Application of Stochastic Calculus to Price a Quanto Spread

Transcription

1 Application of Stochastic Calculus to Price a Quanto Spread Christopher Ting Algorithmic Quantitative Finance July 15, 2017 Christopher Ting July 15, /33

2 Table of Contents Introduction Theory Information Contents Co-Variance Swaps Takeaways Christopher Ting July 15, /33

3 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, /33

4 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 at 3:12 p.m. HK/SGP, falling from an earlier high of 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, /33

5 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, /33

6 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, /33

7 Spread Trading Strategy Christopher Ting July 15, /33

8 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 x S t = A t $5 x + $5 x = A t. Christopher Ting July 15, /33 S t

9 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, /33

10 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 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, /33

11 P&L in Different Scenarios for Q = 20 Christopher Ting July 15, /33

12 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, /33

13 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 V t( dxt ) 2 = Vt µ dt 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, /33

14 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, /33

15 Drift Vector, Volatility Matrix, Market Price of Risk Drift vector ( ν + 1 Θ = 2 σ2 2 + u r ) µ + ν σ2 1 + ρ σ 1σ 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, /33

16 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, /33

17 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, /33

18 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, /33

19 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, /33

20 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 = Q t /F,t τ. (8) ω t := sign ( γ t ) γt. (9) The other solution γ t τ = Q 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, /33

21 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, /33

22 Total Number of Contracts Traded by Maturity Maturity NIY NKD JY ,883 27, ,430 21, , ,825 25, , ,526 39, , ,394 42, , ,419 53, , ,415 48, , ,756 84, , , , , , , , , , , , , , , , , , , , , ,151 1,415, , ,649 1,442,391 Christopher Ting July 15, /33

23 Total Number of Contracts Traded by Maturity (cont d) Maturity NIY NKD JY , ,836 1,636, , ,772 1,934, , ,665 1,862, , ,419 2,684, , ,139 1,765, , ,567 1,630, , ,205 1,786, , ,614 1,948, , ,175 1,908, , ,617 2,577, , ,527 2,413, , ,425 2,210,321 Christopher Ting July 15, /33

24 Average Quanto Spreads Maturity Days Mean Std Min Med Max Christopher Ting July 15, /33

25 Average Quanto Spreads (cont d) Maturity Days Mean Std Min Med Max Christopher Ting July 15, /33

26 Daily Time Series of Average Quanto Spreads of December 2010 Maturity Christopher Ting July 15, /33

27 Daily Time Series of Implied Co-Volatility of December 2010 Maturity Christopher Ting July 15, /33

28 Christopher Ting July 15, /33

29 Rolling Historical Daily Co-Volatility Christopher Ting July 15, /33

30 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 < τ < τ < τ < τ < τ < τ < τ < τ < τ < τ < Christopher Ting July 15, /33

31 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, /33

32 Average P&L of A Long Position in Co-Variance Swap Christopher Ting July 15, /33

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, /33