Pricing and Hedging of European Plain Vanilla Options under Jump Uncertainty by Olaf Menkens School of Mathematical Sciences Dublin City University (DCU) Financial Engineering Workshop Cass Business School, City University of London London, 19 th October 216
Pricing and Hedging of Plain Vanilla Options 1 A Quotation Two men are preparing to go hiking. While one is lacing up hiking boots, he sees that the other man is forgoing his usual boots in favor of sporty running shoes. Why the running shoes? he asks. The second man responds, I heard there are bears in this area and I want to be prepared. Puzzled, the first man points out, But even with those shoes, you can t outrun a bear. The second man says, I don t have to outrun the bear, I just have to outrun you. (See Hubbard (29), p. 157/158)
Pricing and Hedging of Plain Vanilla Options 2 Outline 1. Literature Review 2. Worst Case Option Pricing 3. Superhedging Strategy 4. Model Calibration 5. Conclusion
Pricing and Hedging of Plain Vanilla Options 3 1. Literature Review Option Pricing Some References: Bachelier (19) [ Derivative Pricing using Brownian Motion], Black and Scholes (1973) [ Reference Model for Option Pricing, No Jumps], Cox (1975) [ CEV Model], Heston (1993) [ Stochastic Volatility Model], Madan, Carr, and Chang (1998) [ Variance Gamma Model], Kou (et al.) (22 24, 28) [ Jump Diffusion Model with exponential tails], See Cont and Tankov (24) or Rebonato (24) for more details and for other models as well.
Pricing and Hedging of Plain Vanilla Options 4 Worst Case Scenario Optimization Some References: Hua and Wilmott (1997) [ Binomial Model Derivative Pricing], Korn and Wilmott (22), [ Portfolio Optimisation], Mönnig (212), [ Stochastic Target Approach], Belak, M. (216) [ BSDE Approach]. Remark: The worst case scenario optimisation problem is also known as Wald s Maximin approach (Wald 1945, 195), which is a well known concept in decision theory. There, this approach is known as robust optimisation (e.g. Bertsimas et al. (211)) [ usually involves optimisation procedure done by a computer]. Mataramvura and Oksendal (28), Oksendal and Sulem (26, 29, 211) [ Compute optimal strategies directly]. [ parameter uncertainty, perturbation analysis].
Pricing and Hedging of Plain Vanilla Options 5 Interpretation of Worst Case Scenarios E [ ln ( X t,x,π,τ,k (T) )] π 2 π 1 Merton approach π 3 WC(π 2 ) WC(π 1 ) ( ( τ (π 1 ),k (π 1) ) τ (π 2 ),k (π 2) ) probability free approach (à la de Finetti) WC(π 3 ) (τ,k)
Pricing and Hedging of Plain Vanilla Options 6 2. Worst Case Scenario Option Pricing Consider the initial model with one bond and one risky asset. The aim is to price a contingent claim ξ. Definition 2.1 (Worst-case price; superhedging strategy) (see Belak and M. (216)) The worst-case price V 1 (t;ξ) of ξ at time t [,T] is defined as { V 1 (t;ξ) essinf x L + t : (ζ 1,ζ ) A 1 (t,x) A (ζ 1 ) s.t. X ζ 1,ζ,ϑ t,x (T) ξ ( P (T),P ϑ (T) ) for all ϑ B(t) Furthermore, a strategy (ζ 1,ζ ) A 1 (t,x) A (ζ 1 ) is referred to as a superhedging strategy against ξ if X ζ 1,ζ,ϑ t,x (T) ξ(p (T),P ϑ (T)) for all ϑ B(t). }. We let ξ be a European call option with strike price K >, i.e. ξ(p) = [p K] +.
Pricing and Hedging of Plain Vanilla Options 7 It is well-known (see Black and Scholes (1973)) that the fair price V is given by V (t,p) = pφ(d 1 (K,t,p)) Ke r[t t] (d 2 (K,t,p)) with log ( p K ) ] + [r + σ2 2 [T t] d 1 (K,t,p) = σ T t d 2 (K,t,p) = (d 1 (K,t,p)) σ T t,, and where Φ denotes the standard normal cumulative distribution function. Equivalently, the fair price is given as the unique classical solution of the Black-Scholes PDE t V (t,p) rp p V (t,p) σ2 2 p2 2 p 2V (t,p)+rv (t,p) =, V (T,p) = [p K] +.
Pricing and Hedging of Plain Vanilla Options 8 In the jump-threatened market, we assume the minimum and maximum jump sizes to be given by constants β D ( 1,] and β U [, ). With this, the pricing PDE for the worst-case price V 1 (t,p) can be written as min { t V 1(t,p) rp p V 1(t,p) σ2 V 1 (t,p) max β {β D,β U } [ 2 p2 2 p 2V 1(t,p)+rV 1 (t,p), V (t,[1+β]p) βp p V 1(t,p) ]} =, which is the pricing PDE obtained in both Mönnig (212) and Belak and M. (216). Notice that we have used the strict convexity of V to replace the supremum over all β [β D,β U ] with the maximum over β D and β U. In a similar fashion, the terminal condition can be written as min { V 1 (T,p) max β {β D,β U } [ [(1+β)p K] + βp p V 1(T,p) V 1 (T,p) [p K] +} =. ],
Pricing and Hedging of Plain Vanilla Options 9 We define the constants L := α D/U := K [1+β D ][1+β U ], β2 D/U η D/U (t) := exp K L β U β D 1+β D/U r σ2 2β D/U 1 β D and 1+ 1 β D/U [T t]. The terminal condition can be computed explicitly. Lemma 2.2 (Explicit Formula for the Terminal Condition) Let β D β U. Then the unique solution of (1) is given by V 1 (T,p) = α D p 1 β D 1l {p<l} + [ α U p 1 β U +p K ] 1l {p L}.
Pricing and Hedging of Plain Vanilla Options 1 12 The Terminal Boundary Worst Case Boundary Function with Up and Downward Jump No Jump 1 Downward jump of max. size is worst case. 8 Payoff 6 4 2 Upward jump of max. size is worst case. 2 4 6 8 1 12 14 16 18 2 Risky Asset Price This Figure is plotted assuming β U = β D =.5 and K = 1. The used values are as follows: β U =.5, β D =.5, K = 1.
Pricing and Hedging of Plain Vanilla Options 11 16 The Terminal Boundary Worst Case Boundary Function with Downward Jump No Jump 14 12 Downward jump of max. size is worst case. Payoff 1 8 6 4 2 No jump is worst case. 5 1 15 2 25 Risky Asset Price This Figure is plotted assuming β D =.5, β U =, and K = 1. The used values are as follows: β U =.5, β D =.5, K = 1.
Pricing and Hedging of Plain Vanilla Options 12 12 The Terminal Boundary Worst Case Boundary Function with Upward Jump No Jump 1 No jump is worst case. Upward jump of max. size is worst case. 8 Payoff 6 4 2 2 4 6 8 1 12 14 16 18 2 Risky Asset Price This Figure is plotted assuming β D =, β U =.5, and K = 1. The used values are as follows: β U =.5, and K = 1.
Pricing and Hedging of Plain Vanilla Options 13 Theorem 2.3 (Explicit Solution for the Worst Case Option Price) For (t,p) [,T) R +, the worst-case price V 1 in a Black-Scholes market with constant minimum and maximum jump sizes is given explicitly as V 1 (t,p) = pφ(d 1 (L,t,p)) Ke r[t t] Φ(d 2 (L,t,p)) (1) +α D η D (t)p β 1 ( D Φ d 2 (L,t,p)+ σ ) T t β D +α U η U (t)p β 1 ( U Φ d 2 (L,t,p) σ ) T t. β U
Pricing and Hedging of Plain Vanilla Options 14 This price can be decomposed to: one gap option (with strike K and trigger L) plus α D number of short standard power gap put options (where the standard power option is defined in Haug (27)) with strike and trigger L, and plus α U number of standard power gap call options with strike and trigger L, where these latter three options live in the underlying Black Scholes market (that is without jump risk).
Pricing and Hedging of Plain Vanilla Options 15 Option Price 12 1 8 6 4 2 Payoff Structure Black Scholes Price Worst Case Price Distorted Black Scholes Price with β D Options Prices 2 4 6 8 1 12 14 16 18 2 p Initial Price of the Risky Asset This Figure is plotted assuming K = 1, σ =.4, r =.3, β D =.5, β U =, and T = 1. The used values are: K = 1, r 1 =.3, r =.3, T = 1, σ 1 =.4, σ =.4, and β D =.5. Thus, K D = 2.
Pricing and Hedging of Plain Vanilla Options 16 Option Price 12 1 8 6 4 2 Payoff Structure Black Scholes Price Worst Case Price Distorted Black Scholes Price with β U Options Prices 2 2 4 6 8 1 12 14 16 18 2 p Initial Price of the Risky Asset This Figure is plotted assuming K = 1, σ =.4, r =.3, β D =, β U =.5, and T = 1. The used values are: K = 1, r 1 =.3, r =.3, T = 1, σ 1 =.4, σ =.4, and β U =.5. Thus, K U = 66.6667.
Pricing and Hedging of Plain Vanilla Options 17 Theorem 2.4 (Greeks) p V 1 (t,p) = Φ(d 1 (L)) α Dη D p 1 β 1 D Φ ( d 2 (L)+ σ T t β D β D p 2V 1 (t,p) = 1+β D β 2 D t V 1 (t,p) = α Uη U p 1 β 1 U Φ (d 2 (L) σ T t β U β U + 1+β U [ β 2 U r σ2 2β D [ α D η D p 1 β D 2 Φ ( d 2 (L)+ σ β D T t α U η U p β 1 2 U Φ (d 2 (L) σ T t β U ][ 1+ 1 ][ β D + r σ2 1+ 1 2β U β U rke r[t t] Φ(d 2 (L)), ] ( ), ) ) +, ) + α D η D p β 1 D Φ d 2 (L)+ σ T t β ] D α U η U p β 1 ( U Φ d 2 (L) σ T t β U ) ) + +
Pricing and Hedging of Plain Vanilla Options 18 σ V 1 (t,p) = σk[t t] β U β D η D + βd V 1 (t,p) = α D β 2 D + [ L σk[t t] η U β U β D η D p 1 β D Φ ( p ] 1 βd Φ ( d 2 (L)+ σ [ L p { βd β U β U β D +σ T t β D T t ] 1 βu Φ (d 2 (L) σ T t β U d 2 (L)+ σ T t β D β D [1+β U ] β U [β U β D ][1+β D ] α Uη U p [ ) d 2 (L) σ T t β D 1 β U Φ 1+β U β U β D σ T tle r[t t] φ(d 2 (L)). ( ) ) ]} +, + d 2 (L) σ β U T t ) +
Pricing and Hedging of Plain Vanilla Options 19 1 Delta Delta.9.8.7.6.5.4.3.2.1 Payoff Delta Black Scholes Delta Worst Case Delta 2 4 6 8 1 12 14 16 18 2 p Initial Price of the Risky Asset This Figure is plotted assuming K = 1, σ =.4, r =.3, β U = β D =.5, and T = 1. The used values are: K = 1, r 1 =.3, r =.3, σ 1 =.4, σ =.4, β D =.5, and β U =.5, T = 1.
Pricing and Hedging of Plain Vanilla Options 2 Gamma.15.1.5 Gamma Black Scholes Gamma Worst Case Gamma Worst Case Gamma if β U = Worst Case Gamma if β D = 2 4 6 8 1 12 14 16 18 2 p Initial Price of the Risky Asset This Figure is plotted assuming K = 1, σ =.3, r =.3, β U = β D =.25, and T = 1. The used values are: K = 1, r 1 =.3, r =.3, σ 1 =.3, σ =.3, β D =.25, and β U =.25, T = 1.
Pricing and Hedging of Plain Vanilla Options 21 Theta 1 2 3 4 5 6 7 Theta Black Scholes Theta Worst Case Theta Worst Case Theta if β U = Worst Case Theta if β D = 8 2 4 6 8 1 12 14 16 18 2 p Initial Price of the Risky Asset This Figure is plotted assuming K = 1, σ =.3, r =.3, β U = β D =.25, and T = 1. The used values are: K = 1, r 1 =.3, r =.3, σ 1 =.3, σ =.3, β D =.25, and β U =.25, T = 1.
Pricing and Hedging of Plain Vanilla Options 22 Vega 4 35 3 25 2 15 1 5 Vega Black Scholes Vega Worst Case Vega Worst Case Vega if β U = Worst Case Vega if β D = 2 4 6 8 1 12 14 16 18 2 p Initial Price of the Risky Asset This Figure is plotted assuming K = 1, σ =.3, r =.3, β U = β D =.25, and T = 1. The used values are: K = 1, r 1 =.3, r =.3, σ 1 =.3, σ =.3, β D =.25, and β U =.25, T = 1.
Pricing and Hedging of Plain Vanilla Options 23 Beta 35 3 25 2 15 1 5 Beta Worst Case Beta wrt β D Worst Case Beta wrt β U Worst Case Beta if β U = Worst Case Beta if β D = 2 4 6 8 1 12 14 16 18 2 p Initial Price of the Risky Asset This Figure is plotted assuming K = 1, σ =.3, r =.3, β U = β D =.25, and T = 1. The used values are: K = 1, r 1 =.3, r =.3, σ 1 =.3, σ =.3, β D =.25, and β U =.25, T = 1.
Pricing and Hedging of Plain Vanilla Options 24 Implied Volatility Surface 6 5 4 Implied Volatility 3 2 1.2.4.6 t Time to Expiry.8 1 2 4 6 8 1 12 K Strike of the Option This Figure is plotted assuming σ =.3, r =.3, β U = β D =.25, T = 1. 14 16 18 2
Pricing and Hedging of Plain Vanilla Options 25 Implied Volatility Surface for β U = 4 3.5 3 Implied Volatility 2.5 2 1.5 1.5.2.4.6 t Time to Expiry.8 1 6 7 8 9 1 11 K Strike of the Option This Figure is plotted assuming σ =.3, r =.3, β D =.25, β U =, T = 1. 12 13 14
Pricing and Hedging of Plain Vanilla Options 26 Implied Volatility Surface for β D = 2.2 2 1.8 1.6 Implied Volatility 1.4 1.2 1.8.6.4.2.2.4.6 t Time to Expiry.8 1 2 4 6 8 1 12 K Strike of the Option This Figure is plotted assuming σ =.3, r =.3, β U =.25, β D =, T = 1. 14 16 18 2
Pricing and Hedging of Plain Vanilla Options 27 3. Superhedging Strategy Define H(t,p;β) V 1 (t,p)+βp p V 1(t,p) V (t,[1+β]p). (2) Observe that H is the value of a portfolio. This portfolio consists of one call option and delta shares of the underlying risky asset hence this is the classical delta hedge of Black Scholes for a plain vanilla call option. H is the value of this portfolio if at time t a jump with jump size β happens and the price of the risky asset is p (just prior to the jump).
Pricing and Hedging of Plain Vanilla Options 28 Theorem 3.5 (Superhedging Strategy) One has that [ H(t,p;β) = 1 β ] α D η D p 1 ( β D Φ d 2 (L)+ σ ) T t + (3) β D β [ D + 1 β ] α U η U p 1 ( β U Φ d 2 (L) σ ) T t + β U β U +[1+β]pΦ(d 1 (L)) Ke r[t t] Φ(d 2 (L))+ where [1+β]pΦ ( d 1 ( K 1+β )) +Ke r[t t] Φ ( d 2 ( K 1+β H(t,p;β) for all t [,T],p (, ) and β [β D,β U ] ; (4) and equality holds at least for one (t,p,β). )),
Pricing and Hedging of Plain Vanilla Options 29 2.5 2 1.5 β * 1.5.5 1.8.6.4 t Time to Expiry.2 2 15 1 5 p Initial Price of the Risky Asset The best jump size for a European call option with a possible jump in both directions.
Pricing and Hedging of Plain Vanilla Options 3 p Price of the Risky Asset 113 112 111 11 19 18 17 β * = β U β * = β D 16.1.2.3.4.5.6.7.8.9 1 t Time to Expiry The worst jump size (bottom) for a European call option with a possible jump in both directions. The used values are: K = 1, r 1 =.3, r =.3, σ 1 =.3, σ =.3, β D =.25, and β U
Pricing and Hedging of Plain Vanilla Options 31 2 1.5 1 min H.5.5 2 15 1 p Initial Price of the Risky Asset 1.9.8.7.6.5 5.4.3.2.1 t Time to Expiry H(t,p,β) for a European call with β = β with a possible jump in both directions.
Pricing and Hedging of Plain Vanilla Options 32 14 12 1 H(.,.;β * ) 8 6 4 2 2 15 1 p Initial Price of the Risky Asset 5.2.4.6.8 t Time to Expiry H(t,p,β) for a European call with β = β with a possible jump in both directions. 1
Pricing and Hedging of Plain Vanilla Options 33 5 H min (.,.;2.*β * ) 1 15 2 25 2 15 1 p Initial Price of the Risky Asset 5.2.4.6.8 t Time to Expiry H(t,p,β) for a European call with β = 2β (bottom) with a possible jump in both directions. 1
Pricing and Hedging of Plain Vanilla Options 34 4. European Plain Vanilla Put Now, let ξ be a European put option with strike price K >, i.e. ξ(p) = [K p] +. The boundary condition (1) for a European plain vanilla put writes to { min P 1 (T,p) (K p) +, P 1 (T,p) sup β [β D,β U ] [ (K ) ] + } (1+β)p βp p P 1(T,p) =.(5) It is straightforward to verify (either by direct computation or by using the Put Call Parity (see e.g. Seydel (26, Exercise 1.1, p. 52) or Cont and Tankov (24, p. 356))) that the solution is given by
Pricing and Hedging of Plain Vanilla Options 35 Corollary 4.6 P 1 (T,p) = [ α D p 1 β D +K p ] 1l {p L} α U p 1 β U 1l {p>l}, (6) where p (, ), Moreover, one has the following Corollary 4.7 The worst case price of a European plain vanilla call is given by P 1 (t,p) = Ke r[t t] Φ( d 2 (L)) pφ( d 1 (L))+ (7) +α D η D p β 1 ( D Φ d 2 (L)+ σ ) T t + β D +α U η U p β 1 ( U Φ d 2 (L) σ ) T t. β U
Pricing and Hedging of Plain Vanilla Options 36 Furthermore, H P (t,p;β) = [ 1 β [ β D ] α D η D p β 1 ( D Φ d 2 (L)+ σ β D T t ) +(8) + 1 β ] α U η U p 1 ( β U Φ d 2 (L) σ T t + β U β U +Ke r[t t] Φ( d 2 (L)) [1+β]pΦ( d 1 (L))+ ( ( )) Ke r[t t] K Φ d 2 + 1+β ( ( )) K +[1+β]pΦ d 1. 1+β )
Pricing and Hedging of Plain Vanilla Options 37 5. Model Calibration Calibrated implied volatilities for maturities T =.712 (next slide) and T = 1.1452 (second next slide). The blue circles are the implied volatilities observed in the market while the blue dash dotted lines are the interpolation of the blue circles. The black solid lines give the implied volatility of the worst case option price formula where the parameters have been calibrated using the market data with penalty a = 1 3. In particular, note that the calibration is done in such a way that the calibrated curve (black solid lines) should be greater or equal the curve plotted from market data (blue dash dotted lines). For comparison reasons, the usual calibration (that is without penalty, meaning a = ) is given as well (green dashed lines). T =.712 T =.1479 T =.3973 T =.6466 T = 1.1452 σ.24.9452.12341.43.2529 β D.9958.1259.17554.21514.21988 β U.21213.3289.44456.49974.53858 O( ).381.12859.9537.1415.4946
Pricing and Hedging of Plain Vanilla Options 38 1.4 T =.712 1.3 1.2 1.1 Implied Volatility 1.9.8.7.6 1 15 2 25 3 35 4 K Strike of the Option
Pricing and Hedging of Plain Vanilla Options 39 1.3 T =.1479 1.2 1.1 1 Implied Volatility.9.8.7.6.5 1 15 2 25 3 35 4 K Strike of the Option
Pricing and Hedging of Plain Vanilla Options 4.9 T =.3973.85.8.75 Implied Volatility.7.65.6.55.5 1 15 2 25 3 35 4 45 5 55 6 K Strike of the Option
Pricing and Hedging of Plain Vanilla Options 41.8 T =.6466.75.7 Implied Volatility.65.6.55.5.45 1 15 2 25 3 35 4 K Strike of the Option
Pricing and Hedging of Plain Vanilla Options 42.62 T = 1.1452.6.58.56 Implied Volatility.54.52.5.48.46.44 1 2 3 4 5 6 7 8 K Strike of the Option
Pricing and Hedging of Plain Vanilla Options 43 6. Conclusion To summarize, one has the following properties: jumps are not averaged out but are fully taken into account (compare with liability insurance), first explicit non trivial superhedging price and superhedging strategy, it explains the volatility smile (as well as the smirk), and the closed form solution is numerically of the same level as the solution of Black and Scholes.
Pricing and Hedging of Plain Vanilla Options 44 Thank you very much for your attention! The corresponding paper can be downloaded from SSRN: http://ssrn.com/abstract=2773246