Different types of swaps and generalized pricing structure - correlation swap, variance swap, volatility swap, gamma swap, etc

Question

I am very new to derivatives pricing, and I am currently trying to learn these on my own.

As far as I can tell, most of the derivatives that are simple (in the sense of having a constant strike that is linear with respect to the underlying in its relation), the options may be priced as:

$$ (S_T - K)^+ $$

for a call option, and the other way around for a put option $(K - S_T)^+$, to put if very simply (in the case of European style options).

Now, the swaps (in a general, conceptual way) is similar, in the sense that there is a strike and an underlying that determines the payoff. Now, on top of this, swaps tend to have the payoff of

$$ N (S - K) $$

where $N$ is the notional. Now, this means that for the swaps such as variance swap, volatility swap, and correlation swap, the following holds:

variance swap: $$ N_{\text{var}} (\sigma^2_{\text{realized}} - \sigma^2_{K}) $$

volatility swap: $$ N_{\text{vol}} (\sigma_{\text{realized}} - \sigma_{K}) $$

correlation swap: $$ N_{\text{corr}} (\rho_{\text{realized}} - \rho_{K}) $$

interest rate swap: $$ N (r_{\text{fixed}} - r_{\text{float}}) $$

where the $K$ in each swap indicates the corresponding strike asset (variance/volatility/correlation/etc.). This holds in general for most swap products, or even for each subsection or leg of a credit default swap as well.

Now, I would like to know if there is some type of overarching theory or concept that can be derived for swap-like derivatives, and if there is a similar plug-and-chug parametric formulae for swaps in general (as there is with options via the Black-Scholes). This does not seem to be the case, since pricing swaps seems to be dependent on the underlying (correlation, volatility, variance, interest rate differential, etc.).

Are there some heuristic approaches that do provide an overarching framework for pricing swaps, or does this have to be done on a case-by-case basis depending on the underlying?

Answer 1

For Variance Swaps (and Vol swaps with some caveats), the Black Scholes model is the main tool used for pricing. It is just less obvious.

Using your example, options are not priced with S-K or K-S either. That is simply an algebraic expression of parts of the contract. Pricing involves finding a value for this.

There is no assumption on pricing configuration or even numeraire. Therefore, you cannot simply use Black Scholes. It depends largely on the underlying how you price this. Is it equity, an index, a commodity (mostly futures), an exchange rate, a bond? Or whatever other underlying you can think of. Does the stock pay dividends? In the case of FX, what currency is your notional and premium in (Garman Kohlhagen assumes notional in ccy1 and premium in ccy2, everything else requires (simple) adjustments to the formula).

Just like options where you have the underlying price versus a strike, all swaps share a common payoff, they exchange cashflows (side note, a CDS Swap is a bit of a misnomer as they are really options from a pricing perspective, with upfront payments).

Generally, with swaps, one distinguishes pricing from valuation:

• pricing involves determining the appropriate price (or rate) when initiating the contract which makes the swap typically zero cost at initiation
• valuation involves determining the appropriate value of the commitment (typically after it has been initiated)

For vanilla fixed float IRS swaps, the par swap rate is the coupon of an interest rate swap that makes the market value of the swap equal to zero (the fixed rate that makes the value of the fixed leg equal to the value of the floating leg).

For variance swaps, the fair rate is such that the contract is also zero at the initiation.

That is all there is in common though.

IRS require interest rate curves built from instruments (cash, FRAs or futures and swaps) to correctly price and value them. Essentially you need discount factors and forward rates, but the actual process of curve building is very involved, and nowadays involve usually multiple curves being stripped simultaneously. Curve selection and the like make this almost more art than science.

Variance swaps have a theoretical replication. A vanilla option trader following a delta-hedging strategy is essentially replicating the payoff of a weighted variance swap where the daily squared returns are weighted by the option’s dollar gamma. Taking this argument one step further, a fair variance swap can be shown to equal the integral of weighted prices of out-of-the-money options over all strikes. These weights are being inversely proportional to squared strikes, an application of the Black Scholes closed-form formula for gamma.

One obvious problem here is that options markets are composed of a discrete set of option prices for a given maturity. Therefore, it is common to first compute a vol surface, usually using Black Scholes again (ignoring complications involved in creating vol surfaces, like de-Americanizing option prices, finding implied forwards and dividends and the like if we think of index or equity VS, FX, for example, is generally quoted in vol which makes surface construction easier). Practically, you may also want to limit the integration region (strike range) to avoid issues with the weights (especially very small strikes are a concern due to the weighting).

Due to practical difficulties in replicating the actual log payout across strikes, the market for equity index Var swaps usually trades at a basis to the replicating portfolio.

For Vol Swaps, things are a bit messier. Simplified, a Vol swap is a Var swap - convexity adjusted and the convexity adjustment can be replicated with a portfolio of options on var. So, you essentially have 2 replicating portfolios.

There are two documents from JP Morgan Variance Swaps and Just what you need to know about Variance Swaps. On a side remark, the way delta and gamma are defined here is a simplification. It will only work intraday. Ideally, the Greeks are directly derived from the replicating portfolio. However, such a full decomposition is not something (many) vendors offer, and mainly tier 1 banks have implemented. Towards a Theory of Volatility Trading by Peter Carr et al. is probably the best paper to read.

An intuitive graphical representation of a variance swap computation can be found here.

Correlation swaps are a beast of their own. I cannot comment much on them as it's beyond my knowledge. They are frequently priced with MC based on LV or SLV but neither will price them properly for numerous reasons. There are some models like Local Vol Local Correlation (LVLC) that may be a bit better but at the end of the day, these are very exotic. Since you wrote you are very new to derivatives pricing I would avoid looking into them as long as possible. Chances are you will never need to know what goes on here (unless you are a (physics) PhD specifically hired as a quant for these products). There is a good tweet I stumbled upon some time ago. Many people tend to think if a tool gives a price it works (here Monte Carlo Local Vol). However, that is classic GIGO. MCLV will not even price a vanilla VS properly (calibration is never perfect; MC time steps are limited...).

Answer 2

I think the $^+$ was just a typo. Nice question! I'll try to make this point in the case of interest-rates, but the argument is general.

To some extent it’s case by case, but the general feature of a swap is to be fair - that is, worth $0$ - at inception, say $t=0$. This can be achieved setting to an appropriate value the fixed rate/vol/var etc..

Now, prices are (risk-neutral) expectations, therefore the fixed part need to be set to an expectation of the floating part for the swap to be fair at inception.

The simplest case is probably the case of a FRA (an interest-rate swap with a single payment if you want) which pays the spot rate for $S$ read in $T$, say $r_{floating}(T,S)$. The expectation is under the forward measure $Q^{S}$ associated with the zcb $P(t,S)$ as numeraire (I'll use everywhere notation from Brigo-Mercurio book). Such that, for the FRA to be fair at inception

$$ V^{FRA}(0)=0 \iff r_{fixed} = {\mathbb E}^{Q^{S}}[r_{floating}(T,S) | {\mathbb F}_0] = {\mathbb E}^{Q^{S}}[F(T; T,S) | {\mathbb F}_0] = F(0; T,S) $$

where $F(t; T,S)$ is the forward rate for period $[T,S]$. Now, what if you were long this FRA (that is, you were committed to pay $r_{fixed})$ and want to close this position at time $t>0$? Well, easy, you enter in another FRA in which you commit to pay the floating rate. This second FRA will be fair as well, thus it shall have a $r_{fixed}$ set to the current forward rate: $F(t; T,S)$. The resulting $P\&L$ at $t$ from this strategy coincides with the value of the long position of the original FRA (it's not hard to understand why, see Hull's book for example):

$$ V^{FRA}(t)= N P(t,S) \tau(T,S) (F(t; T,S) - F(0; T,S)) $$

that is, being long a FRA means being long the corresponding forward rate.

The argument can be extended to a fixed-for-floating interest rate swap (IRS), with first reset date $T_{\alpha}$ and last payment date $T_{\beta}$. Let

$$ C_{\alpha, \beta}(t) = \sum_{i=\alpha+1}^{\beta} \tau_i P(t, T_i) $$

be the portfolio of zero coupon bonds and let

$$ S_{\alpha, \beta}(t) = \frac{P(t,T_{\alpha}) - P(t, T_{\beta})}{C_{\alpha, \beta}(t) } $$

be the swap rate. Then (e.g. evaluating the IRS as a portfolio of FRA), the IRS is fair at inception as long as its fixed rate is set to

$$ V^{IRS}(0)=0 \iff r_{fixed} = S_{\alpha, \beta}(0) $$

and we have the analogy (without math, think which $P\&L$ would you realize if you were in a swap paying the fixed $S_{\alpha, \beta}(0)$ and close the position at time $t>0$...)

$$ V^{IRS}(t)= N C_{\alpha, \beta}(t) (S_{\alpha, \beta}(t) - S_{\alpha, \beta}(0)) $$

Again, you can see that paying the fixed rate means being long a particular forward rate, the forward swap rate $S_{\alpha, \beta}(t)$.

This

pay fixed $\iff $ be long the forward swap rate

is another common feature of swaps, deduced by the fairness at inception.