Comparing Probability-Space and Payoff-Space Option Calibration

# Optimising in Probability Space vs Payoff Space


# Optimising in Probability Space vs Payoff Space












We model log-returns under real probability, using historical data:

$$ r_{t2} \sim \text{StudentT}(\mu_{t2}, \sigma_{t2}, \nu) $$

predicting mean and variance in future $t2$ using variance and risk free rate at current time $t$:

$$ \begin{aligned} \sigma_{t2} &= \exp(v_0 + v_1 \cdot \mathrm{var}_t) \\ \mu_{t2} &= m_1 + m_2 r_{rf,t} + m_3 \sigma_{t2}^2 + m_4 \log \sigma_{t2} \end{aligned} $$

It's clear how to optimise it in the Probability Space with maximum likelihood, computationally efficient, one call of StudentT Pdf per data point.

In Payoff Space, we target pricing accuracy of eu options, under the same real-world model.

Goal, informal description - there are N strikes $K_i \in [1.1,\, 1.2,\, \dots,\, 2.0]$. For each strike, you start with zero, trade millions of C(K), and in the end you should end up with zero. Each strike C(K) traded independently, to avoid mixing errors for different strikes. And same for puts with reciprocal strikes $1/K_i$.

More formally, for each data point:

- $N$ calls with strikes $K_i \in [1.1,\, 1.2,\, \dots,\, 2.0]$

- $N$ puts with reciprocal strikes $1/K_i$

- Use model to compute option premiums: $$ C_{t,i}^{\text{model}} = \mathbb{E}[(e^{r_{t2}} - K_i)^+], \quad P_{t,i}^{\text{model}} = \mathbb{E}[(K_i - e^{r_{t2}})^+] $$

- Compute realised premiums using actual returns $r_{t2}$

- Define the loss, for each strike $K$, define separate relative pricing errors for calls and puts:

$$ \begin{aligned} \varepsilon_{\text{call},K} &= \frac{\sum_t C_{t,K}^{\text{model}}}{\sum_t C_{t,K}^{\text{real}}} \\ \varepsilon_{\text{put},K} &= \frac{\sum_t P_{t,K}^{\text{model}}}{\sum_t P_{t,K}^{\text{real}}} \end{aligned} $$

- Then define the total loss, penalising largest pricing error across strikes, calls and puts:

$$ \mathcal{L} = || \varepsilon_{call,K},\varepsilon_{put,K} ||_2 $$

- Strike levels are scaled per each data point, depending stock volatility at time $t$, to reflect volatility — narrower strikes for quiet, wider for volatile ones:

$$ K_i = K_i \mathrm{var}_t C_{\text{some constant}} $$

This is computationally expensive - $2N$ numerical integrals per data point.

#### Question

Can this payoff-based loss be reformulated more efficiently — to retain its meaning but avoid explicit option integration?

And does such approach used at all?