Out of Money Lookback S(T) - 1.1Min(S(t), 0<t<T) ?

[quotes]

Hi All.

There is formula ready for the lookback call option whose payoff at the maturity is

S(T) - Min(S(t), 0<t<T)

where S(T) is the terminal underlying price, and S(t) is the underlying price at time t, then the second term means the minimum of underlying price along the duration of the Call.

Of course this lookback call ends in the money with probability almost 1, the remaining chance at the money.

Now let us twist the payoff just a little bit, the terminal price has to be 10% more than the running minimum to be in the money. This new out of money Lookback has the terminal payoff

S(T) - 1.1Min(S(t), 0<t<T)

How can we solve this out? Good ideas?

 

[Adrián Facchi]

Hi,
Musiela & Rutkowski 2n ed pg 202 gives a result for the Lookback Call when the underlying stock follows a geometric Brownian motion
http://postimage.org/image/a42rz30zp/
The proof takes 2 pages and I think that it applies as well for the leveraged minimum part replacing 'm' by 'L*m' where L is the leverage (I didn't checked all the steps of the proof though so I'm not sure).

This is ok as an academic exercise but in practice you should consider Monte Carlo simulations or PDE solver with a stochastic volatility. This is because this product is depends strongly on the future conditional asset distributions.

For Monte Carlo simulations you should consider also the simulations of the running minimum given the initial and final points of a time step. Those minimums between time steps can be simulated using http://postimage.org/image/9cdw4wb4j/ where 'sigma' is the volatility used to simulate Si+1 from Si (not necessarily local) and U is a uniform (0,1).

 

[quotes]

I am so glad that finally someone replied my post. Thank you FKaria.
I have found that Marc Yor and Geman Yor's book Mathematical Methods in Financial Market has a tedious proof.

The reason why I need an analytic solution is I need to delta hedge and create the derivative. I don't know why a Monte-Carlo simulation provides more information than the analytic formula. Besides, I don't know how to incorporate the Heston stochastic volatility formula into hedging.

Could you be more specific?

 

[koupparis]

Rewrite the option as:

(S_T-(1+\alpha) \min(S_t) = S_T -\min(S_t) -\alpha \min(S_t) +\alpha S_T - \alpha S_T)
(\Rightarrow (1+\alpha)(S_T-\min(S_t)) -\alpha S_T).

Pricing is then trivially a combination of weighted lookback and future contracts.

 

[quotes]

Rewrite the option as:

(S_T-(1+\alpha) \min(S_t) = S_T -\min(S_t) -\alpha \min(S_t) +\alpha S_T - \alpha S_T)
(\Rightarrow (1+\alpha)(S_T-\min(S_t)) -\alpha S_T).

Pricing is then trivially a combination of weighted lookback and future contracts.

Brilliant! The most traditional way- synthetic option!

 

[Adrián Facchi]

I am so glad that finally someone replied my post. Thank you FKaria.
I have found that Marc Yor and Geman Yor's book Mathematical Methods in Financial Market has a tedious proof.

The reason why I need an analytic solution is I need to delta hedge and create the derivative. I don't know why a Monte-Carlo simulation provides more information than the analytic formula. Besides, I don't know how to incorporate the Heston stochastic volatility formula into hedging.

Could you be more specific?

The analytical formula would be just fine for a Black-Scholes world, i.e. for academic proposes. However, in practice Vanilla Option prices obtained from the market (American options for stocks, European for Indices) the implied volatility is not constant. That is, the volatility that you should put in the Black-Scholes formula is not the same for each price. This yields the famous implied volatility surface (2 dimensions: strikes and maturities).

Pricing a Vanilla option with strike K and maturity T with an analytic formula is feasible because the unique requirement is to put on the formula the implied volatility obtained from market prices for that strike K and maturity T (this gives the market price therefore there are no arbitrage opportunities). The problem with lookback options is that you don't know the strike level beforehand so you don't know what is the correct volatility to put in the formula (whatever analytic formula you could use). This is why I say that a Monte Carlo simulation using the Local Volatility model would be more recommendable.

Although, Black-Scholes and Local Volatility rely on the market-implied distributions (especulations of the future volatility) of the asset conditioned to the current spot value and not at the future value (at time t) when the fixing takes place. You will notice that the Delta of the lookback option is 0 before the fixing date. At the fixing date the delta will jump and the value of the lookback should coincide perfectly with an At-the-money call option. Notice that this happens when you make t tend to zero.

Since there is no way to guarantee a correct price (meaning for a correct price the one that coincides with an ATM call option the fixing date) in practice, you should consider the pricing of forward starting options (lookbacks) using a Heston model with Monte Carlo simulation to incorporate some future implied distributions to the model. If you use Monte Carlo simulation the Greeks can be computed using finite differences, although there will be a lot of noise.

 

[quotes]

The analytical formula would be just fine for a Black-Scholes world, i.e. for academic proposes. However, in practice Vanilla Option prices obtained from the market (American options for stocks, European for Indices) the implied volatility is not constant. That is, the volatility that you should put in the Black-Scholes formula is not the same for each price. This yields the famous implied volatility surface (2 dimensions: strikes and maturities).

Pricing a Vanilla option with strike K and maturity T with an analytic formula is feasible because the unique requirement is to put on the formula the implied volatility obtained from market prices for that strike K and maturity T (this gives the market price therefore there are no arbitrage opportunities). The problem with lookback options is that you don't know the strike level beforehand so you don't know what is the correct volatility to put in the formula (whatever analytic formula you could use). This is why I say that a Monte Carlo simulation using the Local Volatility model would be more recommendable.

Although, Black-Scholes and Local Volatility rely on the market-implied distributions (especulations of the future volatility) of the asset conditioned to the current spot value and not at the future value (at time t) when the fixing takes place. You will notice that the Delta of the lookback option is 0 before the fixing date. At the fixing date the delta will jump and the value of the lookback should coincide perfectly with an At-the-money call option. Notice that this happens when you make t tend to zero.

Since there is no way to guarantee a correct price (meaning for a correct price the one that coincides with an ATM call option the fixing date) in practice, you should consider the pricing of forward starting options (lookbacks) using a Heston model with Monte Carlo simulation to incorporate some future implied distributions to the model. If you use Monte Carlo simulation the Greeks can be computed using finite differences, although there will be a lot of noise.

Thank you for the reply, FKaria.

The main reason for the volatility smile, so far as I understand, is that the distribution of the rate of return is not Normal. The heavy tail property of the real distribution in contrast to the theoretical Normal distribution, shows a lower normal standard deviation around the center and a higher standard deviation at tails if we fit normal density to the data.

If instead we use a time dependent volatility model, we get a cumulative volatility till maturity as the average volatility* Time to maturity before we plug it into BS framework ST=S0*exp(aT-ssT/2+sWT). This still does not solve the problem, unless the cumulative distribution till maturity exhibits a heavy tail property.