Application of Stochastic Calculus to Price a Quanto Spread Christopher Ting http://www.mysmu.edu/faculty/christophert/ Algorithmic Quantitative Finance July 15, 2017 Christopher Ting July 15, 2017 1/33
Table of Contents Introduction Theory Information Contents Co-Variance Swaps Takeaways Christopher Ting July 15, 2017 2/33
Introduction Cross-asset investment and trading strategies Management of currency risks Quanto spread trading of NKD-NIY pair Implied co-volatility is a superior forecast of future co-volatility Co-variance swap Christopher Ting July 15, 2017 3/33
A Market Behavior CNBC Asian markets finished mostly higher on (this) Monday, searching for direction... Japan s Nikkei 225 index finished near flat at 19,251.08, reversing earlier losses of nearly 0.6 percent as the yen weakened against the dollar to trade at 113.12 at 3:12 p.m. HK/SGP, falling from an earlier high of 112.75... A stronger yen generally weighs on export-oriented stocks in Japan as it affects their overseas profit margins when funds are converted to the local currency. Christopher Ting July 15, 2017 4/33
A Very Brief Review Variance swap: trade volatility as an asset class Co-variance risk: in crisis, markets tend to move in the same direction: Less research on co-variance swap Carr and Madan (1999) Carr and Corso (2001) Da Fonseca, Grasselli, and Ielpo (2011) What about using forwards or futures to price and hedge co-variance swaps? Christopher Ting July 15, 2017 5/33
Quanto Futures & Spread On CME, two different futures contracts, NIY and NKD, on the same underlying Nikkei 225 index are traded. The quanto spread is defined as Q t = F $,t F,t. Short position in NKD NIY spread on day t is, in dollars, A t = $5 F $,t + R 500 F,t S t, (1) where the spot exchange rate S t is yens per dollar. Christopher Ting July 15, 2017 6/33
Spread Trading Strategy Christopher Ting July 15, 2017 7/33
Proposition 1 Assuming that S t remains constant, the quanto spread position is hedged against Nikkei index movements if and only if the ratio is given by R = S t 100. Proof: By substituting this R into equation 1, we obtain A t = $5 F $,t + $5 F,t = $5 ( F,t Q t + F,t ) = $5 Q t. Suppose both NKD and NIY move up by x index points. The net notional amount becomes B t (x), and B t (x) = $5 ( F $,t + x ) + R 500 ( F,t + x ) = A t $5 x + S t 100 500 x S t = A t $5 x + $5 x = A t. Christopher Ting July 15, 2017 8/33 S t
Marked-to-Market Value The quanto spread seller is directly exposed to currency risk. If yen weakens, equivalently dollar strengthens, i.e., S u > S t, then the quanto seller may end up having sold the NKD-NIY spread at a low value of Q t in comparison to Q u + (1 S t /S u )F,u, where Q u = F $,u F,u, u > t. ( A u = $5 F $,u + S ) ( t F,u = $5 F,u Q u + S ) t F,u S u S ( ( u = $5 Q u + 1 S ) ) t F,u. S u Christopher Ting July 15, 2017 9/33
P&L of a Quanto Spread Seller By unwinding the position at a later time u: ( ( P&L(t, u) = 5 Q t Q u 1 S ) ) t F,u S u (2) for every NKD contract. If the short quanto spread is held to maturity, ( P&L(t, T) = 5Q t + 5 1 S ) t ) (NT t F,t S T (3) where N T is the value of Nikkei 225 Index at maturity T. Christopher Ting July 15, 2017 10/33
P&L in Different Scenarios for Q = 20 Christopher Ting July 15, 2017 11/33
Proposition 2 Nikkei index N t and the exchange rate U t (dollars per yen) are taken to be geometric Brownian motions, which correlate with a correlation coefficient of ρ: N t = N 0 exp ( µ t + σ 1 W 1 (t) ), (4) ( U t = U 0 exp ν t + ρ σ 2 W 1 (t) + ) 1 ρ 2 σ 2 W 2 (t). (5) Under the assumptions that N t and U t are correlated geometric Brownian motions as in equations 4 and 5, the theoretical (fair) price of the quanto futures F $,t maturing on day T is given by F $,t = exp ( C τ ) F,t, (6) where τ = T t, and C is the rate of co-variance. Christopher Ting July 15, 2017 12/33
Geometric Brownian Motion If an asset V t follows the geometric Brownian motion V t = V 0 exp ( ) µ dt + σ W t, then the log price X t = ln(v t ) is an arithmetic Brownian motion dx t = µ dt + σ dw t. Treat V t = exp(x t ) as a function of X t. By Itô s formula, dv t = V t dx t + 1 2 V t( dxt ) 2 = Vt µ dt + 1 2 V t σ 2 dt + V t σ dw t, leading to the stochastic differential equation, dv t = (µ + 12 ) V σ2 dt + σ dw t. t Christopher Ting July 15, 2017 13/33
Proof Setup Risk-free rate r for money market account and yen money market account D t = exp(ut) of risk-free rate u for yen Yen cash bond in dollars U t D t, and the Nikkei index in dollars, U t N t. Discounted money market account and Nikkei 225 Index in dollars: Y t = e rt U t D t Z t = e rt U t N t Stochastic differential equations dy t = ( ν + σ 2 2 Y /2 + u r) dt + ρ σ 2 dw 1 (t) + 1 ρ 2 σ 2 dw 2 (t), t dz t = ( µ + ν + σ 2 1 Z /2 + ρ σ 1σ 2 + σ 2 2 /2 r) dt + ( σ 1 + ρ σ 2 )dw 1 (t) t + 1 ρ 2 σ 2 dw 2 (t). Christopher Ting July 15, 2017 14/33
Drift Vector, Volatility Matrix, Market Price of Risk Drift vector ( ν + 1 Θ = 2 σ2 2 + u r ) µ + ν + 1 2 σ2 1 + ρ σ 1σ 2 + 1 2 σ2 2 r Volatility matrix ( ) ρ σ2 1 ρ2 σ Σ = 2 σ 1 + ρ σ 2 1 ρ2 σ 2 Two-dimensional market price of risk m = Σ 1( Θ r1 ) = ( m1 m 2 ) Christopher Ting July 15, 2017 15/33
Let Martingales Girsanov s Theorem d W i (t) = dw i (t) + m i dt, for i = 1, 2 dy t Y t = ρ σ 2 d W 1 (t) + 1 ρ 2 σ 2 d W 2 (t). dz t Z t = (σ 1 + ρ σ 2 )d W 1 (t) + 1 ρ 2 σ 2 d W 2 (t). By Girsanov s Theorem, there exists an equivalent risk-neutral measure Q associated with these martingales. Christopher Ting July 15, 2017 16/33
Nikkei 225 Index Under Q The stochastic differential equation for N t is dn t N t = ( µ + σ 2 1 /2) dt + σ 1 dw 1 (t). Under Q with d W 1 (t) = dw 1 (t) + m 1 dt, this stochastic differential equation becomes dn t N t = σ 1 d W 1 (t) ( µ + σ 2 1 /2 + ρ σ 1σ 2 u ) dt + ( µ + σ 2 1 /2) dt = σ 1 d W 1 (t) + ( u ρ σ 1 σ 2 ) dt. The solution is N t = N 0 exp ( σ 1 W1 (t) + ( ) u ρ σ 1 σ 2 σ 2 1 /2) t. Christopher Ting July 15, 2017 17/33
Finally At time t = 0, the theoretical forward price of Nikkei index is F,0 = N 0 exp ( ut ). The solution is re-written as N T = exp ( ρ σ 1 σ 2 T ) ) F,T exp (σ 1 W1 (T) σ 2 1 T/2. Let F $,0 be the forward price of the dollar-denominated forward. The value v 0 of this quanto forward at initiation is zero. Namely v 0 = E Q ( NT F $,0 ) = ( exp( ρ σ1 σ 2 T)F,0 F $,0 ) = 0. Consequently, under the risk-neutral measure Q, F $,0 = exp ( ρ σ 1 σ 2 T ) F,0. Since C = ρ σ 1 σ 2, the pricing formula of the quanto forward for any t T is therefore given by F $,t = exp ( C (T t) ) F,t. Christopher Ting July 15, 2017 18/33
Corollary The theoretical value Q th t of the quanto spread is well approximated by Q th t = ( Cτ + 12 ) (Cτ)2 F,t. (7) Given the theoretical formula for the quanto spread (equation 7), one can define the notion of implied co-variance γ t by using the observed quanto spread traded in the futures market. Namely Q t = ( γ t τ + 12 ) (γ tτ) 2 F,t. Here F,t is the observable market price of NIY. Christopher Ting July 15, 2017 19/33
Implied Co-Variance This equation is rewritten as 1 2 (γ tτ) 2 γ t τ Q t F,t = 0. Solving this quadratic equation with respect to γ t τ results in Implied co-volatility ω t : γ t = 1 1 + 2Q t /F,t τ. (8) ω t := sign ( γ t ) γt. (9) The other solution γ t τ = 1 + 1 + 2Q t /F,t is not admissible because it is strictly larger than 1, which is incompatible with the fact that the magnitude of the co-variance between two returns on financial assets is usually less than 1. Christopher Ting July 15, 2017 20/33
Co-Variance Swap Consider the returns of two assets and their respective volatilities σ 1 and σ 2. The covariance is C = ρσ 1 σ 2 Covariance swap is similar to variance swap: Covariance realized over the tenor T Risk-Neutral Covariance Key idea: use the quanto spread traded in the market to back out a risk-neutral co-variance with the proposed model. Christopher Ting July 15, 2017 21/33
Total Number of Contracts Traded by Maturity Maturity NIY NKD JY 200403 998 1,883 27,614 200406 13,430 21,846 203,838 200409 10,825 25,237 240,925 200412 19,526 39,314 325,080 200503 21,394 42,195 447,976 200506 30,419 53,178 513,089 200509 30,415 48,366 607,083 200512 47,756 84,908 559,602 200603 73,295 128,210 619,772 200606 100,900 142,182 913,141 200609 102,623 154,178 653,179 200612 78,135 114,720 675,552 200703 87,147 132,390 803,930 200706 117,905 155,687 850,254 200709 176,610 222,151 1,415,020 200712 181,700 204,649 1,442,391 Christopher Ting July 15, 2017 22/33
Total Number of Contracts Traded by Maturity (cont d) Maturity NIY NKD JY 200803 254,782 278,836 1,636,035 200806 211,164 278,772 1,934,074 200809 249,755 291,665 1,862,708 200812 479,269 466,419 2,684,007 200903 260,835 266,139 1,765,825 200906 257,066 257,567 1,630,273 200909 233,385 223,205 1,786,246 200912 241,637 189,614 1,948,927 201003 196,682 155,175 1,908,279 201006 349,340 284,617 2,577,168 201009 271,686 214,527 2,413,097 201012 262,293 201,425 2,210,321 Christopher Ting July 15, 2017 23/33
Average Quanto Spreads Maturity Days Mean Std Min Med Max 200403 8 2.50 6.38-8.33 3.75 10.00 200406 56 1.28 4.42-7.5 0.73 13.33 200409 55 3.93 8.53-6.36 2.69 57.50 200412 63 2.54 2.31-3.01 2.32 9.62 200503 62 2.34 2.28-2.97 2.18 8.57 200506 67 2.42 1.84-5.83 2.27 8.33 200509 66 2.55 0.94 0.53 2.61 4.17 200512 67 2.72 1.40-0.44 2.62 7.71 200603 66 5.99 3.72-3.09 5.56 15.00 200606 70 6.93 5.02-2.48 5.07 19.69 200609 68 4.66 4.99-3.34 4.30 18.72 200612 67 6.40 3.63-0.62 6.13 16.16 200703 67 6.83 3.54-1.75 6.48 17.58 200706 71 11.00 5.39-1.77 12.00 22.66 200709 74 8.17 5.31 0.88 7.28 20.00 200712 71 24.97 11.03 0.47 27.42 48.92 Christopher Ting July 15, 2017 24/33
Average Quanto Spreads (cont d) Maturity Days Mean Std Min Med Max 200803 68 27.67 14.92 1.41 27.69 61.34 200806 70 44.86 26.21 0.62 44.83 111.38 200809 69 39.89 32.93-0.22 39.25 117.85 200812 72 69.33 40.74 0.53 64.93 182.94 200903 67 82.28 55.39-0.61 85.42 189.84 200906 70 50.03 34.33-0.21 46.71 120.16 200909 71 47.00 22.20 0.23 50.43 87.92 200912 67 29.63 25.30 0.26 18.15 96.76 201003 69 37.26 23.22 0.15 36.98 81.82 201006 71 27.46 19.64-0.25 20.36 75.00 201009 68 22.14 16.50-0.13 18.66 57.65 201012 69 23.79 13.85 0.27 19.20 54.32 Christopher Ting July 15, 2017 25/33
Daily Time Series of Average Quanto Spreads of December 2010 Maturity Christopher Ting July 15, 2017 26/33
Daily Time Series of Implied Co-Volatility of December 2010 Maturity Christopher Ting July 15, 2017 27/33
Christopher Ting July 15, 2017 28/33
Rolling Historical Daily Co-Volatility Christopher Ting July 15, 2017 29/33
Realized Co-Volatility Regressed on ICV t and HCV t σ t (τ) = a + b 1 ICV t (τ) + b 2 HCV t (τ) + u t. Days to Ave Num Adjusted maturity (τ) τ Obs a t-stat b 1 t-stat b 2 t-stat R 2 45 τ <50 46.7 69-1.617-0.89 0.264 1.83 0.603 4.00 0.6150 50 τ <55 51.3 85-0.213-0.11 0.311 1.67 0.692 3.95 0.6299 55 τ <60 57.5 95 0.298 0.14 0.580 2.50 0.445 3.25 0.5450 60 τ <65 62.8 59 1.780 0.79 0.777 2.67 0.334 2.39 0.5858 65 τ <70 65.9 62 0.485 0.17 0.698 1.83 0.343 2.06 0.5045 70 τ <75 72.0 105-0.800-0.34 0.453 1.51 0.469 2.65 0.4244 75 τ <80 78.0 61-1.215-0.47 0.626 2.23 0.283 1.79 0.3673 80 τ <85 81.8 57-1.956-1.00 0.422 2.00 0.334 2.24 0.3849 85 τ <90 86.5 93-2.315-1.01 0.455 1.97 0.216 1.23 0.3327 90 τ <95 92.0 72-1.528-0.66 0.572 2.65 0.143 0.81 0.3351 45 τ <95 69.3 758-0.898-0.82 0.516 3.99 0.359 3.65 0.4708 Christopher Ting July 15, 2017 30/33
Co-Variance Swap On the maturity date T, the payoff to the swap buyer is Payoff = Ĉ(t, T) γ t. (10) Here the fixed leg is the risk-neutral implied co-variance γ t, which is determined by equation 8, and the floating leg is the co-variance Ĉ(t, T) realized over the swap tenor. Ĉ(t, T) = 365 T t 1 T i=t+1 R N,i R U,i R N R U, Christopher Ting July 15, 2017 31/33
Average P&L of A Long Position in Co-Variance Swap Christopher Ting July 15, 2017 32/33
Takeaways A method to obtain implied co-variance from a pair of forwards Implied co-volatility is a superior forecast compared to historical forecast Shorting quanto spread on average does not gain over the sample period 2005 through 2010 Co-variance swaps are fair to both buyers and sellers A new method to price and hedge co-variance swaps. Christopher Ting July 15, 2017 33/33