If a firm sells such a contract and wishes to hedge the risk, one idea would be to buy \(\Delta\) units of the stock \(S_t\), thus the net protfolio value would be: \[Q_t= S_t \Delta - V_t \] To find out the exposure ( sensitivity ) of the portfolio to the stock would could take the partial derivative wrt \(S_t\): \[\frac{\partial Q_t}{ \partial S_t} = \Delta - \frac{\partial V_t}{\partial S_t} \] So if \[ \Delta = \frac{\partial V_t}{\partial S_t} \] then \[\frac{\partial Q_t}{ \partial S_t} = 0\] In which case, at that instance we're hedged against small moves in the stock price.
Leaving aside any formal mathematics for a moment...
Broadly speaking when the payoff \((P(S_T))\) has bumps and kinks we find that the contract valuation \(V_t(S_t)\) is a significantly more smooth function, since the value \(V_t\) is an expectation, so the kinks get averaged out. That said: \[\lim_{t\to T}V_t(S_t)=P(S_T)\] and also \[\lim_{t\to T}\Delta_t=\lim_{t\to T}\frac{\partial V_t}{\partial S_t}=\frac{\partial P}{\partial S_T}\] Thus when we have constraints on the delta ( first derivative ) and indeed gamma ( second derivative) of \(V_t\), we need to be sure that the constraints hold for the payoff \(P(S_T)\).
One practical problem with having high gamma is that it means the delta will change a lot for small changes in the underlying. So if a trader is attempting to keep a portfolio delta neutral, he'll need to rebalance frequently and the rebalances will be large. In such a scenario there is the potential to loose a significant amount of money on transaction costs. Also high gamma normally comes with high theta ( time sensitivity ) and vega ( sensitivity to volatility).
Suppose the contract that we wish to hedge is a barrier option with payout at time T: \[P(S_T)= \begin{cases} 0 \ \ \ \ \ when S_T < K \\ N \ \ \ when \ S_T \geq K \end{cases} \] This is a step function. It is not continuous and the partial derivative \(\frac{\partial P}{\partial S_T}\) does not exist at \(S_T=K\).
Thus we it seems we won't be able to hedge away the exposure to \(S_T\).
So what do we do?
Rather than hedge the actual contract, we could make a synthetic contract which does have nicely behaved partial derivatives. Consider a payoff which consists of the real payoff \(P(S_T)\) plus an "overhedge" \(H(S_T)\), such that \[H(S_T) \geq 0\ \ \forall S_T\] Note that the client does not receive \(H(S_T)\) but the contract we will hedge does include a contribution from it. When we work out a price that includes \(H(S_T)\) it will be higher than it would have been without the overhedge. But to be able to hedge the risks we need a "reasonable" contract, which is nicely behaved.
What we want is a minimal overhedge \(H(S_T)\) such that the delta and gamma are contained.
Let the synthetic contract with the overhedge have payoff: \[\Phi (S_T)= P(S_T) + H(S_T)\] (Why did we choose that Greek letter? Well adding P and H we get PHI )
We're going to insist that the first derivative ( \(\frac{\partial \Phi}{\partial S_T}\) )is continuous, because we want the second derivative ( gamma ) to exist and not be infinite.
Let's suppose we have the following caps and floors as constraints: \[-\Delta_F \leq \frac{\partial \Phi}{\partial S_T} \leq \Delta_C\] and \[-\Gamma_F \leq \frac{\partial ^2 \Phi}{\partial S_T^2} \leq \Gamma_C\]
With \(\Delta_F \geq 0 \), \(\Delta_C \geq 0 \)
and \(\Gamma_F \geq 0 \), \(\Gamma_C \geq 0 \)
So now the question is: what is the minimal overhedge which obeys those constraints?
Well for an analogy, suppose S was time t and H was position.
We are starting at rest at position N.
We wish to get home in our car as quickly as possible to position H=0
and we have a maximum acceleration, a maximum deceleration and a speed limit.
Then what should we do?
Clearly we should use maximum acceleration until we reach the speed limit.
We should then continue at the speed limit for as long as possible until near home
and then apply the breaks, using maximum deceleration until we neatly come to rest at home.
Similarly in this we case we break H up into 3 sections:
\(H_1(S_T)\) is the accelerating phase, ( quadratic in \(S_T\))
\(H_2(S_T)\) is the linear phase ( i.e. at the speed limit).
Then \(H_3(S_T)\) is the decelerating phase.
After (quite) a bit of algebra we find:
\(H_1(S_T)=N-(S_T-K)^2\ \Gamma_F\ \ \ \ \ \ \ \ \ \ \ \ \) in the domain: \(K \leq S_T < \Lambda_1\)
\(H_2(S_T)= H_1(\Lambda_1)-(S_T-\Lambda_1)\Delta_F\ \ \ \ \) in the domain: \(\Lambda_1 \leq S_T < \Lambda_2\)
\(H_3(S_T)=(S_T-\Lambda_3)^2\ \Gamma_C\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \) in the domain: \(\Lambda_2 \leq S_T \leq \Lambda_3\)
With:
\[\Lambda_1=K+\frac{\Delta_F}{2 \Gamma_F}\] \[\Lambda_2=\Lambda_1 + \frac{H_1(\Lambda_1)}{\Delta_F}-\frac{\Delta_F}{4\Gamma_C}\] \[\Lambda_3=\Lambda_1 + \frac{H_1(\Lambda_1)}{\Delta_F}+\frac{\Delta_F}{4\Gamma_C}\] We can see that: \[\Lambda_3=\Lambda_2+\frac{\Delta_F}{2\Gamma_C}\]
Note that \(H(S_T)\) is zero before K and after \(\Lambda_3\).
To derive the above equations we use the fact that, when we set: \[H_2(S_T)-H_3(S_T)= 0\] we have a quadratic in \(S_T\) and that quadratic should have a single root at \(S_T=\Lambda_2\).
I.e. the parabola \(H_3(S_T)\) is a tangent to the line \(H_2(S_T)\) at \(S_T=\Lambda_2\).
Returning to the analogy of using the car to get home quickly: Suppose you start off near home and the acceleration in the car is not very high. Then perhaps you'll need to start decelerating before you reach the speed limit.
In our case that happens when \[\Lambda_1 > \Lambda_2 \] So now \(H_2\) doesn't come into play and we need to find where is the transition from \(H_1\) to \(H_3\). I'll leave it as an exercise for the reader to work it out.
Question for the reader:
If you make some "reasonable" assumptions on the distribution of \(S_T\)
and given that \[\frac{\partial^2 P}{\partial S_T} \leq \Gamma_C \ \ \ \ \ \forall S_T\] then show that \[\frac{\partial^2 V_t}{\partial S_t} \leq \Gamma_C \ \ \ \ \ \forall S_t \ and \ t\] What assumptions did you need to make?
Suggestion: you might consider starting with the log-normal distribution case without drift or interest rates, then generalize.
Summary:
So we have derived the optimal overhedge for a barrier option subject to constraints on delta and gamma. And the synthetic contract which contains the overhedge is continuous and its first derivative is continuous.
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