# How does a pricing model 'understand' the cost of hedging?

Suppose I am pricing a multi asset at the expiry payoff. Theoretically I define their joint distributions in the risk neutral measure, and price using expectation. However, how do I know that the model has accounted for cost of vega hedging? Cost of delta hedging is baked into the marginal distributions, but how to account for cost of vega hedging? How does the model 'know' this cost? I suppose this is somehow 'implied' by the 'joint distribution' part, but that begs the question, do I not need a term structure model (i.e. evolve the vol surface over time) to be accurately able to take care of that cost?

I'm not sure I understand the question, but I'll give it a try anyway.

The mean and variance specified for the terminal distribution $$S_T$$ are dependent on current asset price, $$S_0$$, and implied volatility, $$\sigma_i$$ (which needs to come from the market via hopefully same pricer that one uses).

The expectation of a payoff, function $$f(S_T)$$, is hence a function of $$S_0$$ and $$\sigma_i$$, $$V(0, S_0, \sigma_i)$$. All one can do at this point is compute delta and vega. No hedging so far. Only pricing.

Hedging comes in when one is interested in the terminal $${\rm PnL}_T$$ of the (delta hedged) derivative product.

For this, one has to imagine a process behind $$S_T$$ (martingale representation theorems come to mind) say of the form $$dS_t/S_t = ...dt +\sigma_t dW_t, S_0$$
with $$\sigma_t$$ the 'true' vol along the asset path.

Assuming delta hedging is done at $$\sigma_i$$ over the life of the product (see this link for assumptions and details for hedging vol different from implied vol etc.), the terminal PnL is:

$${\rm PnL}_T = \int_0^T {\rm e}^{-rT}(\sigma_i^2 - \sigma_t^2) \frac{1}{2}S_t^2 \frac{\partial^2 }{\partial S^2} V(t,S_t, \sigma_i) dt$$

which bakes in the assumed variance of the the terminal asset, $$\sigma_i^2$$, but also the realized volatility and Gamma along the asset path. (Gamma is related to Vega; under Black-Scholes assumptions, for European option payoffs, the relationship is explicit: $${\rm Vega} = \sigma_i \tau S^2 {\rm Gamma}$$.)

Edit: It is Feynman-Kac theorem (or rather its reciprocal) that says that

$$u(x,t) = E^Q \left[{\rm e}^{r(T-t)}\psi(X_T) | X_t=x \right]$$

is the solution of the standard parabolic PDE with terminal condition $$u(x,T)=\psi(x)$$

which reveals the delta and gamma terms used in hedging (PDE does 'understand hedging').

• Although i understand what you said, I don't believe your claim that you can price without talking about hedging, since price modeled as expectation in unique risk neutral measure depends upon the assumption of a complete market i.e. replication. It is this subtlety I'm trying to understand. Can I, from the risk neutral distribution, comment on the hedging costs or the self financing strategy that underlies it? – Arshdeep Singh Duggal Jun 28 at 3:02
• @Arshdeep Singh Duggal Sorry, I wasn't making any claim. I was just redirecting the discussion towards an object I thought was more relevant, the terminal PnL. Regarding your question, maybe it is Feynman-Kac theorem that you are looking for, as it establishes the link between (conditional) expectation of terminal payoff and the PDE which 'understands hedging'. – ir7 Jun 28 at 4:02
• Oh yes, I see it now. Thanks a lot for that statement on Feynman Kac. – Arshdeep Singh Duggal Jun 28 at 5:17
• @ArshdeepSinghDuggal I’m glad it helps. – ir7 Jun 28 at 10:49

A model 'understands' the price of risks that are assumed to exist. For example, the Black-Scholes model undertands the cost of delta-hedging, but not of vega-hedging. Hence we have stochastic volatility models: these understand the cost of delta-hedging and volatility hedging. However, none of these models take into account transaction costs. Hence you could also argue that none of these models really understand the cost of hedging in practice. The fact that transaction costs are usually not part of a pricing model makes the need for (semi-) static replication of claims even greater. If possible you should always try to have a (semi-)static replication strategy. Of course you can always model transaction costs if semi-static hedging is not possible, but that entails additional computational costs.

• Your point makes sense but my question is: How? Say I tell you that the market is complete, and I give you the distribution of a stock in the terminal measure. You can for sure price it, but I didn't have to speak of cost of replication at all. – Arshdeep Singh Duggal Jun 23 at 18:57
• Let's take the BS model and forget about transaction costs for the moment. In the derivation of the BS price formula you take into account that you construct a self-financing replicating portfolio by dynamically rebalancing stocks (delta-hedging). The BS price formula is the solution to the resulting PDE which has this delta-hedging embedded in it. I.e. the solution of the PDE understands that hedging is involved. And since the distribution can be epressed in terms of call prices (Breeden-Litzenberger), the distribution understands the cost of replication. Hope it's clearer now. – ilovevolatility Jun 24 at 14:01
• Yes I agree that the distribution somehow understands, as price is just the cost of replication. What I am trying to understand is, given a distribution in the risk neutral measure, what can I say about the replicating portfolio that underlies it? From what I know, I can only say I've assumed there exists one. – Arshdeep Singh Duggal Jun 28 at 3:04
• I am not sure anymore what your question is really about. But since you accepted ir7's answer, which is actually about delta-hedging p/l with a constant vol, I take it that you have your answer. – ilovevolatility Jun 29 at 9:31
• Actually he answered it in the comments; I was trying to relate the terminal distribution and the hedging costs. He pointed me towards the Feynman Kac theorem which takes us from forming a self financing strategy to a terminal distrbution. – Arshdeep Singh Duggal Jun 29 at 11:15