The following is a series of conversations between O1 Pro & myself with questions asked by X user rafkom01.

Prompt:

Replicate the forward staring variance swap in SPX, using SPX options, derive mathematically.

Below is a common approach (and the associated math) for showing how a forward-starting variance swap on an equity index (e.g. the S&P 500) can be replicated using ordinary (“spot-starting”) variance swaps (which, in turn, can be statically replicated via SPX options across strikes).

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1. Forward-Starting Variance Swap Definition

A forward-starting variance swap from T1T_{1} to T2T_{2} is a contract that will pay the (annualized) realized variance of the underlying StS_{t} (here, the S&P 500 index) over the future time interval [T1,T2][T_{1}, T_{2}]. Mathematically, its payoff at time T2T_{2} is PayofffwdVar=(T2−T1) 1T2−T1∫T1T2(dln⁡St)2=∫T1T2(dln⁡St)2,\text{Payoff}_{\mathrm{fwdVar}} = (T_{2} – T_{1}) \,\frac{1}{T_{2} – T_{1}} \int_{T_{1}}^{T_{2}} \bigl(d \ln S_{t}\bigr)^{2} = \int_{T_{1}}^{T_{2}} \bigl(d \ln S_{t}\bigr)^{2},

where (dln⁡St)2  ≈  (ln⁡(St+ΔtSt))2(d \ln S_{t})^{2} \;\approx\; \Bigl(\ln\bigl(\tfrac{S_{t+\Delta t}}{S_{t}}\bigr)\Bigr)^{2}

in a discrete approximation.

Typically, the contract is quoted by its fair strike of forward-start variance: KfwdVar(T1,T2)  =  EQ ⁣[1T2−T1∫T1T2σt2 dt  ∣  F0],K_{\mathrm{fwdVar}}(T_{1}, T_{2}) \;=\; \mathbb{E}^{\mathbb{Q}}\!\Bigl[ \frac{1}{T_{2} – T_{1}} \int_{T_{1}}^{T_{2}} \sigma_{t}^{2}\,dt \;\Big|\;\mathcal{F}_{0} \Bigr],

where σt2\sigma_{t}^{2} is the instantaneous variance process under the risk-neutral measure Q\mathbb{Q}.

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2. Key Idea: Forward Variance as a Difference of Spot-Starting Variances

A crucial (and very elegant) relationship is that the realized variance from T1T_{1} to T2T_{2} can be seen as the difference between:

1. The realized variance from 00 to T2T_{2}, and
2. The realized variance from 00 to T1T_{1},

adjusted for the respective time lengths. Concretely, ∫T1T2(dln⁡St)2  =  ∫0T2(dln⁡St)2  −  ∫0T1(dln⁡St)2.\int_{T_{1}}^{T_{2}} \bigl(d \ln S_{t}\bigr)^{2} \;=\; \int_{0}^{T_{2}} \bigl(d \ln S_{t}\bigr)^{2} \;-\; \int_{0}^{T_{1}} \bigl(d \ln S_{t}\bigr)^{2}.

When annualized (i.e., dividing each by its interval length) and taken in expectation under the risk-neutral measure, one obtains the well-known result that KfwdVar(T1,T2)  =  (T2  Kvar(0,T2))  −  (T1  Kvar(0,T1))T2−T1,K_{\mathrm{fwdVar}}(T_{1}, T_{2}) \;=\; \frac{ (T_{2}\;K_{\mathrm{var}}(0, T_{2})) \;-\; (T_{1}\;K_{\mathrm{var}}(0, T_{1})) } { T_{2} – T_{1} },

where Kvar(0,T)K_{\mathrm{var}}(0, T) is the fair strike of a spot-starting variance swap running from now (t=0)(t=0) until TT.

Hence, buying a variance swap from 00 to T2T_{2} and shorting a variance swap from 00 to T1T_{1} (in the correct notional proportions) effectively carves out the realized variance over the forward interval [T1,T2][T_{1}, T_{2}].

In other words, to replicate a forward-starting variance swap on [T1,T2][T_{1}, T_{2}], you can simply hold a portfolio of two spot-starting variance swaps:Long  1  unit of VarSwap(0,T2)andShort  T2−T1T1  units of VarSwap(0,T1).\text{Long}\;1\;\text{unit of VarSwap}(0, T_{2}) \quad\text{and}\quad \text{Short}\;\frac{T_{2}-T_{1}}{T_{1}}\;\text{units of VarSwap}(0, T_{1}).

In practice, you adjust notional scalars appropriately, but conceptually it is the difference of two standard variance swaps.

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3. Replicating Each Spot-Starting Variance Swap with SPX Options

Since a forward-start variance swap can be built from two ordinary (spot-start) variance swaps, we now recall how a standard variance swap from 00 to TT is replicated with SPX options. The classical result (in a simplified setting with zero rates/dividends for readability) is that the fair strike of a spot-start variance swap is given by Kvar(0,T)  =  2T  erT ∫0∞1K2 [P0(K,T) 1{ K<F0}  +  C0(K,T) 1{ K≥F0}] dK,K_{\mathrm{var}}(0, T) \;=\; \frac{2}{T} \; e^{rT}\, \int_{0}^{\infty} \frac{1}{K^{2}}\, \Bigl[ P_{0}(K, T)\,\mathbf{1}_{\{\,K<F_{0}\}} \;+\; C_{0}(K, T)\,\mathbf{1}_{\{\,K\ge F_{0}\}} \Bigr] \,dK,

where

• F0=S0 erTF_{0} = S_{0}\,e^{rT} is the (no-dividend) forward price of the index for maturity TT.
• P0(K,T)P_{0}(K, T) and C0(K,T)C_{0}(K, T) are the time‑00 prices of European put and call options with strike KK and maturity TT.
• rr is the risk-free rate (if nonzero).
• The factor erTe^{rT} arises to convert forward to spot, depending on the precise convention used.

Sketch of the Replication Argument

1. Realized variance as path integral.
   The payoff of the variance swap is proportional to ∫0T(dln⁡St)2\int_{0}^{T} (d\ln S_{t})^{2}.
2. “Log Contract” approach.
   One can show that a “log payoff” contract log⁡(ST)\log(S_{T}) (plus some bond position) is closely related to ∫(dln⁡St)2\int (d\ln S_{t})^{2} upon continuous rebalancing in the underlying.
3. Static replication of log⁡(ST)\log(S_{T}) via a continuum of OTM calls and puts.
   It is a known result that log⁡(ST)  =  log⁡+(ST−K)  −  (integrals of calls and puts over K)  +  …,\log(S_{T}) \;=\; \log^{+}(S_{T} – K) \;-\; \text{(integrals of calls and puts over \(K\))} \;+\; \ldots, which becomes the continuous‑strike integral.
4. Collect terms to show the fair strike.
   Summing up the cost of these integrals across all strikes leads to the standard closed‑form representation for Kvar(0,T)K_{\mathrm{var}}(0, T).

The end result is that a variance swap can be statically replicated by holding a continuum of out-of-the-money SPX calls and puts across all strikes (plus some dynamic adjustments in the underlying to hedge the drift in ln⁡S\ln S). In practice, traders approximate this by a strip of options at many strikes.

Hence, each of the two pieces we need—VarSwap(0,T2)(0, T_{2}) and VarSwap(0,T1)(0, T_{1})—can be implemented by trading in SPX puts and calls (plus minimal dynamic hedges).

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4. Putting It All Together

1. Spot-starting VarSwaps as building blocks.
   • Buy the standard variance swap from 00 to T2T_{2}.
   • Sell the standard variance swap from 00 to T1T_{1}.
2. Net payoff at time T2T_{2}.
   • The long position (VarSwap(0,T2)(0, T_{2})) pays: (T2)×(realized var over [0,T2])  −  (constant strike)×T2.(T_{2}) \times \bigl(\text{realized var over }[0,T_{2}]\bigr) \;-\; \text{(constant strike)} \times T_{2}.
   • The short position (VarSwap(0,T1)(0, T_{1})) unwinds at T1T_{1}. But effectively, from a final‑payoff viewpoint, it removes the realized variance on [0,T1][0,T_{1}].
3. Result: the difference between realized variance on [0,T2][0,T_{2}] and on [0,T1][0,T_{1}] equals realized variance on [T1,T2][T_{1},T_{2}]. After correct notional scaling, that difference is precisely the forward-start realized variance.
4. Cost at inception.
   The cost (or fair strike) of the forward-start variance (KfwdVar(T1,T2))\bigl(K_{\mathrm{fwdVar}}(T_{1},T_{2})\bigr) must be the difference in the costs of those two spot variance swaps, scaled appropriately by T2−T1T2\tfrac{T_{2}-T_{1}}{T_{2}} vs. T1T2\tfrac{T_{1}}{T_{2}}. Concretely, KfwdVar(T1,T2)  =  T2 Kvar(0,T2)  −  T1 Kvar(0,T1)T2−T1.K_{\mathrm{fwdVar}}(T_{1},T_{2}) \;=\; \frac{T_{2}\,K_{\mathrm{var}}(0,T_{2}) \;-\; T_{1}\,K_{\mathrm{var}}(0,T_{1})} {T_{2}-T_{1}}.
5. Hence, replication via SPX options.
   Since each Kvar(0,T)K_{\mathrm{var}}(0,T) is itself replicable via a continuum of SPX options (puts and calls) at maturity TT, the difference of those strips—one for (0,T2)(0,T_{2}) and one for (0,T1)(0,T_{1})—replicates the forward-start variance swap.

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5. Final “Derivation” in a Nutshell

1. Standard VarSwap formula (from 0 to TT) in terms of SPX options: Price of VarSwap(0,T)  =  2 erTT[∫0F0P(K,T)K2 dK  +  ∫F0∞C(K,T)K2 dK].\text{Price of VarSwap}(0,T) \;=\; \frac{2\,e^{rT}}{T} \left[ \int_{0}^{F_{0}} \frac{P(K,T)}{K^{2}}\,dK \;+\; \int_{F_{0}}^{\infty} \frac{C(K,T)}{K^{2}}\,dK \right].
2. Forward Var from T1T_{1} to T2T_{2}: KfwdVar(T1,T2)  =  T2  Kvar(0,T2)  −  T1  Kvar(0,T1)T2−T1.K_{\mathrm{fwdVar}}(T_{1},T_{2}) \;=\; \frac{ T_{2}\;K_{\mathrm{var}}(0,T_{2}) \;-\; T_{1}\;K_{\mathrm{var}}(0,T_{1}) }{ T_{2}-T_{1} }.
3. Implementation:
   • Take the difference of the two integrals for Kvar(0,T2)K_{\mathrm{var}}(0,T_{2}) and Kvar(0,T1)K_{\mathrm{var}}(0,T_{1}).
   • This difference, scaled by 1T2−T1\frac{1}{T_{2}-T_{1}}, is the cost of the forward-start variance swap.
   • In practice, you go long a strip of SPX options with maturity T2T_{2} (in the weights that replicate a 0→T20\to T_{2} VarSwap) and short a strip of SPX options with maturity T1T_{1} (the weights for a 0→T10\to T_{1} VarSwap). After netting, you end up with an economic exposure to realized variance solely on the interval [T1,T2][T_{1},T_{2}].

Hence, the forward-starting variance swap on [T1,T2][T_{1}, T_{2}] can be viewed and priced as the “carve out” (difference) between two standard variance swaps—each of which is itself replicable from a static continuum of SPX puts and calls across strikes.

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Complete Summary

1. Forward-Start vs. Spot-Start VarSwaps
   • Spot-start variance swap from now to TT: pays out the realized variance of the index over [0,T][0,T].
   • Forward-start variance swap from T1T_{1} to T2T_{2}: pays out the realized variance over [T1,T2][T_{1},T_{2}].
2. Key Decomposition (Realized Var over [T1,T2])  =  (Realized Var over [0,T2])  −  (Realized Var over [0,T1]).\bigl(\text{Realized Var over }[T_{1},T_{2}]\bigr) \;=\; \bigl(\text{Realized Var over }[0,T_{2}]\bigr) \;-\; \bigl(\text{Realized Var over }[0,T_{1}]\bigr). Taking expectation and dividing by (T2−T1)(T_{2}-T_{1}) yields the forward-start variance in terms of standard variances.
3. Price / Fair Strike KfwdVar(T1,T2)  =  T2 Kvar(0,T2)  −  T1 Kvar(0,T1)T2−T1.K_{\mathrm{fwdVar}}(T_{1},T_{2}) \;=\; \frac{ T_{2}\,K_{\mathrm{var}}(0,T_{2}) \;-\; T_{1}\,K_{\mathrm{var}}(0,T_{1}) }{ T_{2}-T_{1} }.
4. Replication via SPX Options
   • Each spot-start var swap Kvar(0,T)K_{\mathrm{var}}(0,T) can be replicated by a static position in out-of-the-money SPX calls and puts maturing at TT.
   • Therefore, the forward-start var swap is replicated by the difference of these two strips (one for maturity T2T_{2} and one for maturity T1T_{1}).

Mathematically, this completes the derivation and shows explicitly how to replicate the forward-starting variance payoff using exchange-traded SPX options.

Prompt 2:

Consider the VIX’s relationship with the S&P 500’s implied volatility & tenor. Using VIX futures options replicate the same S&P 500 forward start variance swap.

Below is a sketch of how one can “carve out” the same forward‐starting variance on the S&P 500 (SPX)—i.e.\ the realized variance over [T1,T2][T_{1},T_{2}]—but now using VIX futures (and their options) rather than directly trading SPX options across strikes.

The essential idea is that the VIX (squared) measures the risk‐neutral expectation of short‐dated implied variance on the S&P 500. A standard (spot‐starting) S&P variance swap from 00 to TT can be approximated by integrating VIX2^2 over [0,T][0,T]. Hence, just as a forward‐starting S&P variance over [T1,T2][T_{1},T_{2}] can be replicated by differences of spot‐starting variance swaps in SPX, so too can it be replicated by differences of appropriate positions in VIX‐linked derivatives (VIX futures options), which in turn approximate that same variance exposure.

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1. Quick Review: VIX and S&P 500 Variance

1. VIX2^2 as implied variance
   By construction, the VIX index at time tt, VIXt\mathrm{VIX}_{t}, is (roughly) the annualized risk‐neutral expectation of the 30‑day forward variance of the S&P 500. More precisely (ignoring second‐order terms and convexity nuances): VIXt2  ≈  (annualized)  ×  EQ ⁣[ ⁣∫t t+30dσu2 du  ∣  Ft]. \mathrm{VIX}_{t}^{2} \;\approx\; \text{(annualized)}\;\times\; \mathbb{E}^{\mathbb{Q}}\!\Bigl[\!\int_{t}^{\,t+30\text{d}} \sigma_{u}^{2}\,du \;\Big|\;\mathcal{F}_{t}\Bigr]. While the exact CBOE formula uses a strip of SPX options to back out this 30‑day implied volatility, conceptually VIXt2\mathrm{VIX}_{t}^{2} is “the market’s short‐term implied variance” from tt to t+1t+1 month.
2. Integrated VIX2^2 and realized variance
   Over a longer horizon, one can approximate the S&P’s realized variance from 00 to TT by integrating the short‐dated implied variance term structure. Symbolically, ∫0Tσu2 du  ≈  ∫0TVIXu2 (time fraction) du,\int_{0}^{T} \sigma_{u}^{2}\,du \;\approx\; \int_{0}^{T} \mathrm{VIX}_{u}^{2}\,\bigl(\text{time fraction}\bigr)\,du, plus adjustments for the fact that VIX is always a “30‑day look‐ahead.” In practice, traders piece together a rolling series of short‐maturity VIX quotes/futures to approximate the path of implied variance.
3. VIX Futures and VIX Futures Options
   • A VIX future expiring at TT settles to VIXT\mathrm{VIX}_{T}, i.e.\ the VIX index value on the expiration date TT.
   • A VIX futures option (call or put) on that contract will have a payoff that depends on VIXT\mathrm{VIX}_{T}.
   • One can construct a “variance‐type” exposure to VIXT2\mathrm{VIX}_{T}^{2} by holding a continuum (or strip) of out‐of‐the‐money calls and puts on that VIX future, much like how one constructs a log‐contract on an underlying by a continuum of OTM options.

Hence, in the same way that a spot‐starting S&P variance swap can be statically replicated by SPX options across strikes, it can also be approximated (less perfectly, but still quite closely over short time windows) by a dynamic or static strategy in VIX futures and VIX options—because VIX2^2 is directly tied to the S&P’s implied variance.

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2. Forward‐Start S&P Variance From T1T_{1} to T2T_{2}

2.1 “Difference of Two Spot Variances” — the Usual Decomposition

Recall the key relationship in variance‐swap space: Realized Var over [T1,T2]  =  ∫0T2σu2 du⏟Var over [0,T2]  −  ∫0T1σu2 du⏟Var over [0,T1] .\text{Realized Var over }[T_{1},T_{2}] \;=\; \underbrace{\int_{0}^{T_{2}} \sigma_{u}^{2}\,du}_{\text{Var over }[0,T_{2}]} \;-\; \underbrace{\int_{0}^{T_{1}} \sigma_{u}^{2}\,du}_{\text{Var over }[0,T_{1}]}\,.

In expectation (risk‐neutral), dividing by (T2−T1)(T_{2} – T_{1}) yields the forward‐start variance swap rate: KfwdVar(T1,T2)  =  T2 Kvar(0,T2)  −  T1 Kvar(0,T1)T2−T1,K_{\mathrm{fwdVar}}(T_{1}, T_{2}) \;=\; \frac{ T_{2}\,K_{\mathrm{var}}(0, T_{2}) \;-\; T_{1}\,K_{\mathrm{var}}(0, T_{1}) }{ T_{2} – T_{1} },

where Kvar(0,T)K_{\mathrm{var}}(0, T) is the fair strike of an S&P variance swap from now until TT.

2.2 Expressing S&P Var in Terms of VIX2^2

The spot‐starting S&P var swap on [0,T][0,T] can be approximated by integrating the instantaneous forward variance. Over short sub‐intervals, that forward variance is well‐proxied by VIXu2\mathrm{VIX}_{u}^{2}. Symbolically: Kvar(0,T)  ≈  1T∫0TVIXu2 du,K_{\mathrm{var}}(0, T) \;\approx\; \frac{1}{T} \int_{0}^{T} \mathrm{VIX}_{u}^{2}\,du,

ignoring the difference between the 30‑day implied window and instantaneous variance. Thus, ∫0Tσu2 du  ≈  ∫0TVIXu2 du.\int_{0}^{T} \sigma_{u}^{2}\,du \;\approx\; \int_{0}^{T} \mathrm{VIX}_{u}^{2}\,du.

Hence, the forward‐start variance on [T1,T2][T_{1},T_{2}] is (approximately) the difference: ∫0T2VIXu2 du⏟Var [0,T2]  −  ∫0T1VIXu2 du⏟Var [0,T1]  =  ∫T1T2VIXu2 du.\underbrace{\int_{0}^{T_{2}} \mathrm{VIX}_{u}^{2}\,du}_{\text{Var}\,[0,T_{2}]} \;-\; \underbrace{\int_{0}^{T_{1}} \mathrm{VIX}_{u}^{2}\,du}_{\text{Var}\,[0,T_{1}]} \;=\; \int_{T_{1}}^{T_{2}} \mathrm{VIX}_{u}^{2}\,du.

At a high level, this is the payoff you want to replicate in order to capture the realized variance of SPX from T1T_{1} to T2T_{2}.

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3. Replicating via VIX Futures Options

3.1 Replicating (Spot) ∫0TVIXu2 du\int_{0}^{T} \mathrm{VIX}_{u}^{2}\,du

If you wanted to replicate a standard (spot‐starting) S&P variance swap from 00 to TT using VIX futures & options, the approximate logic is:

1. Partition the interval [0,T][0,T] into small subintervals Δt\Delta t.
2. At each sub‐date tkt_{k}, hold exposure to VIXtk2\mathrm{VIX}_{t_{k}}^{2} for the next Δt\Delta t. Because the VIX2^2 is the short‐term implied variance, it proxies the actual realized variance over that small window.
3. Implementation: a position in VIX futures options that replicates the payoff (VIXtk2−something)×Δt(\mathrm{VIX}_{t_{k}}^{2} – \text{something})\times \Delta t. In practice, you might “roll” a short‐dated VIX options position every month, because each monthly VIX future’s final settlement is VIXexpiration≈implied var over next 30 days\mathrm{VIX}_{\text{expiration}} \approx \sqrt{\text{implied var over next 30 days}}.
4. Sum (integrate) over kk. Summation of these monthly pieces approximates ∫0TVIXu2 du\int_{0}^{T} \mathrm{VIX}_{u}^{2}\,du.

This is not a perfect 1–1 mechanical replication (the direct replication in the SPX option market is more precise), but it is commonly used by volatility desks to get “variance‐type” exposure via VIX derivatives. In effect, the net P/L from rolling short‐dated VIX trades tracks the realized variance of SPX over [0,T][0,T].

3.2 “Carving Out” the Forward Interval [T1,T2][T_{1},T_{2}]

Once you accept that each “spot‐starting S&P variance” on [0,T][0,T] can be approximated by trading a (rolling) strip of VIX futures options out to TT, it follows exactly the same difference argument as in the SPX‐option world:

1. VarSwap(0,T2)(0,T_{2}): replicate it in the VIX‐options market by “rolling” your VIX exposure from 00 out to T2T_{2}.
2. VarSwap(0,T1)(0,T_{1}): replicate it similarly, rolling from 00 out to T1T_{1}.

By going long the first and short the second (in the correct notional), you remove any net variance exposure from [0,T1][0,T_{1}]. What remains is effectively the S&P realized variance over [T1,T2][T_{1},T_{2}]. Thus, your net position in VIX‐related instruments (the difference of those two rolling exposures) is an approximation to the forward‐starting variance swap on the S&P over [T1,T2][T_{1},T_{2}].

Key point: Instead of booking two S&P variance swaps in the SPX market, you book two rolling VIX variance exposures (one for [0,T2][0,T_{2}] and one for [0,T1][0,T_{1}] with opposite sign) in the VIX futures‐and‐options market. The net effect “carves out” the forward interval [T1,T2][T_{1},T_{2}].

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4. Mathematical “Replication” Sketch with VIX Options

To be more explicit about the option‐based replication on VIX:

1. Payoff for a standard S&P var swap (from 0 to TT): ∫0T(dln⁡St)2⏟≈∫0Tσt2 dt  −  Kvar(0,T)×T.\underbrace{\int_{0}^{T} (d \ln S_{t})^{2}}_{\approx \int_{0}^{T} \sigma_{t}^{2}\,dt} \;-\; K_{\mathrm{var}}(0,T) \times T. We want to replicate ∫0Tσt2 dt\int_{0}^{T} \sigma_{t}^{2}\,dt.
2. Approximate with VIX2^2: ∫0Tσt2 dt  ≈  ∫0TVIXt2 dt.\int_{0}^{T} \sigma_{t}^{2}\,dt \;\approx\; \int_{0}^{T} \mathrm{VIX}_{t}^{2}\,dt.
3. Construct a “VIX2\mathrm{VIX}^{2} payoff” each sub‐period with VIX futures options.
   • A continuum of OTM VIX calls and puts on the near‐maturity VIX future can produce a terminal payoff VIXtk2\mathrm{VIX}_{t_{k}}^{2} (modulo some small adjustments).
   • Roll that position monthly (or, in discrete steps) from t=0t=0 to t=Tt=T.
4. Net the integrated cost over [0,T][0,T]. That total is your “synthetic” or “replicated” ∫0TVIXt2 dt\int_{0}^{T} \mathrm{VIX}_{t}^{2}\,dt.

Because VIXt2\mathrm{VIX}_{t}^{2} is the “short‐term implied variance,” the actual P/L from that strategy tracks the realized S&P variance. Thus, you have effectively built a variance swap on the S&P out to TT from VIX derivatives.

Forward Start via Two Such Positions

To isolate the interval [T1,T2][T_{1},T_{2}], simply do:

• Long the above “VIX variance replication” from 00 to T2T_{2}.
• Short the same replication from 00 to T1T_{1}.

Everything from [0, T1][0,\,T_{1}] cancels out (long–short). You are left with a net position that only has exposure to integrated variance in [T1,T2][T_{1},T_{2}]. In other words, you have replicated a forward‐start S&P variance swap purely in the VIX derivatives market.

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5. Caveats and Practical Points

1. Exact vs. Approximate
   • In the SPX‐option replication, the payoff is exactly pinned to realized variance on the S&P.
   • In the VIX‐option approach, you rely on the approximation σt2≈VIXt2\sigma_{t}^{2} \approx \mathrm{VIX}_{t}^{2}. Over short windows (30 days) it is typically decent, but it is not a perfect identity (term‐structure “slopes,” jumps, skew dynamics, etc.).
2. 30‐day “look‐ahead”
   • The VIX is a 30‐day implied vol measure. If your forward start from T1T_{1} to T2T_{2} is, say, 6 months wide, you need a rolling approach (each month you use that month’s VIX contract). This is standard practice in “volatility‐of‐volatility” or “vol carry” trading strategies.
3. Liquidity
   • SPX index options remain the primary direct route to trade large notional variance.
   • VIX options/futures can be more liquid at times on certain maturities, but large lumps of forward variance may be more easily sourced directly in the SPX “variance swap” or “vol swap” OTC market.

Nonetheless, from a theoretical perspective, the same forward‐start payoff on S&P realized variance over [T1,T2][T_{1},T_{2}] can indeed be replicated by carefully combining VIX futures options with different maturities and rolling them appropriately—mirroring the well‐known decomposition that forward S&P variance is the difference of two (spot‐start) S&P variances, each of which can be approximated by integrating VIX2^2.

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Final Summary

1. Forward‐start S&P variance from T1T_{1} to T2T_{2} is the difference between the spot‐start variances on [0,T2][0,T_{2}] and [0,T1][0,T_{1}].
2. Each spot‐start S&P variance can be approximated by a rolling strategy in short‐dated VIX derivatives, since VIXt2\mathrm{VIX}_{t}^{2} reflects the local implied variance of the S&P 500.
3. Hence, buy the “VIX‐based replication” for [0,T2][0,T_{2}] and short the “VIX‐based replication” for [0,T1][0,T_{1}]. Their difference is effectively the forward variance over [T1,T2][T_{1},T_{2}].
4. In practice, this means trading strips of VIX futures options (calls/puts) at successive maturities, constructing payoffs proportional to VIXt2\mathrm{VIX}_{t}^{2}, and rolling them up to times T1T_{1} and T2T_{2}. The net effect approximates the same payoff one would get from an SPX forward‐start variance swap.

Prompt 3:

Is there an arbitrage possibility between these two replication methods? How would that work?

Below is a conceptual discussion of whether there could be an arbitrage between:

1. Forward‐start SPX variance constructed via SPX options (the “direct” method), and
2. Forward‐start SPX variance approximated via VIX futures & options (the “VIX‐based” method).

We’ll see that, in theory, a mispricing between these two methods could be exploited in a risk‐free way; however, in practice, there are important nuances (approximation errors, basis risk, liquidity frictions) that usually prevent a pure, no‐risk arbitrage. Instead, such misalignments often become “relative value” trades or “basis” trades rather than pure, certain‐profit arbitrages.

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1. The Basic Arbitrage Argument (Idealized)

1.1 If They Truly Replicate the Same Payoff…

Both strategies intend to replicate the same final payoff: ∫T1T2(dln⁡St)2(the realized variance of SPX over [T1,T2]).\int_{T_{1}}^{T_{2}} (\mathrm{d}\ln S_{t})^{2} \quad \Bigl(\text{the realized variance of SPX over }[T_{1},T_{2}]\Bigr).

• Method A (direct SPX): Long (spot) VarSwap(0, T2T_{2}) and short VarSwap(0, T1T_{1}) with correct notional.
• Method B (VIX‐based): Rolling positions in VIX futures & VIX options to produce the same realized‐variance payoff, again in difference form for [T1,T2][T_{1},T_{2}].

If both methods perfectly replicate the same final cash flow—no path‐dependence mismatch, no date mismatch, etc.—then they should, in a frictionless market, cost the same at inception. If the two are priced differently, one could buy the cheaper replication and sell (short) the more expensive replication, in principle locking in a riskless profit. That position would have:

• Zero net cost (or net credit) at inception if you do it at the correct ratio.
• Exactly zero net payoff at expiration if, in fact, they replicate the same payoff but with opposite sign.

The result is an arbitrage if that payoff is guaranteed to be zero (and the upfront net is non‐negative).

1.2 How the Arbitrage Would Be Done

Suppose you observe that the “VIX‐based forward var” is priced below the direct SPX forward var for the same interval [T1,T2][T_{1},T_{2}]. Then the arbitrage strategy would be:

1. Buy the cheap version (the VIX‐based forward var replication).
2. Sell the expensive version (the direct SPX forward‐start variance swap).

If indeed they converge to the same final variance payoff, you have locked in a guaranteed spread. The payoff difference at maturity (assuming perfect replication) would be zero, so you face no net risk. If you received net premium up front, that’s a pure arbitrage profit.

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2. The Practical Reality: Basis Risk & Approximations

In practice, however, there is almost never a true, guaranteed “locked‐in” arbitrage because:

1. VIX2^2 vs. Instantaneous SPX Variance
   • The “VIX‐based” replication relies on VIXt2\mathrm{VIX}_{t}^{2} (a 30‐day implied measure) to track the instantaneous variance σt2\sigma_{t}^{2} of SPX. They do not move identically; there is a term‐structure effect, jumps, skew changes, etc.
   • As a result, the VIX strategy only approximately replicates the realized variance payoff of SPX. There is basis risk between actual realized variance of SPX and the path integral of short‐dated implied variance (VIX).
2. Discrete Rolling vs. Continuous
   • The VIX approach is typically implemented by rolling short‐maturity VIX futures/options each month. That introduces roll timing and any “gap” between monthly resets can lead to differences versus actual daily path variance of the S&P 500.
3. Liquidity and Market Frictions
   • SPX options and VIX futures/options can have different liquidity conditions, margin requirements, or time‐zone coverage.
   • Bid/ask spreads, collateral costs, capital charges, and the fact that large notional trades move markets can all eat into any “arbitrage” spread.
4. Model Dependence
   • Even if the payoffs line up in a “theoretical” sense, certain dynamic hedging steps for the VIX approach might not be feasible in continuous time or might incur large transaction costs.

In other words, the forward‐start variance you get from VIX derivatives is not 100% identical to the forward‐start variance you get from SPX options. It’s close, but it’s not a perfect replication. Hence, any difference in pricing between these two “replications” will generally reflect:

• The “convenience premium” (or discount) in one market.
• The risk that the two payoffs won’t line up perfectly in stressed conditions.
• Supply/demand imbalances or hedging constraints in the VIX vs. SPX markets.

Such differences are more akin to a relative‐value or basis trade rather than a pure arbitrage.

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3. Relative‐Value Trades in Practice

Because of these mismatches, desks do sometimes trade forward variance on SPX vs. a portfolio of VIX‐linked instruments, aiming to capture any mispricing. However, they treat it as a basis or RV (relative‐value) trade rather than a guaranteed free lunch. The reason is:

• They expect that as time evolves, the difference between realized SPX variance and the integral of VIX2^2 (and the embedded forward intervals) will remain small or mean‐revert.
• They demand a certain “spread” or “edge” in the initial pricing to compensate for the risk that a shock in the volatility term structure could cause realized variance on SPX to diverge from what was implied by rolling short‐dated VIX futures.

If all goes well, the spreads converge, and the trade yields a profit. If the mismatch widens or a volatility event breaks the relationship, it can incur losses.

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4. Final Answer

• Yes, in an ideal, frictionless world, if both methods truly replicated exactly the same final payoff (the same forward‐start SPX realized variance) yet were priced differently, there would be a risk‐free arbitrage by going long the cheaper replication and short the more expensive one.
• In real markets, the VIX‐based forward variance replication is not a perfect, one‐for‐one match for SPX realized variance. It is an approximation, subject to term‐structure and rolling issues, basis risk, and market frictions. Therefore, any pricing difference between the two is usually not a pure arbitrage but rather a source of relative‐value or basis trading opportunities. Traders can attempt to capture it, but it carries risk that the two payoffs will not perfectly offset under all market conditions.

Hence, you can think of it this way:

• If you believe the VIX replication and the SPX forward‐start variance payoff are “close enough,” and you see a big pricing discrepancy, you might do that trade.
• But you are exposed to the “basis risk” that the relationship breaks down (VIX2^2 vs. realized var diverges).
• Thus it’s not a pure arbitrage in the strict sense—it is more a relative‐value proposition with potential correlation or basis risk.