17. The volatility Smile and its Implied TreeMA6622, Ernesto Mordecki, CityU, HK, 2006.Reference for this Lecture:E. Derman and I. Kani (Course web site),The Volatility Smile and Its Implied Tree.Also as: Riding on a Smile, Risk Journal, 1994, 7, 32.1Plan of Lecture 17(17a) The implied tree.(17b) Calibration of the first node.(17c) Calibration of the second node.(17d) Calibration of the third node.217a. The Implied treeIn order to construct a model consistent with the volatilitysmile at each maturity, i.e. consistent with the volatilitymatrix, Derman and Kani (1994) proposed to calibrate atime-space dependent binomial tree, taking into accountthe smile at each step.The binomial tree is implied by the option prices at eachtime step and at each node.In order to examine the proposal, we construct a two stepimplied tree for the HSI, with dates June 8, June 29, andJuly 28.3Jn 8 Jn 29 Pr. Jl 28 Pr.SCp1p2SAp115816.5 S0p1(1 − p2) + (1 − p1)p3SB1 − p1SD(1 − p1)(1 − p3)The idea is to determine:• SA, SBand p1from the forward price of the HSI, andfrom an at the money option price, plus a centering con-dition (first node)• SCand p2from a forward price and an at the moneycall option price (second node).• SDand p3from a forward price and an at the moneyput option price (third node)417b. Calibration of the first nodeTime period: June 8 - J une 29.From S0the model assumes either:• An upward movement to a value SA, with probabilityp1,• A downward movement to a value SBwith probability1 − p,• SA, SB, p1to be determined.We need three conditions in order to find the parametersSA, SB, p.5• First: Risk Neutral condition: The expected value of thestock must be equal to the forward price.p1SA+ (1 − p1)SB= S0exp(rT ) = F1,Observe that this condition can be transformed intop1=F1− SBSA− SB.• Second: Centering condition:SA× SB= (S0)2.• Third: Calibration with a call option price at the money.Now the numerical computations:6• For the forward price we use the risk-free interest rate ofUS Bonds (remember the observations of Lec ture 16).For the time period considered:r(June 8,June 29) = 0.047.As we have 21 calendar days in 360, we obtainF1= 15816.5 exp0.047 × (21/360)= 15859.9• Combining the centering condition we eliminate SB:p1=F1− (S20/SA)SA− (S20/SA)=F1SA− S20(SA− S0)(SA+ S0).In particularp1(SA− S0) =F1SA− S20SA+ S0(1)7• We compute the price of an at t he money option.We interpolate in the smile:Month Strike Price r(t, T ) % σ(t, T ) %June, 29 15800 335 4.70 19.54June, 29 15816.5 326.35 4.70 19.534June, 29 16000 240 4.70 19.47We use 15 t rading days in 247. We interpolate to findthe volatility for our at the money option:σ = 19.54 + 16.5h19.47 − 19.54200i= 19.528The option price in the Binomial model isCall = e−rTp(SA− S0),and using (1), we obtainF SA− S20SA+ S0= erTCall,(that only depends on SA).Numerical results follow:• SA=S0erT+Call1−(Call/S0)= 16438.7,• SB=S20SA= 15217.8,• p1=F −SBSA−SB= 0.526.9The status of our implied tree is:Jn 8 Jn 29 Pr. Jl 28 Pr.SCp1p216438.7 0.52615816.5 S0p1(1 − p2) + (1 − p1)p315217.8 0.474SD(1 − p1)(1 − p3)1017c. Calibration of the second nodeTime period June 30 - July 28. The term rate for thisperiod isr(June 30, July 28) = 0.04724,as computed in Lecture 16.Observe that the model assumes that the downward valueis S0(centering condition).We determine SCand p2, based on• The price F2of a forward during June 30, July 28.• The price of an option during June 9, July 28, struck atSA= 16438.7.11• We c ompute the forward (using calendar days):r(Jn30, Jl28) =7 × 0.0470 + 22 × 0.04767 + 22= 0.04746andF2= SAexp0.0476(29/360)= 16501.9From the risk neutral equation:p2SC+ (1 − p2)S0= F2we obtainp2=F2− S0SC− S0.• We c ompute the option price:Month Strike Price r(t, T ) % σ(t, T ) %July, 28 16400 246 4.724 18.09July, 28 16438.7 231.17 4.724 17.97July, 28 16600 185 4.724 17.4912We interpolate to fi nd the implied volatilityσ = 18.09 +38.7200(17.49 − 18.09) = 17.97• The price of the option in the Binomial model isCall = e−r(J n8,Jl28)Tp1p2(SC− SA).Numerical values follow:– SC= 17569.9.– p2=F2−S0SC−S0= 0.3910713The status of our implied tree is:Jn 8 Jn 29 Pr. Jl 28 Pr.17569.9 0.20616438.7 0.52615816.5 15816.5 0.320 + (1 − p1)p315217.8 0.474SD(1 − p1)(1 − p3)1417c. Calibration of the third nodeWe must determine SDand p3, using• The forward price (the interest rate is the same as in theprevious node) for SB,• A put option with dates June 8, July 28, struck at SB.The calculations are:• The forward price isF3= 15217.8 exp0.0476(29/360)= 15276.3.giving the risk neutral equation for the probability1 − p3=S0− F3S0− SDWe compute the price of the Put through put call paritiy.The implied volatility for a strike of 15200 is 0.195252, that15gives a Put price of1P ut = 191.027The price of this put under the binomial model isP ut = e−r(J n8,Jl28)T(1 − p1)(1 − p3)(SB− SD)= e−r(J n8,Jl28)T(1 − p1)S0− F3S0− SD(SB− SD).Here the only unknown value is SD, obtainingSD= 13412.5The probability isp3= 0.775288.1If we take Put quoted prices we can not calibrate the model, giving rise to arbitrage opor-tunities.16The final implied tree is:Jn 8 Jn 29 Pr. Jl 28 Pr.17569.9 0.20616438.7 0.52615816.5 15816.5 0.68815217.8 0.47413412.5 0.107Application: Based on the information on t his impliedtree, we can price a Bermudean option, i.e. a call option,written on June 8, that can be executed on June 29, or onJuly 28.The procedure is the s ame as in the usual american options,we compare the expected reward of holding the option onJune 29, with the reward of executing the