Forward implied volatility

Question

Can one price accurately by only using vanilla options a derivative that is exposed/sensitive mainly to the forward volatility ?

If it is impossible, why do we hear sometimes "being long a long dated straddle and short a short dated straddle" is being exposed to forward vol ?

Here are some examples :

a) In equity markets :

        - pricing a volatility swap starting in 1y and expiring 1y later.

        - pricing a forward starting option with the strike determined in 1y as 100% of the spot and expiring in 5y.

b) In rates markets : (FVA swaption) a 1y5y5y Swaption, which is 6y5y swaption with the strike determined in 1y.

In the equity world, a way to express the question is : If we use a sufficiently rich model like Stochastic Local Volatility model (SLV) where the local component of the model is calibrated on vanillas (hence the price of any vanillas will be unique regardless of the choice of the stochastic part). Would our model provide a unique price of the above instruments regardless of the stochastic component choice ?

Answer 1

From an equities perspective, there are two concepts that should not be confused in my opinion and context should make the distinction self-explicit:

• Forward variance swap volatility (A)
• Forward implied volatility smile (B)

I really recommend reading Bergomi's "Stochastic Volatility Modeling" which is an excellent book for equity practitioners. The topics you mentioned are discussed in a great amount of details.

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To give more of a theoretical insight, in what follows I'll assume that the underlying follows a pure diffusion process (i.e. no jumps). I'll also consider two types of instruments:

1. Idealised variance swaps that payout at maturity $$ \phi_{VS}(T) = A \,\,\underbrace{\frac{1}{N-1}\sum_{i=1}^N \ln\left(\frac{S_{t_i}}{S_{t_{i-1}}}\right)^2}_{\text{realised $\delta t = T/N$ returns variance}} - \underbrace{\hat{\sigma}^2}_{\text{VS variance}}$$ where $t_0 = 0 < \dots < t_i < \dots < t_N = T$ represents a partition of the horizon $[0,T]$, $A = N/T$ is an annualisation factor and where by idealised I mean that $N \to \infty$ such that the realised returns variance over the horizon may be replaced by the quadratic variation of the log price process over $[0,T]$ such that $$ \phi_{VS}(T) = \frac{1}{T}\langle \ln S \rangle_T - \hat{\sigma}^2_T $$ Since the variance swap is entered at zero cost at inception we have that \begin{align} \hat{\sigma}^2_T &= \frac{1}{T} \Bbb{E}_0^\Bbb{Q} \left[ \langle \ln S \rangle_T \right] \\ &= -\frac{2}{T} \Bbb{E}_0^\Bbb{Q} \left[ \ln\left( \frac{S_T}{F_T} \right) \right] \tag{1} \end{align} see for instance this excellent answer by Gordon.

2. Forward start options, e.g. a forward start call paying out: $$ \phi_{FS}(T=T_2) = \left( \frac{S_{T_2}/F_{T_2}}{S_{T_1}/F_{T_1}} - k \right)^+ $$ for a forward forward starting date $T_1$ and tenor $\tau = T_2-T_1$ with fixed moneyness $k$. To make my point, I'll further consider a homogeneous diffusion model such that we can successively write the price of the forward start option as \begin{align} V(k,T_1,T_2) &= \Bbb{E}_0^\Bbb{Q}\left[ \left( \frac{S_{T_2}/F_{T_2}}{S_{T_1}/F_{T_1}} - k \right)^+ \right] \\ &= \Bbb{E}_0^\Bbb{Q}\left[ \Bbb{E}_1^\Bbb{Q}\left[ \left( \frac{S_{T_2}/F_{T_2}}{S_{T_1}/F_{T_1}} - k \right)^+ \right] \right] \\ &= \Bbb{E}_0^\Bbb{Q}\left[ \frac{F_{T_1}}{S_{T_1} F_{T_2}} C\left( S_{T_1}, K=k S_{T_1} \frac{F_{T_2}}{F_{T_1}} , \tau=T_2-T_1; \hat{\sigma}^{T_1T_2}_k \right) \right] \\ &= \frac{F_{T_1}}{F_{T_2}} \Bbb{E}_0^\Bbb{Q}\left[ C\left(1, K=k \frac{F_{T_2}}{F_{T_1}} , \tau=T_2-T_1; \hat{\sigma}^{T_1 T_2}_k \right) \right] \tag{2} \end{align} where we have defined $\hat{\sigma}^{T_1T_2}_k$ as the future implied volatility at $T_1$ of options of tenors $T_2-T_1$ and moneyness $k$.

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As you might have guessed, the two "forward volatility" concepts (A) and (B) I've introduced in the beginning are related to the two former instruments as follows:

1. Forward VS volatility as seen of $T_1$ for the tenor $T_2-T_1$, which I will denote by $\hat{\sigma}_{T_1 T_2}$, can be defined as $$ \hat{\sigma}_{T_1 T_2}^2 = \frac{1}{T_2-T_1} \Bbb{E}_0^\Bbb{Q}\left[ \int_{T_1}^{T_2} d\langle \ln S \rangle_t \right] = \frac{T_2 \hat{\sigma}_{T_2}^2 - T_1 \hat{\sigma}_{T_1}^2}{T_2-T_1} $$ As you can see this amounts to trading in a calendar spread of fresh-start VS of maturities $(T_1,T_2)$. As per formula $(1)$, the price of each of these fresh-start VS only depends on the unconditional distribution of $S_t$ at $t = T_1$ and $t=T_2$. By the Breeden-Litzenberger identity this means that as soon as you know (or have a model calibrated to perfectly match) the prices of all vanillas at maturities $T_1$ and $T_2$ then forward VS will be priced unequivocally. Consequently, both a LV and a SV model perfectly calibrated to the vanilla market will yield the same prices for these instruments under the modelling assumptions I've made (pure diffusion + idealised variance swaps).

2. As per formula $(2)$ you see that for a forward start option, the real underlying of the option is not 'the stock' itself but rather the future implied volatility $\sigma_k^{T_1 T_2}$, an information which is simply not encoded in a European vanilla option. As such, forward implied volatilities $\sigma_k^{T_1 T_2}$ cannot in general be determined from the now-prevailing vanilla smile. The dynamics of $\sigma_k^{T_1 T_2}$ is rather "embedded" in the model (or equivalently the assumptions you are willing to use as Attack68 mentions in his answer). This means that a LV and a SV lodel both perfectly calibrated to the vanilla market will in general yield different implied volatility dynamics, hence different forward start option prices.

Some additional remarks:

• In LSV models as you described you'll generally set up the parameters of the SV layer to tune the dynamics of the model (i.e. conditional distributions) and calibrate the LV layer to make sure that the statics of your model is consistent with the now-prevailing smile (i.e. unconditional distributions). At least this is what a tractable LSV model should allow you to do. So the answer is no, if your only target is calibrating your model to the vanilla market then definitely the price of forward start options will not be unique because there are infinite many ways to do this.
• The calendar spread of straddles you mention is similar to the calendar spread of variance swaps in the sense that its price will unequivocally be determined by the unconditional distribution of the underlying (it's a calendar spread of European vanilla instruments). So yes, it gives you exposure to what you could call "forward vol" but it is more akin to a "forward VS vol" above than to a forward start option.

Answer 2

It is possible, yes, but it requires assumptions. But, philosophically speaking, this is the case as with all pricing, of any instrument. For example, given only the price of a 6Y and 7Y IRS can you correctly price the 6.5Y IRS rate? Well, yes you can, but it depends upon your assumptions about interpolation which is a subjective choice.

Lets look specifically at your swaption question:

Can one price the 5Y5Y vol 1Y forward, denoted $\sigma^{5Y5Y\_1Y}$?

Component 1: Forward Volatility

The two components I need to price this forward volatility are:

• The 6Y5Y vol (6y expiry 5y swap),
• The 1Y5Y5Y vol (1y expiry 5Y5Y swap).
• Modelling assumptions.

You now have a framework to equate this mathematically by modelling the assumptions about market movements. If, for example you model with normal distributions of market movements the resultant formula is fairly generic:

$$\sigma^{5Y5Y\_1Y} = \sqrt{\frac{6(\sigma^{6Y5Y})^2-1(\sigma^{1Y5Y5Y})^2}{6-1}}$$

Graphically you have:

+-------------------------+------------------+
|      6Y EXPIRY          |     5Y SWAP      |   =  Benchmark price
+-------------------------+------------------+
| 1Y EXP. |  5Y FWD       |     5Y SWAP      |   =  Composit price
+-------------------------+------------------+
| 1Y FWD  |  5Y EXPIRY    |     5Y SWAP      |   =  Implied, required price.
+-------------------------+------------------+

Component 2: Volatility of a forward, i.e. midcurve

You will observe that the above used the vol info on the 1Y5Y5Y. However, this doesn't exist as a benchmark product. In fact it isn't even a vanilla traded swaption.

To calculate this price you need the information about:

• 1Y5Y vol (1y expiry 5y swap)
• 1Y10Y vol (1y expiry 10y swap)
• the expected correlation between the above rates in the next year.
• some modelling assumptions about change in deltas and discount factors modelled over all scenarios.

Graphically you have:

+-------------------------+
| 1Y EXP. |  5Y SWAP      |                      =  Benchmark price
+-------------------------+------------------+
| 1Y EXP. |         10Y SWAP                 |   =  Benchmark price 
+-------------------------+------------------+
| 1Y EXP. |  5Y FWD       |     5Y SWAP      |   =  Composit Price
+-------------------------+------------------+

The correlation component can sometimes be inferred from exotic swaption markets where curve spread options are priced, eg a call on 5s10s curve for example.

Conclusion

I started this answer with it is possible, yes but in light of the complexity I can see why many people simply say no because the variance of accuracy, subject to all of the model assumptions, leading to weak confidence levels on the price is far from the confidence of pricing a 6.5Y swap from 6Y and 7Y IRS price.

As for trading the risk exposure to specifically this component, I don't know the answer but I seriously doubt it is possible, lest it be very complicated with some mechanical process that is far to expensive constantly hedging the changes in exotic exposures.

References:

This material is better explained and clearer in Darbyshire: Pricing and Trading Interest Rate Derivatives.

Answer 3

The procedure outlined by @attack68 is correct for estimating forward vol assuming you are in a world where volatility is deterministic and uncorrelated with the underlying. If these assumptions are not valid, the situation is more complicated.

Taking his (or her) example, suppose you sell a usd100mm forward Vol contract on a 5yr 5yr swaption straddle, settling in 1yr from now, at a normalized volatility of 70bp per annum. This means in one year you will sell to your client $100mm of a 5yr 5yr swaption straddle struck at the then ATM 5yr 5yr forward rate. As a hedge , you buy usd100 mm of a 6yr 5yr swaption straddle and sell usd100mm of a 1 yr option on a 5yr5yr rate , both struck at today's forward rate (say 3pct ). What happens ? If over the next year , the market migrates a long way from 3pct (say 5pct), your hedge is in fact equal to the value of usd200mm of a then 5yr 5yr 3pct receiver swaption, which doesn't have much vega exposure , certainly much less than the trade you are trying to hedge.

Thus, in order to keep the hedge vega neutral versus the trade , you need to acquire more hedge as the underlying moves away from 3pct. The new hedge depends where implied volatility is on those future dates. Hence you are very dependent on the path of volatility versus rates , and the interrelationship between them.

So the simple forward vol calculation by @attack68 works as a good estimate of the market expectation of forward vol, but it doesn't form a static hedge so it can't be used to actually lock in the forward vol.