ∗ n , τ )] − V n ≤ sup τ ∈T T E P (cid:20)(cid:12)(cid:12)(cid:12)(cid:12) f (cid:18) ϕ n ( τ ) h, S θ ( n ) ϕn ( τ ) (cid:19) − f ( τ, S τ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) + E P (cid:20)(cid:12)(cid:12)(cid:12)(cid:12) g (cid:18) ζ ∗ n h, S θ ( n ) ζ ∗ n (cid:19) − g (cid:0) γ ∗ n , S γ ∗ n (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:21) = O ( n − / ) . (2.10)Next, define the map ˜ ϕ n ( γ ) : T T → S n by˜ ϕ n ( γ ) = (cid:16) n ∧ min { k : θ ( n ) k ≥ γ } (cid:17) I γ Consider the local volatility model which is given by dS t S t = min (cid:32) . , max (cid:32) . , √ S t (cid:33)(cid:33) dW t . B. Gottesman Berdah Table 1. We provide numerical results for game call optionswith the above parameters, with different initial stock prices. Thenumber of steps in the trinomial approximations denoted by n .Game Call Option Prices S n = 400 n = 700 n = 1200 n = 200080 6.8637 6.8357 6.8081 6.782385 8.2609 8.2221 8.1884 8.143390 9.6056 9.5534 9.5407 9.508395 10.9539 10.9332 10.9123 10.8943100 12 12 12 12105 17 17 17 17110 22 22 22 22This model can be viewed as a truncated version of the CEV model ([3]). Theprocess S t , t ≥ r = 0 . Game Call Options. Consider a game call option with strike price K = 100and constant penalty δ = 12. Namely, the discounted payoff is given by H ( γ, τ ) := e − . γ ∧ τ ) (cid:0) ( S γ ∧ τ − + + 12 I γ<τ (cid:1) . We assume that the maturity date is T = 2.First, by applying the constructed above trinomial trees, we compute (Table(3.1)) the option prices for different initial stock prices.Next, we calculate numerically the stopping regions. For American call optionsthe discounted payoff is a sub–martingale (under the martingale measure) and sothe buyer stopping time is τ ≡ T .It remains to treat the seller. Set, V call ( u, x ) := inf γ ∈T u sup τ ∈T u E (cid:104) e − . γ ∧ τ ) (cid:0) ( S γ ∧ τ − + + 12 I γ<τ (cid:1)(cid:105) , u, x > S = x .We observe that the optimal stopping time for the seller is given by (recall that S is the discounted stock price) γ ∗ = T ∧ inf { t : e . t S t ∈ D call } where D call = { ( t, x ) : V call ( T − t, x ) = ( x − + + 12 } . We obtain numerically (Figure 1) that the structure of the stopping region D call is of the form D = { ( t, x ) : t ∈ [0 , T ] , K ≤ x ≤ b call ( t ) } (cid:91) { [ T , T ] × { K }} where T < T < T and b call : [0 , T ] → [ K, ∞ ). ecombining tree approximations for Game Options in Local Volatility models 7 Figure 1. We consider a game call option with the above pa-rameters. We take n = 2000 and compute numerically the stop-ping region for the seller. We get that for t ∈ [0 , . 93] the sellershould exercise at the first moment when the stock price is be-tween the strike price and the value given by the blue curve. For t ∈ [0 . , . 33] the seller stops at the first moment the stock priceequals to the strike price. After the time t = 1 . 33 the investorshould not exercise (before the maturity date). Table 2. We provide numerical results for game put optionswith the above parameters, with different initial stock prices. Thenumber of steps in the trinomial approximations denoted by n .Game Put Option Prices S n = 400 n = 700 n = 1200 n = 200080 22.6243 22.6312 22.6341 22.618485 19.7150 19.6465 19.6027 19.584890 16.9593 16.9420 16.9110 16.896995 14.4512 10.4282 14.4104 14.3933100 12 12 12 12105 11.1356 11.0930 11.0841 11.0378110 10.1854 10.1368 10.0905 10.0385115 9.1742 9.1132 9.0713 9.0622120 8.3025 8.2646 8.2529 8.2378 B. Gottesman Berdah Figure 2. We consider a game put option with the above param-eters. We take n = 2000 and compute numerically the stoppingregions for the buyer. We get that the holder should exercise atthe first moment when the stock price is below the value given bythe blue curve.3.2. Game Put Options. We consider a game put option with strike price K =100 and constant penalty δ = 12. Thus, the discounted payoff is given by H ( γ, τ ) := e − . γ ∧ τ ) (cid:0) (100 − S γ ∧ τ ) + + 12 I γ<τ (cid:1) . As before we take the maturity T = 2.In Table (3.2) we compute the option prices for different initial stock prices.Finally, we calculate numerically the stopping regions. We start with the seller.In [11] the authors showed that for finite horizon game put options in the Black–Scholes model the stopping time for the seller is of the form γ ∗ = T ∧ inf { t ∈ [0 , t ∗ ] : e rt S t = K } for some t ∗ which the authors characterize. In fact, their arguments are valid for anylocal volatility model. Of course, the characterization of t ∗ is more complicated inmodels with non constant parameters. By applying our trinomial models we shownumerically that the seller stopping time is given by γ ∗ = T ∧ inf { t ∈ [0 , . 33] : e rt S t = 100 } . Namely, after time t ∗ = 1 . 33 the seller wait for the maturity date. Observe thatfor game call options (with the same parameters) we also obtained numericallythat after time 1 . 33 the seller wait for the maturity date. An open question, which ecombining tree approximations for Game Options in Local Volatility models 9 we leave for future research is to understand whether this is just a coincidence orwhether there is some connection between game call options and game put options? It remains to compute numerically the holder stopping time. Roughly speaking,the holder will reason in the same way as he would for the associated Americanput. That is to make a compromise between the stock reaching a prescribed lowvalue and not waiting too long.Thus, we expect that the holder stopping time will of the form τ ∗ = T ∧ inf { t : e rt S t ∈ D put } where the stopping region D put is of the form D = { ( t, x ) : t ∈ [0 , , x ≤ φ (2 − t ) } where φ : [0 , → (0 , Acknowledgments I would like to cordially thank my adviser and teacher, Yan Dolinsky, for guidingme and, for the long and fruitful discussions on this topic. References [1] E. Baurdoux and A.E. 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