Pricing, Delta Hedging, and Volatility Risk in Derivative Models

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


# 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?

## Answer by ir7 (score 2, accepted)

https://quant.stackexchange.com/a/55193

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').

## Answer by user34971 (score 0)

https://quant.stackexchange.com/a/55143

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.