Towards a Theory of Volatility Trading. by Peter Carr. Morgan Stanley. and Dilip Madan. University of Maryland

Transcription

1 owards a heory of Volatility rading by Peter Carr Morgan Stanley and Dilip Madan University of Maryland

2 Introduction hree methods have evolved for trading vol:. static positions in options eg. straddles. delta-hedged option positions 3. volatility swaps he purpose of this talk is to explore the advantages and disadvantages of each approach. I'll show how the rst two methods can be combined to create the third. I'll also show the link between some \exotic" volatility swaps and some recent work by Dupire[3] and Derman, Kani, and Kamal[].

3 Part I Static Positions in Options

4 rading Vol via Static Positions in Options he classic position for trading vol is an at-the-money straddle. Unfortunately, the position loses sensitivity to vol as the underlying moves away from the strike. Is there a static options position which maintains its sensitivity to vol as the underlying moves? o answer this question, we rst need to develop a theory of static replication using options. We assume the following:. frictionless markets. no arbitrage 3. underlying futures matures at 4. continuum of European futures option strikes; single maturity Note that we do not restrict the price process in any way! 3

5 Spanning a Payo Consider a terminal cash ow f(f ), which is a twice dierentiable function of the nal futures price F. We will show that only the second derivative of the payo is relevant for generating volatility-based payos. Accordingly, we can restrict attention to payos whose value and slope vanish at an arbitrary point. he paper proves that for any such payo: f(f ) = Z f (K)(K? F ) + dk + Z f (K)(F? K) + dk: In words, to create a twice dierentiable payo f() with value and slope vanishing at a given point, buy f (K)dK puts at all strikes K less than and buy f (K)dK calls at all strikes K greater than. he absence of arbitrage requires that the initial value V f ( ) of the nal payo f() can be expressed in terms of the initial prices of puts P (K; ), and calls C (K; ) respectively: V f ( ) = Z f (K)P (K; )dk + Z f (K)C (K; )dk: 4

6 Variance of erminal Futures Price he variance of the terminal futures price is: Var (F ) = E f[f? E (F )] g: If we use risk-neutral expectations with the money market account as numeraire, then all futures prices are martingales, and so: E (F ) = F : hus, the variance of F is just the futures price of the portfolio of options which pays o [F? F ] at (see Figure.): Payoff for Variance of erminal Futures Payoff Futures Price Figure.: Payo for Variance of erminal Futures Price(F = ).

7 Variance of erminal Futures Price(con'd) Recall that the spot value of an arbitrary payo f() with value and slope vanishing at some point was given by: V ( ) = Z f (K)P (K; )dk + Z f (K)C (K; )dk: For f(f ) = (F? F ), the value and slope vanish at F and f (K) = : hus, the risk-neutral variance of the terminal futures price can be expressed in terms of the futures prices ^P and ^C of puts and calls respectively: Var (F ) = " Z F ^P (K; )dk + Z F # ^C (K; )dk : We can similarly calculate the risk-neutral variance of the log futures price relative by nding the futures price of the portfolio of options which pays o ln F? E F ln F, F where E ln F F is the futures price of the portfolio of options which pays o ln F F at. 6

8 Advantages and Disadvantages of Static Positions in Options When compared to an at-the-money straddle, the quadratic payos have the advantage of maintaining their sensitivity to volatility (suitably de- ned), as the underlying moves away from its initial level. Unfortunately, like straddles, the quadratic payos will have non-zero delta once the underlying moves away from its initial level. he solution to this problem is to delta-hedge with the underlying. 7

9 Part II Delta-Hedging Options Positions 8

10 Review of Delta-hedging in a Constant Vol World he Black model assumes continuous trading, a constant interest rate, and a continuous futures price process with constant volatility. Let's review delta-hedging of European-style claims in this model. For future use, we assume that even though the current time is t =, the claim is sold at t = and that the hedge occurs over (; ), where is the maturity of the claim. Let V (F; t) be any function of the futures price and time. Applying It^o's Lemma to V (F; t)e r(?t) gives: V (F ; ) = V (F ; )e r(? ) + Z + Z er(?t) 6 @t (F t; t) + F (F t; t)df 3 (F 7 t; t)? rv (F t ; t) dt 9

11 Review of Delta-hedging in a Constant Vol World (con'd) Recall that for any function V (F; t): V (F ; ) = V (F ; )e r(? ) + Z + Z er(?t) 6 @t (F t; t) + F t Now consider a function V (F; t; ) which (F t; t)df 3 (F 7 t; t)? rv (F t ; t) F (F; t; ) (F; t; )? rv (F; t; ) = ; V (F; ; ) = f(f ): Substitution gives: f(f ) = V (F ; ; )e r(? ) + (F t; t; )df t : Evidently, the payo f(f ) at can be created by investing V (F ; ; ) dollars in the riskless asset at and always t; t; ) futures contracts over the time interval (; ) (assuming continuous marking-tomarket).

12 Delta-Hedging at a Constant Vol in a Stochastic Vol World Now continue to assume that the price process is continuous, but assume that the true vol is given by some unknown stochastic process t. Assume that the claim is sold for an implied vol of h and that delta-hedging is conducted using the Black model delta evaluated at this constant hedge vol. Let V (F; t; h ) be a function satisfying the terminal condition V (F; ; h ) = f(f ) and the Black p.d.e. with constant volatility h. hen the paper shows that: f(f ) + P &L = V (F ; ; h )e r(? ) + Z where: P &L = Z er(?t) F (F t; t; h )df t (F t; t; h )( h? t )dt: In words, when we sell the claim for an implied vol of h at, the instantaneous P&L rate from delta-hedging with the constant vol h over (; ) is half the dollar gamma weighted average of the dierence between the hedge variance and the true variance. Note that the P&L vanishes if t = h. (F t ; t; h ) as is true for options, and if t > h for all t [; ], then you sold the claim for too low a vol and a loss results, regardless of the path. Conversely, if you manage to sell the claim for an implied vol h which dominates the subsequent realized vol at all times, then delta-hedging at h guarantees a positive P&L.

13 Advantages and Disadvantages of Delta-hedging Options When compared with static options positions, delta-hedging appears to have the advantage of being insensitive to the price of the underlying. However, recall the expression for the P&L at : P &L = Z er(?t) F (F t; t; h )( h? t )dt: In general, this expression depends on the path of the price. One solution is to use a stochastic vol model to conduct the delta-hedging. However, this requires specifying the volatility process and dynamic trading in options. A better solution is to choose the payo function f(), so that the path dependence can be removed or managed. For example, Neuberger[4] recognized that if f(f ) = ln F, V t ; t; h ) = e?r(?t)? and the cumulative P&L at is the payo of F t a variance swap R ( t? h)dt.

14 Part III Volatility Contracts 3

15 Delta-hedging with Zero Vol Recall the expression for the nal portfolio value when delta-hedging at a constant vol h : f(f ) + P &L = V (F ; ; h )e r(? ) + Z where: P &L Z Setting h = implies: er(?t) F (F t; t; h )df t (F t; t; h )( h? t )dt: V (F; t; ) = e?r(?t) (F; t; ) = e?r(?t) f (F V (F; t; ) = e?r(?t) f (F Substituting into the top equation and re-arranging gives: Z f (F t ) F t t dt = f(f )? f(f )? Z f (F t )df t : 4

16 Delta-hedging with Zero Vol (con'd) Recall the expression for the hedging error/p&l when delta-hedging at zero vol: Z f (F t ) F t t dt = f(f )? f(f )? Z f (F t )df t : he left hand side is a payo dependent on both the realized instantaneous volatility t and the futures price F t. he dependence on the payo f(f ) occurs only through its second derivative. hus, we can and will restrict attention to payos whose value and slope vanish at a given point. he right hand side results from adding the following three payos:. he payo from a static position in options maturing at paying f(f ) at.. he payo from a static position in options maturing at paying?e?r(? ) f(f ) and future-valued to 3. he payo from maintaining a dynamic position in?e?r(?t) f (F t ) futures contracts (assuming continuous marking-to-market).

17 hree Interesting Vol Contracts Recall the equivalence between a volatility-based payo and 3 price-based payos: Z f (F t )F t t dt = f(f )? f(f )? Z f (F t )df t : We can choose f() so that the dependence of the volatility-based payo on the price path is to our liking. We next consider the following 3 second derivatives of payos at and work out the f() which leads to them: f (F t ) Payo at F t F t [F t (? 4; + 4)] (F t? ) R R R t dt [F t (? 4; + 4)] t dt (F t? ) t dt. 6

18 Contract Paying Future Variance Recall the following equivalence between a volatility-based payo and 3 price-based payos: Z f (F t )F t t dt = f(f )? f(f )? Z f (F t )df t : Consider the following function (F ) (see Figure.): (F t ) = A + F 3 t F t? where is an arbitrary nite positive number. ; 4 Payoff to Delta Hedge to Create Variance Payoff Futures Price Figure.: Payo to Delta-Hedge to Create Contract Paying Variance ( = ). he rst derivative is (F t ) =? : F t Note that the value and slope vanish at F =. he second derivative is (F t ) = : F t Substitution gives R t dt = F A + F 3?? A + F 3 F?? Z 4? F t 3 df t : 7

19 Contract Paying Future Variance (Con'd) Recall the following equivalence between the variance over (; ) and 3 price-based payos: Z t dt = 4ln? F A + F 4? 3 F t 3? df t : Since the value and slope of vanish at : (F ) = Z? (K)(K? F ) + dk + Z Since (F ) = F ; substitution gives: Z t dt = Z + Z? K (K? F )+ dk + Z Z K (K? F ) + dk + Z 4? 3 F t df t : A + F 3 F? (K)(F? K) + dk: K (F? K)+ dk K (F? K) + dk 8

20 Contract Paying Future Variance(Con'd Again) Recall the decomposition: Z t dt = Z K (K? F )+ dk + Z K (F? K)+ dk + Z K (K? F ) + dk + Z K (F? K) + dk Z? 4? 3 df t : F t o create the contract paying R t dt at, at t =, buy: R?e?r(? ) [ R P K (K; )dk + R P K (K; )dk + R K C (K; )dk K C (K; )dk]: At t =, borrow to nance the payout of e?r(? ) ln F + F? from having initially written the maturity options. Also start a dynamic strategy in futures, holding?e?r(?t)? F t futures for each t [; ]. he net payo at is: as desired. Z + Z? K (K? F )+ dk + Z Z K (K? F ) + dk + Z 4? 3 F t df t = Z t dt; K (F? K)+ dk K (F? K) + dk 9

21 Contract Paying Future Corridor Variance Consider a corridor (?4; +4) and suppose that we wish to generate a payo at of R [F t (? 4; + 4)] dt. t Dene: F t max[? 4; min(f t ; + 4)] as the futures price oored at? 4 and capped at + 4 (see Figure.3): Capped and Floored Futures Price Payoff Futures Price Figure.3: Futures Price Capped and Floored( = ; 4 = :). Note that lim 4" F t = F and lim 4# F t =.

22 Contract Paying Future Corridor Variance(Con'd) Recall the payo which generates the future variance when delta hedged at zero vol: (F t ) = A + F 3 t F t? = A + t F t? 3 A : F t Consider the following generalization of this payo () (see Figure.4): 4 (F t ) = A F + t t? A F t 3 :.7 Payoff to Delta Hedge to Create Corridor Variance.6. Payoff Futures Price Figure.4: rimming the Log Payo ( = ; 4 = :). he rst derivative is 4 (F t) =? F : Once again, the value and t slope vanish at F =. he second derivative is 4 (F t) = [F F t t (? 4; + 4)]: Substitution gives R t [F t (? 4; + 4)]dt = 6 4ln A F +? 7? 3 6 F 3 F A +? 7? Z 4? F t 3 df t :

23 Contact Paying Future Corridor Variance (Con'd) Recall the decomposition of the corridor variance: = Z 6 4ln? t [F t (? 4; + 4)]dt Z A F +? 7? 4? F t 3 df t : 3 6 4ln Since the value and slope of 4 vanish at : 4 (F ) = Z 4 (K)(K? F )+ dk + F F A +? (K)(F? K)+ dk: Since 4 (F ) = F [F t (? 4; + 4)]dt, substitution gives: Z = Z?4 t [F t (? 4; + 4)]dt K (K? F )+ dk + Z +4?e?r(? ) [ Z? Z?4 4? 3 F t df t : K (K? F ) + dk + Z K (F? K)+ dk +4 K (F? K) + dk]

24 Contact Paying Future Corridor Variance (Con'd Again) Recall the decomposition of the corridor variance: Z t [F t (? 4; + 4)]dt K (K? F )+ dk + Z +4 K (F? K)+ dk?e?r(? ) [ Z?4 K (K? F ) + dk + Z +4 K (F? K) + dk] Z? 4? 3 df F t : t = Z?4 hus, to create the contract paying R t [F t (? 4; + 4)]dt at, at t =, buy and sell options struck within the corridor: R?4?e?r(? ) [ R?4 P K (K; )dk + R +4 P K (K; )dk + R +4 K C (K; )dk K C (K; )dk]: At t =, borrow to nance the payout of e?r(? ) ln F + F? F from having initially written the maturity options. Also start a dynamic strategy in futures, holding?e?r(?t)? F futures for each t [; ]. t he net payo at is: Z = Z?4 K [K? F ]+ dk + Z +4?e?r(? ) [ Z? Z?4 4? 3 F t K [K? F ] + dk + Z df t t [F t (? 4; + 4)]dt; K [F? K]+ dk +4 K [F? K] + dk] as desired. 3

25 Contract Paying Variance Along a Strike Recall that we created a contract paying R t [F t (? 4; + 4)]dt at by initially spreading options struck within the corridor: R?4?e?r(? ) [ R?4 P K (K; )dk + R +4 P K (K; )dk + R +4 K C (K; )dk K C (K; )dk]: Suppose we re-scale everything by 4. he payo at would instead be: Z [F t (? 4; + 4)] t 4 dt: By letting 4 #, the variance received can be localized in the spatial dimension: R (F t? ) t dt: Since only options struck within the corridor are used, the initial cost of creating this localized cash ow is: [V (; )? e?r(? ) V (; )]; where V (; ) is the cost of a straddle struck at and maturing at : V (; ) = P (; ) + C (; ): As usual, at t =, borrow to nance the payout of jf?j from having initially written the maturity straddle. One can work out that the dynamic strategy in futures initiated at involves holding? e?r(?t) sgn(f t? ) futures contracts, where sgn(x) is the sign function: sgn(x) 8 >< >:? if x < ; if x = ; if x >. he dynamic futures strategy is known as the (deferred) stop-loss start-gain strategy investigated by Carr and Jarrow[]. 4

26 Advantages and Disadvantages of Volatility Contracts When compared to delta-hedging options, volatility contracts oer the user control over the sensitivity to the path. Not all volatility based payos can be spanned unless one is willing to specify or derive a risk-neutral volatility process. It is an open question as to which volatility payos can be spanned by static positions in options combined with dynamic trading in the underlying.

27 Part IV Connection to Recent Work on Stochastic Vol 6

28 Contract Paying Local Variance Recall that we were able to create a contract paying the variance along a strike, R (F t? ) t dt, by initially buying a (ratioed) calendar spread of straddles, [V (; )? e?r(? ) V (; )]: Suppose we further re-scale this payo by 4 where 4?. he payo at would instead be: Z (F t? ) t 4 dt: he cost of creating this position would be: 6 4 V (; )? e?r(? ) V (; ) By letting 4 #, one gets the beautiful result of Dupire[3] (; ) + (; ) is the cost of creating the payment (F? ) at. 4 As shown in Dupire, the forward local variance can be dened as the number of buttery spreads paying (F? ) at one must sell in order to nance the above option position initially. A discretized version of this result can be found in Derman et. al. []. 3 7 : 7

29 Summary One can go on to impose a stochastic process on the spot or forward price of local variance as in Dupire[3] and in Derman et. al.[]. he approach taken here is to examine the theoretical underpinnings of all such stochastic processes for volatility. It is interesting to note how naturally options arise as part of the analysis. Copies of the overheads can be downloaded from 8

30 Bibliography [] Carr P. and R. Jarrow, 99, \he Stop-Loss Start-Gain Strategy and Option Valuation: A New Decomposition into Intrinsic and ime Value", Review of Financial Studies, 3, 469{49. [] Derman E., I. Kani, and M. Kamal, 997. \rading and Hedging Local Volatility" Journal of Financial Engineering, October. [3] Dupire B., 996, \A Unied heory of Volatility", Paribas working paper. [4] Neuberger, A. 99, \Volatility rading", London Business School working paper. 9