# Models Used in Variance Swap Pricing

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1 Models Used in Variance Swap Pricing Final Analysis Report Jason Vinar, Xu Li, Bowen Sun, Jingnan Zhang Qi Zhang, Tianyi Luo, Wensheng Sun, Yiming Wang Financial Modelling Workshop 2011 Presentation Jan 15, 2011 Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47

2 Report Summary Variance Swap Introduction (Tianyi Luo) Definition and Pay-off Pricing Methods Intro: Rule of Thumb, Replication and Simulation Models of Volatility Skew: Stochastic Volatility Inspired and 7-Parameter Skew Model Parametrization (Tianyi Luo) Constructing The Volatility Surface (Wensheng Sun) Analysis of Rule of Thumb Pricing (Jingnan Zhang) Analysis of Continuous and Discrete Replication (Bowen Sun) Analysis of Pricing Using Simulation (Qi Zhang, Yiming Wang) Discussion of Future Work (Xu Li) Variance Swap with Stochastic Interest Rate The Bound of Value of Variance Swap Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47

3 What is variance swap? A variance swap is an instrument which allows investors to trade future realized volatility against current implied volatility. It is a popular way to add volatility exposure to a portfolio since It comes without the directional risk of the underlying security. The variance swap quotes are based on the implied volatility while the final pay-off is based on the realized volatility. The additivity of variance allows the investor to easily take a forward volatility position. Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47

4 The Pay-off The pay-off of the variance swap can be expressed as: P = N(σ 2 R K var ) where N is notional amount, K var is the strike quoted in annualized variance, and σr 2 is the realized variance over the life of the contract defined as σr 2 = 252 D (ln S i ) 2 D S i 1 with D being the number of trading days during the contract. i=1 Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47

5 The Pay-off Continued Note the pay-off of the variance swap is a convex function of the realized volatility. Here is an example of the pay-off of a variance swap with notional amount \$ and strikes at Payoff of Variance Swap Strikes at Payoff Realized Volatility Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47

6 Pricing Method: Rule of Thumb The rule of thumb pricing includes the following two methods: 90% put: take 90% of the forward price as the strike and plug that into the volatility function. 25% delta put: search for the strike that has a -25% delta (on the put side) and plug that strike into the volatility function. These two methods are empirically based and were justified by practitioners for non-steep volatility skew. They are used as quick ways to estimate the fair strike of variance swap. Under different scenarios, these rule of thumb might not be accurate enough thus other pricing methods would be suggested. Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47

7 Pricing Method: Discrete Replication Theoretically, the pay-off of variance swap can be replicated statically using infinite many puts and calls (some of them are not traded in the market). So we also used two replication methods to compute the fair strike. Using weights that are inversely proportional to the square of the strike produces a constant dollar gamma. In a discrete setting the fair strike has the form: K vs = 2er N put T T [ N put(k) call call(k) w i + w i ] F F i=1 where w i = c with c = 1 k 2 N 1, k = K F, put(k) is the put price and call(k) is the call price. This approach uses a range of percent strikes (of the forward) to find the dollar strike, e.g. 50% to 150% by 5% increments. i=1 Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47

8 Pricing Method: Continuous Replication In a continuous setting, the fair strike is calculated as the following: 2e K vs = rt F T [ 1 put(k) 1 call(k) 0 k 2 dk + F F k 2 dk] F This approach requires a parametrized volatility surface, gaps and non-exist strikes should be filled and extrapolated. Intuitively, the accuracy of this approach depends highly on the reliability of volatility surface. The derivation of the theoretical strike will be included in our final report. Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47

9 Pricing Method: Local Volatility & Monte Carlo Simulation With a parametrized implied volatility surface, we can construct the local volatility surface used Monte Carlo simulation. Conceptually, this is defined as: σ local (s, t) = f (σ imp (K, T )) After the translation, we simulate the underlying index with daily time steps. Using the fact that payoff = N i=1 (ln S i+1 S i ) 2 K vs and at time zero, the contract has no value, we can solve for a fair strike. We will be giving more on this method in the later part of our presentation. Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47

10 Modelling Vol Skew: SVI We will use Gatheral s Stochastic Volatility Inspired (SVI) and 7-Parameter Skew Model to model the volatility skew. Define k = ln(k/f ), where K is the strike and F is the forward price. Gatheral s SVI Model reads a + b[ρ(k m) + (k m) σ imp (F, T, K) = 2 + σ 2 ] T 0.65 Illustration of the SVI Model Implied Vol Moneyness k=log(k/f) This is an illustration with a = 0.026, m = 0, b = 0.1, ρ = 0.6, σ = 0.6 Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47

11 Modelling Vol Skew: 7-Parameter Skew The 7-Parameter Skew Model reads A L e λlk + β 4 σ imp (F, T, K) = β 1 + kβ 2 T + k2 R β 3 2T A R e λrk + β 6 if k < k L if k L k K R if k R < k Illustration of the Skew Model Implied Vol Moneyness k=log(k/f) This is an illustration with β 1 = 20% implied vol at the at the money forward, β 2 = 0.1, β 3 = 0.5, β 4 = 50%, β 5 = 0.75, β 6 = 15%, β 7 = 1. Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47

12 Data Set Structure & Model Parametrization The data set structure: Market data on 20 market dates For each market date, we have 15 maturities For each maturity, we have options with different strikes 300 realized market fair strike as benchmark Which means: we have 300 volatility skews to calibrate, 20 volatility surface to construct. We followed the these steps to calibrate both the SVI and skew models: Loss function is defined as Σ N i=1 (σmodel i σ imp i ) 2. Constrains are added to both the set of parameters. Minimize the loss function using Matlab function lsqnonlin.m under prescribed constrains. Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47

13 Illustration of Parametrization Here, we show a typical fit of these two models. This is a fit for a relatively short maturity A Parameterization of SVI & Skew Model on Short Maturity SVI Fit Market Skew Fit 0.6 Implied Vol Moneyness k=log(k/f) Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47

14 Illustration of Parametrization This is a fit for a relatively long maturity A Parameterization of SVI & Skew Model on Long Maturity SVI Fit Market Skew Fit 0.4 Implied Vol Moneyness k=log(k/f) Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47

15 Conclusion of Parametrization After hundreds of tests of fit, we come to the following conclusion about the parametrization: The accuracy of both models is higher for shorter maturity and lower for longer maturity. Generally, 7-Parameter Skew model performs better (with accuracy of the order 10 6 ) than SVI does (with accuracy of the order 10 4 ). Please note: our data exhibit volatility skew other than smile which means conclusion can reverse when data shows a smile! We will use these parametrization to construct our volatility surface and later local volatility surface. Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47

16 Constructing The Volatility Surface How to interpolate the volatility surface? Since the variance is additive, we used the forward volatility formula to interpolate the desired implied volatility to construct the surface: σ 2 2T 2 = σ 2 1T 1 + σ 2 F (T 2 T 1 ) for T 2 > T 1 Then given T star, T 1 T star T 2, define dt = T star T 1, we have the interpolated volatility: σ1 2 σ star = T 1 + σf 2 dt T 1 + dt Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47

17 Constructing The Volatility Surface A Visual Illustration: Step 1 σ k T Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47

18 Constructing The Volatility Surface A Visual Illustration: Step 2 σ T_star k T Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47

19 Constructing The Volatility Surface A Visual Illustration: Step 3 σ T_star T_lower k T Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47

20 Constructing The Volatility Surface A Visual Illustration: Step 4 σ T_star T_lower T_upper k T Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47

21 Constructing The Volatility Surface A Visual Illustration: Step 5 σ T_star k T Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47

22 Constructing The Volatility Surface A Visual Illustration: Step 6 σ T_star k T Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47

23 Constructing The Volatility Surface A Visual Illustration: Step 7 Surface σ k T Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47

24 Constructing The Volatility Surface Extrapolate Out of The Maturity Range In constructing local volatility surface, we need to use daily time step which might be less than the smallest maturity and our desired variance swap maturity might be longer than the longest the data provided, so we need to extrapolate out of the maturity range the data specified. We did the following: Assign σ(k star, T star ) = σ(k star, T (1)) if T star < T (1) Assign σ(k star, T star ) = σ(k star, T (end)) if T star > T (end) Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47

25 Constructing The Volatility Surface We built a MATLAB function to calculate the volatility surface: surface = vol surf (para, T, Model, T star, K star) para: parameter set from the calibration (a matrix) T: data maturity range (a column vector) Model: SVI or skew (string) T star: specified maturity (an arbitrary m by 1 vector) k star: specified moneyness (an arbitrary n by 1 vector) surface: interpolated volatility surface (an m by n matrix) In conclusion, this function searches the neighbouring maturity slices T 1 and T 2, evaluates σ 1 and σ 2 and then interpolates in between to give σ(k star, T star ). Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47

26 Constructing The Volatility Surface This is how the volatility surface look like: Volatility Surface Implied Volatility Moneyness = log(k/f) Maturity Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47

27 Constructing The Volatility Surface To verify this is a non-arbitrage volatility surface, we look at the total variance surface. Total Variance = σ 2 impt 2 Total Variance Surface 1.5 Implied Volatility Maturity Moneyness = log(k/f) Unlike the implied vol, the total variance is an increasing function of T for any k, so this volatility surface implies the non-arbitrage assumption. Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47

28 Test of Rule of Thumbs To get a quick view of the fair strike for variance swaps, one can use either of the following rule of thumb calculations. We can price variance swaps using rule of thumb: 90% put: take 90% of the forward strike as the strike and plug that into the volatility function 25% delta put: search for the strike that has a -25% delta (on the put side) and plug that strike into the volatility function Review that we already have two models: Gatherals SVI Model Skew Model We are going to show comparisons of these roles of thumb calculations to the market prices in hopes of creating a guide line for the use of these two methods. Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47

29 Test of Rule of Thumbs Next, we apply these two rules into SVI Model and Skew Model Respectively and compare model data with market data over both maturity and calendar time (market date), then do statistics analysis. The guideline will cover the following dimensions: Time to maturity or Calendar Time Type of volatility surface Rule of Thumb Method Review that we already have two models: Gatherals SVI Model Skew Model After determining the guidelines, we also wanted to check the appropriateness of the 90% and 25% values used in the respective methods and make a recommendation for better levels given the time series data available. Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47

30 Test of Rule of Thumbs Here we calculate volatility derived from market data of 10/27/2010 as example: Figure: Rule of thumb in SVI model (left) and skew model (right) Two Figures above have applied rule of thumb into SVI Model and Skew Model with same market date, therefore the same strike is taken. 1 1 Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47

31 Test of Rule of Thumbs From previous graphs, we find: The shape of two Figures is similar which means choice of volatility surface does not seem to show a big difference, and Market Fair Price is almost higher than other two ways, we notice when the Maturity goes increasing, both rules approximate to same volatility. It shows how each method performs relative to the time to maturity for fixed market date. 25% Delta put performs very well for maturities less than 2 years and turns out not so well when maturity is increasing to large period of time. Moreover, the 90% Strike does not work well at all due to the poor match of the market data. At 10 years maturity, the difference is 4 vol points of 25% Delta Put, however at 2 years, the difference is very tiny, how does this spread? We need to do different version for calendar time of market date instead of Maturity. Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47

32 Test of Rule of Thumbs Expiration date of 12/21/2012 = 2Y Expiration date of 12/21/2013 = 3Y 1 Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47

33 Test of Rule of Thumbs We did similar plot on longer maturity. The conclusion is: When maturity goes further, 90% Forward rules seems plausible than 25% rule. While, for short maturities, the 25% Delta Put matches market better. 90% forward rule is good for long maturity, but lower than market. 25% Delta Put is good for short maturity, but lower than market. There are no significant differences between Skew Model and SVI Model using these two rules of thumb. Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47

34 Test of Rule of Thumbs We also sampled from the market fair strike and our model strike price randomly and then F-test is performed on the random samples. And we find that the statistical analysis is in line with our assumption from graphs before. The 90% strike does not seems to work for any maturity in our data sample though it does perform better than 25% for the long maturity, but not good enough to use. The 25% delta put works very well for short term maturity variance swaps, but becomes ineffective when the maturity goes longer than 2 years. Given the reservation stated in 1 and 2, each pricing method works equally well with either volatility model. Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47

35 Conclusion of Discrete & Continuous Replication Discrete and continuous replication is theoretically true to price the variance swap. Commonly, discrete replication uses a range of percent strikes to find the dollar strike. Usually we use 50%-150%. We also did trials on 10%-190%, 30%-170%, 70%-130% and 90%-110%. Statistical analysis shows that: Discrete Replication + SVI is better than Discrete Replication + Skew Continuous Replication + SVI and Continuous Replication + skew are equivalent. Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47

36 Conclusion of Discrete & Continuous Replication Discrete and continuous replication is theoretically true to price the variance swap. Commonly, discrete replication uses a range of percent strikes to find the dollar strike. Usually we use 50%-150%. We also did trials on 10%-190%, 30%-170%, 70%-130% and 90%-110%. Statistical analysis shows that: Discrete Replication + SVI is better than Discrete Replication + Skew Continuous Replication + SVI and Continuous Replication + skew are equivalent. Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47

37 Transfer Implied Volatility to Local Volatility The formula that transfer implied volatility to local volatility reads: σ(s, t) 2 loc = 1 k w w T w k ( w + k2 w 2 )( w k ) w k 2 where k = ln( S F ) and w = σ2 impt. Derivatives are found numerically: we use forward approximation to calculate the first order derivative and central approximation to calculate the second derivative. Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47

38 Monte Carlo Simulation We assume underlying asset follows the stochastic equation: ds t S t = µ(t)dt + σ(s, t)dw t where µ(t) = r(t) q(t) and σ(s, t) is the local volatility. Every step of simulation we generate a new normal distributed random number and plug in local volatility and local drift term to produce a new underlying price. And the final price can be obtained using: payoff = N i=1 (ln S i+1 S i ) 2 K vs Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47

39 Monte Carlo Simulation & Error Analysis To reach a more accurate pay-off, we should simulate large number of paths of price evolution. Then, take the expectation of the total number of pay-off, which will be the fair volatility strike that let the initial value of the variance swap equal to zero. We finally compare the simulated value to market price: Percentage Error = abs(simulated value-market data) / market data Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47

40 Monte Carlo Simulation & Error Analysis Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47

41 Discussion of Future Work: Variance Swap with Stochastic Rate So far, the work we have been done is under the assumption that he underlying price process is an Itô process with a deterministic short rate. we have shown that rule of thumbs and the discrete model and continuous model work well in this setting for a relative short maturity, that is, less than 2 years for rule of thumbs and 3 to 5 years for the discrete and continuous models. Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47

42 Variance Swap with Stochastic Interest Rate The market has a stochastic interest rate; The value of an equity variance swap depends on the interest rate volatility; Long-dated variance swaps will be sensitive to the interest rate volatility; Long-dated variance swaps do happen; Price becomes more complicated. Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47

43 Variance Swap with Stochastic Interest Rate The rest is an upper bound of the fair strike developed by Per Hörfelt and Olaf Torn. P(t, T ) bond price at time t that matures at T. S(t) stock price at time t. dp(t, T ) P(t, T ) = r(t)dt + σ P(t, T ) dw 1 (t), (1) ds(t) S(t) = r(t) dt + σ S(t) dw 2 (t). (2) r(t) is the continuous compounded short rate at time t, Q is a risk-neutral measure, (W 1, W 2 ) is a 2-dim Q-Wiener process with joint correlation ρ. Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47

44 Discussion of Future Work: The Bound of Value of Variance Swap Introduce The solution of (1) is t β(t) = exp( r(θ) dθ), 0 P(t, T ) t = β(t) exp( σ P (θ, T )dw 1 (θ) 1 t σp 2 (θ, T ) dθ). (3) P(0, T ) It follows β(t) = 1 t P(0, t) exp( σ P (θ, T )dw 1 (θ) + 1 t σp 2 (θ, T ) dθ). (4) Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47

45 The Bound of Value of Variance Swap Define T-forward measure Q T by dq T dq = P(t, T ) Ft P(0, T )β(t) Then Girsanov theorem implies that W T defined by dw T 1 (t) = dw 1 (t) σ P (t, T ) dt, dw T 2 (t) = dw 2 (t) ρσ P (t, T ) dt, is a Wiener process with respect to Q T. Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47

46 The Bound of Value of Variance Swap Let V vs be the fair variance, V 0 vs be the fair variance in a market with zero bond volatility. V bvs the total bond variance. Some simple calculation yields V vs = E Q T [ 1 T ] [ σs 2 T dt = Vvs 0 + E Q T 1 T ] 2ρσ P σ S σp 2 0 T dt 0 V 0 vs + 2ρV 1 2 vs V 1 2 bvs V bvs. χ ρ Vvs χ ρ ( V 0 vs (1 ρ 2 )V bvs + ρ V bvs ) where χ ρ is the sign function of ρ. Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47

47 References More Than You Ever Wanted To Know About Volatility Swaps Just What You Need To Know About Variance Swaps The Volatility Surface: A practitioner s guide The Fair Value of a Variance Swap - a question of interest Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47

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