Absolute Dividends

Convexity adjustment for volatility swaps Chrif YOUSSFI Global Equity Linked Products Outline • • • • • • • Generalities about volatility/variance swaps. Intuition and motivation The framework of stochastic volatility. Convexity adjustment under stochastic volatility. Convexity adjustment and current smile. Numerical results. Conclusions. Volatility and Variance Swaps • A volatility swap is a forward contract on the annualized volatility that delivers at maturity: N . Vol  KVol  • A variance contract pays at maturity: N . Var  KVar  • The annualized volatility is defined as the square root of the variance: 2 Vol  Var  1   Si 1     ln  S    n i 0   i      n 1 1  Si 1  Si   S   n i 0  i   n 1 2 where S i is the closing price of the underlying at the ith business day and (n+1) is the total number of trade days. Hedge and Valuation • When there no jumps, the variance swaps valuation and hedging are model independent. • The vega-hedge portfolio for variance swaps is static and the value is directly calculated from the current smile. • The valuation of a volatility swap is model dependent and the pricing requires model calibration and simulations. • The vega hedge portfolio is not static. Motivation Smile of volatility 80.00% 70.00% 60.00% Volatility 50.00% 40.00% 30.00% 20.00% 10.00% 0.00% Implied 0 50 100 150 200 250 300 350 Strik es Smile of volatility generated by a stochastic volatility model where:spot is 100,maturity 1y and correlation is estimated at -70% • What is the price of ATM option? By considering the linearity of the option price w.r.t. volatility, the price is approximately 14.11% • What is the value of the variance swap? 16.17%. • What is the value of the volatility Swap? More difficult. Intuition • p(.) is the spot density at maturity T and  (.) the diffusion factor which be stochastic:  1 T 2 1 T 2 Var  E (   t dt )   E (   t dt | S  K ) p( K )dK 0 T 0 T 0 • A rough estimation of the volatility swap: 2 Var    implied (T , K ) p( K )dK 0  The weighting is not exact. Question: What can the moments of the implied volatility teach us about the value of volatility swap? MIV: Moment of Implied Volatility • We define by MIV (n) as the nth moment of the implied volatility weighted by the risk neutral density.   2C (T , K )   p( K )    K 2    n MIV (n)    implied (T , K ) p ( K )dK 0  • We define the smile convexity by: impvol  MIV(2)  MIV(1) • The convexity adjustment for the volatility swaps: vol / var  E(Var)  E(Vol) • Question: What is the relation between  vol / var and  implied? Stochastic volatility assumptions dSt • The underlying dynamics are:    t dWt1 St d • The volatility itself is log-normal: t   v dWt 2 t with the initial condition ( S 0 , 0 ) and  dWt1 , dWt 2   dt • The dynamics correspond to the short time analysis and the factor can be considered proportional to the square root  of time to maturity. (Patrick Hagan Model (1999)) Forward and backward equations • The backward equation for the call prices is V (t, S , , T , K ) : 1 2 2 2 Vt    S VSS  2  v S VS ,  v 2 V,  0 2   • The transition probability  (t , S0 ,  0 , T , K , ) from the state (t , S 0 ,  0 ) to (T , S , ) satisfies the forward equation (FPDE) : 1 2 T    2 S 2  2 Smile effect The curve    2  v 2 S  SS    v 2  2 S ,    0  , • When integrating the Forward PDE (Tanaka’s formula): 1 2 2 T  2 V (t , S0 ,  0 , T , K )  ( S0  K )   K     (t , S0 ,  0 , T , K , )dT d t 0 2  Intrinsic Integral over calendar spreads Call Price and Implied Volatility • Define by: S0 ) K ;   T  t; z A( z )  1  2vz   2v 2 z 2  0 log( and X • The solution of the system (S) is: V (t , S , 0 , T , K )  ( S0  K )  B( z, 0 , v,  )X 2  2   2 h  3 2 1 A( z )    vz log{ } v 1  exp(q)q dq B ( z ,  0 , v,  )  1 1   2   0 | z |  (1   2  0 z 2 ) KS 0   O( 4 ) 4   24  h • In the BS case we have a similar formula with v  0 :  1  2 1  vz  6 v0  v 2 (2  3 2 )   2v 2 z 2 (2  3 2 )   O( 4 )  BS (T , K )   0   24  2  1 2 { 0  v 2 (2  3 2 )  6 v 0 }X 2 24 Volatility swap convexity adjustment • The expected variance under the model assumptions: 2  S  1 2  log( T )    20 1   2v 2  O( 3 ) . E (Var)   E   T  S0  2    • The value of the expected volatility: 1    1 T 2 2 2   1  E (Vol )  E      t dt      0 1   2 v 2  O( 3 )     12   T 0   It follows that the convexity adjustment is: Var /Vol 1   0v 2 3  O( 4 ) 6 Smile convexity • By considering the value of the log-contract and the square of the log-profile: E( z)   2 S 1 1 1 E (log(  ))    0 (1   2 v 2 )  O( 4 )  0 S0 2 2 1 2 2 2 1 2 2 2 E ( z )     0    v   O( 3 ) 4 2 • The smile convexity of the implied volatility is: implied 1 3   0v 2  2  O( 4 ) 8 Convexity adjustment • As long as the  2 is large enough (which is satisfied in the equity markets), to the leading orders show that the relation between the two convexities is very simple: Var / Vol  4 3  implied  O( 4 ) 2 • There no dependencies on maturity and volatility of the volatility. • The value of the volatility swap does not depend on the correlation, however the implied volatility depends on  and therefore intuitively we need to strip off this dependency. Numerical Results (4) Convexity adjustment vs Smile Convexity 4.50% 4.00% 3.50% Adjustment 3.00% 2.50% 2.00% 1.50% 1.00% 0.50% 0.00% 0% 10% 20% 30% 40% 50% 60% 70% 80% Vol=40% Rho=-90% T=1 yr Epsilon=1 MC Formula voVol Numerical Results (5) Convexity adjustment vs Smile Convexity 3.50% 3.00% Adjustment 2.50% 2.00% 1.50% 1.00% 0.50% 0.00% 0% 10% 20% 30% 40% 50% 60% 70% 80% Vol=30% Rho=-90% T=1 yr Epsilon=1 MC Formula voVol Numerical Results(6): Heston Convexity adjustment vs Smile Convexity 3.00% 2.50% 2.00% 1.50% 1.00% 0.50% 0.00% 0% 10% 20% 30% 40% 50% 60% 70% 80% Vol=40% Rho=-70% T=1 yr Lambda=80% Vbar=16% MC Theory Numerical Results (7): Heston Convexity adjustment vs Smile Convexity 3.00% 2.50% 2.00% 1.50% 1.00% 0.50% 0.00% 0% 10% 20% 30% 40% 50% 60% 70% 80% Vol=40% Rho=-90% T=1 yr Lambda=80% Vbar=16% MC Theory Conclusion • This analysis shows that option prices can be very insightful to estimate the convexity adjustment. • Even though the results are derived in the case of Hagan model, they can be extended to other models of stochastic volatility (Heston) as long as the correlation is in an appropriate range. • It sheds some light on the importance of the curve factors to decide the value of a volatility swap.