User:EdwardLockhart

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1 Implied Trinomial Trees
2 The Local Volatility Formula
3 Local Volatility as a forward-starting conditional corridor variance swap strike
4 ni * (H − L)
5 2 * St − H − L
6 Variance Swap as a strip of Band Crossing Derivatives
7 Forward starting conditional corridor variance swap

[edit] Implied Trinomial Trees

The market implied Black-Scholes volatilities for an underlying often vary across a range of strikes (skew) and term structures. These implied volatilities for a given underlying are known as the implied volatility surface. For options where only the terminal distribution is important this volatility surface tell us all we need to know in order to calculate the price. However this assumes that returns on the underlying evolve with a constant volatility. This can be pictured using a simple trinomial tree

%ATTACHURL%/lowvol.PNG

In continuous terms the evolution is described by the stochastic differential equation

$dS_t = \mu S_t\,dt + \sigma S_t\,dW_t \,$

And so if we price an option on the same underlying but with a different strike and maturity the trinomial tree will evolve with a different volatility

%ATTACHURL%/highvol.PNG

This means that the two trees are inconsistent in the areas that they both cover. For options that aren’t path dependent that doesn’t matter, but what about those that are. In that case we would want to price it in a way that was consistent with both the above trees. In order to do so we have to create an ‘implied tree’. An implied tree will evolve with a varying volatility over the life of the option.

%ATTACHURL%/treewithtime.PNG

and if there is skew present then the evolution will also need to be consistent at various spot levels i.e. the volatility must also vary with spot level %ATTACHURL%/treewithtimeandspot.PNG

The volatility at each point is in effect the ‘local volatility’. In continuous terms the evolution is described by the stochastic differential equation

$dS_t = \mu S_t\,dt + \sigma(S,t) S_t\,dW_t \,$

In many ways this process of using local volatilities can be likened to the process for calculating forward interest rates, we are using the market implied volatilities to calculate the arbitrage free value for volatility at some point in the future in the same way that we can calculate forward interest rates in such a way as to make them arbitrage free. So in the same way that you can lock in forward rates by buying a long dated bond and selling a short dated one you can lock in forward (local) volatility by buying a calendar spread and selling butterfly spreads with a zero net cost.

N.B. It should be noted that in practice it is usual to change the vol of the evolution by keeping the same grid points and changing the weights rather than changing the points and keeping the weights constant as shown above.

[edit] The Local Volatility Formula

The Local volatility formula in the absence of dividends and rates illustrates this point

$\sigma(K,t)^2 = \frac{\frac{\partial C}{\partial T}}{\frac{1}{2}K^2\frac{\partial ^2 C}{\partial K^2}}$

However this is not stable at low or high strikes but since in the Black-Scholes world we can think of option prices as implied volatilities this equation can be transformed into a more stable form using implied volatilities

$\sigma(K,t)^2 = \frac{\sigma^2 + 2\sigma(T-t)\frac{\partial \sigma}{\partial T} + 2r \sigma K(T-t)\frac{\partial \sigma}{\partial K}}{(1 + K d_1 \sqrt{T-t}\frac{\partial \sigma}{\partial K})^2 + \sigma(T-t)K^2(\frac{\partial^2 \sigma}{\partial K^2} - d_1 (\frac{\partial \sigma}{\partial K})^2\sqrt{T-t})}$

Where

$d_1 = \frac{\ln(S/K) + (r + \sigma^2/2)(T-t)}{\sigma\sqrt{T-t}}$

[edit] Local Volatility as a forward-starting conditional corridor variance swap strike

Imagine a ‘Band Crossing Derivative’ (BCD) which defines two levels, let’s call them H(igh) and L(ow). Assume that the underlying asset we write the BCD on starts below the level of L and that for every complete crossing of the band (from L or below to H or above and from above H or above to L or below) we pay (H-L) i.e. the value of the width of the band.

The payoff of the BCD is therefore

[edit] $n i * (H - L)$

where $n i$ is the number of times the underlying asset crosses the band during the term of the BCD.

For the sake of simplicity we will also assume there are no dividends or carry on the underlying and that interest rates are zero. We will also assume that the underlying asset trades continuously and without jumps.

One possible way to hedge this might be to sell to shares of underlying asset when it reaches H and to buy them back when it reaches L. If we do this and the underlying asset has crossed the band an even number of times when the BCD expires then we will have replicated the payoff exactly (making no profit or loss). However if the underlying asset has crossed the band an odd number of times at expiry we sell out of our short position at whatever level the underlying asset is at when the BCD expires (which must by definition be greater than L) we have a difference between the profit or loss made on our hedge and the payoff of the BCD. If the underlying asset finishes in the bottom half of the band then we may make a small profit on our hedge strategy (which will always be less than the width of the band (H-L)). If the underlying asset finishes in the top half of the band or above H then we will have made a loss on our hedge strategy (this loss will be smaller than the width of the band (H-L) if the underlying asset finishes in the top half of the band and the loss will be greater than the width of the band (H-L) if the underlying asset finishes above H).

The difference in the case of an odd number of crossings is given by

[edit] $2 * S t - H - L$

where $S t$ is the level of the underlying asset at the time of expiry of the BCD.

If we wanted to protect against the loss we could buy 2 calls struck in the middle of the band. These would protect us in the case of an odd number of crossings at expiry of the BCD and the underlying asset finishing in the top half of the band or above the band. In the case of the underlying asset finishing in the bottom half of the band we will still make a small profit (less than the width of the band) from our position in the underlying asset and the call options will finish out of the money. We should note that this profit tends to zero as the width of the band tends to zero.

Since we can do this hedge whether we are long or short the BCD this hedge provides us with an arbitrage limit to the value of the BCD, the limit value being that of two calls struck in the middle of the band.

By similar arguments it can be shown that the arbitrage limit value of a BCD where the initial level of the underlying asset is above the band is that of two puts struck in the middle of the band.

[edit] Variance Swap as a strip of Band Crossing Derivatives

A variance swap pays off according to the realised log variance of the underlying asset during the term of the swap. If we note that the expectation of the variance is unaffected by the frequency of the observations if the frequency of the observations taken is sufficiently high and that it is still unaffected if we choose the observation times to correspond to underlying asset levels rather than calendar dates, then we can see that the payoff of the swap is

$\sum_{bands} (ln(\frac{K_{i+1}}{K_i}))^2 * n_i$

Where $n i$ is the number of times the band is crossed and the bands represent the observation levels chosen.

As we have seen the arbitrage limit value of a band crossing derivative is 2 calls struck in the middle of the band if the band is above the initial underlying level and 2 puts struck in the middle of the band if the band is beneath the initial underlying level. Recalling the payoff of the BCD above, if we split the bands of this variance swap into ‘upside’ bands (above the initial level of the underlying) and ‘downside’ bands (below the initial level of the underlying asset) then we can re-write the equation above as

$\sum_{upsidebands} (ln(\frac{K_{i+1}}{K_i}))^2. \frac{2Call((K_i + K_{i+1})/2)}{K_{i+1} - K_i}+\sum_{downsidebands} (ln(\frac{K_{i+1}}{K_i}))^2. \frac{2Put((K_i + K_{i+1})/2)}{K_{i+1} - K_i}$

If we now assume that the observation levels are chosen to be equally spaced then we can re-write this as

$\sum_{upsidebands} (ln(\frac{K_i + \Delta K}{K_i}))^2 .\frac{2Call(K_i + \Delta K/2)}{\Delta K} + \sum_{downsidebands} (ln(\frac{K_i + \Delta K}{K_i}))^2 .\frac{2Put(K_i + \Delta K/2)}{\Delta K}$

Assuming the observation levels are close enough together we can approximate this as

$\sum_{upsidebands} 2\frac{\Delta K}{K_i ^2}Call(K_i) + \sum_{downsidebands} 2\frac{\Delta K}{K_i ^2}Put(K_i)$

This is the variance swap hedge portfolio.

[edit] Forward starting conditional corridor variance swap

Imagine a variance swap in which only variance from within a set band of underlying asset levels counts towards the payoff (a corridor variance swap) and where the fixed leg also only accrues when the underlying asset level is within the band (a conditional corridor variance swap).

For a forward starting conditional corridor variance swap with a narrow band centred on K, the expected value of the floating leg is the calendar spread of the term variances

$2\frac{\Delta K}{K^2}(Call(K, T + \Delta T)-Call(K, T))$

As $Δ T$ tends to zero this becomes

$2\frac{\Delta K}{K^2}\frac{\partial Call}{\partial T}\Delta T$

For the fixed leg we know that the expected value will be the strike multiplied by the probability of being in the band multiplied by the $Δ T$ . If we assume $Δ T$ is suitably small and that $Δ K$ is also suitably small then the probability of being in the band can be approximated by the second derivative of the value of the Call with respect to strike multiplied by the width of the band i.e.

$\frac{\partial^2Call}{\partial K^2}\Delta K$